Download Goals and Objectives

DAMIEN HIGH SCHOOL
Mathematics & Computer Science
Department Handbook
Revised February 2013
Table of Contents
Philosophy……………………………………………………………………
2
Goals and Objectives…………………………………………………………
2
Organization………………………………………………………………….
3
Sequence of Courses…………………………………………………………
4
Instructional Procedures……………………………………………………...
5
Course Descriptions (Curriculum Maps)
Algebra 1A……………………………………………………….…...
6
Algebra 1B……………………………………………………………
15
Algebra 1……………………………………………………………..
25
Geometry………………………………………….………………….
41
Honors Geometry…………………………………………………….
58
Algebra 2………………………………………………..…………....
74
Honors Algebra 2………………………………………..………..….
87
Pre-Calculus………………………………………………..………..
101
Statistics…………………………………………………….……….
115
Advanced Placement Statistics……………………………………...
124
Advanced Placement Calculus AB……………………………….…
142
Advanced Placement Calculus BC………………………………..…
153
Differential Equations…………………………………………….…
166
Introduction to Computer Science…………………………………..
174
Introduction to Game Design………………………………….…….
181
Advanced Placement Computer Science……………………...…..…
189
1
Philosophy
The philosophy of the Mathematics Department at Damien High School is to
promote and encourage each student, appropriate to his needs and abilities, to achieve a
level of competence in applying mathematical concepts and procedures. The department
offers a vast range of course sequences to successfully complete steps to arrive at the
individual’s goals. Course sequences guide and prepare each student to function
responsibly in an increasingly technological society and provide him with a mathematical
background required for entrance into any of the major universities.
Goals and Objectives
A. Goals
1. To have students acquire the mathematical knowledge necessary to function
mathematically after graduation.
2. To provide the mathematical training for those students who plan to pursue a
career in mathematics or fields that entail higher mathematics.
3. To have students gain an appreciation for the processes used in mathematics.
4. To have students realize the importance and necessity of mathematics as it
relates to solving problems which arise in everyday, real-life situations.
B. Objectives
The students will attain these goals through a number of ways:
1. Seeing real-life applications of algebra and geometry in their classes.
2. Learning the necessary deductive reasoning through solving problems and
proving theorems.
3. Nightly homework assignments that challenge the student’s problem solving
skills.
4. Major examinations that test the student’s knowledge of necessary material.
2
Organization
A. Chairperson’s Duties
(See Faculty Handbook)
B. Meetings
Department meetings are formally held bi-monthly and informally among
teachers on a consistent basis. The department chairperson makes the agenda with input
from department members. Copies of department meeting minutes are on file with the
Principal, with the Assistant Principal-Academic Affairs, and at the department’s
resource room.
C. Budget Process
The needs of the Department are determined during department meetings and
submitted to the Principal in terms of a budget request.
D. Instructional Materials
The Department has an extensive collection of mathematics texts that are stored in
the Department’s resource room located between rooms 301 and 303. Each math
classroom is equipped with a projector screen, DVD player, projector, and digital
drawing tablet.
3
Sequence of Courses
The Department has the following common sequences available depending on the
student’s ability level in mathematics. The student must complete 4 years of
Mathematics to fulfill the graduation requirements. To move into another sequence, the
student must successfully pass a mathematics course in summer school.
Sequence I
Sequence II *
-Algebra 1A
-Algebra 1B
-Geometry
-Algebra 2
-Algebra 1
-Geometry
-Algebra II
-Pre-Calculus/Statistics
Sequence III * **
Sequence IV * **
-Honors Geometry
-Honors Algebra 2/Algebra 2
-AP Calculus AB/Precalculus
-AP Calculus BC/AP Statistics
-Honors Algebra 2
-AP Calculus AB
-AP Calculus BC/AP Statistics
-Differential Equations
* To take Algebra 1 as a freshman, the student must earn a minimum grade of 50 on
both parts of the Damien Algebra 1 Placement Test. To take Geometry as a
freshman, the student must earn a minimum grade of 70 on both parts of the Algebra
1 Placement Test. To take Honors Algebra 2 as a freshman, the student must pass the
Geometry Placement test with a minimum grade of 70.
** Advanced Placement Statistics may be taken concurrent with AP Calculus AB as a
junior. As a senior, AP Statistics can be taken after passing Precalculus as a junior.
4
Instructional Procedures
A. Methodologies
Depending on the course, the Department employs several methodologies of
instruction. These include the following:
1.
2.
3.
4.
5.
Lecture
Class Participation
Demonstration
In-class Written Work
Student Collaboration
6. Projects
B. Evaluation
Depending on the course, instructors employ the following means on evaluating
students:
1.
2.
3.
4.
5.
6.
Homework
Quizzes
Major Tests
Cumulative Common Finals
Class Participation
Group/Class Projects
C. Controversial Topics
When controversial topics arise, the matter is referred to the Department
chairperson. If the chairperson cannot resolve the problem, then the matter is referred to
the Assistant Principal-Academic Affairs.
5
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Algebra 1A
This course is required for all students that do not qualify for Algebra 1
Yes – Category D
Year
Brief Course Description
This course is designed to be a full year course covering the material from the first semester of Algebra 1. The course will provide an introduction to, development of, and
study of the basic skills and concepts of elementary algebra concepts. Topics include: fundamental operations of real numbers, evaluating algebraic expressions, simple linear
equations and inequalities, solving systems of linear equations and inequalities, operations on polynomials – including factoring, integer exponents, introduction to square
roots, introduction to graphing, ratio and proportion, applications (word problems), introduction to algebraic rational expressions and equations, and solving quadratic
equations by factoring, the quadratic formula, and completing the square.
Assigned Textbook(s)
Allan E. Bellman, Sadie Chavis Bragg, Randall I. Charles, Basia Hall, William G. Handlin, Sr., Dan Kennedy;
Algebra ; Pearson Prentice Hall; 2007
Algebra 1 Workbook; Pearson
PHSchool.com
Supplemental Material(s)
Common Assessments Utilized
Common Final each semester
In-Class Work
Homework
Quizzes
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
6
Major Content Outcomes
Major Skill Outcomes (include Labs???)
Upon completion of this course the student should be able to:
I. Variables, Functions, Patterns, and Graphs
A. Using Variables
B. Exponents and Order of Operations
C. Exploring Real Numbers
D. Patterns and Functions
II. Rational Numbers
A. Adding, Subtracting, Multiplying and Dividing Rational Numbers
B. The Distributive Property
C. Properties of Numbers
1.
Perform calculations on real numbers and use order of operations.
2.
Simplify and evaluate expressions using multiple algebraic expressions.
3.
Solve linear equations and simple quadratic equations.
4.
Solve application problems (word problems) using linear equations.
5.
Graph linear equations in two variables by using x- and y-intercepts.
6.
Add, subtract, multiply and factor polynomials.
7.
Use the rules/properties of exponents.
Unit 1
Students should be able to define variables and write an equation to model the relationship in a table.
Students should be able to simplify and evaluate an expression.
Students should be able to name the set(s) of numbers to which a number belongs.
Students should be able to identify the independent and dependent variables and write a function rule to
describe the situation.
III. Solving Equations
A. Solving Two-Step and Multi-Step Equations
B. Equations With Variables on Both Sides
C. Ratio and Proportion
D. Equations and Problem Solving
E. Percent of Change
Unit 2
Students should be able to simplify an expression.
Students should be able to find each sum or difference of rational numbers.
Students should be able to evaluate an expression.
Students should be able to decide which property the equation illustrates.
IV. Solving Inequalities
A. Inequalities and Their Graphs
B. Solving Inequalities Using Addition, Subtraction, Multiplication and Division
C. Solving Multi-Step Inequalities
D. Compound Inequalities
E. Absolute Value Equations and Inequalities
Unit 3
Students should be able to solve an equation.
Students should be able to define a variable, write an equation, and then solve.
Students should be able to solve a proportion.
Unit 4
Students should be able to determine whether a number is a solution of the given inequality.
Students should be able to define a variable and write an inequality to model a situation.
Students should be able to solve an equation or inequality and then graph the solution.
Students should be able to solve a compound inequality and graph the solutions.
V. Graphs and Functions
A. Relations and Functions
B. Function Rules, Tables, and Graphs
C. Writing a Function Rule
D. Direct Variation and Inverse Variation
Unit 5
Students should be able to determine whether a relation is a function and if it is a function, state the domain
and range.
Students should be able to find the range of a function given the domain.
Students should be able to model a rule with a table of values and a graph.
Students should be able to write a function rule to describe a statement.
Students should be able to write a function rule for a table of values.
Students should be able to write an equation of the direct variation that includes the given point.
Students should be able to determine whether a graph shows a direct variation.
Students should be able to find the constant of variation for an inverse function.
VI. Linear Equations and Their Graphs
A. Rate of Change and Slope
B. Slope-Intercept Form and Standard Form
C. Point-Slope Form and Writing Linear Functions
D. Parallel and Perpendicular Lines
Unit 6
Students should be able to find the slope of a line that passes through a pair of points.
Students should be able to graph a linear equation.
7
Students should be able to write the equation of a line in standard form and in slope-intercept form.
Students should be able to find the x- and y-intercepts of a line.
Students should be able to write the equation of a line in point-slope form given slope and through a given
point.
Students should be able to write an equation in point-slope form through the given points.
Students should be able to write an equation in slope-intercept form for a line the passes through the given
point and is parallel or perpendicular to the given line.
8
Algebra 1A
Unit 1 – Variables, Function Patterns, and Graphs
Content Outcomes
Students will understand the use of variables.
Essential Questions
How are phrases in English written as algebraic
expressions?
Key Concepts
Modeling relationships with variables.
Modeling relationships with equations.
Common Core: 6.EE-2
How are situations modeled with equations?
Students will learn about exponents and order of
operations.
How is order of operations used to simplify
expressions?
How is substitution used to evaluate expressions?
Students will learn about real numbers.
How are real numbers classified?
Simplifying expressions using order of operations
PEMDAS:
Parentheses, Exponents, Multiplication or Division,
Addition or Subtraction.
Evaluating expressions using substitution.
California: CA 17
Classifying real numbers.
California: CA 17
How are real numbers compared?
Students will learn about patterns and functions.
Standards Addressed
California: CA 17
Common Core:
Common Core: 8.NS-1
How are dependent and independent quantities
identified?
Writing Function Rules.
Understanding dependent and independent
quantities.
How are relationships written as functions?
9
California: CA 17
Common Core: 6.EE-9
Algebra 1A
Unit 2 – Rational Numbers
Content Outcomes
Students will understand how to add rational
numbers.
Essential Questions
What are real numbers?
Key Concepts
Adding numbers with the same sign.
Adding numbers with different signs.
How are real numbers added?
Students will understand how to subtract rational
numbers.
What are rational numbers?
Common Core: 7.NS-1
Subtracting numbers.
How are rational numbers subtracted?
Students will understand how to multiply and divide
rational numbers.
How are real numbers multiplied?
How are real numbers divided?
Students will understand the Distributive Property.
What is the Distributive Property?
What are the properties of real numbers?
California: CA 1
Common Core: 7.NS-1c
Multiplying and dividing numbers with the same
sign.
Multiplying and dividing numbers with different
signs.
California: CA 1
The Distributive Property.
California: CA 1
How is the Distributive Property used to simplify
expressions?
Students will learn about the properties of numbers.
Standards Addressed
California: CA 1
Common Core: 7.NS-2
Common Core: 7.NS-2a
Properties of real numbers.
How are the properties of real numbers identified?
California: CA 1
Common Core: 7.NS-1d, 7.NS-2c
10
Algebra 1A
Unit 3 – Solving [Simple] Equations
Content Outcomes
Students will understand how to solve two-step
equations.
Essential Questions
What are inverse operations?
Key Concepts
Solving two-step-equations using inverse
operations.
How are inverse operations used to solve two-stepequations?
Students will understand how to solve multi-step
equations.
How is Distributive Property used to combine like
terms?
Standards Addressed
California: CA 4,5
Common Core: A-REI-3
Using Distributive Property to combine like terms.
Using Distributive Property to solve equations.
California: CA 4,5
Common Core: A-REI-3
How is Distributive Property used to solve
equations?
Students will understand how to solve equations
with variables on both sides.
How are equations with variables on both sides
solved?
Solving equations with variables on both sides.
Identifying equations that are identities and have no
solution.
California: CA 4,5
Solving algebraic proportions.
California: CA 4,5
Common Core: A-REI-3
How are equations that are identities or have no
solution identified?
Students will learn about ratios and proportions.
What are proportions?
How are algebraic proportions solved?
Students will understand how to solve real-world
problems.
How is a variable defined in terms of another
variable?
Common Core: A-REI-3
Defining a variable in terms of another variable.
Solving distance-rate-time problems.
Solving consecutive integer problems.
How are distance-rate-time problems solved?
How are consecutive integer problems solved?
11
California: CA 4,5
Common Core: A-REI-3
Algebra 1A
Unit 4 – Solving Inequalities
Content Outcomes
Students will learn about inequalities and their
graphs.
Essential Questions
How are solutions of inequalities identified?
How are solutions of inequalities graphed?
Students will understand how to solve inequalities
using addition and subtraction.
How are inequalities solved using addition?
Key Concepts
Identifying solutions of inequalities using
substitution.
Graphing inequalities.
Standards Addressed
California: CA 3 - 5
Using addition to solve inequalities.
Using subtraction to solve inequalities.
California: CA 3 - 5
How are inequalities solved using subtraction?
Students will understand how to solve inequalities
using multiplication and division.
How are inequalities solved using multiplication?
Common Core: A-REI-3
Using multiplication to solve inequalities.
Using division to solve inequalities.
How are inequalities solved using division?
Students will understand how to solve multi-step
inequalities.
How are multi-step inequalities with variables on
one side solved?
How are multi-step inequalities with variables on
both sides solved?
Students will understand how to solve compound
inequalities.
How are inequalities containing “and” solved and
graphed?
How are equations that involve absolute value
solved?
California: CA 3 - 5
Common Core: A-REI-3
Solving multi-step inequalities with variables on
one side.
Solving multi-step inequalities with variables on
both sides.
California: CA 3 - 5
Solving and graphing inequalities containing “and”.
Solving and graphing inequalities containing “or”.
California: CA 3 - 5
Common Core: A-REI-3
Common Core: A-REI-3
How are inequalities containing “or” solved and
graphed?
Students will understand how to solve absolute
value equations and inequalities.
Common Core:
Solving equations that involve absolute value.
Solving inequalities that involve absolute value.
California:
Common Core:
How are inequalities that involve absolute value
solved?
12
Algebra 1A
Unit 5 – Solving Inequalities
Content Outcomes
Students will learn about relations and functions.
Essential Questions
What is a function?
Key Concepts
Identifying relations and functions.
Evaluating functions.
How are relations and functions identified?
Standards Addressed
California: CA 16, 17
Common Core: F-IF-1
How are functions evaluated?
Students will understand how to use function rules,
tables, and graphs.
How are functions modeled using rules, tables, and
graphs?
Modeling functions using rules, tables, and graphs.
California: CA 16, 17
Common Core:
Students will understand how to write a function
rule.
How is a function rule written given a table or realworld situation?
Writing a function rule given a table or real-world
situation.
California: CA 16, 17
Common Core:
Students will learn about direct variation.
How is a direct variation equation written?
Writing an equation of a direct variation.
Using ratios and proportions with direct variations.
How are ratios and proportions used with direct
variations?
Students will learn about inverse variation.
How are inverse variations solved?
California: CA 16, 17
Common Core:
Solving inverse variations.
Comparing direct and inverse variation.
How do direct variations and inverse variations
compare?
California: CA 16, 17
Common Core:
13
Algebra 1A
Unit 6 – Linear Equations and their Graphs
Content Outcomes
Students will learn about the rate of change and
slope.
Essential Questions
How are rates of change found from tables and
graphs?
What is slope?
How is slope [of a line] found?
Students will learn about the slope-intercept form of
a line.
What is slope-intercept form of a line?
What is a linear equation?
Key Concepts
Finding rates of change from tables and graphs.
Finding slope using the slope formula: y 2  y1 .
x 2  x1

Writing linear equations in slope-intercept form: y =
mx + b.
Graphing linear equations in slope-intercept form.
Standards Addressed
California: CA 6 -8
Common Core:
California: CA 6 -8
Common Core: F-IF-7a
How are linear equations written in slope-intercept
form?
Students will learn about the standard form of a
line.
How are linear equations graphed [using slope and
y-intercept]?
What is an intercept?
Graphing equations using x- and y-intercepts.
Writing equations in standard form: Ax + By = C.
How are linear equations graphed using x- and yintercepts?
California: CA 6 -8
Common Core: F-IF-7a
How are linear equations written in standard form?
Students will learn about the point-slope form of a
line and how to write linear equations.
What is point-slope form?
Graphing and writing linear equations using pointslope form: y – y1 = m(x – x1).
How are linear equations graphed using point-slope
form?
Students will learn about parallel and perpendicular
lines.
How are linear equations written using point-slope
form?
What are parallel lines?
What are perpendicular lines?
California: CA 6 -8
Common Core:
Determining whether the lines are parallel by
comparing slopes.
Determining whether the lines are perpendicular by
comparing slopes.
How are parallel lines determined?
How are perpendicular lines determined?
14
California: CA 6 -8
Common Core:
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Algebra 1B
Completion of Algebra 1A with a minimum grade of C
Yes – Category D
Year
Brief Course Description
This course is designed to be a full year course covering the material from the second semester of Algebra 1. The course will provide an introduction to, development of, and
study of the basic skills and concepts of elementary algebra concepts. Topics include: fundamental operations of real numbers, evaluating algebraic expressions, simple linear
equations and inequalities, solving systems of linear equations and inequalities, operations on polynomials – including factoring, integer exponents, introduction to square
roots, introduction to graphing, ratio and proportion, applications (word problems), introduction to algebraic rational expressions and equations, and solving quadratic
equations by factoring, the quadratic formula, and completing the square.
Assigned Textbook(s)
Allan E. Bellman, Sadie Chavis Bragg, Randall I. Charles, Basia Hall, William G. Handlin, Sr., Dan Kennedy;
Algebra ; Pearson Prentice Hall; 2007
Algebra 1 Workbook; Pearson
PHSchool.com
Supplemental Material(s)
Common Assessments Utilized
Common Final each semester
In-Class Work
Homework
Quizzes
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
15
Major Content Outcomes
Major Skill Outcomes (include Labs???)
Upon completion of this course the student should be able to:
I. Systems of Equations and Inequalities
A. Solving Systems by Graphing, Using Substitution, and Using Elimination
B. Applications of Linear Systems
C. Linear Inequalities
D. Systems of Linear Inequalities
8.
Perform calculations on real numbers and use order of operations.
9.
Simplify and evaluate expressions using multiple algebraic expressions.
10. Solve linear equations and simple quadratic equations.
11. Solve application problems (word problems) using linear equations.
12. Graph linear equations in two variables by using x- and y-intercepts.
13. Add, subtract, multiply and factor polynomials.
II. Exponents and Exponential Functions
A. Zero and Negative Exponents
B. Scientific Notation
C. Multiplication and Division Properties of Exponents
D.
Exponential Functions
14. Use the rules/properties of exponents.
Unit 1
Students should be able to solve a system by graphing, using substitution, and using elimination.
Students should be able to define variables, write an equation, and solve.
Students should be able to graph a linear inequality.
Students should be able to solve a system of inequalities by graphing.
III. Polynomials and Factoring
A. Adding and Subtracting Polynomials
B. Multiplying and Factoring
C. Multiplying Binomials and Special Cases
D. Factoring Trinomials and Special Cases
E. Factoring by Grouping
Unit 2
Students should be able to simplify the expression.
Students should be able to evaluate exponential expressions and functions.
Students should be able to write a number in scientific notation.
Students should be able to graph exponential functions.
IV. Quadratic Equations and Functions
A. Exploring Quadratic Graphs
B. Quadratic Functions
C. Solving Quadratic Equations by Factoring, Completing the Square, and Using the Quadratic
Formula
D. Using the Discriminant
Unit 3
Students should be able to write a polynomial in standard form and name the polynomial based on its
degree and number of terms.
Students should be able to simply and write the answer in standard form.
Students should be able to simplify each product and write in standard form.
Students should be able to find the GCF of the terms of the polynomial.
Students should be able to factor an expression.
Students should be able to factor completely.
V. Radical Expressions and Equations
A. Simplifying Radicals
B. Operations With Radical Expressions
C. Solving Radical Equations
D. Graphing Square Root Functions
Unit 4
Students should be able to find the equation of the axis of symmetry and the coordinates of the vertex of the
graph of a function.
Students should be able to graph a quadratic function, label the axis of symmetry and the vertex.
Students should be able to graph a quadratic inequality.
Students should be able to find the number of x-intercepts of each function.
Students should be able to solve quadratic equations.
Students should be able to find the number of solutions of a quadratic equation using the discriminant.
Students should be able to solve the quadratic equation using the quadratic formula.
Students should be able to write a quadratic equation in standard form and then solve by factoring.
Students should be able to solve a quadratic equation by completing the square.
VI. Optional
E. Trigonometric Ratios
F. Angles of Elevation and Depression
VII. Rational Expressions and Functions
A. Graphing Rational Functions
Unit 5
Students should be able to simplify a radical expression.
Students should be able to solve a radical equation.
Students should be able to make a table and graph a radical function.
16
B.
C.
D.
Simplifying, Multiplying, Dividing, Adding, and Subtracting Rational Expressions
Dividing Polynomials
Solving Rational Equations
Students should be able to find the domain of a radical function and then graph.
Unit 6 - Optional
Students should be able to find the measure of the other sides of a right triangle given the length of a side
and the measure of an angle.
Students should be able to use the distance formula to find the distance between two points.
Students should be able to use the midpoint formula to find the midpoint between two points.
Unit 7
Students should be able to simplify, multiply, divide, add, and subtract rational expressions.
Students should be able to solve a rational equation.
17
Algebra 1B
Unit 1 – Systems of Equations and Inequalities
Content Outcomes
Students will understand how to solve systems by
graphing.
Essential Questions
What is a system of linear equations?
How are systems of linear equations solved by
graphing?
Key Concepts
Solving systems of linear equations by graphing.
Determining possible number of solutions of a
system of linear equations.
Standards Addressed
California: CA 6, 9, 15
Solving systems of equations using substitution.
California: CA 6, 9, 15
Common Core: A-REI-6
What is the possible number of solutions of a
system of linear equations?
Students will understand how to solve systems
using substitution.
How are systems of linear equations solved using
substitution?
Common Core:
Students will understand how to solve systems
using elimination.
How are systems of linear equations solved using
elimination?
Solving systems of linear equations using
elimination.
California: CA 6, 9, 15
Common Core:
Students will understand how to apply linear
systems to solve real-world problems.
How are real-world problems written as a system of
linear equations?
Writing systems of linear equations given realworld problems.
California: CA 6, 9, 15
Common Core: A-CED-2
Students will learn about linear inequalities.
How are linear inequalities graphed?
Graphing linear inequalities.
California: CA 6, 9, 15
Common Core:
Students will understand how to solve systems of
linear inequalities.
How are systems of linear inequalities graphed?
Graphing systems of linear inequalities.
California: CA 6, 9, 15
Common Core:
18
Algebra 1B
Unit 2 – Exponents and Exponential Functions
Content Outcomes
Students will understand how to use zero and
negative exponents.
Essential Questions
How are expressions with zero and negative
exponents simplified?
Key Concepts
Simplifying Expressions with Zero and Negative
0
How are exponential expressions evaluated?
Exponents: a =1; a
-n
n
1 1  n
= n ;   =a
a a 
Standards Addressed
California: CA 6, 9, 15
Common Core:
Evaluating Exponential expressions.
Students will learn about scientific notation.
What is scientific notation?
Writing number in scientific [a x 10n] and standard
notation.
 
What is standard notation?
California: CA 6, 9, 15
Common Core: 8.EE-3, 8.EE-4
How are numbers written in scientific and standard
notation?
Students will understand how to use multiplication
properties of exponents.
How is scientific notation used?
What is a power?
How are powers multiplied?
Multiplying powers: am x an = am+n
Raising power to a power: (am)n = amn
Raising a product to a power: (ab)n = anbn
California: CA 6, 9, 15
a m m–n
Dividing powers with the same base:
=a
an
a n a n
Raising a quotient to a power:   = n
b  b .
California: CA 6, 9, 15
Common Core:
How is a power raised to a power?
How is a product raised to a power?
Students will understand how to use division
properties of exponents.
How are powers with the same base divided?
How is a quotient raised to a power?
Students will learn about exponential functions.
How are exponential functions evaluated?
Evaluating exponential
functions.
Graphing exponential functions.
How are exponential functions graphed?

19

Common Core:
California: CA 6, 9, 15
Common Core: F-IF-7e
Algebra 1B
Unit 3 – Polynomials and Factoring
Content Outcomes
Students will understand how to add and subtract
polynomials.
Essential Questions
What is a polynomial?
How is a polynomial named based on its degree and
the number of its terms?
Key Concepts
Describing polynomials by its degree and the
number of its terms.
Adding and subtracting polynomials.
Standards Addressed
California: CA 10, 11
Multiplying a polynomial by a monomial.
Factoring a monomial from a polynomial.
California: CA 10, 11
Common Core: A-APR-1
How are polynomials added and subtracted?
Students will understand how to multiply and factor
polynomials.
How is a polynomial multiplied by a monomial?
How is a monomial factored from a polynomial?
Students will understand how to multiply binomials.
Common Core: A-APR-1
What is a binomial?
What is a trinomial?
Multiplying binomials using FOIL and distributive
property.
Multiplying trinomials and binomials.
California: CA 10, 11
Factoring trinomials of the type x2 + bx + c.
California: CA 10, 11
Common Core: A-APR-1
How are binomials multiplied?
Students will understand how to factor trinomials of
the type x2 + bx + c.
How is a binomial multiplied by a trinomial?
How is a trinomial of the type x2 + bx + c factored?
Common Core:
Students will understand how to factor trinomials of
the type ax2 + bx + c.
How is a trinomial of the type ax2 + bx + c factored?
Factoring trinomials of the type ax2 + bx + c.
California: CA 10, 11
Common Core:
Students will understand how to factor by grouping.
How are polynomials with four terms factored?
Factoring polynomials with four terms by grouping.
California: CA 10, 11
Common Core:
20
Algebra 1B
Unit 4 – Quadratic Equations and Functions
Content Outcomes
Students will learn about quadratic graphs.
Essential Questions
How are quadratic functions of the form y=ax2 and y
= ax2 + c graphed?
Key Concepts
Graphing quadratic functions of the form y=ax2 and
y = ax2 + c using tables.
Standards Addressed
California: CA 14, 19 -22
Common Core: F-IF-7a
Students will learn about quadratic functions.
What is a quadratic function?
Graphing quadratic functions of the form
How are quadratic functions of the form y = ax2 +
bx + c graphed?
2a .
How is a quadratic equation solved by graphing?
Solving quadratic
 equations by graphing.
Solving quadratic equations using square roots.
How is a quadratic equation solved using square
roots?
Students will understand how to solve a quadratic
equation using factoring.
California: CA 14, 19 -22
Common Core: F-IF-7a
Graphing quadratic inequalities.
How are quadratic inequalities graphed?
Students will understand how to solve a quadratic
equation.
y = ax2 + bx + c using x = b
How is a quadratic equation solved by factoring?
California: CA 14, 19 -22
Common Core: A-REI-4b
Solving quadratic equations by factoring.
California: CA 14, 19 -22
Common Core: A-SSE-3a; A-REI-4b
Students will understand how to solve a quadratic
equation by completing the square.
How is a quadratic equation solved by completing
the square?
Solving quadratic equations by completing the
square.
California: CA 14, 19 -22
Common Core: A-REI-4b
Students will learn about the quadratic formula.
Students will learn about the discriminant.
How is the quadratic formula used to solve
quadratic equations?
What is a discriminant?

Using the quadratic formula, x =
California: CA 14, 19 -22
b  b2  4ac , to solve quadratic equations.
2a
Common Core: A-REI-4b
Finding the number of solutions of a quadratic
equation using the discriminant: b2 – 4ac.
How is the number of solutions of a quadratic
equation found?
California: CA 14, 19 -22
Common Core:
21
Algebra 1B
Unit 5 – Radical Expressions and Equations
Content Outcomes
Students will understand how to simplify radicals.
Essential Questions
What is a radical?
Key Concepts
Simplifying radicals involving products:
Standards Addressed
California: CA 2
How are radicals involving products simplified?
ab a b .
Simplifying radicals involving quotients:
Common Core:
a

b
How are radicals involving quotients simplified?

Students will understand how to use operations with
radical expressions.
How are sums and differences of radicals
simplified?

How are products and quotients of radicals
simplified?
Students will understand how to solve radical
equations.
How are equations containing radicals solved?
a
b.
Simplifying sums and differences of radicals by
combining like radicals.
Simplifying products of radicals by FOIL or
distributive property.
Simplifying quotients of radicals by rationalizing
the denominator.
California: CA 2
Solving equations containing radicals
Identifying extraneous solutions.
California: CA 2
How are extraneous solutions identified?
Students will understand how to graph square root
functions.
How are square root functions graphed?
Common Core:
Common Core: A-REI 2
Graphing square root functions by finding the
domain and making a table.
California: CA 2
Common Core: F-IF-7b
22
Algebra 1B
Unit 6 – Optional Lessons
Content Outcomes
Students will learn about the Pythagorean Theorem.
Essential Questions
How are real-world problems solved using the
Pythagorean Theorem?
Key Concepts
Pythagorean Theorem: a2 + b2 = c2
Standards Addressed
California:
Common Core:
How are right triangles identified?
Students will understand how to use the distance
formula.
How is the distance formula used to find the
distance between two points?
Distance formula: d =
x2 x1 y2 y1
2
2
California:
Common Core: 8.G-8

Students will understand how to use the midpoint
formula.
How is the midpoint formula used to find the
midpoint of a line segment?
x  x y  y 
Midpoint formula: M = 
 1 2 , 1 2 
 2
2 .
:
California:
Common Core:

Students will learn about trigonometric ratios.
How are trigonometric ratios found?
Finding trigonometric ratios using SOHCAHTOA.
California:
Common Core:
Students will learn about angles of elevation and
angles of depression.
How are trigonometric ratios used?
Solving problems using trigonometric ratios.
California:
Common Core:
23
Algebra 1B
Unit 7 – Rational Explanations and Functions
Content Outcomes
Students will understand how to graph rational
functions.
Essential Questions
What are rational functions?
Key Concepts
Graphing rational functions using the vertical
asymptote and making a table.
How are rational functions graphed?
Standards Addressed
California: 10, 12, 13
Common Core: F-IF-7d
Identifying types of functions by families.
How are types of functions identified?
Students will understand how to simplify rational
expressions.
How are rational expressions simplified?
Simplifying rational expressions.
California: 10, 12, 13
Common Core:
Students will understand how to multiply and divide
rational expressions.
How are rational expressions multiplied?
Multiplying rational expressions.
Dividing rational expressions.
How are rational expressions divided?
Students will understand how to divide
polynomials.
How are polynomials divided?
California: 10, 12, 13
Common Core: A-APR-7
Dividing polynomials using long division.
California: 10, 12, 13
Common Core: A-APR-6
Students will understand how to add and subtract
rational expressions.
How are rational expressions with like
denominators added and subtracted?
How are rational expressions with unlike
denominators added and subtracted?
Students will understand how to solve rational
equations.
How are rational equations solved?
How are proportions solved?
Adding and subtracting rational expressions with
like denominators.
Adding and subtracting rational expressions with
unlike denominators.
California: 10, 12, 13
Solving rational equations by multiplying the
equation with the LCD.
Solving proportions using cross-multiplication.
California: 10, 12, 13
24
Common Core: A-APR-7
Common Core: A-REI 2
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Algebra 1
Average of 60 or higher on the Quantitative and Mathematics portions of the HSPT or Passage of Part I of the Algebra Qualifying Exam or a “B” or higher in Damien’s
Summer School Pre-Algebra course
Yes – Category D
Year
Brief Course Description
This course is designed to provide an introduction to, development of, and study of the basic skills and concepts of elementary algebra concepts. Topics include: fundamental
operations of real numbers, evaluating algebraic expressions, simple linear equations and inequalities, solving systems of linear equations and inequalities, operations on
polynomials – including factoring, integer exponents, introduction to square roots, introduction to graphing, ratio and proportion, applications (word problems), introduction
to algebraic rational expressions and equations, and solving quadratic equations by factoring, the quadratic formula, and completing the square.
Assigned Textbook(s)
Allan E. Bellman, Sadie Chavis Bragg, Randall I. Charles, Basia Hall, William G. Handlin, Sr., Dan Kennedy;
Algebra ; Pearson Prentice Hall; 2007
Algebra 1 Workbook; Pearson
PHSchool.com
Supplemental Material(s)
Common Assessments Utilized
Common Final each semester
In-Class Work
Homework
Quizzes
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
25
Major Content Outcomes
Major Skill Outcomes (include Labs???)
Upon completion of this course the student should be able to:
I. Variables, Functions, Patterns, and Graphs
E. Using Variables
F. Exponents and Order of Operations
G. Exploring Real Numbers
H. Patterns and Functions
15. Perform calculations on real numbers and use order of operations.
16. Simplify and evaluate expressions using multiple algebraic expressions.
17. Solve linear equations and simple quadratic equations.
18. Solve application problems (word problems) using linear equations.
19. Graph linear equations in two variables by using x- and y-intercepts.
II. Rational Numbers
D. Adding, Subtracting, Multiplying and Dividing Rational Numbers
E. The Distributive Property
F. Properties of Numbers
III. Solving Equations
F. Solving Two-Step and Multi-Step Equations
G. Equations With Variables on Both Sides
H. Ratio and Proportion
I.
Equations and Problem Solving
J. Percent of Change
IV. Solving Inequalities
F. Inequalities and Their Graphs
G. Solving Inequalities Using Addition, Subtraction, Multiplication and Division
H. Solving Multi-Step Inequalities
I.
Compound Inequalities
J. Absolute Value Equations and Inequalities
V. Graphs and Functions
E. Relations and Functions
F. Function Rules, Tables, and Graphs
G. Writing a Function Rule
H. Direct Variation and Inverse Variation
VI. Linear Equations and Their Graphs
E. Rate of Change and Slope
F. Slope-Intercept Form and Standard Form
G. Point-Slope Form and Writing Linear Functions
H. Parallel and Perpendicular Lines
VII. Systems of Equations and Inequalities
E. Solving Systems by Graphing, Using Substitution, and Using Elimination
F. Applications of Linear Systems
20. Add, subtract, multiply and factor polynomials.
21. Use the rules/properties of exponents.
Unit 1
Students should be able to define variables and write an equation to model the relationship in a table.
Students should be able to simplify and evaluate an expression.
Students should be able to name the set(s) of numbers to which a number belongs.
Students should be able to identify the independent and dependent variables and write a function rule to
describe the situation.
Unit 2
Students should be able to simplify an expression.
Students should be able to find each sum or difference of rational numbers.
Students should be able to evaluate an expression.
Students should be able to decide which property the equation illustrates.
Unit 3
Students should be able to solve an equation.
Students should be able to define a variable, write an equation, and then solve.
Students should be able to solve a proportion.
Unit 4
Students should be able to determine whether a number is a solution of the given inequality.
Students should be able to define a variable and write an inequality to model a situation.
Students should be able to solve an equation or inequality and then graph the solution.
Students should be able to solve a compound inequality and graph the solutions.
Unit 5
Students should be able to determine whether a relation is a function and if it is a function, state the domain
and range.
Students should be able to find the range of a function given the domain.
Students should be able to model a rule with a table of values and a graph.
Students should be able to write a function rule to describe a statement.
Students should be able to write a function rule for a table of values.
Students should be able to write an equation of the direct variation that includes the given point.
Students should be able to determine whether a graph shows a direct variation.
Students should be able to find the constant of variation for an inverse function.
Unit 6
Students should be able to find the slope of a line that passes through a pair of points.
Students should be able to graph a linear equation.
Students should be able to write the equation of a line in standard form and in slope-intercept form.
Students should be able to find the x- and y-intercepts of a line.
26
G.
H.
Students should be able to write an equation of a line in point-slope form given slope and a point.
Students should be able to write an equation in point-slope form through the given points.
Students should be able to write an equation in slope-intercept form for a line the passes through the given
point and is parallel or perpendicular to the given line.
Linear Inequalities
Systems of Linear Inequalities
VIII. Exponents and Exponential Functions
E. Zero and Negative Exponents
F. Scientific Notation
G. Multiplication and Division Properties of Exponents
H. Exponential Functions
Unit 7
Students should be able to solve a system by graphing, using substitution, and using elimination.
Students should be able to define variables, write an equation, and solve.
Students should be able to graph a linear inequality.
Students should be able to solve a system of inequalities by graphing.
Unit 8
Students should be able to simplify the expression.
Students should be able to evaluate exponential expressions and functions.
Students should be able to write a number in scientific notation.
Students should be able to graph exponential functions.
IX. Polynomials and Factoring
F. Adding and Subtracting Polynomials
G. Multiplying and Factoring
H. Multiplying Binomials and Special Cases
I.
Factoring Trinomials and Special Cases
J. Factoring by Grouping
X. Quadratic Equations and Functions
E. Exploring Quadratic Graphs
F. Quadratic Functions
G. Solving Quadratic Equations by Factoring, Completing the Square, and Using the Quadratic
Formula
H. Using the Discriminant
XI. Radical Expressions and Equations
G. Simplifying Radicals
H. Operations With Radical Expressions
I.
Solving Radical Equations
J. Graphing Square Root Functions
XI. Optional
K. Trigonometric Ratios
L. Angles of Elevation and Depression
XII. Rational Expressions and Functions
E. Graphing Rational Functions
F. Simplifying, Multiplying, Dividing, Adding, and Subtracting Rational Expressions
G. Dividing Polynomials
H. Solving Rational Equations
Unit 9
Students should be able to write a polynomial in standard form and name the polynomial based on its
degree and number of terms.
Students should be able to simply and write the answer in standard form.
Students should be able to simplify each product and write in standard form.
Students should be able to find the GCF of the terms of the polynomial.
Students should be able to factor an expression.
Students should be able to factor completely.
Unit 10
Students should be able to find the equation of the axis of symmetry and the coordinates of the vertex of the
graph of a function.
Students should be able to graph a quadratic function, label the axis of symmetry and the vertex.
Students should be able to graph a quadratic inequality.
Students should be able to find the number of x-intercepts of each function.
Students should be able to solve quadratic equations.
Students should be able to find the number of solutions of a quadratic equation using the discriminant.
Students should be able to solve the quadratic equation using the quadratic formula.
Students should be able to write a quadratic equation in standard form and then solve by factoring.
Students should be able to solve a quadratic equation by completing the square.
Unit 11
Students should be able to simplify a radical expression.
Students should be able to solve a radical equation.
Students should be able to make a table and graph a radical function.
Students should be able to find the domain of a radical function and then graph.
Unit 11 - Optional
Students should be able to find the measure of the other sides of a right triangle given the length of a side
and the measure of an angle.
Students should be able to use the distance formula to find the distance between two points.
Students should be able to use the midpoint formula to find the midpoint between two points.
Unit 12
Students should be able to simplify, multiply, divide, add, and subtract rational expressions.
Students should be able to solve a rational equation.
27
Algebra 1
Unit 1 – Variables, Function Patterns, and Graphs
Content Outcomes
Students will understand the use of variables.
Essential Questions
How are phrases in English written as algebraic
expressions?
Key Concepts
Modeling relationships with variables.
Modeling relationships with equations.
Common Core: 6.EE-2
How are situations modeled with equations?
Students will learn about exponents and order of
operations.
How is order of operations used to simplify
expressions?
How is substitution used to evaluate expressions?
Students will learn about real numbers.
How are real numbers classified?
Simplifying expressions using order of operations
PEMDAS:
Parentheses, Exponents, Multiplication or Division,
Addition or Subtraction.
Evaluating expressions using substitution.
California: CA 17
Classifying real numbers.
California: CA 17
How are real numbers compared?
Students will learn about patterns and functions.
How are dependent and independent quantities
identified?
Standards Addressed
California: CA 17
Common Core:
Common Core: 8.NS-1
Writing Function Rules.
Understanding dependent and independent
quantities.
How are relationships written as functions?
28
California: CA 17
Common Core: 6.EE-9
Algebra 1
Unit 2 – Rational Numbers
Content Outcomes
Students will understand how to add rational
numbers.
Essential Questions
What are real numbers?
Key Concepts
Adding numbers with the same sign.
Adding numbers with different signs.
How are real numbers added?
Students will understand how to subtract rational
numbers.
What are rational numbers?
Common Core: 7.NS-1
Subtracting numbers.
How are rational numbers subtracted?
Students will understand how to multiply and divide
rational numbers.
How are real numbers multiplied?
How are real numbers divided?
Students will understand the Distributive Property.
What is the Distributive Property?
What are the properties of real numbers?
California: CA 1
Common Core: 7.NS-1c
Multiplying and dividing numbers with the same
sign.
Multiplying and dividing numbers with different
signs.
California: CA 1
The Distributive Property.
California: CA 1
How is the Distributive Property used to simplify
expressions?
Students will learn about the properties of numbers.
Standards Addressed
California: CA 1
Common Core: 7.NS-2
Common Core: 7.NS-2a
Properties of real numbers.
How are the properties of real numbers identified?
California: CA 1
Common Core: 7.NS-1d, 7.NS-2c
29
Algebra 1
Unit 3 – Solving [Simple] Equations
Content Outcomes
Students will understand how to solve two-step
equations.
Essential Questions
What are inverse operations?
Key Concepts
Solving two-step-equations using inverse
operations.
How are inverse operations used to solve two-stepequations?
Students will understand how to solve multi-step
equations.
How is Distributive Property used to combine like
terms?
Standards Addressed
California: CA 4,5
Common Core: A-REI-3
Using Distributive Property to combine like terms.
Using Distributive Property to solve equations.
California: CA 4,5
Common Core: A-REI-3
How is Distributive Property used to solve
equations?
Students will understand how to solve equations
with variables on both sides.
How are equations with variables on both sides
solved?
Solving equations with variables on both sides.
Identifying equations that are identities and have no
solution.
California: CA 4,5
Solving algebraic proportions.
California: CA 4,5
Common Core: A-REI-3
How are equations that are identities or have no
solution identified?
Students will learn about ratios and proportions.
What are proportions?
How are algebraic proportions solved?
Students will understand how to solve real-world
problems.
How is a variable defined in terms of another
variable?
Common Core: A-REI-3
Defining a variable in terms of another variable.
Solving distance-rate-time problems.
Solving consecutive integer problems.
How are distance-rate-time problems solved?
How are consecutive integer problems solved?
30
California: CA 4,5
Common Core: A-REI-3
Algebra 1
Unit 4 – Solving Inequalities
Content Outcomes
Students will learn about inequalities and their
graphs.
Essential Questions
How are solutions of inequalities identified?
How are solutions of inequalities graphed?
Students will understand how to solve inequalities
using addition and subtraction.
How are inequalities solved using addition?
Key Concepts
Identifying solutions of inequalities using
substitution.
Graphing inequalities.
Standards Addressed
California: CA 3 - 5
Using addition to solve inequalities.
Using subtraction to solve inequalities.
California: CA 3 - 5
How are inequalities solved using subtraction?
Students will understand how to solve inequalities
using multiplication and division.
How are inequalities solved using multiplication?
Common Core: A-REI-3
Using multiplication to solve inequalities.
Using division to solve inequalities.
How are inequalities solved using division?
Students will understand how to solve multi-step
inequalities.
How are multi-step inequalities with variables on
one side solved?
How are multi-step inequalities with variables on
both sides solved?
Students will understand how to solve compound
inequalities.
How are inequalities containing “and” solved and
graphed?
How are equations that involve absolute value
solved?
California: CA 3 - 5
Common Core: A-REI-3
Solving multi-step inequalities with variables on
one side.
Solving multi-step inequalities with variables on
both sides.
California: CA 3 - 5
Solving and graphing inequalities containing “and”.
Solving and graphing inequalities containing “or”.
California: CA 3 - 5
Common Core: A-REI-3
Common Core: A-REI-3
How are inequalities containing “or” solved and
graphed?
Students will understand how to solve absolute
value equations and inequalities.
Common Core:
Solving equations that involve absolute value.
Solving inequalities that involve absolute value.
California:
Common Core:
How are inequalities that involve absolute value
solved?
31
Algebra 1
Unit 5 – Solving Inequalities
Content Outcomes
Students will learn about relations and functions.
Essential Questions
What is a function?
Key Concepts
Identifying relations and functions.
Evaluating functions.
How are relations and functions identified?
Standards Addressed
California: CA 16, 17
Common Core: F-IF-1
How are functions evaluated?
Students will understand how to use function rules,
tables, and graphs.
How are functions modeled using rules, tables, and
graphs?
Modeling functions using rules, tables, and graphs.
California: CA 16, 17
Common Core:
Students will understand how to write a function
rule.
How is a function rule written given a table or realworld situation?
Writing a function rule given a table or real-world
situation.
California: CA 16, 17
Common Core:
Students will learn about direct variation.
How is a direct variation equation written?
Writing an equation of a direct variation.
Using ratios and proportions with direct variations.
How are ratios and proportions used with direct
variations?
Students will learn about inverse variation.
How are inverse variations solved?
California: CA 16, 17
Common Core:
Solving inverse variations.
Comparing direct and inverse variation.
How do direct variations and inverse variations
compare?
California: CA 16, 17
Common Core:
32
Algebra 1
Unit 6 – Linear Equations and their Graphs
Content Outcomes
Students will learn about the rate of change and
slope.
Essential Questions
How are rates of change found from tables and
graphs?
What is slope?
How is slope [of a line] found?
Students will learn about the slope-intercept form of
a line.
What is slope-intercept form of a line?
What is a linear equation?
Key Concepts
Finding rates of change from tables and graphs.
Finding slope using the slope formula: y 2  y1 .
x 2  x1

Writing linear equations in slope-intercept form: y =
mx + b.
Graphing linear equations in slope-intercept form.
Standards Addressed
California: CA 6 -8
Common Core:
California: CA 6 -8
Common Core: F-IF-7a
How are linear equations written in slope-intercept
form?
Students will learn about the standard form of a
line.
How are linear equations graphed [using slope and
y-intercept]?
What is an intercept?
Graphing equations using x- and y-intercepts.
Writing equations in standard form: Ax + By = C.
How are linear equations graphed using x- and yintercepts?
California: CA 6 -8
Common Core: F-IF-7a
How are linear equations written in standard form?
Students will learn about the point-slope form of a
line and how to write linear equations.
What is point-slope form?
Graphing and writing linear equations using pointslope form: y – y1 = m(x – x1).
How are linear equations graphed using point-slope
form?
Students will learn about parallel and perpendicular
lines.
How are linear equations written using point-slope
form?
What are parallel lines?
What are perpendicular lines?
California: CA 6 -8
Common Core:
Determining whether the lines are parallel by
comparing slopes.
Determining whether the lines are perpendicular by
comparing slopes.
How are parallel lines determined?
How are perpendicular lines determined?
33
California: CA 6 -8
Common Core:
Algebra 1
Unit 7 – Systems of Equations and Inequalities
Content Outcomes
Students will understand how to solve systems by
graphing.
Essential Questions
What is a system of linear equations?
How are systems of linear equations solved by
graphing?
Key Concepts
Solving systems of linear equations by graphing.
Determining possible number of solutions of a
system of linear equations.
Standards Addressed
California: CA 6, 9, 15
Solving systems of equations using substitution.
California: CA 6, 9, 15
Common Core: A-REI-6
What is the possible number of solutions of a
system of linear equations?
Students will understand how to solve systems
using substitution.
How are systems of linear equations solved using
substitution?
Common Core:
Students will understand how to solve systems
using elimination.
How are systems of linear equations solved using
elimination?
Solving systems of linear equations using
elimination.
California: CA 6, 9, 15
Common Core:
Students will understand how to apply linear
systems to solve real-world problems.
How are real-world problems written as a system of
linear equations?
Writing systems of linear equations given realworld problems.
California: CA 6, 9, 15
Common Core: A-CED-2
Students will learn about linear inequalities.
How are linear inequalities graphed?
Graphing linear inequalities.
California: CA 6, 9, 15
Common Core:
Students will understand how to solve systems of
linear inequalities.
How are systems of linear inequalities graphed?
Graphing systems of linear inequalities.
California: CA 6, 9, 15
Common Core:
34
Algebra 1
Unit 8 – Exponents and Exponential Functions
Content Outcomes
Students will understand how to use zero and
negative exponents.
Essential Questions
How are expressions with zero and negative
exponents simplified?
Key Concepts
Simplifying Expressions with Zero and Negative
0
How are exponential expressions evaluated?
Exponents: a =1; a
-n
n
1 1  n
= n ;   =a
a a 
Standards Addressed
California: CA 6, 9, 15
Common Core:
Evaluating Exponential expressions.
Students will learn about scientific notation.
What is scientific notation?
Writing number in scientific [a x 10n] and standard
notation.
 
What is standard notation?
California: CA 6, 9, 15
Common Core: 8.EE-3, 8.EE-4
How are numbers written in scientific and standard
notation?
Students will understand how to use multiplication
properties of exponents.
How is scientific notation used?
What is a power?
How are powers multiplied?
Multiplying powers: am x an = am+n
Raising power to a power: (am)n = amn
Raising a product to a power: (ab)n = anbn
California: CA 6, 9, 15
a m m–n
=a
an
a n a n
Raising a quotient to a power:   = n
b  b .
California: CA 6, 9, 15
Common Core:
How is a power raised to a power?
How is a product raised to a power?
Students will understand how to use division
properties of exponents.
How are powers with the same base divided?
How is a quotient raised to a power?
Students will learn about exponential functions.
How are exponential functions evaluated?
Dividing powers with the same base:
Evaluating exponential
functions.
Graphing exponential functions.
How are exponential functions graphed?

35

Common Core:
California: CA 6, 9, 15
Common Core: F-IF-7e
Algebra 1
Unit 9 – Polynomials and Factoring
Content Outcomes
Students will understand how to add and subtract
polynomials.
Essential Questions
What is a polynomial?
How is a polynomial named based on its degree and
the number of its terms?
Key Concepts
Describing polynomials by its degree and the
number of its terms.
Adding and subtracting polynomials.
Standards Addressed
California: CA 10, 11
Multiplying a polynomial by a monomial.
Factoring a monomial from a polynomial.
California: CA 10, 11
Common Core: A-APR-1
How are polynomials added and subtracted?
Students will understand how to multiply and factor
polynomials.
How is a polynomial multiplied by a monomial?
How is a monomial factored from a polynomial?
Students will understand how to multiply binomials.
Common Core: A-APR-1
What is a binomial?
What is a trinomial?
Multiplying binomials using FOIL and distributive
property.
Multiplying trinomials and binomials.
California: CA 10, 11
Factoring trinomials of the type x2 + bx + c.
California: CA 10, 11
Common Core: A-APR-1
How are binomials multiplied?
Students will understand how to factor trinomials of
the type x2 + bx + c.
How is a binomial multiplied by a trinomial?
How is a trinomial of the type x2 + bx + c factored?
Common Core:
Students will understand how to factor trinomials of
the type ax2 + bx + c.
How is a trinomial of the type ax2 + bx + c factored?
Factoring trinomials of the type ax2 + bx + c.
California: CA 10, 11
Common Core:
Students will understand how to factor by grouping.
How are polynomials with four terms factored?
Factoring polynomials with four terms by grouping.
California: CA 10, 11
Common Core:
36
Algebra 1
Unit 10 – Quadratic Equations and Functions
Content Outcomes
Students will learn about quadratic graphs.
Essential Questions
How are quadratic functions of the form y=ax2 and y
= ax2 + c graphed?
Key Concepts
Graphing quadratic functions of the form y=ax2 and
y = ax2 + c using tables.
Standards Addressed
California: CA 14, 19 -22
Common Core: F-IF-7a
Students will learn about quadratic functions.
What is a quadratic function?
Graphing quadratic functions of the form
How are quadratic functions of the form y = ax2 +
bx + c graphed?
2a .
How is a quadratic equation solved by graphing?
Solving quadratic
 equations by graphing.
Solving quadratic equations using square roots.
How is a quadratic equation solved using square
roots?
Students will understand how to solve a quadratic
equation using factoring.
California: CA 14, 19 -22
Common Core: F-IF-7a
Graphing quadratic inequalities.
How are quadratic inequalities graphed?
Students will understand how to solve a quadratic
equation.
y = ax2 + bx + c using x = b
How is a quadratic equation solved by factoring?
California: CA 14, 19 -22
Common Core: A-REI-4b
Solving quadratic equations by factoring.
California: CA 14, 19 -22
Common Core: A-SSE-3a; A-REI-4b
Students will understand how to solve a quadratic
equation by completing the square.
How is a quadratic equation solved by completing
the square?
Solving quadratic equations by completing the
square.
California: CA 14, 19 -22
Common Core: A-REI-4b
Students will learn about the quadratic formula.
Students will learn about the discriminant.
How is the quadratic formula used to solve
quadratic equations?
What is a discriminant?

Using the quadratic formula, x =
California: CA 14, 19 -22
b  b2  4ac , to solve quadratic equations.
2a
Common Core: A-REI-4b
Finding the number of solutions of a quadratic
equation using the discriminant: b2 – 4ac.
How is the number of solutions of a quadratic
equation found?
California: CA 14, 19 -22
Common Core:
37
Algebra 1
Unit 11 – Radical Expressions and Equations
Content Outcomes
Students will understand how to simplify radicals.
Essential Questions
What is a radical?
Key Concepts
Simplifying radicals involving products:
Standards Addressed
California: CA 2
How are radicals involving products simplified?
ab a b .
Simplifying radicals involving quotients:
Common Core:
a
a

b
b.
How are radicals involving quotients simplified?

Students will understand how to use operations with
radical expressions.
How are sums and differences of radicals
simplified?

How are products and quotients of radicals
simplified?
Students will understand how to solve radical
equations.
How are equations containing radicals solved?
Simplifying sums and differences of radicals by
combining like radicals.
Simplifying products of radicals by FOIL or
distributive property.
Simplifying quotients of radicals by rationalizing
the denominator.
California: CA 2
Solving equations containing radicals
Identifying extraneous solutions.
California: CA 2
How are extraneous solutions identified?
Students will understand how to graph square root
functions.
How are square root functions graphed?
Common Core:
Common Core: A-REI 2
Graphing square root functions by finding the
domain and making a table.
California: CA 2
Common Core: F-IF-7b
38
Algebra 1
Unit 11 – Optional Lessons
Content Outcomes
Students will learn about the Pythagorean Theorem.
Essential Questions
How are real-world problems solved using the
Pythagorean Theorem?
Key Concepts
Pythagorean Theorem: a2 + b2 = c2
Standards Addressed
California:
Common Core:
How are right triangles identified?
Students will understand how to use the distance
formula.
How is the distance formula used to find the
distance between two points?
Distance formula: d =
x2 x1 y2 y1
2
2
California:
Common Core: 8.G-8

Students will understand how to use the midpoint
formula.
Students will learn about trigonometric ratios.
How is the midpoint formula used to find the
midpoint of a line segment?
How are trigonometric ratios found?
x  x y  y 2 
Midpoint formula: M = 
 1 2 , 1

 2
2 .

Finding trigonometric ratios using SOHCAHTOA.
:
California:
Common Core:
California:
Common Core:
Students will learn about angles of elevation and
angles of depression.
How are trigonometric ratios used?
Solving problems using trigonometric ratios.
California:
Common Core:
39
Algebra 1
Unit 12 – Rational Explanations and Functions
Content Outcomes
Students will understand how to graph rational
functions.
Essential Questions
What are rational functions?
Key Concepts
Graphing rational functions using the vertical
asymptote and making a table.
How are rational functions graphed?
Standards Addressed
California: 10, 12, 13
Common Core: F-IF-7d
Identifying types of functions by families.
How are types of functions identified?
Students will understand how to simplify rational
expressions.
How are rational expressions simplified?
Simplifying rational expressions.
California: 10, 12, 13
Common Core:
Students will understand how to multiply and divide
rational expressions.
How are rational expressions multiplied?
Multiplying rational expressions.
Dividing rational expressions.
How are rational expressions divided?
Students will understand how to divide
polynomials.
How are polynomials divided?
California: 10, 12, 13
Common Core: A-APR-7
Dividing polynomials using long division.
California: 10, 12, 13
Common Core: A-APR-6
Students will understand how to add and subtract
rational expressions.
How are rational expressions with like
denominators added and subtracted?
How are rational expressions with unlike
denominators added and subtracted?
Students will understand how to solve rational
equations.
How are rational equations solved?
How are proportions solved?
Adding and subtracting rational expressions with
like denominators.
Adding and subtracting rational expressions with
unlike denominators.
California: 10, 12, 13
Solving rational equations by multiplying the
equation with the LCD.
Solving proportions using cross-multiplication.
California: 10, 12, 13
40
Common Core: A-APR-7
Common Core: A-REI 2
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Geometry
“C” or better in Algebra 1
Yes – Category D
Year
Brief Course Description
This course covers the foundations of geometrical figures and their measurement. Beginning with the component part of geometrical
figures – points, lines, and planes – and through the use of reasoning and proof, the course encompasses the study of triangles,
quadrilaterals, other polygons, circles, and solids. Through the study of definitions, postulates, and theorems, in addition to other related
mathematical topics, the properties of these figures are incorporated into an understanding and ability to construct and measure both plane
figures and solids. Major topics in the course include deductive and inductive reasoning, triangle relationships and congruence, right
triangle trigonometry, similarity, areas of plane figures, and surface areas and volumes of solids.
Assigned Textbook(s)
Laurie Bass and Art Johnson; Geometry; Prentice Hall
Accompanying Workbook; Geometry Workbook; Prentice Hall
Supplemental Material(s)
Videos: topical episodes of a Caltech produced series entitled Mathematics
(Similarity, the Pythagorean Theorem, and Pi)
Common Assessments Utilized
Common Final each semester
In-Class Worksheets & Problems
Quizzes
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
Overview of Course / Skill Outcomes
41
Major Content Outcomes
Students will be introduced to inductive and deductive forms of reasoning and use of this
reasoning in arriving at the fundamental proofs of geometry.
Major Skill Outcomes
Students will be able to prove basic theorems, and prove triangle congruence and similarity.
Students will be able to measure in the coordinate plane.
Students will be able to find measures of areas, surface areas, and volumes of plane and space figures.
Students will be able to construct basic geometrical figures with a straight edge and compass.
Students will be able to apply geometric knowledge in solving selected “real-world” problems modeled
with geometric figures, particularly in areas related to trigonometry and the Pythagorean Theorem.
Students will be able to use properties of geometric figures to determine equations and solve for unknown
dimensions of the figures.
Students will be introduced to the fundamental properties of geometric figures and the use of
them in constructing and working with them.
Students will be introduced to the principles of measuring components of geometric figures
such as segments and angles.
Students will be introduced to triangle relationships, including congruence, similarity, and right
triangle trigonometry.
Students will be introduced to the measurement of areas of plane figures and surface area and
volume of solids.
Unit 1
Students should be able to describe a pattern and find the next term in the sequence.
Students should be able to name collinear and coplanar points.
Students should be able to find the intersection of a line and a plane.
Students should be able to identify opposite rays.
Students should be able to find the length of segments using Algebra.
Students should be able to find the length of segments based on distance from the midpoint.
Students should be able to find the area of a square, rectangle, and circle.
Students should be able to identify complementary angles, supplementary angles, and perpendicular
bisectors of a given figure.
Students should be able to correctly name lines, segments, rays, and angles using correct notation.
Unit 2
Students should be able to identify the hypothesis and conclusion of a statement.
Students should be able to use a statement to write a conditional.
Students should be able to find a counterexample to show a statement is not true.
Students should be able to write the converse of a conditional.
Students should be able to explain why a statement is not a good definition.
Students should be able to identify certain Geometric Properties.
Students should be able to find the measure of each angle in a given figure.
Students should be able to rewrite bi-conditionals as two conditionals and vice versa.
Students should be able to use the Law of Detachment and the Law of Syllogism to draw conclusions from
statements.
Unit 3
Students should be able to classify triangles by their sides and angles.
Students should be able to find the measures of angles based on parallel lines and a transversal.
Students should be able to find the values for a variable for which two lines must be parallel.
Students should be able to write the equation of a line given a slope and a point the line passes through.
Students should be able to write the equation of a line given two points the line passes through.
Students should be able to find the measures of interior/exterior angles in various polygons.
Students should be able to tell whether two lines are parallel or perpendicular based on their slopes.
Unit 4
Students should be able to identify triangle congruency based on various given congruent angles and sides.
Students should be able to identify which postulate can be used to prove triangle congruency based on the
given information.
Students should be able to find the values of sides and angles based on triangle congruency.
Students should be able to identify triangle congruency in overlapping triangles.
42
Unit 5
Students should be able to write the inverse and contrapositive of a statement.
Students should be able to identify statements that contradict one another.
Students should be able to list angles in order of size based on the length of opposite sides.
Students should be able to list sides in order of size based on the length of opposite angles.
Students should be able to find the measure of sides and angles using algebra.
Students should be able to find the center of a circle that can be circumscribed about a triangle.
Unit 6
Students should be able to classify quadrilaterals in as many ways as possible.
Students should be able to find the values of variables based on the properties of various quadrilaterals.
Students should be able to find the measures of angles and sides in parallelograms.
Students should be able to prove a quadrilateral is a parallelogram based on known properties.
Students should be able to find the measure of sides and angles in a rhombus and in a rectangle.
Students should be able to find the measure of sides and angles in a kite and in an isosceles trapezoid.
Students should be able to place quadrilaterals in the coordinate plane and identify coordinates of each
vertex.
Unit 7
Students should be able to write ratios and solve various proportions.
Students should be able to identify similar polygons and give the similarity ratio.
Students should be able to find the values of variables in similar polygons.
Students should be able to prove triangles are similar and write similarity statements.
Students should be able to explain why triangles are similar by using algebra to solve.
Students should be able to find the geometric mean in a pair of numbers.
Students should be able to find the values of variables in right triangles.
Students should be able to use proportions in triangles to solve for variables.
Unit 8
Students should be able to find the lengths of sides of a right triangle using the Pythagorean Theorem.
Students should be able to decide whether a set of numbers form a Pythagorean triple.
Students should be able to determine whether a triangle is a right triangle based on given values.
Students should be able to find the values of variables in 45-45-90 and 30-60-90 right triangles.
Students should be able to write tangent ratios.
Students should be able to find the values of variables based on the tangent ratio.
Students should be able to write sine and cosine ratios.
Students should be able to find the values of variables based on sine and cosine ratios.
Students should be able to identify and find the angles of elevation and depression.
Students should be able to describe vectors as ordered pairs, give the coordinates, and describe the direction
of a vector.
Unit 9
Students should be able to state whether a transformation image appears to be an isometry.
Students should be able to find the image of a figure under a given translation.
Students should be able to find the coordinates of reflection images in the coordinate plane.
Students should be able to draw reflection images across a line of reflection.
Students should be able to draw an image based on a given rotation.
Students should be able to tell what type symmetry can be found in a given figure.
Students should be able to draw the lines of symmetry in a given figure.
Students should be able to describe the dilation image of a figure.
Students should be able to find the image of points in the coordinate plane for a given scale factor.
Students should be able to classify isometries.
43
Students should be able to find the glide reflection image of a given figure in the coordinate plane.
Students should be able to identify whether a figure shows a tessellation of repeating figures.
Students should be able to determine whether a figure will tessellate in a plane.
Students should be able to list the symmetries in each tessellation.
Unit 10
Students should be able to find the area of a parallelogram.
Students should be able to find the area of a triangle.
Students should be able to find the area of a trapezoid.
Students should be able to find the area of a kite.
Students should be able to find the area of a rhombus.
Students should be able to find the area of a regular polygon.
Students should be able to find the measure of various angles in polygons based on given radius and
apothem.
Students should be able to find the perimeters and areas based on ratios of similar figures.
Students should be able to find the areas of regular polygons using trigonometry.
Students should be able to find the circumference of a circle and the measures of arcs in circles.
Students should be able to find the area of an entire circle, a sector of a circle, and a shaded section of a
circle.
Students should be able to find geometric probability in various figures.
Unit 11
Students should be able to find the number of vertices, edges, and faces in a polyhedron.
Students should be able to use Euler’s Formula to find the number of faces, edges, or vertices in a
polyhedron.
Students should be able to describe the cross section of a 3-D figure.
Students should be able to find the surface area of a prism using nets.
Students should be able to find the surface area of a cylinder.
Students should be able to find the surface area and lateral area of a pyramid and a cone.
Students should be able to find the volume of a prism, cylinder, and a composite space figure.
Students should be able to find the volume of a square pyramid and a cone.
Students should be able to find the surface area of a sphere from a given diameter.
Students should be able to find the volume of a sphere and the surface area based on a given volume.
Students should be able to identify similarity in 3-D figures and give the similarity ratio.
Students should be able to use the similarity ratio to find volumes of similar figures.
Unit 12
Students should be able to find the values of variables based on tangent lines and the center of a circle.
Students should be able to determine whether a line on a circle is a tangent line.
Students should be able to find the values of variables based on given chords and arcs of circles.
Students should be able to identify inscribed angles and their intercepted arcs.
Students should be able to find the values of variables of inscribed angles within circles.
Students should be able to find the values of variables based on given angle measures and segment lengths
in circles.
Students should be able to write the standard equation of a circle with a given center.
Students should be able to find the center and radius of a circle then graph the circle in the coordinate plane.
Students should be able to draw and describe a locus in a plane.
Students should be able to draw and describe a locus in the coordinate plane.
44
Geometry
Unit 1 – What are the basic tools of Geometry?
45
Content Outcomes
Students will learn how to use patterns and
inductive reasoning.
Essential Questions
How is inductive reasoning used to make
conjectures?
Key Concepts
Definition of inductive reasoning
Identifying patterns/counterexamples
Standards Addressed
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
What are some ways to find and use patterns?
How can patterns be used to find counterexamples?
Students will understand points, lines, and planes.
What are points, lines, and planes?
Definitions of points, lines, and planes
Using postulates
What are postulates and how are they understood?
Students will understand segments, rays, parallel
lines, and planes.
How are segments and rays identified?
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
Definitions of segments and rays
Definition of parallel lines
How are parallel lines recognized?
Students will understand how to measure segments.
How are the lengths of segments determined?
California: 1.0, 3.0, 8.0, 16.0, 17.0
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
Finding the length of segments
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
Students will understand how to measure angles.
How are the measures of angles determined?
Definition of angles/angle pairs
Measuring angles
How are special angle pairs identified?
Students will understand the coordinate plane.
What is the coordinate plane?
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
Definition of the coordinate plane and its uses
(distance/midpoint)
How is the coordinate plane used to find the
distance between two points?
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
How is the coordinate plane used to find the
midpoint of a segment?
Geometry
Unit 2 – How are the concepts of reasoning and proofs used in Geometry?
46
Content Outcomes
Students will understand how to write and recognize
conditional statements.
Essential Questions
How are conditional statements recognized?
How are the converses of conditional statements
written?
How is the hypothesis and conclusion in a
conditional statement identified?
What are counterexamples and how are they found?
Key Concepts
Definition of a conditional statement (if-then
statement)
Definition of converse
Identifying the hypothesis and conclusion in a
conditional statement
Identifying counterexamples
to
conditional
statements
Definition/identification of Venn Diagram
Identifying truth value of conditional statements
Standards Addressed
California: 1.0, 2.0, 4.0, 13.0
Definition of bi-conditionals
Splitting bi-conditionals into two separate
statements
Recognizing good definitions (necessary
requirements of good definitions)
California: 1.0, 2.0, 4.0, 13.0
Definition of the Law of Detachment
Definition of the Law of Syllogism
California: 1.0, 2.0, 4.0, 13.0
Common Core: G-CO.9
How are Venn Diagrams used?
Students will understand how to write biconditionals and definitions.
How is the truth value of a conditional statement
found?
How are bi-conditionals written?
How are bi-conditionals split into separate parts?
How are good definitions recognized?
Students will understand deductive reasoning.
How are bi-conditionals written from definitions?
What is the Law of Detachment and how is it used?
What is the Law of Syllogism and how is it used?
Students will understand reasoning in algebra.
How is reasoning in algebra connected to reasoning
in geometry?
Common Core: G-CO.9
Connecting algebra to geometry
How are theorems about angles used and applied in
Geometry?
California: 1.0, 2.0, 4.0, 13.0
Common Core: G-CO.9
What tools are used to justify steps in solving
equations?
Students will understand how to prove angles
congruent.
Common Core: G-CO.9
Definition of theorem
Vertical Angles Theorem
California: 1.0, 2.0, 4.0, 13.0
Common Core: G-CO.9
What is the Vertical Angles theorem?
47
Content Outcomes
Students will understand the properties of parallel
lines.
Students will understand how to prove lines
parallel.
Students will understand parallel and perpendicular
lines.
Geometry
Unit 3 – How are parallel and perpendicular lines identified and used in Geometry?
Essential Questions
Key Concepts
What is a transversal?
Definition of a transversal
Identifying angle pairs
How are angles formed by two lines and transversal
identified?
What are some properties of parallel lines and how
are they used?
How is a transversal used to prove lines parallel?
What are perpendicular lines?
How are parallel and perpendicular lines related?
Students will understand parallel lines and the
Triangle Angle-Sum Theorem.
What are triangles and how are they classified?
How are the measures of the angles in a triangle
found?
Standards Addressed
California: 1.0, 7.0, 12.0, 13.0, 16.0
Common Core: G-CO.9
Corresponding Angles Postulate
Alternate Interior Angles Theorem
Same-Side Interior Angles Theorem
Alternate Exterior Angles Theorem
Same-Side Exterior Angles Theorem
California: 1.0, 2.0, 4.0, 13.0
Converse of the Corresponding Angles Postulate
Converse of The Alternate Interior Angles Theorem
Converse of the Same-Side Interior Angles
Theorem
Converse of the Alternate Exterior Angles Theorem
Converse of the Same-Side Exterior Angles
Theorem
Multiple Parallel/Perpendicular Lines Theorem
California: 1.0, 2.0, 4.0, 13.0
Triangle Angle-Sum Theorem
Definitions of equiangular, acute, right, obtuse,
equilateral, isosceles, and scalene triangles
Triangle Exterior Angle Theorem
California: 1.0, 2.0, 4.0, 13.0
Definition of a polygon (concave/convex)
Polygon Angles-Sum Theorem
Polygon Exterior Angle-Sum Theorem
California: 1.0, 2.0, 4.0, 13.0
Slope-intercept Form
Standard Form of a Linear Equation
Point-Slope Form
California: 1.0, 2.0, 4.0, 13.0
Common Core: G-CO.9
Common Core: G-CO.9
Common Core: G-CO.9
How are the exterior angles of triangles used?
Students will understand the Polygon Angle-Sum
Theorem.
What is a polygon and how are they classified?
How are the sums of the measures of the
interior/exterior angles of a polygon found?
Students will understand how to use lines in the
coordinate plane.
Given their equations, how are lines graphed in the
coordinate plane?
What are slope-intercept form, standard form, and
point-slope form of linear equations?
Students will understand the relationship of slopes
of parallel and perpendicular lines.
What are the equations for horizontal/vertical lines?
How are slope and parallel/perpendicular lines
related?
Slopes of Parallel Lines
Slopes of Perpendicular Lines
How are the equations of parallel/perpendicular
lines written?
48
Common Core: G-CO.9
Common Core: G-CO.9
Content Outcomes
Students will understand congruent figures.
Geometry
Unit 4 – What are the different ways to identify triangle congruency?
Essential Questions
Key Concepts
What are congruent figures and how are they Definitions of Congruent Figures
recognized?
Triangle 3rd Angle Congruency Theorem
Common Core: G-CO.7, G-CO.8, G-CO.10
How are the corresponding parts of congruent
figures identified and named?
Students will understand how to prove triangle
congruency by SSS and SAS.
How are triangles proved congruent by SSS?
Standards Addressed
California: 1.0, 4.0, 5.0
Side-Side-Side (SSS) Postulate
Side-Angle-Side (SAS) Postulate
How are triangles proved congruent by SAS?
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
What are some ways to tell if two triangles are
congruent?
Students will understand how to prove triangle
congruency by ASA and AAS.
How are triangles proved congruent by ASA?
Angle-Side-Angle (ASA) Postulate
Angle-Angle-Side (AAS) Postulate
How are triangles proved congruent by AAS?
Students will understand how to use congruent
triangles and CPCTC.
How is triangle congruence and CPCTC used to
prove other parts of two triangles congruent?
Common Core: G-CO.7, G-CO.8, G-CO.10
Corresponding Parts of Congruent Triangles are
Congruent (CPCTC)
What are the properties of isosceles triangles and
how are they applied?
What are the properties of equilateral triangles?
Students will understand how to prove congruency
in right triangles.
What is the HL Theorem?
Isosceles Triangle Theorem
Converse of Isosceles Triangle Theorem
Isosceles Triangle Bisector Theorem
Definition of a corollary
Corollary to Isosceles Triangle Theorem
Corollary to Converse of Isosceles Triangle
Theorem
Hypotenuse-Leg (HL) Theorem
How is the HL Theorem used to prove triangles
congruent?
Students will understand how to use corresponding
parts of congruent triangles.
How are congruent overlapping triangles identified?
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
What is CPCTC and how is it applied to congruent
triangles?
Students will understand isosceles and equilateral
triangles.
California: 1.0, 4.0, 5.0
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
Identifying overlapping congruent triangles
Using two pairs of triangles
How are common parts within overlapping triangles
identified and used?
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
How can two triangles be proven congruent using
another pair of triangles?
What are ways to separate overlapping triangles?
49
Content Outcomes
Students will understand mid-segments of triangles.
Geometry
Unit 5 – What are the various relationships within a Triangle?
Essential Questions
Key Concepts
What is the mid-segment of a triangle?
Definition of a Midsegment
Triangle Midsegment Theorem
How are the properties of mid-segments used to Definition of a Coordinate Proof
solve problems in geometry?
Standards Addressed
California: 1.0, 6.0, 12.0
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
What is a coordinate proof and how is it used?
Students will understand bisectors in triangles.
What are the properties of perpendicular bisectors
and how are they applied?
What are the properties of angle bisectors and how
are they applied?
Students will understand concurrent lines, medians,
and altitude.
How are the properties of perpendicular bisectors
identified?
How are the properties of angle bisectors identified?
What are the circumcenter, incenter, and
orthocenter of a triangle?
Perpendicular Bisector Theorem
Converse of the Perpendicular Bisector Theorem
Angle Bisector Theorem
Converse of the Angle Bisector Theorem
Definition of concurrent/points of concurrency
California: 1.0, 6.0, 12.0
Concurrent Perpendicular Bisector Theorem
Concurrent Angle Bisector Theorem
Definition of circumcenter/incenter/orthocenter of a
triangle
Triangle Medians Theorem
Triangle Altitudes Theorem
California: 1.0, 6.0, 12.0
Definition of contrapositive
Application of contrapositive to a statement
Indirect Proofs
California: 1.0, 6.0, 12.0
Comparison Property of Inequality
Corollary to the Triangle Exterior Angle Theorem
Unequal Triangle Sides Theorem
Unequal Triangle Angles Theorem
Triangle Inequality Theorem
California: 1.0, 6.0, 12.0
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
How are the properties of the medians of a triangle
identified?
How are the properties of altitudes of triangles
identified?
Students will understand inverses, contrapositives,
and indirect reasoning.
How is the negation of a statement written?
How are the inverse and contrapositive of a
conditional statement written?
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
How is indirect reasoning used in proofs?
Students will understand inequalities in triangles.
How are inequalities used when involving angles of
triangles?
How are inequalities used when involving sides of
triangles?
What is the Triangle Inequality Theorem and how is
it applied to triangles?
50
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
Content Outcomes
Students will understand how to classify
quadrilaterals.
Students will understand the properties of
parallelograms.
Students will understand how to prove that a
quadrilateral is a parallelogram.
Students will understand special parallelograms.
Students will understand trapezoids and kites.
Students will understand how to place figures in the
coordinate plane.
Students will understand proofs using coordinate
geometry.
Geometry
Unit 6 – What are the various Quadrilaterals and their properties?
Essential Questions
Key Concepts
How are quadrilaterals defined?
Definitions/classifications of a parallelogram,
How are quadrilaterals classified?
rhombus, rectangle, square, kite, trapezoid, and
isosceles trapezoid
Standards Addressed
California: 1.0, 7.0, 12.0
Common Core: G.CO-11, G-GPE.4, G-GPE.7
What are the relationships among the sides of
parallelograms and how are they used?
What are the relationships among the angles of
parallelograms and how are they used?
How are the relationships involving diagonals of
parallelograms or transversals used?
Opposite Sides of a Parallelogram Theorem
Opposite Angles of a Parallelogram Theorem
Diagonals of a Parallelogram Theorem
Parallel Lines and Transversals Theorem
California: 1.0, 7.0, 12.0
How is it determined whether a quadrilateral is a
parallelogram?
How are the values for parallelograms found?
Quadrilateral Opposite Sides Theorem
Quadrilateral Opposite Angles Theorem
Quadrilateral Bisecting Diagonals Theorem
Quadrilateral Opposite Sides Parallel Theorem
California: 1.0, 7.0, 12.0
What are the properties of diagonals of rhombuses
and rectangles and how are they used?
How is it determined whether a parallelogram is a
rhombus or rectangle?
Rhombus Diagonal Bisector Theorem
Rhombus Perpendicular Diagonals Theorem
Rectangle Diagonals Theorem
Parallelogram Diagonal Bisector Theorem
Parallelogram Perpendicular Diagonals Theorem
Parallelogram Diagonal Congruency Theorem
California: 1.0, 7.0, 12.0
What are the properties of trapezoids and kites?
Isosceles Trapezoid Base Angles Theorem
Isosceles Trapezoid Diagonals Theorem
Kite Perpendicular Diagonals Theorem
California: 1.0, 7.0, 12.0
Naming coordinates of special figures
Definition of the midsegment of a trapezoid
Trapezoid Midsegment Theorem
California: 1.0, 7.0, 12.0
Coordinate proofs
California: 1.0, 7.0, 12.0
How can the properties of special figures be used to
name their coordinates in the coordinate plane?
How can coordinate geometry be used to prove
theorems?
What is needed to plan a coordinate geometry
proof?
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
51
Content Outcomes
Students will understand ratios and proportions.
Students will understand similar polygons.
Geometry
Unit 7 – What is Similarity and how it is used in Geometry?
Essential Questions
Key Concepts
How are ratios written?
Definition of a ratio
Definition of a proportion/extended proportion
How are proportions solved?
Properties of Proportions (Cross-Product Property)
Definition of scale/scale drawings
What are the properties of proportions?
What are scale and scale drawings?
What are similar polygons?
How are similar polygons identified and how are
they used in geometry?
Students will understand how to prove triangles
similar.
What are AA, SAS, and SSS similarity statements
and how are they applied?
How can indirect measurement be used to measure
objects that are otherwise difficult to measure?
Students will understand similarity in right
triangles.
What are the relationships in similar right triangles
and how are they used?
What is the geometric mean?
.
Students will understand proportions in triangles
What is the Side-Splitter Theorem and how is it
applied?
Standards Addressed
California: 1.0, 4.0
Common Core: G-CO.6, G-SRT.2 - 5
Definition of similar/similarity ratio
Definition of golden rectangle/golden ratio
Using/identifying similar polygons
California: 1.0, 4.0
Angle-Angle Similarity (AA~) Postulate
Side-Angle-Side Similarity (SAS~) Theorem
Side-Side-Side Similarity (SSS~) Theorem
Definition of an indirect measurement
California: 1.0, 4.0
Right Triangle Altitude Hypotenuse Theorem
Definition of Geometric Mean
First Corollary to Right Triangle Altitude
Hypotenuse Theorem
Second Corollary to Right Triangle Altitude
Hypotenuse Theorem
California: 1.0, 4.0
Side-Splitter Theorem
Corollary to Side-Splitter Theorem
Triangle-Angle-Bisector Theorem
California: 1.0, 4.0
What is the Triangle-Angle-Bisector Theorem and
how is it applied?
52
Common Core: G-CO.6, G-SRT.2 - 5
Common Core: G-CO.6, G-SRT.2 - 5
Common Core: G-CO.6, G-SRT.2 - 5
Common Core: G-CO.6, G-SRT.2 - 5
Content Outcomes
Students will understand the Pythagorean Theorem
and its converse.
Students will understand special right triangles.
Students will understand the tangent ratio.
Geometry
Unit 8 – What is the relationship between Right Triangles and Trigonometry?
Essential Questions
Key Concepts
What is the Pythagorean Theorem and how is it Pythagorean Theorem
applied in right triangles?
Definition of Pythagorean triple
Converse of Pythagorean Theorem
What is the Converse of the Pythagorean Theorem Obtuse Triangle Theorem
and how is it used in right triangles?
Acute Triangle Theorem
What other theorems can be used to identify obtuse
and acute triangles?
What are the properties of 45-45-90 triangles, and
how can they be used to find the length of the
hypotenuse and legs of a triangle?
What are the properties of 30-60-90 triangles, and
how can they be used to find the length of the
hypotenuse and legs of a triangle?
What is a tangent ratio and how are they written?
45-45-90 Triangle Theorem
30-60-90 Triangle Theorem
What are sine and cosine ratios and how are they
written?
Definition of tangent
Writing tangent ratios
What is an angle of elevation?
California: 1.0, 14.0, 15.0, 18.0 - 20.0
California: 1.0, 14.0, 15.0, 18.0 - 20.0
Common Core: G-SRT.6 - 11
Definition of sine
Definition of cosine
Writing sine/cosine ratios
California: 1.0, 14.0, 15.0, 18.0 - 20.0
Definition of angle of elevation
Definition of angle of depression
California: 1.0, 14.0, 15.0, 18.0 - 20.0
How are sine and cosine used to determine side
lengths in triangles?
Students will understand angles of elevation and
depression.
Common Core: G-SRT.6 - 11
Common Core: G-SRT.6 - 11
How are tangent ratios used to determine side
lengths in triangles?
Students will understand sine and cosine ratios.
Standards Addressed
California: 1.0, 14.0, 15.0, 18.0 - 20.0
What is an angle of depression?
Common Core: G-SRT.6 - 11
Common Core: G-SRT.6 - 11
How are angles of elevation and depression used to
solve problems in Geometry?
Students will understand vectors.
What is a vector?
What is a vector direction?
Definition of a Vector
Definition of magnitude/initial point/terminal
point/resultant
Property of Adding Vectors
How are problems that involve vector addition
solved?
53
California: 1.0, 14.0, 15.0, 18.0 - 20.0
Common Core: G-SRT.6 - 11
Content Outcomes
Students will understand translations.
Geometry
Unit 9 – What are the various Transformations and their uses?
Essential Questions
Key Concepts
What is an isometry?
Definitions of transformation/pre-image/image
Definition of isometry
How is the transformation of a geometric figure Definition of translation
identified?
Definition of composition
Standards Addressed
California: 1.0, 11.0, 22.0
Common Core: G-CO.2 -5, G-SRT.1, 2
What is a translation image and how is it found?
Students will understand reflections.
What is a reflection image and how is it found?
Definition of reflection
Finding reflections
California: 1.0, 11.0, 22.0
Common Core: G-CO.2 -5, G-SRT.1, 2
Students will understand rotations.
Students will understand symmetry.
What are rotation images and how are they drawn
and identified?
What is symmetry?
What are the various types of symmetry in figures
and how are they identified in a figure?
Students will understand dilations.
Students will understand compositions of
reflections.
What is a dilation image and how are they located?
What is a composition of reflections?
How is composition of reflections used?
What are glide reflections?
Definition of rotation
Drawing rotation images
Definition of the center of an image
California: 1.0, 11.0, 22.0
Definition of symmetry
Definitions of reflectional symmetry/line
symmetry/rotational symmetry/point symmetry
Identifying symmetry
California: 1.0, 11.0, 22.0
Definition of dilation
Definitions of enlargement/reduction
Translation or Rotation Theorem
California: 1.0, 11.0, 22.0
Parallel Lines Composition of Reflections Theorem
Intersection Lines Composition of Reflections
Theorem
Fundamental Theorem of Isometries
Definition of glide reflection
Isometry Classification Theorem
California: 1.0, 11.0, 22.0
Definition of tessellation/tiling
Triangle Tessellation Theorem
Quadrilateral Tessellation Theorem
Definition of translational symmetry
Definition of glide reflectional symmetry
California: 1.0, 11.0, 22.0
Common Core: G-CO.2 -5, G-SRT.1, 2
Common Core: G-CO.2 -5, G-SRT.1, 2
Common Core: G-CO.2 -5, G-SRT.1, 2
Common Core: G-CO.2 -5, G-SRT.1, 2
How are glide reflections identified?
Students will understand tessellations.
What is a tessellation?
How are transformations in tessellations identified?
How is it known if figures will tessellate?
How are symmetries in tessellations identified?
54
Common Core: G-CO.2 -5, G-SRT.1, 2
Content Outcomes
Students will understand how to find areas of
parallelograms and triangles.
Students will understand how to find areas of
trapezoids, rhombuses, and kites.
Students will understand how to find areas of
regular polygons.
Geometry
Unit 10 – How are the Areas of Various Polygons calculated?
Essential Questions
Key Concepts
What formula is used to find the area of a rectangle? Area of a Rectangle Theorem
Definition of base/altitude/height of a parallelogram
What formula is used to find the area of a Area of a Parallelogram Theorem
parallelogram?
Definition of base/height of a triangle
Area of a Triangle Theorem
What is the formula to find the area of a triangle?
What formula is used to find the area of a
Definition of height of a trapezoid
trapezoid?
Area of a Trapezoid Theorem
Area of a Rhombus or a Kite Theorem
What formula is used to find the area of a rhombus?
What formula is used to find the area of a kite?
What formula is used to find the area of a regular
polygon?
Definition of radius/apothem of a regular polygon
Area of a Regular Polygon Theorem
Standards Addressed
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
Students will understand how to find perimeters and
areas of similar figures.
What formula is used to find the perimeter of
similar figures?
Perimeters and Areas of Similar Figures Theorem
Common Core: G-MG.1 - 3
What formula is used to find the area of similar
figures?
Students will understand the relationship between
trigonometry and area.
Students will understand circles and arcs.
How can trigonometry be used to find the area of a
regular polygon?
Relationship between trigonometry and area of
regular polygons
How can trigonometry be used to find the area of a
triangle?
Area of a Triangle Given SAS Theorem
What is an arc?
Definitions of circle/circle center/central angle
Definitions of radius/diameter
Definitions of semicircle/minor arc/major arc /
adjacent arcs
Arc Addition Postulate
Definition of circumference/concentric circles
Circumference of a Circle Theorem
Definition of arc length
Arc Length Theorem
Area of a Circle Theorem
Definition of a sector of a circle
Area of a Sector of a Circle Theorem
Definition of a segment of a circle
What is the central angle of a circle?
How are the measures of central angles and arcs
found?
Students will understand how to find the areas of
circles and sectors.
How is circumference and arc length found in
circles?
What is the sector of a circle?
What equation is used to find the area of a circle?
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
What equation is used to find the area of a circle
sector?
What equation is used to find the area of a circle
segment?
55
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
Content Outcomes
Students will understand space figures and cross
sections.
Students will understand how to find the surface
areas of prisms and cylinders.
Geometry
Unit 11 – How are Surface Area and Volume of various 3-D Figures calculated?
Essential Questions
Key Concepts
What is a polyhedron?
Definition of a polyhedron
Definitions of face/edge/vertex
How are polyhedral and their parts recognized?
Euler’s Formula
Definition of cross section
What are cross sections of space figures?
Drawing cross sections
How are cross sections of space figures visualized?
What is a prism and what formula is used to find the
surface area of a prism?
What is a cylinder, and what formula is used to find
the surface area of a cylinder?
Students will understand how to find the surface
areas of pyramids and cones.
What is a pyramid, and what formula is used to find
the surface area of a pyramid?
What is a cone, and what formula is used to find the
surface area of a cone?
Students will understand how to find the volumes of
prisms and cylinders.
What is volume?
What formula is used to find the volume of a prism?
What formula is used to find the volume of a
cylinder?
Students will understand how to find the volumes of
pyramids and cones.
What formula is used to find the volume of a
pyramid?
What is a sphere?
What formula is used to find the surface area of a
sphere?
Common Core: G-GMD.1 -4, G-MG.1 - 3
Definition of a prism
Definitions of bases/lateral faces/altitude/height
Definitions of right/oblique prisms
Lateral and Surface Areas of Prisms Theorem
Definition of cylinder
Definitions of right/oblique cylinders
Lateral and Surface Areas of Prisms Theorem
California: 1.0, 8.0, 9.0, 11.0
Definition of a pyramid
Definitions of regular pyramid/slant height
Lateral and Surface Areas of a Regular Pyramid
Theorem
Definition of a cone
Lateral and Surface Areas of a Cone Theorem
Definition of volume
Cavalieri’s Principle Theorem
Volume of a Prism Theorem
Volume of a Cylinder Theorem
Definition of a composite space figure
California: 1.0, 8.0, 9.0, 11.0
Volume of a Pyramid Theorem
Volume of a Cone Theorem
California: 1.0, 8.0, 9.0, 11.0
Common Core: G-GMD.1 -4, G-MG.1 - 3
Common Core: G-GMD.1 -4, G-MG.1 - 3
California: 1.0, 8.0, 9.0, 11.0
Common Core: G-GMD.1 -4, G-MG.1 - 3
Common Core: G-GMD.1 -4, G-MG.1 - 3
What formula is used to find the volume of a cone?
Students will understand how to find the surface
areas and volumes of spheres.
Standards Addressed
California: 1.0, 8.0, 9.0, 11.0
Definition of a sphere
Definitions of great spheres/hemispheres
Surface Area of a Sphere Theorem
Volume of a Sphere Theorem
California: 1.0, 8.0, 9.0, 11.0
Definition of similar solids
Definition of similarity ratio
Areas and Volumes of Similar Solids Theorem
California: 1.0, 8.0, 9.0, 11.0
Common Core: G-GMD.1 -4, G-MG.1 - 3
What formula is used to find the volume of a
sphere?
Students will understand how to find the areas and
volumes of similar solids.
What are similar solids?
What formulas are used to find the relationships
between the ratios of the areas and volumes of
similar solids?
56
Common Core: G-GMD.1 -4, G-MG.1 - 3
Content Outcomes
Students will understand tangent lines.
Students will understand chords and arcs.
Students will understand inscribed angles.
Students will understand how to find angle
measures and segment lengths.
Geometry
Unit 12 – What are the properties of Circles and how are they applied?
Essential Questions
Key Concepts
What is the tangent to a circle?
Definition of a tangent to a circle
Definition of the point of tangency
How is the relationship between a radius and a Perpendicular Relationship of tangent to radius
tangent used in Geometry?
Theorem
Perpendicular to Radius Endpoint Theorem
How is the relationship between two tangents from Definitions of inscribed in/circumscribed about
one point used in Geometry?
Dual Tangent Segment Congruency Theorem
What is a chord?
Definition of a chord
Congruent Central Angles/Chords/Arcs
How are chords used in conjunction with arcs and
Relationships Theorem
central angles in circles?
Equidistant Chords Theorem
Perpendicular Diameter to Chord Bisector Theorem
What are the various properties of lines through the
Diameter Bisecting Chord Theorem
center of a circle?
Perpendicular Bisector of a Chord Circle Center
Theorem
What is an inscribed angle?
Definition of an Inscribed Angle
Inscribed Angle Theorem
How is the measure of an inscribed angle found?
Three Corollaries to the Inscribed Angle Theorem
Tangent/Chord Angle Measure Theorem
What technique is used to find the measure of an
angle formed by a tangent and a chord?
What is a secant?
What technique is used to find the measures of
angles formed by chords, secants, and tangents?
Standards Addressed
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
Definition of a secant
Angle Measure of Intersecting Lines Inside/Outside
of a Circle Theorem
Product of Segment Length for a Given Point and
Circle Theorem
California: 1.0, 7.0, 17.0, 21.0
Standard Equation of a Circle Theorem
Naming a circle’s radius/center
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
How are the lengths of segments associated with
circles found?
Students will understand using circles in the
coordinate plane.
How is the equation of a circle written?
What technique is used to find the center and radius
of a circle?
Students will understand the concept of locus.
What is a locus?
How is a locus drawn?
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
Definition of a locus
Drawing a locus
Describing a locus
How is a locus described?
57
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Honors Geometry
Must be in 9th grade and have shown Algebra proficiency on placement exam
Yes – Category D
Year
Brief Course Description
This course covers the foundations of geometrical figures and their measurement. Beginning with the component part of geometrical
figures – points, lines, and planes – and through the use of reasoning and proof, the course encompasses the study of triangles,
quadrilaterals, other polygons, circles, and solids. Through the study of definitions, postulates, and theorems, in addition to other related
mathematical topics, the properties of these figures are incorporated into an understanding and ability to construct and measure both plane
figures and solids. Major topics in the course include deductive and inductive reasoning, triangle relationships and congruence, right
triangle trigonometry, similarity, areas of plane figures, and surface areas and volumes of solids. The course covers the same material as a
standard geometry course, but with additional emphasis on application of the concepts. Since this course is open only to freshmen, an
evaluation of important algebra 1 skills is incorporated to review key concepts that students tested out of through the placement exam.
Assigned Textbook(s)
Laurie Bass and Art Johnson; Geometry; Prentice Hall
Accompanying Workbook; Geometry Workbook; Prentice Hall
Supplemental Material(s)
Videos: topical episodes of a Caltech produced series entitled Mathematics
(Similarity, the Pythagorean Theorem, and Pi)
Common Assessments Utilized
Common Final each semester
In-Class Worksheets & Problems
Quizzes
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
58
Major Content Outcomes
Students will be introduced to inductive and deductive forms of reasoning and use of this
reasoning in arriving at the fundamental proofs of geometry.
Major Skill Outcomes
Students will be able to prove basic theorems, and prove triangle congruence and similarity.
Students will be able to measure in the coordinate plane.
Students will be able to find measures of areas, surface areas, and volumes of plane and space figures.
Students will be able to construct basic geometrical figures with a straight edge and compass.
Students will be able to apply geometric knowledge in solving selected “real-world” problems modeled
with geometric figures, particularly in areas related to trigonometry and the Pythagorean Theorem.
Students will be able to use properties of geometric figures to determine equations and solve for unknown
dimensions of the figures.
Students will be introduced to the fundamental properties of geometric figures and the use of
them in constructing and working with them.
Students will be introduced to the principles of measuring components of geometric figures
such as segments and angles.
Students will be introduced to triangle relationships, including congruence, similarity, and right
triangle trigonometry.
Students will be introduced to the measurement of areas of plane figures and surface area and
volume of solids.
Students will review and practice algebra topics in the applications of these geometric concepts.
Unit 1
Students should be able to describe each pattern and find the next terms in each sequence.
Students should be able to name collinear and coplanar points.
Students should be able to find the intersection of a line and a plane.
Students should be able to identify opposite rays.
Students should be able to find the length of segments using Algebra.
Students should be able to find the length of segments based on distance from the midpoint.
Students should be able to solve linear equations.
Students should be able to find the area of a square, rectangle, and circle.
Students should be able to identify complementary angles, supplementary angles, and perpendicular
bisectors of a given figure.
Students should be able to correctly name lines, segments, rays, and angles using correct notation.
Unit 2
Students should be able to identify the hypothesis and conclusion of a statement.
Students should be able to use a statement to write a conditional.
Students should be able to find a counterexample to show a statement is not true.
Students should be able to use algebraic properties and reasoning to solve equations and inequalities.
Students should be able to write the converse of a conditional.
Students should be able to explain why a statement is not a good definition.
Students should be able to identify certain Geometric Properties
Students should be able to find the measure of each angle in a given figure.
Students should be able to rewrite bi-conditionals as two conditionals and vice versa.
Students should be able to use the Law of Detachment and the Law of Syllogism to draw conclusions from
statements.
Unit 3
Students should be able to classify triangles by their sides and angles.
Students should be able to find the measures of angles based on parallel lines and a transversal.
Students should be able to find the values for a variable for which two lines must be parallel.
Students should be able to calculate the slope of a line on a coordinate plane.
Students should be able to write the equation of a line given a slope and a point the lines passes through.
Students should be able to write the equation of a line given two points the line passes through.
Students should be able to find the measures of interior/exterior angles in various polygons.
Students should be able to tell whether two lines are parallel or perpendicular based on their slopes.
Unit 4
Students should be able to identify triangle congruency based on various given congruent angles and sides.
Students should be able to identify which postulate can be used to prove triangle congruency based on the
given information.
Students should be able to find the values of sides and angles based on triangle congruency.
Students should be able to solve systems of linear equations using graphing, substitution, and elimination.
59
Students should be able to identify triangle congruency in overlapping triangles.
Unit 5
Students should be able to write the inverse and contrapositive of a statement.
Students should be able to identify statements that contradict one another.
Students should be able to list angles in order of size based on the length of opposite sides.
Students should be able to list sides in order of size based on the length of opposite angles.
Students should be able to find the measure of sides and angles using algebra.
Students should be able to solve linear inequalities.
Students should be able to find the center of a circle that can be circumscribed about a triangle.
Unit 6
Students should be able to classify quadrilaterals in as many ways as possible.
Students should be able to find the values of variables based on the properties of various quadrilaterals.
Students should be able to find the measures of angles and sides in parallelograms.
Students should be able to prove a quadrilateral is a parallelogram based on known properties.
Students should be able to find the measure of sides and angles in a rhombus and rectangle.
Students should be able to find the measure of sides and angles in a kite and isosceles trapezoid.
Students should be able to place quadrilaterals in the coordinate plane and identify vertices.
Students should be able to solve quadratic equations by factoring.
Unit 7
Students should be able to write ratios and solve various proportions.
Students should be able to identify similar polygons and give the similarity ratio.
Students should be able to solve quadratic equations using the quadratic formula.
Students should be able to find the values of variables in similar polygons.
Students should be able to prove triangles are similar and write similarity statements.
Students should be able to explain why triangles are similar by using algebra to solve.
Students should be able to simplify radical expressions.
Students should be able to find the geometric mean in a pair of numbers.
Students should be able to find the values of variables in right triangles.
Students should be able to use proportions in triangles to solve for variables.
Unit 8
Students should be able to find the lengths of sides of a right triangle using the Pythagorean Theorem.
Students should be able to decides whether a set of numbers form a Pythagorean triple.
Students should be able to determine whether a triangle is a right triangle based on given values.
Students should be able to find the values of variables in 45-45-90 and 30-60-90 right triangles.
Students should be able to write tangent ratios.
Students should be able to find the values of variables based on the tangent ratio.
Students should be able to write sine and cosine ratios.
Students should be able to find the values of variables based on sine and cosine ratios.
Students should be able to identify and find the angles of elevation and depression.
Students should be able to describe vectors as ordered pairs, give the coordinates, and describe the direction
of a vector.
Unit 9
Students should be able to state whether a transformation image appears to be an isometry.
Students should be able to find the image of a figure under a given translation.
Students should be able to find the coordinates of reflection images in the coordinate plane.
Students should be able to draw reflection images across a line of reflection.
Students should be able to draw an image based on a given rotation.
60
Students should be able to tell what type symmetry can be found in a given figure.
Students should be able to draw the lines of symmetry in a given figure.
Students should be able to describe the dilation image of a figure.
Students should be able to find the image of points in the coordinate plane for a given scale factor.
Students should be able to classify isometries.
Students should be able to find the glide reflection image of a given figure in the coordinate plane.
Students should be able to identify whether a figure shows a tessellation of repeating figures.
Students should be able to determine whether a figure will tessellate a plane.
Students should be able to list the symmetries in each tessellation.
Unit 10
Students should be able to find the area of a parallelogram.
Students should be able to find the area of a triangle.
Students should be able to find the area of a trapezoid.
Students should be able to find the area of a kite.
Students should be able to find the area of a rhombus.
Students should be able to find the area of a regular polygon.
Students should be able to apply the law of sines and cosines to solve for unknown values in a triangle.
Students should be able to find the measure of various angles in polygons based on given radii and apothem.
Students should be able to find the perimeters and areas based on ratios of similar figures.
Students should be able to find the areas of regular polygons using trigonometry.
Students should be able to find the circumference of a circle and the measures of arcs in circles.
Students should be able to find the area of a circle, a sector of a circle, and a shaded section of a circle.
Students should be able to find geometric probability in various figures.
Unit 11
Students should be able to find the number of vertices, edges, and faces in a polyhedron.
Students should be able to use Euler’s Formula.
Students should be able to describe the cross section of a 3-D figure.
Students should be able to find the surface area of a prism using nets.
Students should be able to find the surface area of a cylinder.
Students should be able to find the surface area and lateral area of a pyramid and cone.
Students should be able to find the volume of a prism, cylinder, and composite space figure.
Students should be able to find the volume of a square pyramid and cone.
Students should be able to find the surface area of a sphere from a given diameter.
Students should be able to find the volume of a sphere and surface area based on a given volume.
Students should be able to identify similarity in 3-D figures and give the similarity ratio.
Students should be able to use the similarity ratio to find volumes of similar figures.
Unit 12
Students should be able to find the values of variables based on tangent lines and the center of a circle.
Students should be able to determine whether a line on a circle is a tangent line.
Students should be able to find the values of variables based on given chords and arcs of circles.
Students should be able to identify inscribed angles and their intercepted arcs.
Students should be able to find the values of variables of inscribed angles within circles.
Students should be able to find the values of variables given angle measures and segment lengths in circles.
Students should be able to write the standard equation of a circle with a given center.
Students should be able to find the center and radius of a circle then graph the circle in the coordinate plane.
Students should be able to draw and describe a locus in a plane.
61
Content Outcomes
Students will learn how to use patterns and
inductive reasoning.
Honors Geometry
Unit 1 – What are the basic tools of Geometry?
Essential Questions
Key Concepts
How is inductive reasoning used to make
Definition of inductive reasoning
conjectures?
Identifying patterns/counterexamples
Standards Addressed
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
What are some ways to find and use patterns?
How can patterns be used to find counterexamples?
Students will understand points, lines, and planes.
What are points, lines, and planes?
Definitions of points, lines, and planes
Using postulates
What are postulates and how are they understood?
Students will understand segments, rays, parallel
lines and planes.
How are segments and rays identified?
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
Definitions of segments and rays
Definition of parallel lines
How are parallel lines recognized?
Students will understand how to measure segments.
How are the lengths of segments determined?
California: 1.0, 3.0, 8.0, 16.0, 17.0
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
Finding the length of segments
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
Students will understand how to measure angles.
How are the measures of angles determined?
Definition of angles/angle pairs
Measuring angles
How are special angle pairs identified?
Students will understand the coordinate plane.
What is the coordinate plane?
Definition of the coordinate plane and its uses
(distance/midpoint)
How is the coordinate plane used to find the
distance between two points?
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
How is the coordinate plane used to find the
midpoint of a segment?
Students will understand how to find perimeter,
circumference, and area.
How are the perimeters of rectangles and squares
calculated?
Formulas for perimeter/area of squares and
rectangles
Formulas for circumference/area of circles
How is the circumference of a circle calculated?
How are the areas of squares, rectangles, and circles
calculated?
62
California: 1.0, 3.0, 8.0, 16.0, 17.0
Common Core: G-C0.1, G-CO.12, G-GPE.6, GGPE.7, G-GMD.1
Content Outcomes
Students will understand how to write and recognize
conditional statements.
Honors Geometry
Unit 2 – How are the concepts of reasoning and proofs used in Geometry?
Essential Questions
Key Concepts
How are conditional statements recognized?
Definition of a conditional statement (if-then
statement)
How are the converses of conditional statements
Definition of converse
written?
Identifying the hypothesis and conclusion in a
conditional statement
How is the hypothesis and conclusion in a Identifying counterexamples
to
conditional
conditional statement identified?
statements
Definition/identification of Venn Diagram
What are counterexamples and how are they found?
Identifying truth value of conditional statements
Standards Addressed
California: 1.0, 2.0, 4.0, 13.0
Common Core: G-CO.9
How are Venn Diagrams used?
Students will understand how to write biconditionals and definitions.
How is the truth value of a conditional statement
found?
How are bi-conditionals written?
How are bi-conditionals split into separate parts?
How are good definitions recognized?
Students will understand deductive reasoning.
How are bi-conditionals written from definitions?
What is the Law of Detachment and how is it used?
Definition of bi-conditionals
Splitting bi-conditionals into two separate
statements
Recognizing good definitions (necessary
requirements of good definitions)
California: 1.0, 2.0, 4.0, 13.0
Definition of the Law of Detachment
Definition of the Law of Syllogism
California: 1.0, 2.0, 4.0, 13.0
What is the Law of Syllogism and how is it used?
Students will understand reasoning in algebra.
How is reasoning in algebra connected to reasoning
in geometry?
Common Core: G-CO.9
Connecting algebra to geometry
How are theorems about angles used and applied in
Geometry?
California: 1.0, 2.0, 4.0, 13.0
Common Core: G-CO.9
What tools are used to justify steps in solving
equations?
Students will understand how to prove angles
congruent.
Common Core: G-CO.9
Definition of theorem
Vertical Angles Theorem
California: 1.0, 2.0, 4.0, 13.0
Common Core: G-CO.9
What is the Vertical Angles theorem?
63
Content Outcomes
Students will understand the properties of parallel
lines.
Students will understand how to prove lines
parallel.
Students will understand parallel and perpendicular
lines.
Honors Geometry
Unit 3 – How are parallel and perpendicular lines identified and used in Geometry?
Essential Questions
Key Concepts
What is a transversal?
Definition of a transversal
Identifying angle pairs
How are angles formed by two lines and transversal
identified?
What are some properties of parallel lines and how
are they used?
How is a transversal used to prove lines parallel?
What are perpendicular lines?
How are parallel and perpendicular lines related?
Students will understand parallel lines and the
Triangle Angle-Sum Theorem.
What are triangles and how are they classified?
How are the measures of the angles in a triangle
found?
Standards Addressed
California: 1.0, 7.0, 12.0, 13.0, 16.0
Common Core: G-CO.9
Corresponding Angles Postulate
Alternate Interior Angles Theorem
Same-Side Interior Angles Theorem
Alternate Exterior Angles Theorem
Same-Side Exterior Angles Theorem
California: 1.0, 2.0, 4.0, 13.0
Converse of the Corresponding Angles Postulate
Converse of The Alternate Interior Angles Theorem
Converse of the Same-Side Interior Angles
Theorem
Converse of the Alternate Exterior Angles Theorem
Converse of the Same-Side Exterior Angles
Theorem
Multiple Parallel/Perpendicular Lines Theorem
California: 1.0, 2.0, 4.0, 13.0
Triangle Angle-Sum Theorem
Definitions of equiangular, acute, right, obtuse,
equilateral, isosceles, and scalene triangles
Triangle Exterior Angle Theorem
California: 1.0, 2.0, 4.0, 13.0
Definition of a polygon (concave/convex)
Polygon Angles-Sum Theorem
Polygon Exterior Angle-Sum Theorem
California: 1.0, 2.0, 4.0, 13.0
Slope-intercept Form
Standard Form of a Linear Equation
Point-Slope Form
California: 1.0, 2.0, 4.0, 13.0
Common Core: G-CO.9
Common Core: G-CO.9
Common Core: G-CO.9
How are the exterior angles of triangles used?
Students will understand the Polygon Angle-Sum
Theorem.
What is a polygon and how are they classified?
How are the sums of the measures of the
interior/exterior angles of a polygon found?
Students will understand how to use lines in the
coordinate plane.
Given their equations, how are lines graphed in the
coordinate plane?
What are slope-intercept form, standard form, and
point-slope form of linear equations?
What are the equations for horizontal and vertical
lines?
Students will understand the relationship of slopes
of parallel and perpendicular lines.
How are slope and parallel/perpendicular lines
related?
Slopes of Parallel Lines
Slopes of Perpendicular Lines
How are the equations of parallel/perpendicular
lines written?
64
Common Core: G-CO.9
Common Core: G-CO.9
Content Outcomes
Students will understand congruent figures.
Honors Geometry
Unit 4 – What are the different ways to identify triangle congruency?
Essential Questions
Key Concepts
What are congruent figures and how are they Definitions of Congruent Figures
recognized?
Triangle 3rd Angle Congruency Theorem
Common Core: G-CO.7, G-CO.8, G-CO.10
How are the corresponding parts of congruent
figures identified and named?
Students will understand how to prove triangle
congruency by SSS and SAS.
How triangles are proved congruent by SSS?
Standards Addressed
California: 1.0, 4.0, 5.0
Side-Side-Side (SSS) Postulate
Side-Angle-Side (SAS) Postulate
How triangles are proved congruent by SAS?
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
What are some ways to tell if two triangles are
congruent?
Students will understand how to prove triangle
congruency by ASA and AAS.
How triangles are proved congruent by ASA?
Angle-Side-Angle (ASA) Postulate
Angle-Angle-Side (AAS) Postulate
How triangles are proved congruent by AAS?
Students will understand how to use congruent
triangles and CPCTC.
How is triangle congruence and CPCTC used to
prove other parts of two triangles congruent?
Common Core: G-CO.7, G-CO.8, G-CO.10
Corresponding Parts of Congruent Triangles are
Congruent (CPCTC)
What are the properties of isosceles triangles and
how are they applied?
What are the properties of equilateral triangles?
Students will understand how to prove congruency
in right triangles.
What is the HL Theorem?
Isosceles Triangle Theorem
Converse of Isosceles Triangle Theorem
Isosceles Triangle Bisector Theorem
Definition of a corollary
Corollary to Isosceles Triangle Theorem
Corollary to Converse of Isosceles Triangle
Theorem
Hypotenuse-Leg (HL) Theorem
How is the HL Theorem used to prove triangles
congruent?
Students will understand how to use corresponding
parts of congruent triangles.
How are congruent overlapping triangles identified?
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
What is CPCTC and how is it applied to congruent
triangles?
Students will understand isosceles and equilateral
triangles.
California: 1.0, 4.0, 5.0
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
Identifying overlapping congruent triangles
Using two pairs of triangles
How are common parts within overlapping triangles
identified and used?
California: 1.0, 4.0, 5.0
Common Core: G-CO.7, G-CO.8, G-CO.10
How can two triangles be proven congruent using
another pair of triangles?
What are ways to separate overlapping triangles?
65
Content Outcomes
Students will understand mid-segments of triangles.
Honors Geometry
Unit 5 – What are the various relationships within a Triangle?
Essential Questions
Key Concepts
What is the mid-segment of a triangle?
Definition of a Midsegment
Triangle Midsegment Theorem
How are the properties of mid-segments used to Definition of Coordinate Proof
solve problems in geometry?
Standards Addressed
California: 1.0, 6.0, 12.0
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
What is a coordinate proof and how is it used?
Students will understand bisectors in triangles.
What are the properties of perpendicular bisectors
and how are they applied?
What are the properties of angle bisectors and how
are they applied?
Students will understand concurrent lines, medians,
and altitude.
How are the properties of perpendicular bisectors
identified?
How are the properties of angle bisectors identified?
What are the circumcenter, incenter, and
orthocenter of a triangle?
Perpendicular Bisector Theorem
Converse of the Perpendicular Bisector Theorem
Angle Bisector Theorem
Converse of the Angle Bisector Theorem
Definition of concurrent/points of concurrency
California: 1.0, 6.0, 12.0
Concurrent Perpendicular Bisector Theorem
Concurrent Angle Bisector Theorem
Definition of circumcenter/incenter/orthocenter of a
triangle
Triangle Medians Theorem
Triangle Altitudes Theorem
California: 1.0, 6.0, 12.0
Definition of contrapositive
Application of contrapositive to a statement
Indirect Proofs
California: 1.0, 6.0, 12.0
Comparison Property of Inequality
Corollary to the Triangle Exterior Angle Theorem
Unequal Triangle Sides Theorem
Unequal Triangle Angles Theorem
Triangle Inequality Theorem
California: 1.0, 6.0, 12.0
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
How are the properties of the medians of a triangle
identified?
How are the properties of altitudes of triangles
identified?
Students will understand inverses, contrapositives,
and indirect Reasoning.
How is the negation of a statement written?
How are the inverse and contrapositive of a
conditional statement written?
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
How is indirect reasoning used in proofs?
Students will understand inequalities in triangles.
How are inequalities used when involving angles of
triangles?
How are inequalities used when involving sides of
triangles?
What is the Triangle Inequality Theorem and how is
it applied to triangles?
66
Common Core: G-C.3, G-CO.9, 10, 13, G-GPE.7
Content Outcomes
Students will understand how to classify
quadrilaterals.
Students will understand the properties of
parallelograms.
Students will understand how to prove that a
quadrilateral is a parallelogram.
Students will understand special parallelograms.
Students will understand trapezoids and kites.
Students will understand how to place figures in the
coordinate plane.
Students will understand proofs using coordinate
geometry.
Honors Geometry
Unit 6 – What are the various Quadrilaterals and their properties?
Essential Questions
Key Concepts
How are quadrilaterals defined?
Definitions/classifications of a parallelogram,
How are quadrilaterals classified?
rhombus, rectangle, square, kite, trapezoid, and
isosceles trapezoid
Standards Addressed
California: 1.0, 7.0, 12.0
Common Core: G.CO-11, G-GPE.4, G-GPE.7
What are the relationships among the sides of
parallelograms and how are they used?
What are the relationships among the angles of
parallelograms and how are they used?
How are the relationships involving diagonals of
parallelograms or transversals used?
Opposite Sides of a Parallelogram Theorem
Opposite Angles of a Parallelogram Theorem
Diagonals of a Parallelogram Theorem
Parallel Lines and Transversals Theorem
California: 1.0, 7.0, 12.0
How is it determined whether a quadrilateral is a
parallelogram?
How are the values for parallelograms found?
Quadrilateral Opposite Sides Theorem
Quadrilateral Opposite Angles Theorem
Quadrilateral Bisecting Diagonals Theorem
Quadrilateral Opposite Sides Parallel Theorem
California: 1.0, 7.0, 12.0
What are the properties of diagonals of rhombuses
and rectangles and how are they used?
How is it determined whether a parallelogram is a
rhombus or rectangle?
Rhombus Diagonal Bisector Theorem
Rhombus Perpendicular Diagonals Theorem
Rectangle Diagonals Theorem
Parallelogram Diagonal Bisector Theorem
Parallelogram Perpendicular Diagonals Theorem
Parallelogram Diagonal Congruency Theorem
California: 1.0, 7.0, 12.0
What are the properties of trapezoids and kites?
Isosceles Trapezoid Base Angles Theorem
Isosceles Trapezoid Diagonals Theorem
Kite Perpendicular Diagonals Theorem
California: 1.0, 7.0, 12.0
Naming coordinates of special figures
Definition of the midsegment of a trapezoid
Trapezoid Midsegment Theorem
California: 1.0, 7.0, 12.0
Coordinate proofs
California: 1.0, 7.0, 12.0
How can the properties of special figures be used to
name their coordinates in the coordinate plane?
How can coordinate geometry be used to prove
theorems?
What is needed to plan a coordinate geometry
proof?
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
Common Core: G.CO-11, G-GPE.4, G-GPE.7
67
Content Outcomes
Students will understand ratios and proportions.
Students will understand similar polygons.
Honors Geometry
Unit 7 – What is Similarity and how it is used in Geometry?
Essential Questions
Key Concepts
How are ratios written?
Definition of a ratio
Definition of a proportion/extended proportion
How are proportions solved?
Properties of Proportions (Cross-Product Property)
Definition of scale/scale drawings
What are the properties of proportions?
What are scale and scale drawings?
What are similar polygons?
How are similar polygons identified and how are
they used in geometry?
Students will understand how to prove triangles
similar.
What are AA, SAS, and SSS similarity statements
and how are they applied?
How can indirect measurement be used to measure
objects that are otherwise difficult to measure?
Students will understand similarity in right
triangles.
What are the relationships in similar right triangles
and how are they used?
What is the geometric mean?
.
Students will understand proportions in triangles
What is the Side-Splitter Theorem and how is it
applied?
Standards Addressed
California: 1.0, 4.0
Common Core: G-CO.6, G-SRT.2 - 5
Definition of similar/similarity ratio
Definition of golden rectangle/golden ratio
Using/identifying similar polygons
California: 1.0, 4.0
Angle-Angle Similarity (AA~) Postulate
Side-Angle-Side Similarity (SAS~) Theorem
Side-Side-Side Similarity (SSS~) Theorem
Definition of an indirect measurement
California: 1.0, 4.0
Right Triangle Altitude Hypotenuse Theorem
Definition of Geometric Mean
First Corollary to Right Triangle Altitude
Hypotenuse Theorem
Second Corollary to Right Triangle Altitude
Hypotenuse Theorem
California: 1.0, 4.0
Side-Splitter Theorem
Corollary to Side-Splitter Theorem
Triangle-Angle-Bisector Theorem
California: 1.0, 4.0
What is the Triangle-Angle-Bisector Theorem and
how is it applied?
68
Common Core: G-CO.6, G-SRT.2 - 5
Common Core: G-CO.6, G-SRT.2 - 5
Common Core: G-CO.6, G-SRT.2 - 5
Common Core: G-CO.6, G-SRT.2 - 5
Content Outcomes
Students will understand the Pythagorean Theorem
and its converse.
Students will understand special right triangles.
Students will understand the tangent ratio.
Honors Geometry
Unit 8 – What is the relationship between Right Triangles and Trigonometry?
Essential Questions
Key Concepts
What is the Pythagorean Theorem and how is it Pythagorean Theorem
applied in right triangles?
Definition of Pythagorean triple
Converse of Pythagorean Theorem
What is the Converse of the Pythagorean Theorem Obtuse Triangle Theorem
and how is it used in right triangles?
Acute Triangle Theorem
What other theorems can be used to identify obtuse
and acute triangles?
What are the properties of 45-45-90 triangles, and
how can they be used to find the length of the
hypotenuse and legs of a triangle?
What are the properties of 30-60-90 triangles, and
how can they be used to find the length of the
hypotenuse and legs of a triangle?
What is a tangent ratio and how are they written?
45-45-90 Triangle Theorem
30-60-90 Triangle Theorem
What are sine and cosine ratios and how are they
written?
Definition of tangent
Writing tangent ratios
What is an angle of elevation?
California: 1.0, 14.0, 15.0, 18.0 - 20.0
California: 1.0, 14.0, 15.0, 18.0 - 20.0
Common Core: G-SRT.6 - 11
Definition of sine
Definition of cosine
Writing sine/cosine ratios
California: 1.0, 14.0, 15.0, 18.0 - 20.0
Definition of angle of elevation
Definition of angle of depression
California: 1.0, 14.0, 15.0, 18.0 - 20.0
How are sine and cosine used to determine side
lengths in triangles?
Students will understand angles of elevation and
depression.
Common Core: G-SRT.6 - 11
Common Core: G-SRT.6 - 11
How are tangent ratios used to determine side
lengths in triangles?
Students will understand sine and cosine ratios.
Standards Addressed
California: 1.0, 14.0, 15.0, 18.0 - 20.0
What is an angle of depression?
Common Core: G-SRT.6 - 11
Common Core: G-SRT.6 - 11
How are angles of elevation and depression used to
solve problems in Geometry?
Students will understand vectors.
What is a vector?
What is a vector direction?
Definition of a Vector
Definition of magnitude/initial point/terminal
point/resultant
Property of Adding Vectors
How are problems that involve vector addition
solved?
69
California: 1.0, 14.0, 15.0, 18.0 - 20.0
Common Core: G-SRT.6 - 11
Content Outcomes
Students will understand translations.
Honors Geometry
Unit 9 – What are the various Transformations and their uses?
Essential Questions
Key Concepts
What is an isometry?
Definitions of transformation/pre-image/image
Definition of isometry
How is the transformation of a geometric figure Definition of translation
identified?
Definition of composition
Standards Addressed
California: 1.0, 11.0, 22.0
Common Core: G-CO.2 -5, G-SRT.1, 2
What is a translation image and how is it found?
Students will understand reflections.
What is a reflection image and how is it found?
Definition of reflection
Finding reflections
California: 1.0, 11.0, 22.0
Common Core: G-CO.2 -5, G-SRT.1, 2
Students will understand rotations.
Students will understand symmetry.
What are rotation images and how are they drawn
and identified?
What is symmetry?
What are the various types of symmetry in figures
and how are they identified in a figure?
Students will understand dilations.
Students will understand compositions of
reflections.
What is a dilation image and how are they located?
What is a composition of reflections?
How is composition of reflections used?
What are glide reflections?
Students will understand tessellations.
How are glide reflections identified?
What is a tessellation?
How are transformations in tessellations identified?
How is it known if figures will tessellate?
Definition of rotation
Drawing rotation images
Definition of the center of an image
California: 1.0, 11.0, 22.0
Definition of symmetry
Definitions of reflectional symmetry/line
symmetry/rotational symmetry/point symmetry
Identifying symmetry
California: 1.0, 11.0, 22.0
Definition of dilation
Definitions of enlargement/reduction
Translation or Rotation Theorem
California: 1.0, 11.0, 22.0
Parallel Lines Composition of Reflections Theorem
Intersection Lines Composition of Reflections
Theorem
Fundamental Theorem of Isometries
Definition of glide reflection
Isometry Classification Theorem
California: 1.0, 11.0, 22.0
Definition of tessellation/tiling
Triangle Tessellation Theorem
Quadrilateral Tessellation Theorem
Definition of translational symmetry
Definition of glide reflectional symmetry
California: 1.0, 11.0, 22.0
How are symmetries in tessellations identified?
70
Common Core: G-CO.2 -5, G-SRT.1, 2
Common Core: G-CO.2 -5, G-SRT.1, 2
Common Core: G-CO.2 -5, G-SRT.1, 2
Common Core: G-CO.2 -5, G-SRT.1, 2
Common Core: G-CO.2 -5, G-SRT.1, 2
Content Outcomes
Students will understand how to find areas of
parallelograms and triangles.
Students will understand how to find areas of
trapezoids, rhombuses, and kites.
Students will understand how to find areas of
regular polygons.
Honors Geometry
Unit 10 – How are the Areas of Various Polygons calculated?
Essential Questions
Key Concepts
What formula is used to find the area of a rectangle? Area of a Rectangle Theorem
Definition of base/altitude/height of a parallelogram
What formula is used to find the area of a Area of a Parallelogram Theorem
parallelogram?
Definition of base/height of a triangle
Area of a Triangle Theorem
What is the formula to find the area of a triangle?
What formula is used to find the area of a
Definition of height of a trapezoid
trapezoid?
Area of a Trapezoid Theorem
Area of a Rhombus or a Kite Theorem
What formula is used to find the area of a rhombus?
What formula is used to find the area of a kite?
What formula is used to find the area of a regular
polygon?
Definition of radius/apothem of a regular polygon
Area of a Regular Polygon Theorem
Standards Addressed
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
Students will understand how to find perimeters and
areas of similar figures.
What formula is used to find the perimeter of
similar figures?
Perimeters and Areas of Similar Figures Theorem
Common Core: G-MG.1 - 3
What formula is used to find the area of similar
figures?
Students will understand the relationship between
trigonometry and area.
Students will understand circles and arcs.
How can trigonometry be used to find the area of a
regular polygon?
Relationship between trigonometry and area of
regular polygons
How can trigonometry be used to find the area of a
triangle?
Area of a Triangle Given SAS Theorem
What is an arc?
Definitions of circle/circle center/central angle
Definitions of radius/diameter
Definitions of semicircle/minor arc/major arc /
adjacent arcs
Arc Addition Postulate
Definition of circumference/concentric circles
Circumference of a Circle Theorem
Definition of arc length
Arc Length Theorem
Area of a Circle Theorem
Definition of a sector of a circle
Area of a Sector of a Circle Theorem
Definition of a segment of a circle
What is the central angle of a circle?
How are the measures of central angles and arcs
found?
Students will understand how to find the areas of
circles and sectors.
How is circumference and arc length found in
circles?
What is the sector of a circle?
What equation is used to find the area of a circle?
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
What equation is used to find the area of a circle
sector?
What equation is used to find the area of a circle
segment?
71
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
California: 1.0, 7.0, 8.0, 10.0, 11.0, 21.0
Common Core: G-MG.1 - 3
Content Outcomes
Students will understand space figures and cross
sections.
Students will understand how to find the surface
areas of prisms and cylinders.
Honors Geometry
Unit 11 – How are Surface Area and Volume of various 3-D Figures calculated?
Essential Questions
Key Concepts
What is a polyhedron?
Definition of polyhedron
Definitions of face/edge/vertex
How are polyhedral and their parts recognized?
Euler’s Formula
Definition of cross section
What are cross sections of space figures?
Drawing cross sections
How are cross sections of space figures visualized?
What is a prism and what formula is used to find the
surface are of a prism?
What is a cylinder, and what formula is used to find
the surface area of a cylinder?
Students will understand how to find the surface
areas of pyramids and cones.
What is a pyramid, and what formula is used to find
the surface area of a pyramid?
What is a cone, and what formula is used to find the
surface area of a cone?
Students will understand how to find the volumes of
prisms and cylinders.
What is volume?
What formula is used to find the volume of a prism?
What formula is used to find the volume of a
cylinder?
Students will understand how to find the volumes of
pyramids and cones.
What formula is used to find the volume of a
pyramid?
What is a sphere?
What formula is used to find the surface area of a
sphere?
Common Core: G-GMD.1 -4, G-MG.1 - 3
Definition of prism
Definitions of bases/lateral faces/altitude/height
Definitions of right/oblique prisms
Lateral and Surface Areas of Prisms Theorem
Definition of cylinder
Definitions of right/oblique cylinders
Lateral and Surface Areas of Prisms Theorem
California: 1.0, 8.0, 9.0, 11.0
Definition of a pyramid
Definitions of regular pyramid/slant height
Lateral and Surface Areas of a Regular Pyramid
Theorem
Definition of a cone
Lateral and Surface Areas of a Cone Theorem
Definition of volume
Cavalieri’s Principle Theorem
Volume of a Prism Theorem
Volume of a Cylinder Theorem
Definition of composite space figure
California: 1.0, 8.0, 9.0, 11.0
Volume of a Pyramid Theorem
Volume of a Cone Theorem
California: 1.0, 8.0, 9.0, 11.0
Common Core: G-GMD.1 -4, G-MG.1 - 3
Common Core: G-GMD.1 -4, G-MG.1 - 3
California: 1.0, 8.0, 9.0, 11.0
Common Core: G-GMD.1 -4, G-MG.1 - 3
Common Core: G-GMD.1 -4, G-MG.1 - 3
What formula is used to find the volume of a cone?
Students will understand how to find the surface
areas and volumes of spheres.
Standards Addressed
California: 1.0, 8.0, 9.0, 11.0
Definition of a sphere
Definitions of great spheres/hemispheres
Surface Area of a Sphere Theorem
Volume of a Sphere Theorem
California: 1.0, 8.0, 9.0, 11.0
Definition of similar solids
Definition of similarity ratio
Areas and Volumes of Similar Solids Theorem
California: 1.0, 8.0, 9.0, 11.0
Common Core: G-GMD.1 -4, G-MG.1 - 3
What formula is used to find the volume of a
sphere?
Students will understand how to find the areas and
volumes of similar solids.
What are similar solids?
What formulas are used to find the relationships
between the ratios of the areas and volumes of
similar solids?
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Common Core: G-GMD.1 -4, G-MG.1 - 3
Content Outcomes
Students will understand tangent lines.
Students will understand chords and arcs.
Students will understand inscribed angles.
Students will understand how to find angle
measures and segment lengths.
Honors Geometry
Unit 12 – What are the properties of Circles and how are they applied?
Essential Questions
Key Concepts
What is the tangent to a circle?
Definition of tangent to a circle
Definition of point of tangency
How is the relationship between a radius and a Perpendicular Relationship of tangent to radius
tangent used in Geometry?
Theorem
Perpendicular to Radius Endpoint Theorem
How is the relationship between two tangents from Definitions of inscribed in/circumscribed about
one point used in Geometry?
Dual Tangent Segment Congruency Theorem
What is a chord?
Definition of a chord
Congruent Central Angles/Chords/Arcs
How are chords used in conjunction with arcs and
Relationships Theorem
central angles in circles?
Equidistant Chords Theorem
Perpendicular Diameter to Chord Bisector Theorem
What are the various properties of lines through the
Diameter Bisecting Chord Theorem
center of a circle?
Perpendicular Bisector of a Chord Circle Center
Theorem
What is an inscribed angle?
Definition of Inscribed Angle
Inscribed Angle Theorem
How is the measure of an inscribed angle found?
Three Corollaries to the Inscribed Angle Theorem
Tangent/Chord Angle Measure Theorem
What technique is used to find the measure of an
angle formed by a tangent and a chord?
What is a secant?
What technique is used to find the measures of
angles formed by chords, secants, and tangents?
Standards Addressed
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
Definition of a secant
Angle Measure of Intersecting Lines Inside/Outside
of a Circle Theorem
Product of Segment Length for a Given Point and
Circle Theorem
California: 1.0, 7.0, 17.0, 21.0
Standard Equation of a Circle Theorem
Naming a circle’s radius/center
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
How are the lengths of segments associated with
circles found?
Students will understand using circles in the
coordinate plane.
How is the equation of a circle written?
What technique is used to find the center and radius
of a circle?
Students will understand the concept of locus.
What is a locus?
How is a locus drawn?
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
Definition of a locus
Drawing a locus
Describing a locus
How is a locus described?
73
California: 1.0, 7.0, 17.0, 21.0
Common Core: G-C.1 - .5; G-CO.13; G-GPE.1 - 3
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Algebra 2
“C” or better in Geometry
Yes – Category D
Year
Brief Course Description
Algebra 2 “complements and expands the mathematical content and concepts of Algebra I and Geometry” (Mathematics Content Standards for California Public Schools).
Major emphasis is placed on learning basic families of functions, their properties, graphs, and algebraic methods for solving equations of these functions. Students expand on
their knowledge of linear, quadratic, other polynomial, radical, exponential, logarithmic, rational, and trigonometric functions. In addition, students expand upon their
abilities to solve problems in a variety of content areas, including systems of equations. Algebra 2 requires students to have a basic knowledge of the properties of real
numbers and the skills to simplify and expand algebraic expressions, solve equations and inequalities, and graph in the xy-coordinate plane.
Assigned Textbook(s)
Supplemental Material(s)
Bellman, etc. al., Algebra 2 © 2007, Prentice Hall.
Algebra 2 Workbook, www.phschool.com, www.hotmath.com,
Common Assessments Utilized
Common Final each semester
In-Class Practice Questions (Free Response and Multiple Choice Questions)
Homework (Free Response and Multiple Choice Questions)
In-Class Quizzes (Free Response and Multiple Choice Questions)
Chapter Exams (Free Response and Multiple Choice Questions)
ISOs Addressed
Which Integral Student Outcomes (ISOs) will this course address – ISOs are the replacement for ESLRs
Be academically prepared for a higher education …
Exhibit community and global awareness …
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Major Content Outcomes
I. Properties of Families of Functions
A. Parent functions: common and standard forms
B. Properties of the parent function: zeros, intercepts, asymptotes, symmetries
C. Graphical transformations (including vertical/horizontal reflections, translations, and stretching)
D. Algebraic methods for solving equations/inequalities involving each function
E. Inverse functions
II. Linear functions
A.
B.
C.
D.
E.
Algebraic forms of linear functions
Direct variations
The relationship between a linear function’s algebraic expression and its graph, including
horizontal, vertical, parallel, and perpendicular lines.
Inverse functions of linear functions.
Graphing and solving absolute value equations/inequalities.
III. Systems of Equations
A.
B.
C.
D.
Independent, Dependent, and Inconsistent systems
Solving by graphing, substitution, and elimination
Linear programming and objective functions
Solving 3-variable systems
IV. Quadratic functions
A. Algebraic forms of quadratic functions
B. The relationship between a quadratic’s algebraic expression and its graph, including axis of
symmetry, vertex, x-intercepts, and transformations of parabolas
C. Imaginary numbers and the complex number system
D. Solving quadratic equations
1. Factoring
2. Taking square roots
3. Completing the Square
4. The Quadratic Formula
V. Polynomial functions
A. Identifying various forms and names of polynomials
B. Synthetic and polynomial long division
C. Sum and Difference of Cube factoring patterns
D. Solving polynomial equations
E. Fundamental Theorem of Algebra, Rational, Irrational and Imaginary Root theorems
F. Pascal’s Triangle and the Binomial Theorem for binomial expansion
VI. Radical functions
A.
Properties of integer exponents
B.
Rational exponents
C.
Properties of radical functions, including adding, subtracting, multiplying and dividing radicals
D.
Solving radical equations
Major Skill Outcomes
Students will extend their skills in solving equations and inequalities.
Students will solve absolute value equations and inequalities using compound inequalities.
Students will simplify and solve one and two-variable expressions, equations and inequalities.
Students will represent function relationships by writing and graphing linear equations and
inequalities.
Students will solve systems of equations and inequalities in two variables algebraically and by
graphing.
Students will graph points and equations in three dimensions.
Students will solve systems of equations in three variables.
Students will use quadratic functions to model real-world data.
Students will graph and solve quadratic equations.
Students will graph complex numbers and use them in solving quadratic equations.
Students will write and graph polynomial functions and solve polynomial equations.
Students will use important theorems about the number of solutions to polynomial equations.
Students will solve problems involving permutations, combinations, and binomial probability.
Students will extend their knowledge of roots to include cube roots, fourth roots, fifth roots, and so on.
Students will add, subtract, multiply, and divide radical expressions, including binomial radical
expressions.
Students will solve radical equations, and graph translations of radical functions and their inverses.
Students will use exponential functions to model real-world data.
Students will graph exponential functions and their inverse, logarithmic functions.
Students will solve exponential and logarithmic equations.
Students will use inverse variations and the graphs of inverse variations to solve real-world problems.
Students will identify and use properties of rational functions.
Students will simplify rational expressions and solve rational equations.
Students will learn how geometric measurement relates to trigonometry.
Students will use radian measure.
Students will write and graph functions that describe periodic data.
Students will solve real-world problems involving right triangles.
Unit 1
Students will solve one-variable equations.
Students will state restrictions on variables.
Students will write and solve equations related to word problems.
Students will solve inequalities and graph their solutions.
Unit 2
Students will find the domain and range of a relation and graph it.
Students will evaluate equations in function notation.
Students will find the slope of a given line.
Students will write in standard form an equation of a line with a given slope through a given point.
Students will write in point-slope form an equation of a line through a given pair of points.
Students will find the constant of variation and evaluate a direct variation at a given value.
Students will graph absolute value equations.
Students will graph linear equations.
Students will describe the transformation of graphs from their equations to graph the equation.
Students will graph two-variable inequalities.
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E.
F.
Inverse functions: the square root and quadratic
The relationship between a radical function’s algebraic expression and its graph, including
properties of translating graphs
VII. Exponential and Logarithmic functions
A. The relationship between both families of functions (graphically and algebraically)
B. Solving exponential and logarithmic functions
C. Properties of logarithmic functions, including the change of base formula
D. The properties of graphing both functions, including asymptotes and translations
E. Common Logarithms
F. The Natural Logarithmic Function
VIII. Rational and Reciprocal functions
A. Inverse variation
B. Graphs of rational functions
C. Vertical, horizontal, and slant asymptotes
D. Simplifying, adding, subtracting, multiplying and dividing rational expressions
E. Solving rational equations
F.
G.
H.
I.
J.
K.
L.
M.
IX. Trigonometric functions
A. Basic trigonometric ratios: sine, cosine, and tangent
B. Unit Circle
C. Radian and degree measure
D. Solving trigonometric equations
E. Laws of Sines and Cosines
Unit 3
Students will classify linear systems without graphing.
Students will solve linear systems by graphing.
Students will solve linear systems using substitution.
Students will solve linear systems using elimination.
Students will graph systems of linear inequalities.
Students will graph a system of constraints to evaluate the minimum/maximum of an objective
function.
Students will solve a system of 3 linear 3 variable equations.
Unit 4
Students will sketch the graph of a parabola with given vertex through a given point.
Students will graph quadratic functions, identifying the axis of symmetry and the coordinates of the
vertex.
Students will evaluate expressions using imaginary numbers.
Students will find the additive inverse of complex numbers.
Students will solve quadratic equations.
Students will write quadratic functions in vertex form.
Students will evaluate the discriminant of quadratic equations to determine the quantity and type of
solutions to the quadratic equation.
Unit 5
Students will write given polynomials in standard form and classify them by degree and number of
terms.
Students will graph polynomial functions and approximate real zeros to the nearest hundredth.
Students will write a polynomial function with rational coefficients in standard form with given zeros.
Students will solve polynomial equations.
Students will divide polynomials using long division.
Students will divide polynomials by linear factors using synthetic division.
Students will use synthetic division and the remainder theorem to evaluate polynomial functions.
Unit 6
Students will simplify radical expressions, using absolute value symbols when needed.
Students will rationalize denominators involving radical expressions.
Students will simplify expressions with fractional exponents.
Students will solve equations with fractional exponents, checking for extraneous solutions.
Students will perform function operations and determine the domain of composite functions.
Students will compose functions inside other functions.
Students will rewrite functions to show the translation of the function.
Students will graph rational functions, identifying domain and range.
Students will find the inverse of a function and determine if it is a function.
Unit 7
Students will evaluate exponential functions to the nearest hundredth and graph the results.
Students will write exponential functions of the form y = a(b)x through given points.
Students will describe how graphs of exponential functions are related to the parent function.
Students will evaluate logarithms.
Students will graph logarithmic functions.
Students will use properties of logarithms to rewrite logarithmic expressions and evaluate them.
Students will solve exponential and logarithmic equations.
Students will use the change of base formula to rewrite logarithms using common bases.
Students will use the natural logarithm in solving equations.
76
Unit 8
Students will model inverse and joint variations.
Students will graph rational functions.
Students will determine any holes, horizontal, or vertical asymptotes a function might have.
Students will simplify rational expressions.
Students will find the least common multiple of polynomial expressions.
Students will find the difference, sum, product, and quotient of rational expressions.
Students will simplify complex fractions.
Students will solve equations involving rational expressions.
Unit 9
Students will determine if a function is periodic or not, and if so, find its period and amplitude.
Students will find the measure of angles co-terminal to a given angle.
Students will convert angles from degree to radian measure.
Students will determine a sine function’s amplitude and period from its graph.
Students will sketch a cycle of the sine graph and sketch sine functions on an interval from 0 to 2pi.
Students will determine a cosine function’s amplitude and period from its graph.
Students will sketch a cycle of the cosine graph and sketch cosine functions on an interval from 0 to
2pi.
Students will write equations of translations of sine and cosine graphs.
Students will evaluate trigonometric functions at specific points along the unit circle.
\
77
Content Outcomes
Students will know Properties of Real Numbers
Students can simplify Algebraic Expression
Students apply skills in Solving Equations
Algebra 2
Unit 1 – What are the fundamental tools needed to be successful in Algebra?
Essential Questions
Key Concepts
How do you graph real numbers?
Real number line
Can you identify the properties of really numbers?
subsets of real numbers
Can you apply the properties of real numbers?
additive inverse
multiplicative inverse
properties of real numbers
absolute values of a real number
How do you evaluate an algebraic expression?
How do you simplify algebraic expressions?
How do you solve one-variable equations?
How can you solve word problems by writing and
solving equations?
Standards Addressed
California: CA A1: 1.0 – 5.0, 15.0, 25.0; CA A2:
1.0, 15.0, 25.0
Common Core: N-RN 3; A-REI 1
National Discipline: NCTM 1 – 4, 7 – 10
variables
algebraic expressions
evaluating a variable
order of operations
algebraic terms and coefficients
properties of algebraic expressions
California: CA A1: 1.0 – 5.0, 15.0, 25.0; CA A2:
1.0, 15.0, 25.0
properties of equality
finding the solution to an equation
solving for unknown values in word problems
California: CA A1: 1.0 – 5.0, 15.0, 25.0; CA A2:
1.0, 15.0, 25.0
Common Core: A-SSE 1a, 1b, 2
National Discipline: NCTM 1 – 4, 7 – 10
Common Core: A-CED 1, 4
National Discipline: NCTM 1 – 4, 7 – 10
Students apply skills in Solving Inequalities
Students apply skills in Absolute Value Equations
and Inequalities
How do you solve and graph single-variable
inequalities?
How can you solve and graph compound
inequalities?
How can you write and graph compound
inequalities?
properties of inequalities
graphing one variable inequalities
compound inequality
and/or
graphing compound inequalities
converting into compound inequality statements
California: CA A1: 1.0 – 5.0, 15.0, 25.0; CA A2:
1.0, 15.0, 25.0
How do you solve equations with absolute values?
How do you solve inequalities with absolute values?
Algebraic definition of absolute values
Solving absolute value equations
extraneous solutions in absolute value equations
Absolute value inequalities “and/or”
solving absolute value inequalities with absolute
values
California: CA A1: 1.0 – 5.0, 15.0, 25.0; CA A2:
1.0, 15.0, 25.0
78
Common Core: A-CED 1; A-REI
National Discipline: NCTM 1 – 4, 7 – 10
Common Core: F-IF 7a
National Discipline: NCTM 1 – 4, 7 – 10
Content Outcomes
Students understand Relations and Functions
Students can model Linear Equations
Students can identify Direct Variations
Algebra 2
Unit 2 – What are the basic properties of linear equations and their graphs?
Essential Questions
Key Concepts
How do we graph a relation?
relation
How do we identify functions?
domain/range
What is function notation?
mapping diagram
definition of function
vertical line test
function notation
How do we graph linear equations?
How do we write the equations of graphed lines?
How do write a direction variation?
How can we use data to write a model for a direct
variation?
Standards Addressed
California: CA A1: 6.0 – 8.0, 15.0 – 18.0
Common Core: F-IF 1, 2, 5
National Discipline: NCTM 2,3, 5- 10
linear function
dependent variable
x- and y-intercepts
Standard form of a linear equation
slope formula
point-slope form
slope-intercept form
converting graphing into equations
vertical and horizontal lines
parallel and perpendicular lines
California: CA A1: 6.0 – 8.0, 15.0 – 18.0
direct variation
constant of variation
determining if data is direct variation
finding the constant of variation
California: CA A1: 6.0 – 8.0, 15.0 – 18.0
Common Core: A-CED 1, 2; F-IF 4,5,6; F-BF 1, FLE 1, 1a
National Discipline: NCTM 2,3, 5- 10
Common Core: F-IF 7a; F-LE 1a, 1b
National Discipline: NCTM 2,3, 5- 10
Students understand when to use Linear Models
How can we apply linear equations to real-world
data?
How can we use linear models to make predictions?
reading word-problems involving linear models
using linear models to predict unknown values
California: CA A1: 6.0 – 8.0, 15.0 – 18.0
Common Core: A-CED 1,2; F-LE 2, 5
National Discipline: NCTM 2,3, 5- 10
Students learn about Absolute Value Functions and
their Graphs
How do we graph absolute value functions?
graphing absolute values using xy-tables
vertex of an absolute value graph
writing absolute value functions as two linear
equations
California: CA A1: 6.0 – 8.0, 15.0 – 18.0
Common Core: F-IF 7a
National Discipline: NCTM 2,3, 5- 10
Students can identify transformations of Families of
Functions
How can we translate graphs of functions?
How can we stretch, shrink, and reflect graphs of
functions?
Knowing parent function
Recognizing horizontal and vertical translations
Recognizing vertical stretch
Recognizing vertical reflection
California: CA A1: 6.0 – 8.0, 15.0 – 18.0
Common Core: F-IF 4, 7a
National Discipline: NCTM 2,3, 5- 10
Students graph Two-Variable Inequalities
How do you graph linear inequalities?
How do you graph absolute value inequalities?
graphing pairs of linear inequalities on same xyplane
finding overlapping region
graphing absolute value inequalities
California: CA A1: 6.0 – 8.0, 15.0 – 18.0
Common Core: F-IF 7a
National Discipline: NCTM 2,3, 5- 10
79
Content Outcomes
Students understand Graphing Systems of Equations
Algebra 2
Unit 3 – How do we solve systems of linear equations?
Essential Questions
Key Concepts
How do you solve a system by graphing?
systems of linear equations
independent systems
dependent systems
inconsistent systems
Standards Addressed
California: CA A1: 9.0, CA LA: 6.0
Common Core: A-REI 6, F-IF 7a
National Discipline: NCTM 2,3, 5, 6, 8 - 10
Students understand Solving Systems Algebraically
How do you solve a system using substitution?
How do you solve a system using elimination?
Solving systems using substitution
Dependent and Inconsistent Systems
Solving systems using elimination
Dependent and Inconsistent Systems
California: CA A1: 9.0, CA LA: 6.0
Common Core: A-REI 5
National Discipline: NCTM 2,3, 5, 6, 8 - 10
Students can graphically solve Systems of
Inequalities
How do you solve systems of linear inequalities?
Graphing linear inequalities
Finding overlapping regions of linear inequalities
California: CA A1: 9.0, CA LA: 6.0
Common Core: F-IF 7a
National Discipline: NCTM 2,3, 5, 6, 8 - 10
Students are introduced to Linear Programming
How do you use systems of equations to solve linear
programming maximum and minimum problems?
Linear Programming
Restraints
Objective Function
Minimizing/Maximizing Objective Function
California: CA A1: 9.0, CA LA: 6.0
Common Core: A-CED 2, 3; F-IF 5
National Discipline: NCTM 2,3, 5, 6, 8 - 10
Students learn about Graphs in Three Dimensions
Students solve Systems with Three Variables
How do you graph points in 3 dimensions?
How do you graph equations in 3 dimensions?
How do you solve systems with 3 variables using
elimination?
How do you solve systems with 3 variables using
substitution?
xyz-Plane
ordered triples
coordinate space
intercepts
traces
California: CA A1: 9.0, CA LA: 6.0
Solving systems using elimination twice
Solving systems using substitution twice
Dependent and Inconsistent Systems
California: CA A1: 9.0, CA LA: 6.0
Common Core:
National Discipline: NCTM 2,3, 5, 6, 8 - 10
Common Core:
National Discipline: NCTM 2,3, 5, 6, 8 - 10
80
Content Outcomes
Students begin Modeling Data with Quadratic
Functions
Students learn about Properties of Parabolas
Students learn Transforming Parabolas
Students practice Factoring Quadratic Expressions
Students learn methods of solving Quadratic
Equations
Students learn about Complex Numbers
Students use Completing the Square to solve
quadratic equations
Students use The Quadratic Formula
Algebra 2
Unit 4 – How can you solve quadratic equations?
Essential Questions
Key Concepts
How can we identify quadratic functions?
Standard form of a Quadratic
What are the properties of quadratic’s graphs?
Finding the vertex of a parabola
How can we use data to create a model of a
Finding the axis of symmetry of a parabola
quadratic function?
Using symmetry to find points on a parabola
Solving a system of 3 variables, 3 equations to
model data.
How can we graph quadratics?
How can we identify the maximum /minimum
values of quadratic functions?
Graphing Quadratics using xy-tables
Identifying maximum and minimum values based
on value of a.
How can we use vertex form to graph a quadratic
function?
Know vertex form of quadratic
Knowing parent graph of a quadratic
Recognizing horizontal and vertical translations in
vertex form
Recognizing vertical stretch in vertex form
Converting from vertex from to standard form
Factoring greatest common factor
Factoring greatest common factor
Difference of Squares
Factoring quadratics trinomial into the product of 2
trinomials.
Can we recognize common factors in a quadratic
expression?
How can we factor quadratic binomials?
How can we factor quadratic trinomials?
How can we solve quadratic equations by graphing?
How can we solve quadratic equations by factoring?
How can we solve quadratic equations by using
square roots?
Graphing a quadratic to find x-intercepts
Know zero-product property
Apply zero-product property to factored quadratic
equations = 0.
Using roots to solve quadratic equations with no
linear term.
What does it represent?
How can we simplify radicals with a negative
radicand?
How can we identify complex numbers?
How can we graph complex numbers?
How do you add, subtract, and multiply complex
numbers by complex numbers?
How can you use “completing the square” to solve
quadratic equations?
How can use “completing the square” to rewrite
quadratics in vertex form?
Square root of negative number is i.
Decomposing radical to separate number and
negative 1
Real and Imaginary components of complex
number
Graphing complex numbers
Adding, subtracting, multiplying complex numbers
Understanding powers of i.
Completing the Square to solve quadratics
Completing the square to create quadratic formula
Completing the square to rewrite standard form into
vertex form.
How can we solve quadratic equations by using the
Quadratic formula?
How can we determine the different types of
solutions a quadratic equation will have by using
the discriminant?
Know quadratic formula and it’s derivation
Apply quadratic formula
Know the discriminant
Know relationship between the discriminant, graph,
number of solutions, and type of solutions.
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Standards Addressed
California: CA A1: 10.0, 11.0, 14.0, 19.0 – 23.0;
CA A2: 4.0 – 6.0, 8.0 – 10.0, 16.0, 17.0; CA Au 2.0
Common Core: N-Q 1, 2
National Discipline: NCTM 1 – 3, 5 – 7, 9 - 10
California: CA A1: 9.0, CA LA: 6.0
Common Core: F-IF 7a,
National Discipline: NCTM 2,3, 5, 6, 8 - 10
California: CA A1: 9.0, CA LA: 6.0
Common Core: F-IF 7a
National Discipline: NCTM 2,3, 5, 6, 8 - 10
California: CA A1: 9.0, CA LA: 6.0
Common Core: A-APR 3
National Discipline: NCTM 2,3, 5, 6, 8 - 10
California: CA A1: 9.0, CA LA: 6.0
Common Core: A-SSE 3a; A-APR 3; A-REI 4b; FIF 8a
National Discipline: NCTM 2,3, 5, 6, 8 - 10
California: CA A1: 9.0, CA LA: 6.0
Common Core: N – CN 1, 2
National Discipline: NCTM 2,3, 5, 6, 8 - 10
California: CA A1: 9.0, CA LA: 6.0
Common Core: N-CN 7; A-SSE 3b; A-REI 4a, 4b
National Discipline: NCTM 2,3, 5, 6, 8 - 10
California: CA A1: 9.0, CA LA: 6.0
Common Core: N-CN 7, A-REI4b
National Discipline: NCTM 2,3, 5, 6, 8 - 10
Content Outcomes
Students classify Polynomial Functions
Algebra 2
Unit 5 – How can we apply our knowledge of linear and quadratic equations to solve higher-order polynomial equations?
Essential Questions
Key Concepts
Standards Addressed
How can we classify different types of polynomials
Standard form of a polynomial
California: CA A1: 10.0, 11.0; CA A2: 4, 18 – 20;
How can we use polynomials functions to model
Degree of Polynomial
CA Au: 4.0
data?
Number of Terms of a Polynomial
Using graphs to classify data into polynomial
Common Core: A-APR 1
functions
Students write Polynomials using Linear Factors
Students use long and synthetic division to Divide
Polynomials
Students engage in Solving Polynomial Equations
Students explore Theorems about Roots of
Polynomial Equations
Students are taught The Fundamental Theorem of
Algebra
How do we factor polynomials?
How do we write a polynomial from its zeros?
How do we divide polynomials using long division?
How do we divide polynomials using synthetic
division?
How can we solve polynomials by graphing?
How can we solve polynomials by factoring?
Factored form of a polynomial
Factor Theorem
Multiplicity of zeros
Different meaning of zeros
Converting zeros to factors
Long Division Algorithm
Remainder Theorem
Synthetic Division
Remainder Theorem
Graphing polynomials and finding x-intercepts
Factoring Polynomials
Sum and Difference of Cubes Factoring
Factoring using quadratic pattern
How can we solve equations using the Rational
Roots Theorem?
How can we use the Irrational Roots Theorem?
How can we use the Imaginary Roots Theorem?
Rational Roots Theorem
Testing Possible Roots
Irrational Roots Theorem
Irrational Conjugates
Imaginary Roots Theorem
Imaginary Conjugates
How can we use the Fundamental Theorem of
Algebra in solving polynomials with complex
roots?
Fundamental Theorem of Algebra and it’s
Corollary
National Discipline: NCTM 1 – 2, 5- 10
California: CA A1: 10.0, 11.0; CA A2: 4, 18 – 20;
CA Au: 4.0
Common Core: A-APR 1
National Discipline: NCTM 1 – 2, 5- 10
California: CA A1: 10.0, 11.0; CA A2: 4, 18 – 20;
CA Au: 4.0
Common Core: A-APR 1, A-APR 2
National Discipline: NCTM 1 – 2, 5- 10
California: CA A1: 10.0, 11.0; CA A2: 4, 18 – 20;
CA Au: 4.0
Common Core: N-CN 7; A-APR 3; F-IF 7c
National Discipline: NCTM 1 – 2, 5- 10
California: CA A1: 10.0, 11.0; CA A2: 4, 18 – 20;
CA Au: 4.0
Common Core: F-IF 7c
National Discipline: NCTM 1 – 2, 5- 10
California: CA A1: 10.0, 11.0; CA A2: 4, 18 – 20;
CA Au: 4.0
Common Core: N-CN 7, N-CN 9 (+)
Students are introduced to Permutations and
Combinations
Students apply The Binomial Theorem
How can we calculate a permutation?
How can we calculate a combination?
How can we use the Pascal’s Triangle for binomial
expansion?
Factorial
Recognizing Permutations
Permutation Formula
Recognizing Combinations
Combinations Formula
Pascal’s Triangle (coefficients of Expansion)
Binomial Theorem
Patterns from Pascal’s Triangle
82
National Discipline: NCTM 1 – 2, 5- 10
California: CA A1: 10.0, 11.0; CA A2: 4, 18 – 20;
CA Au: 4.0
Common Core:
National Discipline: NCTM 1 – 2, 5- 10
California: CA A1: 10.0, 11.0; CA A2: 4, 18 – 20;
CA Au: 4.0
Common Core:
Content Outcomes
Students learn about Roots and Radical Expressions
Algebra 2
Unit 6 – How do we solve functions involving radicals and rational exponents?
Essential Questions
Key Concepts
How do you simplify nth root radicals?
nth roots
identifying radicand, index
principal root
Standards Addressed
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: F-IF 8a
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
Students begin Multiplying and Dividing Radical
Expressions
How do we multiply radical expressions?
How do we divide radical expressions?
multiplying radical expressions
dividing radical expressions
rationalize the denominator
Students are introduced to binomial Radical
Expressions
How do we add and subtract radical expressions?
How do we multiple and divide binomial radical
expressions?
combining like radicals
simplifying radicals
multiplying radicals expressions
multiplying conjugates
rationalizing binomial radical denominators
Students work with Rational Exponents
How do we simplify expressions with radical
exponents?
rational exponents
radical form to exponential form
properties of rational exponents
Students will engage in Solving Square Root and
Other Radical Equations
How do we solve radical equations?
solving square root equations
solving radical equations with rational exponents
checking for extraneous solutions
solving equations with 2 rational exponents
Students are introduced to Function Operations
How do we add, subtract, multiple, and divide
functions?
How can we compose two functions?
function operations
composing functions
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: F-IF 8a
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: F-IF 8a
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: N-RN 1, 2; A-SSE 3c
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: A-REI 2
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: F-IF 1, 2; F-BF 1, 1a, 1b, 1c
Students learn about Inverse Relations and
Functions
How do we find the inverse of a relation or
function?
finding inverse relations graphically
finding inverse relations algebraically
graphing an inverse relation
composition of functions with their inverse
Students begin Graphing Square Root and Other
Radical Functions
How do we graph the radical functions?
Knowing parent graph of a square and cube root
functions
Recognizing horizontal and vertical translations
Recognizing vertical stretch
Recognizing vertical reflection
83
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: F-BF 4a, 4b, 4c, 4d
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: F-IF 7b, 9
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
Content Outcomes
Students begin Exploring Exponential Models
Algebra 2
Unit 7 – How do we solve equations using exponential and logarithmic equations?
Essential Questions
Key Concepts
How is exponential growth modeled?
Exponential growth model
How is exponential decay modeled?
growth factor
Exponential decay model
decay factor
horizontal asymptote
Standards Addressed
California: CA A2: 11.0 – 14.0
Common Core: N-Q 1, 2; F-IF 7E, 8b, 9; F-LE 1b,
1c, 2, 3, 4, 5
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
Students learn Properties of Exponential Functions
Students are introduced to Logarithmic Functions as
Inverses
Students explore Properties of Logarithms
How do we transform the parent graph of an
exponential function?
What is base e?
What is the compound continuous Interest Formula?
How do we convert an exponential equation into a
logarithmic equation?
How do we evaluate a logarithmic expression?
How do you graph a logarithmic function?
What are the laws of logarithms?
Knowing parent graph of an exponential
growth/decay models
Recognizing horizontal and vertical translations
Recognizing vertical stretch
Recognizing vertical reflection
national number e
Continuously Compounded Interest Formula
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Definition of a logarithm
conversion between exponential and logarithmic
equations
common logarithm base
evaluate logarithms using knowledge of exponents
using inverse of exponential growth/decay
Knowing parent graph of an logarithmic function
Recognizing horizontal and vertical translations
Recognizing vertical stretch
Recognizing vertical reflection
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Multiplication to Addition Property
Division to Subtraction Property
Exponent to Coefficient Property
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: F-IF 7e, 8b; F-LE 5
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
Common Core: F-BF 5
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
Common Core: B-BF 5, F-LE 4
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
Students solve Exponential and Logarithmic
Equations
How do we solve exponential equations?
How do we solve logarithmic equations?
conversion to logarithmic equations
change of base formula
conversion to exponential equation
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: A-REI 2, B-BF 5, F-LE 4
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
Students explore Natural Logarithms
What is the natural logarithm?
How do you evaluate a natural logarithmic
expression?
How do you solve equations using national
logarithms?
natural logarithmic base
evaluate logarithms of base e using knowledge of
exponents
conversion to logarithmic equations
change of base formula
conversion to exponential equation
84
California: CA A1: 2.0, CA A 2: 12.0, 15.0, 24.0,
25.0
Common Core: B-BF 5
National Discipline: NCTM 1 – 3, 6, 7, 9, 10
Content Outcomes
Students learn about Inverse Variation
Algebra 2
Unit 8 – How do we solve equations using rational functions?
Essential Questions
Key Concepts
How do we determine an inverse variation?
inverse variation
What other types of variation are there?
joint variation
Standards Addressed
California: CA A1: 10.0 – 13.0; CA A2: 3.0, 4.0,
7.0, 12.0, 15.0; CA Au: 6.0
Common Core: F-IF 7d
National Discipline: NCTM 1 – 3, 6, 10
Students are introduced to The Reciprocal Function
Family
How do we graph reciprocal functions?
How do we translate reciprocal functions?
Knowing parent graph of the sine function
Recognizing horizontal and vertical translations
Recognizing vertical stretch
Recognizing vertical reflection
California: CA A1: 10.0 – 13.0; CA A2: 3.0, 4.0,
7.0, 12.0, 15.0; CA Au: 6.0
Common Core: F-IF 7d
National Discipline: NCTM 1 – 3, 6, 10
Students learn about Rational Functions and their
graphs
Students simplify Rational Expressions
What are the properties of rational functions?
How do we graph a rational function?
How do we simplify rational expressions?
How do we multiply and divide rational
expressions?
Definition of a rational function
Points of Discontinuity
Vertical Asymptotes
Holes Test
Horizontal Asymptotes
California: CA A1: 10.0 – 13.0; CA A2: 3.0, 4.0,
7.0, 12.0, 15.0; CA Au: 6.0
Common Core: F-IF 7d
Simplest Form of rational expression
restrictions
Multiplying Rational expressions
Dividing Rational Expressions
California: CA A1: 10.0 – 13.0; CA A2: 3.0, 4.0,
7.0, 12.0, 15.0; CA Au: 6.0
Common Core:
National Discipline: NCTM 1 – 3, 6, 10
National Discipline: NCTM 1 – 3, 6, 10
Students begin Adding and Subtracting Rational
Expressions
How do we add and subtract rational expressions?
How do we simplify complex fractions?
Adding Rational Expressions
Subtracting Rational Expressions
Writing complex fractions horizontally
California: CA A1: 10.0 – 13.0; CA A2: 3.0, 4.0,
7.0, 12.0, 15.0; CA Au: 6.0
Common Core:
National Discipline: NCTM 1 – 3, 6, 10
Students work on Solving Rational Equations
How do you solve rational equations?
proportions
rational equations
California: CA A1: 10.0 – 13.0; CA A2: 3.0, 4.0,
7.0, 12.0, 15.0; CA Au: 6.0
Common Core: F-IF 7d
National Discipline: NCTM 1 – 3, 6, 10
85
Content Outcomes
Students begin Exploring Periodic Data
Algebra 2
Unit 9 – What are trigonometric functions of the unit circle?
Essential Questions
Key Concepts
How can we determine the periods and cycle of a
periodic functions
periodic function?
cycle
How can we find the amplitude of a periodic
period
function?
amplitude
Standards Addressed
California: CA T: 1.0, 2.0, 4.0, 9.0
Common Core: F-TF 5
National Discipline: NCTM 1 – 5, 7, 8
Students are taught about Angles and the Unit
Circle
Students learn to use Radian Measure
Students are taught The Sine Function
Students are taught The Cosine Function
Students graph by Translating Sine and Cosine
Functions
What is standard position of an angle?
What do the coordinates of a point on the unit circle
represent?
What is radian measure of an angle?
What is the length of an arc on a circle?
What are the properties of the sine function?
How do you graph the sine function?
What are the properties of the cosine function?
How is the cosine function graphed?
How are transformations of trigonometric functions
graphed?
standard position of an angle
initial side and terminal side
positive and negative angles
conterminal angles
unit circle
definition of sine and cosine on unit circle
sine and cosine of values around the unit circle
California: CA T: 1.0, 2.0, 4.0, 9.0
radian measure
central angle
intercepted arc
conversion between radian and degrees
formula for an arc length around a circle
California: CA T: 1.0, 2.0, 4.0, 9.0
parent graph of the sine function
zero-max-zero-min-zero
period and amplitude of parent function
vertical stretch
vertical reflection
California: CA T: 1.0, 2.0, 4.0, 9.0
parent graph of the cosine function
max-zero-min-zero-max
period and amplitude of parent function
vertical stretch
vertical reflection
California: CA T: 1.0, 2.0, 4.0, 9.0
horizontal and vertical translations
California: CA T: 1.0, 2.0, 4.0, 9.0
Common Core: F-TF 1, 2, 3, 4
National Discipline: NCTM 1 – 5, 7, 8
Common Core: F-TF 1, 2, 3, 4
National Discipline: NCTM 1 – 5, 7, 8
Common Core: F-TF 3, 4
National Discipline: NCTM 1 – 5, 7, 8
Common Core: F-TF 1, 2, 3,4, 5
National Discipline: NCTM 1 – 5, 7, 8
Common Core: F-TF 5
National Discipline: NCTM 1 – 5, 7, 8
86
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Brief Course Description
Honors Algebra 2
“B” of better in Honors Geometry or “A” in both Algebra 1 and Geometry
Yes – Category D
Year
A rigorous treatment of intermediate algebra and trigonometric topics which include equations and inequalities, functions and graphs, polynomial and rational functions,
exponential and logarithmic functions, systems of equations, and trigonometric functions and their applications.
Assigned Textbook(s)
Supplemental Material(s)
Algebra and Trigonometry with Analytic Geometry; Swokowski and Cole 12th edition.
TI-89 Calculator
Common Assessments Utilized
Common Final each semester
Homework
In-Class Quizzes
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
87
Major Content Outcomes
I.
II.
Know the basic concepts from Algebra 2.
A The application of exponent and radical rules.
B The simplification of exponential expressions.
1 Factoring by trial and error
2 Factoring using special cases: difference of two squares, perfect trinomials, and the
sum and difference of cubes.
3 Factoring by grouping.
C Rational expressions.
1 Recognizing the limitations on the domain of rational expressions.
2 Multiplication and division of rational expressions.
3 Addition and Subtraction of rational expressions.
4 Simplification of compound fractions.
5 Rationalizing a complex denominator.
D Solving of equations.
1 Solving of equations using the graphing method, completing the
square, and the
quadratic equation.
2
Solving fractional expressions.
3
Solving equations with fractional powers
4
Solving absolute value equations.
E Solving and Graphing of Inequalities
1
Quadratic inequalities
2
Compound inequalities
3
Absolute value inequalities
Know the application and manipulation of functions.
A Definition of a function
B Graphs of functions.
1
Power functions.
2
Piecewise functions.
C Transformations of functions.
1 Even/Odd functions
2
Horizontal and vertical shifts.
3 Vertical and horizontal stretching and compression.
D Quadratic functions.
1
Maximums and minimums.
2
Standard form.
3
Local maxima and minima.
E Combining functions.
1
Addition, subtraction, multiplication, and division
2
Composite functions.
3
Domain of composite functions.
III. Understand the many applications of rational functions.
A The graphs of polynomial functions.
1 End behavior of a polynomial.
2
Intermediate value theorem.
3
Finding zeros and graphing of rational functions.
B Dividing polynomials.
1
Polynomial long division.
2
Synthetic division.
Major Skill Outcomes
By the end of the year students should be able to:
I Know the basic concepts from Algebra 2.
II. Understand the many applications of rational functions.
III. Know the application and manipulation of functions.
VI. Understand how graph and solve exponential and logarithmic equations.
V. Be familiar with all the applications, including graphing and problem solving associated with
the unit circle.
VI. Be familiar with polar coordinates and vectors.
VII. Know systems of equations including 2 by 2 and 3 by 3 matrices.
Unit 1
Students should be able to simplify an expression and rationalize the denominator.
Students should be able to simplify polynomials.
Students should be able to factor polynomials.
Students should be able to simplify expressions.
Students should be able to rewrite an absolute value expression without absolute value symbols.
Unit 2
Students should know how to solve equations.
Students should know how to complete the square.
Students should know how to solve inequalities and place the answer into interval format.
Students should know how to express expressions in a +bi format.
Unit 3
Students should be able to apply the distance formula.
Students should be able to find the equation of lines, quadratics and circles in both standard form.
Students should be able to find the center and radius of a circle.
Students should be able to evaluate functions.
Students should be able to graph functions.
Students should be able to find the domain and range of functions.
Students should be able to find the minimum or maximum values of a quadratic.
Students should be able to find composite functions and their domain.
Unit 4
Students should be able to sketch a polynomial function.
Students should know how to find the quotient and remainder of a function?
Students should know how to use synthetic division to find the quotient and remainder of a function?
Students should know how find a functions given information?
Students should know how to find all solutions of a function.
Students should be able to sketch rational functions
Unit 5
Students should be able to find inverse functions.
Students should be able to sketch exponential and logarithmic functions.
Students should be able to evaluate logarithms.
Students should be able to solve logarithmic and exponential equations.
Students should be able to expand and contract logarithms.
88
IV.
C Rational zeros of a polynomial.
D Complex zeros.
1 Definition of i.
2 Addition, subtraction, multiplication, and division.
E Complex zeros.
1 Multiplicities of zeros
2 Complex factorization.
F Rational functions.
1 Vertical and horizontal asymptotes.
2 Intercepts.
3 Graphing.
Understand how to graph and solve exponential and logarithmic equations.
A Inverse functions.
1 Definition of one to one functions.
2
Computation of inverse functions.
B Exponential functions.
1 Definition and graphing.
2 Natural exponential function.
3
Interest equations.
C Logarithmic functions.
1 Definition.
2 Graphing
3 Common logarithm and natural logarithm.
D Laws of logarithms.
1 Combining logarithmic expressions.
2 Expanding logarithmic expressions.
3 Change of base formula.
F Exponential and logarithmic equations.
V. Be familiar with all the applications, including graphing and problem solving associated with the
unit circle.
A. The unit circle.
1 The relationship between degrees and radians and the ability to change radians to
degrees and degrees to radians.
B Trigonometric functions of real numbers.
1 Even/odd properties.
2 Reciprocal identities.
3 Pythagorean identities.
C The graphing of sine, cosine, tangent, cosecant, secant, and cotangent.
1 Transformations of trigonometric functions.
D Right triangle trigonometry
1 Law of sines.
2 Law of cosines.
3 Heron's formula for triangle area.
E
Trigonometric Identities.
1 Addition and subtraction formulas.
2 Double-angle, half-angle, and product to sum formulas.
F
Inverse trigonometric functions.
1 Arcsine, arccosine, and arctangent.
G Trigonometric equations.
VI.
Unit 6
Students should be able to switch between radians and degrees.
Students should be able to find arc length and sector area.
Students should be able to verify trigonometric identities.
Students should be able to find the exact values of the trigonometric values.
Students should be able sketch and describe the six trigonometric functions including transformations.
Unit 7
Students should be able to verify trigonometric identities.
Students should be able to find all solutions to trigonometric equations.
Students should be able to apply all trigonometric formulas.
Students should be able to graph and solve inverse trigonometric functions.
Unit 8
Students should be able to solve triangles using the law of sines and cosines.
Students should be able to solve for the area of a triangle using Heron’s formula.
Students should be able to sketch vectors.
Students should be able to find the i and j format of a vector.
Students should be able to find the dot product of two vectors.
Students should be able to find scalar vectors and the sum or difference of vectors.
Students should be able to prove vectors either parallel or orthogonal.
Unit 9
Students should be able to solve a system of equations.
Students should be able to sketch a system of equations.
Students should be able to apply the algebra of matrices to express a problem as a single matrix.
Students should be able to find an inverse matrix.
Unit 10
Students should be able to find the vertices and foci of conics.
Students should be able to sketch a conic.
Students should be able to find the equation for a conic that satisfies certain criteria.
Students should be able to switch between polar and rectangular coordinates.
Students should be able to find polar equations.
Be familiar with polar coordinates and vectors.
A Polar coordinates.
1 Definition of polar coordinates.
89
B
C
VII.
2 Converting polar coordinates to rectangular coordinates and rectangular
coordinates to polar coordinates.
3 The relationship between polar equations and rectangular equations.
Sketching polar equations.
Vectors
1 Component form of vectors.
2 Operations on vectors.
3 Magnitude of vectors.
Know systems of equations including 2 by 2 and 3 by 3 matrices.
A Systems of equations.
1 Graphing method.
2 Substitution method.
3 Elimination method and elimination method with addition and subtraction.
B Two variable, two equation systems.
C Three variable, three equation systems.
D Matrices
1 Solving matrices using elimination method.
E
Algebra of matrices.
1 Addition, subtraction, and scalar multiplication of matrices.
2 Matrix multiplication.
F
Inverse matrices.
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Content Outcomes
Students learn about all the properties of real
numbers.
Students will understand the applications of all
exponent rules.
Students will understand the applications of
algebraic expressions.
Students will understand how to simplify fractional
expressions.
Honors Algebra 2
Unit 1 – Fundamental Concepts of Algebra
Essential Questions
Key Concepts
What are the properties of real numbers?
Definitions of different number types.
How are the properties of equality applied?
Properties of real numbers.
What are the properties involving zero?
Properties of equality.
How do the properties of quotients apply to Zero product property and multiplicative property
fractions?
of zero.
How is absolute value used?
Definition of subtraction and division.
How is the distance between two points calculated?
Properties of quotients.
What is scientific notation?
Definition of absolute value.
Definition of distance between points on a
coordinate line.
Definition of scientific notation.
How would the law of exponents be applied to
Laws of exponents.
simplify expressions?
Theorem on negative exponents.
How are negative exponents utilized?
Definition of a radical.
What is a fractional exponent and how to convert
Properties of radicals.
roots to fractional exponents?
Laws of radicals.
How are radicals simplified and rationalized?
Rationalizing denominators of radicals greater than
How are denominators with roots rationalized?
root two.
How are radicals combined into a single root?
Definition of rational exponents.
How are polynomials added and subtracted?
Definitions of algebraic expressions.
How are polynomials multiplied?
Definition of a polynomial.
How are polynomials divided?
Order of operations.
How can a polynomial be multiplied with a
Product formulas.
formula?
Factoring formulas.
How are polynomials factored?
How is trial and error factoring applied to
trinomials?
How are polynomials factored by grouping?
How are rational expressions multiplied and
Reducing common factors.
divided?
Finding a least common denominator.
How are rational expressions added and subtracted?
Finding a least common denominator with binomial
How are complex fractions simplified?
and polynomial denominators.
How are complex denominators and numerators
Use of a conjugate to rationalize.
rationalized?
How are fractional expressions simplified?
91
Standards Addressed
California: CA A2: 1.0, 3.0, 4.0, 7.0, 12.0
Common Core:N-RN-1,2,3
California: CA A2: 1.0, 3.0, 4.0, 7.0, 12.0
Common Core: N-RN-1,2
California: CA A2: 1.0, 3.0, 4.0, 7.0, 12.0
Common Core: A-SSE-1,2
California: CA A2: 1.0, 3.0, 4.0, 7.0, 12.0
Common Core: A-SSE-2
Content Outcomes
Students will understand how to solve equations.
Students will understand how to solve applied
problems.
Students will understand the many ways to solve
quadratic equations.
Students will understand the applications of
complex numbers.
Students will understand how to solve fractional,
radical and other type of equations.
Students will understand how to solve inequalities.
Students will understand how to solve complex
inequalities.
Honors Algebra 2
Unit 2 – Equations and Inequalities
Essential Questions
Key Concepts
How to solve a linear equation?
The properties of equality.
How to solve an equation in combinations with the The order of operations.
order of operations?
Multiplication of binomials.
How to solve fractional equations?
Domain limitations on fractional equations.
How to solve equations with rational expressions?
Finding a LCD with rational expressions.
How to solve a formula for a specific variable?
Factoring and multiplication by a LCD.
Standards Addressed
California: CA A2: 8.0, 15.0
Common Core:A-CED-4, A-REI-1
How to find a simple average and further
applications of the formula?
How to find sales tax and interest problems?
How to complete a mixture problem ?
How to solve rate times time equals distance
problems?
How to solve work rate problems?
How to apply the zero factor theorem?
How to solve an equation by factoring?
How to solve a quadratic by completing the square?
How to apply the quadratic formula?
How to use the order of operations to place isolate a
quadratic on one side?
Critical reading of a problem to determine necessary
information.
Charting and placing information from a problem
into a chart.
Simple interest formula.
Rate x time equals distance.
Work rates.
Zero factor theorem.
Completing the square.
Quadratic Formula.
California: CA A2: 8.0, 15.0
How to apply the properties of i?
How to add and multiply complex numbers?
How to perform order of operations with complex
numbers?
What is a conjugate of a complex number?
How to solve square roots of negative numbers?
How to solve a quadratic equation with complex
solutions?
How to solve equations with complex solutions?
How to solve an equation containing absolute
value?
How to solve an equation by grouping?
How to solve an equation containing rational
exponents?
How to solve an equation containing a radical and
multiple radicals?
How to solve an equation in quadratic form?
How to place inequalities in interval format?
How to solve an inequality?
How to solve ‘or’ and ‘and’ inequalities?
Solving a rational inequality?
How to solve an absolute value inequality?
Properties of i and i squared.
Definition of a complex number and its conjugate.
Properties of conjugates.
Square roots of negatives.
California: CA A2: 8.0, 15.0
Definition of absolute value.
Factoring by grouping.
Factoring out a greatest common factor.
Fractional exponents.
Properties of equality.
Isolating a root.
Using substitution.
California: CA A2: 8.0, 15.0
Interval notation.
Properties of inequalities.
Properties of absolute values.
California: CA A2: 8.0, 15.0
How to solve a quadratic inequality?
How to use a diagram of signs to solve a rational
inequality?
Sign diagrams.
Additional properties of inequalities.
California: CA A2: 8.0, 15.0
Common Core: N-Q-1,2,3
California: CA A2: 8.0, 15.0
Common Core: A-SSE-3, A-REI-3,4
Common Core:N-CN-1,2,3
Common Core: N-CN-7,8,9 A-REI-2
Common Core: A-REI-3
Common Core: A-REI-3
92
Content Outcomes
Students will understand the basics of the
rectangular co-ordinate system.
Honors Algebra 2
Unit 3 – Functions and Graphs
Essential Questions
Key Concepts
How to find the distance between two points?
Distance formula.
How to find a midpoint of a segment?
Midpoint formula.
Standards Addressed
California: CA A2: 9.0, 10.0, 17.0
Common Core:
Students will understand how to graph equations.
Students will understand how to find and the
applications of linear equations.
Students will understand what the definition of and
uses of functions.
Students will understand how functions are graphed.
Students will understand how to graph and solve
quadratic functions.
Students will understand how operations on
functions are carried out.
How to graph functions by plugging in values of x?
How to use and apply x and y intercepts?
How symmetry applies to different functions?
How to find the center and radius of a circle?
How to identify semicircles?
Definition of x and y intercepts.
Symmetries of graphs of equations in x and y.
Standard equation of a circle.
California: CA A2: 9.0, 10.0, 17.0
How to find the slope of a line?
How to sketch linear equations with slope and yintercept?
How to find an equation of a line using the pointslope formula?
How to find an equation of a line using slopeintercept formula?
How to go between general form of a line and
slope-intercept form?
How to apply the theorems on parallel and
perpendicular lines?
How to evaluate a function?
How to determine the domain of a function?
How to apply the vertical line test?
Definition of slope of a line.
Point-slope form of a line.
Standard form of a line.
Slope-intercept form of a line.
Theorem on slopes of parallel lines.
Theorem on slopes of perpendicular lines.
California: CA A2: 9.0, 10.0, 17.0
Definition of a function.
Definition of a graph of a function.
Vertical line test.
Definition of a linear function.
California: CA A2: 9.0, 10.0, 17.0
How to distinguish even and odd functions?
How to vertically and horizontally shift a graph?
How to horizontally and vertically stretch and
compress a graph?
How to graph a piecewise function?
Definition of even and odd functions.
Transformations on graphs.
California: CA A2: 9.0, 10.0, 17.0
How to sketch the graph of a quadratic function?
How to use the standard from of a parabola?
How to find the vertex of a parabola?
How to find the maximum and minimum of a
parabola?
Definition of a quadratic function.
Standard equation of a parabola with a vertical axis.
Theorem for locating the vertex of a parabola.
Theorem on the minimum or maximum value of a
quadratic function.
California: CA A2: 9.0, 10.0, 17.0
How to perform basic operations on functions?
How to find composite functions?
How to find the domain of a composite function?
How to find the composite form of a function?
Sum, difference, produce and quotient of functions.
Definition of a composite function.
California: CA A2: 9.0, 10.0, 17.0
Common Core: F-IF-4,5
Common Core: F-IF-5
Common Core: F-IF-1,2
Common Core: F-IF-4,5,7
Common Core: F-IF-7,8
Common Core: F-IF-2
93
Content Outcomes
Students will understand how to sketch polynomial
functions of degree greater than two.
Honors Algebra 2
Unit 4 – Functions and Graphs
Essential Questions
Key Concepts
How to sketch a polynomial function of degree
Intermediate value theorem.
larger than two?
Standards Addressed
California: CA A2: 3.0, 5.0, 6.0, 8.0, 24.0, 25.0
Common Core: A-APR-3
Students will understand how to apply the
properties of division.
Students will understand how to use zeros of
polynomials in sketching.
Students will understand how to find and use
complex and rational zeros of polynomials.
Students will understand how to sketch rational
functions.
Students will understand the applications of direct
and indirect variation.
How to use polynomial long division?
How to apply the remainder theorem?
How the factor theorem applies to factoring a
polynomial?
How to synthetically divide a polynomial by x-c?
Polynomial long division.
Remainder theorem.
Factor theorem.
Guidelines for synthetic division.
California: CA A2: 9.0, 10.0, 17.0
How to find a polynomial with given zeros?
How to find the multiplicities of the zeros of a
polynomial?
How to find the zeros of a polynomial?
How to use the rule of signs?
The fundamental theorem of algebra.
Complete factorization theorem for polynomials.
Theorem on the maximum number of zeros of a
polynomial.
Theorem on the exact number of zeros of a
polynomial.
Descartes’ rule of signs.
California: CA A2: 9.0, 10.0, 17.0
How to find a polynomial with complex zeros?
How to find all zeros of a polynomial?
How to prove a polynomial has no rational zeros?
How to find all rational zeros of a polynomial?
Theorem on conjugate pair zeros of a polynomial.
Theorem on expressing a polynomial as a product
of linear and quadratic equations.
Theorem on rational zeros of a polynomial.
California: CA A2: 9.0, 10.0, 17.0
How to find the domain of a rational function?
How to find vertical and horizontal asymptotes of a
rational function?
How to sketch a rational function?
How to find an oblique asymptote of a rational
function?
Definition of vertical asymptotes.
Definition of horizontal asymptotes.
Theorem on horizontal asymptotes.
Guidelines for sketching the graph of a rational
function.
California: CA A2: 9.0, 10.0, 17.0
How to find directly and indirectly proportional
variables?
How to combine several methods of variation?
General formula of direct and inverse variation.
Guidelines for solving variation problems.
California: CA A2: 9.0, 10.0, 17.0
Common Core: A-APR-2
Common Core: F-IF7
Common Core: F-IF-7
Common Core: F-IF-7
Common Core:
94
Content Outcomes
Students will understand how to find inverse
functions.
Students will understand how to solve and graph
exponential functions.
Students will understand the natural exponential
function.
Students will understand logarithmic functions.
Students will understand how to apply the
properties of logarithms.
Students will understand how to solve exponential
and logarithmic equations.
Honors Algebra 2
Unit 5 – Inverse, Exponential, and Logarithmic Functions
Essential Questions
Key Concepts
How to determine if a function is one to one?
Theorem on increasing or decreasing functions are
How to apply the horizontal line test to see is a one to one.
function is one to one?
Definition of an inverse function.
How to find an inverse function?
Theorem on inverse functions.
The relationship between the domain and range of
inverse functions.
Guidelines for finding inverse functions.
How to solve an exponential equation?
Theorem on exponential functions being one to one.
How to sketch an exponential function?
Compound interest formula.
How to shift an exponential function using
transformations?
How to find the equation of an exponential equation
that satisfies conditions?
How to apply the compound interest function?
How to use the continuously compounded interest
The number e.
formula?
Definition of the natural exponential function.
How to graph the natural exponential function?
Continuously compounded interest formula.
Standards Addressed
California: CA A2: 11.0, 11.1, 11.2, 12.0 – 15.0
Common Core: F-BF-4
California: CA A2: 11.0, 11.1, 11.2, 12.0 – 15.0
Common Core: F-IF-7, F-BF-5, F-LE-1
California: CA A2: 11.0, 11.1, 11.2, 12.0 – 15.0
Common Core:
How to change an equation from exponential to
logarithmic form?
How to find basic logarithms without a calculator?
How to solve logarithmic equations?
How to graph a logarithmic equation?
How to move a logarithmic equation using
transformations?
How to graph and solve natural logarithms?
Definition of log base a.
Logarithmic and exponential forms of an equation.
Properties of logs.
Definition of a common logarithm.
Definition of a natural logarithm.
California: CA A2: 11.0, 11.1, 11.2, 12.0 – 15.0
How to apply the laws of logarithms?
How to solve a logarithmic equation?
How to sketch the graph of a logarithmic equation?
Laws of logarithms.
California: CA A2: 9.0, 10.0, 17.0
How to solve an exponential equation?
How to use the change of base formula to solve an
exponential equation?
How to use the definition of a logarithm to solve an
exponential equation?
How to solve an equation involving logarithms?
The change of base formula.
Special change of base formulas.
Common Core: F-IF-7, F-BF-5
Common Core: F-BF-5
California: CA A2: 9.0, 10.0, 17.0
Common Core: F-LE-5
95
Content Outcomes
Students will understand radians and degrees.
Students will understand the trigonometric functions
of angles.
Students will understand the trigonometric functions
of real numbers.
Students will understand how to find the values of
the trigonometric functions.
Students will understand how to sketch sine and
cosine.
Honors Algebra 2
Unit 6 – The Trigonometric Functions
Essential Questions
Key Concepts
How to find co-terminal angles?
Definition of a radian measure.
What is the relationship between degree and Relationship between degrees and radians.
radians?
Formula for the length of a circular arc.
How to change radians to degrees and degrees to Formula for the area of a circular sector.
radians?
How to find the measure of a length of a circular arc
and area of circular sector?
How to find the six trigonometric ratios of an angle? Definition of the trigonometric functions of an acute
How to use the fundamental trigonometric
angle of a right triangle.
identities?
Reciprocal identities.
How to verify a trigonometric identity?
The fundamental identities.
How to find the trigonometric function values of an
Definition of the trigonometric functions of any
angle in standard position?
angle.
Standards Addressed
California: CA T: 1.0 – 7.0, 12.0, 19.0
Common Core: F-TF-1
California: CA T: 1.0 – 7.0, 12.0, 19.0
Common Core: F-TF-2
How to find the value of trigonometric functions?
Finding special values of the trigonometric
function?
How to apply the formulas for negatives?
How to graph the six basic trigonometric functions?
Definition of the trigonometric functions in terms of
a unit circle.
Definition of periodic functions.
Formulas for negatives.
Features of the trigonometric functions and their
graphs.
California: CA T: 1.0 – 7.0, 12.0, 19.0
How to find reference angles?
How to find trigonometric ratios using reference
angles?
How to find angles with a calculator?
Definition of a reference angle.
Theorem on reference angles.
California: CA T: 1.0 – 7.0, 12.0, 19.0
How to sketch a transformed sine and cosine graph?
Theorem on amplitudes and periods.
Theorem on amplitudes, periods, and phase shifts.
Common Core: F-TF-3
Common Core: F-TF-3
California: CA T: 1.0 – 7.0, 12.0, 19.0
Common Core: F-IF-7, F-TF-4
Students will understand how to sketch cosecant,
secant, tangent, and cotangent.
How to sketch tangent, cotangent, secant and
cosecant?
Theorem on the graph y=tan(bx+c).
California: CA T: 1.0 – 7.0, 12.0, 19.0
Common Core: F-IF-7 F-TF-4
Students will understand how to apply trigonometry
to triangle problems.
How to solve a right triangle?
No new concepts just application of prior work.
California: CA T: 1.0 – 7.0, 12.0, 19.0
Common Core:
96
Content Outcomes
Students will understand how to verify
trigonometric identities.
Essential Questions
How to verify an identity?
Honors Algebra 2
Unit 7 – Analytical Trigonometry
Key Concepts
No new concepts just application of prior work.
Standards Addressed
California: CA T: 3.0 - 3.2, 8.0 – 11.0, 19.0
Common Core: F-TF-8
Students will understand how to solve trigonometric
equations.
Students will understand how to apply the addition
and subtraction formulas.
Students will understand how to apply the multiple
angle formulas.
Students will understand how to apply the productto-sum and sum-to-product formulas.
Students will understand the inverse trigonometric
functions.
California: CA T: 3.0 - 3.2, 8.0 – 11.0, 19.0
How to solve a trigonometric equation?
How to solve a trigonometric equation involving
multiple angles?
How to solve a trigonometric equation by factoring?
Unit circle.
Coterminal angles.
How to use the subtraction formula?
How to use the addition formulas to find the
quadrant containing an angle?
Subtraction formula for cosines.
Addition formula for cosines.
Cofunction formulas.
Addition and subtraction formulas for sine and
tangent.
California: CA T: 3.0 - 3.2, 8.0 – 11.0, 19.0
How to use the double angle formulas?
How to change the form of a multiple angle
trigonometric function?
How to use the half-angle formulas and double
angle formulas to verify an identity?
Double angle formulas.
Half-Angle identities.
Half-Angle formulas.
Half-Angle formulas for the tangent.
California: CA T: 3.0 - 3.2, 8.0 – 11.0, 19.0
How to express a product as a sum?
How to express a sum or a difference as a product?
How to verify a trigonometric identity using the
sum to product or product to sum formulas?
How to use the sum to product formulas to solve an
equation?
Product-to-sum formulas.
Sum-to-product formulas.
California: CA T: 3.0 - 3.2, 8.0 – 11.0, 19.0
How to find the inverse functions of the six
trigonometric functions?
How to find the values of inverse functions?
How to change a product as a sum?
Relationships between a function and its inverse.
Definition of the inverse sine function.
Properties of inverse sine functions.
Definition of the inverse cosine function.
Properties of the inverse cosine function.
Definition of the inverse tangent function.
Properties of the inverse tangent function.
Common Core: F-TF-7
Common Core: F-TF-9
Common Core: F-TF-9
Common Core: F-TF-9
97
California: CA T: 3.0 - 3.2, 8.0 – 11.0, 19.0
Common Core: F-TF-6,7
Content Outcomes
Students will understand how to apply the law of
sines.
HonorsAlgebra 2
Unit 8 – Applications of Trigonometry
Essential Questions
Key Concepts
How to solve a triangle in ASA form?
The law of sines.
How to solve a triangle in SSA form?
Standards Addressed
California: CA T: 13.0, 14.0, 18.0; CA AN: 1.0
Common Core: G-SRT-10,11
Students will understand how to apply the law of
cosines.
Students will understand vectors and their
applications.
Students will understand the dot product of two
vectors.
How to solve a triangle in SAS form?
How to solve a triangle in SSS form?
How to find the area of a triangle?
The law of cosines.
Area of a triangle.
Heron’s formula.
California: CA T: 13.0, 14.0, 18.0; CA AN: 1.0
How to sketch a vector?
How to sketch scalar vectors?
How to express vectors in i and j format?
Definition of the magnitude of a vector.
Definition of addition of vectors.
Definition of a scalar multiple of a vector.
Definition of 0 and –a.
Properties of addition and scalar multiples of
vectors.
Definition of subtraction of vectors.
Definition of i and j.
i and j form of vectors.
Formulas for horizontal and vertical components of
a vector.
California: CA T: 13.0, 14.0, 18.0; CA AN: 1.0
How to find the dot product of two vectors?
How to find the angle between two vectors?
How to show that two vectors are parallel?
How to show a pair of vectors is orthogonal?
Definition of the dot product.
Properties of the dot product.
Definition of parallel and orthogonal vectors.
Theorem on the dot product.
Theorem on the cosine of the angle between
vectors.
Theorem on orthogonal vectors.
California: CA T: 13.0, 14.0, 18.0; CA AN: 1.0
98
Common Core: G-SRT-10,11
Common Core: N-VM 1,2,3,4,5
Common Core:
Content Outcomes
Students will understand how to solve systems of
equations.
Students will understand how to solve systems of
linear equations in two variables.
Students will understand how to solve systems of
inequalities.
Honors Algebra 2
Unit 9 – Systems of Equalities and Inequalities
Essential Questions
Key Concepts
How to solve a system of two equations?
Guidelines for the method of substitution for two
How to solve a system of two equations using
equations in two variables.
substitution?
How to use the theorem on equivalent systems?
How to solve a system of linear equations with
infinite solutions?
How to solve a system of linear equations with no
solutions?
How to solve a linear system by elimination?
Theorem on equivalent systems.
How to sketch the graph of an inequality?
How to sketch the graph of a linear inequality?
How to sketch the graph of a system?
How to solve a system of linear inequalities?
Guidelines for sketching the graph of an inequality
in x and y.
Standards Addressed
California: CA A2 2.0
Common Core: A-REI-5,6
California: CA A2 2.0
Common Core: A-REI-7,8
California: CA A2 2.0
Common Core: A-REI-10,11,12
:
Students will understand how to solve a problem
with linear programming.
Students will understand how to solve systems of
linear equations in more than two variables.
Students will understand how to apply the algebra
of matrices.
Students will understand how to find the inverse of
a matrix.
How to find the minimum and maximum values of
an objective function?
How to solve a linear programming problem?
Guidelines for solving a linear programming
problem.
How to use elimination to solve a system of linear
equations?
How to use matrices to solve a system of linear
equations?
Definition of a matrix.
Theorem on matrix row transformations.
Echelon form of a matrix.
Guidelines for finding the echelon form of a matrix.
California: CA A2 2.0
How to find the product of two matrices?
How to add and subtract matrices?
Definition of the product of a real number and a
matrix.
Theorem on matrix properties.
Guidelines for finding the product of two matrices.
Definition of the product of two matrices.
California: CA A2 2.0
Definition of the inverse of a matrix
California: CA A2 2.0
How to find the inverse of a two by two matrix?
California: CA A2 2.0
Common Core:
Common Core:N-VM-6,7,8
Common Core: N-VM-8,9,10
Common Core: A-REI-9
99
Content Outcomes
Students will understand how to sketch and apply
ellipses.
Students will understand how to sketch and apply
hyperbolas.
Students will understand the polar coordinate
system.
Honors Algebra 2
Unit 10 – Topics from Analytical Geometry
Essential Questions
Key Concepts
How to sketch an ellipse?
Definition of an ellipse.
How to find an equation of an ellipse given vertices Standard equations of an ellipse with center at the
and foci?
origin.
How to find equations for half-ellipses?
Definition of eccentricity.
How to find the eccentricity of an ellipse?
Standards Addressed
California: CA T: 16.0 – 18.0
Common Core:
How to sketch a hyperbola?
How to find an equation of a hyperbola satisfying
prescribed condition?
How to find the equation of portions of hyperbolas?
Definition of a hyperbola.
Standard equations of a hyperbola with center at the
origin.
California: CA A2: 16.0, 17.0; CA T: 15.0 – 18.0
How to change polar coordinates into rectangular
coordinates?
How to change rectangular coordinates into polar
coordinates?
How to change a rectangular equation into a polar
equation?
Relationship between rectangular and polar
coordinates.
California: CA A2: 16.0, 17.0; CA T: 15.0 – 18.0
Common Core:
Common Core:
:
100
Damien High School
Mathematics and Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Pre-Calculus
“B” or better during both semesters in Algebra 2 or “C” or better in Honors Algebra 2
Yes – Category D
Year
Brief Course Description
This class will deal with the topics introduced in Algebra 2 with an emphasis on the properties and graphs of polynomial, piecewise, absolute value,
rational, logarithmic, exponential, and trigonometric functions. The 2 nd semester focuses on right triangle trigonometry and trigonometric identities. This
course is intended to prepare students for the study of calculus.
Assigned Textbook(s)
Supplemental Material(s)
Pre-Calculus; Stewart, Redlin, and Watson 5th Edition.
Graphing Calculator
Common Assessments Utilized
Common Final each semester
Homework
In-Class Problems
Quizzes
Exams
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
101
Major Content Outcomes
1.
2.
Know the basic concepts from Algebra 2.
a. The application of exponent and radical rules.
b. The simplification of exponential expressions.
i. Factoring by trial and error
ii. Factoring using special cases: difference of two squares,
perfect trinomials, and the sum and difference of cubes.
iii. Factoring by grouping
c. Rational expressions
i. Recognizing the limitations on the domain of rational
expressions.
ii. Multiplication and division of rational expressions.
iii. Addition and Subtraction of rational expressions.
iv. Simplification of compound fractions.
v. Rationalizing a complex denominator.
d. Solving of equations.
i. Solving of equations using the graphing method,
completing the square, and the quadratic equation.
ii. Solving fractional expressions.
iii. Solving equations with fractional powers
iv. Solving absolute value equations.
e. Solving and Graphing of Inequalities
i. Quadratic inequalities
ii. Compound inequalities
iii. Absolute value inequalities
Know the application and manipulation of functions.
a. Definition of a function
b. Graphs of functions.
i. Power functions.
ii. Piecewise functions.
c. Transformations of functions.
i. Even/Odd functions
ii. Horizontal and vertical shifts.
iii. Vertical and horizontal stretching and compression.
d. Quadratic functions.
i. Maximums and minimums.
ii. Standard form.
iii. Local maxima and minima.
e. Combining functions.
i. Addition, subtraction, multiplication, and division
ii. Composite functions.
iii. Domain of composite functions.
Major Skill Outcomes (include Labs???)
Students will review the basic concepts from Algebra 2
Students will know the many applications of rational functions.
Students will know the application and manipulation of functions.
Students will understand how graph and solve exponential and logarithmic equations.
Students will be familiar with all the applications, including graphing and problem solving
associated with the unit circle.
Students will be familiar with polar coordinates and vectors.
Students will know systems of equations including 2 by 2 and 3 by 3 matrices.
Unit 1
Students should be able to express inequalities in terms of interval notation.
Students should be able to evaluate expressions involving radicals, absolute value, and exponents.
Students should be able to simplify expressions involving radicals and exponents.
Students should be able to factor completely.
Students should be able to perform the order of operations.
Unit 2
Students should be able evaluate functions.
Students should be able to apply the vertical line test.
Students should be able to find the domain and range of functions.
Students should be able to sketch functions.
Students should be able to graph functions using transformations.
Students should be able to determine if a function is even or odd.
Students should be able to express a quadratic function in standard form.
Students should be able to find the vertex and the maximum or minimum of a quadratic function.
Students should be able to find composite functions and their domains.
Students should be able to determine if a function is one to one.
Students should be able to find the inverse of a function.
Unit 3
Students should be able to graph a polynomial using its zeros and end behavior.
Students should be able to divide polynomials using long division.
Students should be able to apply the remainder theorem.
Students should be able to apply the factor theorem.
Students should be able to use synthetic division.
Students should be able to evaluate expressions involving complex numbers.
Students should be able to find a polynomial given certain conditions.
Students should be able to graph rational functions.
Unit 4
Students should be able to sketch exponential and logarithmic functions.
Students should be able to find the domain of a logarithmic function.
Students should be able to switch between logarithmic and exponential functions.
Students should be able to evaluate logarithms without a calculator.
Students should be able to expand and contract logarithmic expressions.
Students should be able to solve exponential and logarithmic equations.
Students should be able to use the compound interest formula and other exponential growth and decay
problems.
102
f.
3.
4.
5.
Inverse functions.
i. Definition of one to one functions.
ii. Computation of inverse functions.
Understand the many applications of rational functions.
a. The graphs of polynomial functions.
i. End behavior of a polynomial.
ii. Intermediate value theorem.
iii. Finding zeros and graphing of rational functions.
b. Dividing polynomials.
i. Polynomial long division.
ii. Synthetic division.
c. Rational zeros of a polynomial.
d. Complex zeros.
i. Definition of i.
ii. Addition, subtraction, multiplication, and division.
e. Complex zeros.
i. Multiplicities of zeros
ii. Complex factorization.
f. Rational functions.
i. Vertical and horizontal asymptotes.
ii. Intercepts.
iii. Graphing.
Understand how to graph and solve exponential and logarithmic equations.
a. Exponential functions.
i. Definition and graphing.
ii. Natural exponential function.
iii. Interest equations.
b. Logarithmic functions.
i. Definition.
ii. Graphing
iii. Common logarithm and natural logarithm.
c. Laws of logarithms.
i. Combining logarithmic expressions.
ii. Expanding logarithmic expressions.
iii. Change of base formula.
d. Exponential and logarithmic equations.
Be familiar with all the applications, including graphing and problem solving
associated with the unit circle.
a. The unit circle.
i. The relationship between degrees and radians and the
ability to change radians to degrees and degrees to radians.
b. Trigonometric functions of real numbers.
i. Even/odd properties.
Unit 5
Students should be able to find the six trigonometric functions of an angle measure.
Students should be able to switch between radians and degrees.
Students should be able to find co-terminal and reference angles.
Students should be able to graph the six trigonometric functions.
Students should be able to find the values of remaining trigonometric functions given certain information.
Unit 6
Students should be able to convert degrees to radians and radians to degrees.
Students should be able to find an arc length and sector area of a circle.
Students should be able to solve for missing sides of a triangle using trigonometry.
Students should be able to find exact trigonometric values.
Students should be able to identify the quadrant of an angle based on its trigonometric values.
Students should be able to solve triangles using the law of sines and the law of cosines.
Unit 7
Students should be able to verify trigonometric identities.
Students should be able to solve trigonometric equations over an interval and over all real angle measures.
Students should be able to apply the multiple angle, sum, difference, product to sum, and the sum to
product formulas.
Students should be able to graph the inverse trigonometric functions.
Students should be able to find inverse trigonometric expressions.
Unit 8
Students should be able to understand the polar coordinate system.
Students should be able to convert polar coordinates to rectangular coordinates and rectangular coordinates
to polar coordinates.
Students should be able to convert rectangular equations to polar equations and polar equations to
rectangular equations.
Students should be able to find scalar vectors and add or subtract vectors.
Students should be able to find the magnitude of a vector.
Students should be able to find the dot product of vectors.
Students should be able to find an angle between vectors and state if the vectors are orthogonal or not.
Unit 9
Students should be able to solve a system of equations by graphing, substitution or elimination.
Students should be able to solve a 3 variable 3 equation system of equations.
Students should be able to perform operations on matrices.
Students should be able to find the inverse of a matrix.
Students should be able to solve a system of inequalities.
Unit 10
Students should be able to find the vertices and foci of conics.
Students should be able to sketch a conic.
Students should be able to find the equation for a conic that satisfies certain criteria.
103
6.
7.
ii. Reciprocal identities.
iii. Pythagorean identities.
c. The graphing of sine, cosine, tangent, cosecant, secant, and
cotangent.
i. Transformations of trigonometric functions.
d. Right triangle trigonometry
i. Law of sines.
ii. Law of cosines.
iii. Heron's formula for triangle area.
e. Trigonometric Identities.
i. Addition and subtraction formulas.
ii. Double-angle, half-angle, and product to sum formulas.
f. Inverse trigonometric functions.
i. Arcsine, arccosine, and arctangent.
g. Trigonometric equations.
Be familiar with polar coordinates and vectors.
a. Polar coordinates.
i. Definition of polar coordinates.
ii. Converting polar coordinates to rectangular coordinates and
rectangular coordinates to polar coordinates.
iii. The relationship between polar equations and rectangular
equations.
b. Sketching polar equations.
c. Vectors
i. Component form of vectors.
ii. Operations on vectors.
iii. Magnitude of vectors.
Know systems of equations including 2 by 2 and 3 by 3 matrices.
a. Systems of equations.
i. Graphing method.
ii. Substitution method.
iii. Elimination method and elimination method with addition
and subtraction.
b. Two variable, two equation systems.
c. Three variable, three equation systems.
d. Matrices
i. Solving matrices using elimination method.
e. Algebra of matrices.
i. Addition, subtraction, and scalar multiplication of matrices.
ii. Matrix multiplication.
f. Inverse matrices.
104
Content Outcomes
Students will learn about the properties of real
numbers.
Students will understand the application of
exponent rules and radicals.
Students will understand the applications of
algebraic expressions.
Students will understand the applications of rational
expressions.
Students will understand how to solve linear,
quadratic and rational equations.
Students will understand how to solve an inequality.
Pre-Calculus
Unit 1 – What are the Fundamental Concepts of Algebra?
Essential Questions
Key Concepts
What are the properties of real numbers?
Definitions of different number types.
How are the properties of equality applied?
Properties of real numbers.
What are the properties involving zero?
Properties of negatives.
How do the properties of quotients apply to Properties of fractions.
fractions?
Definition of absolute value.
How is absolute value used?
Definition of distance between points on a
How to find union and intersection sets?
coordinate line.
How to give answers in interval notation?
Definition of union and intersection.
How is the distance between two points calculated?
Interval notation.
How would the law of exponents be applied to
Laws of exponents.
simplify expressions?
Theorem on negative exponents.
How are negative exponents utilized?
Definition of a radical.
What is a fractional exponent and how to convert
Properties of radicals.
roots to fractional exponents?
Laws of radicals.
How are sums, differences, products, and quotients
Rationalizing denominators of radicals greater than
of radicals applied?
root two.
How are denominators with roots rationalized?
Definition of rational exponents.
How are radicals combined into a single root?
Scientific notation.
How are polynomials added and subtracted?
Definitions of algebraic expressions.
How are polynomials multiplied?
Definition of a polynomial.
How are polynomials divided?
Order of operations.
How can a polynomial be multiplied with the
Product formulas.
special product formulas?
Factoring formulas.
How are polynomials factored?
How is trial and error factoring applied to
trinomials?
How are polynomials factored by grouping?
How are rational expressions multiplied and
Reducing common factors.
divided?
Finding a least common denominator.
How are rational expressions added and subtracted?
Finding a least common denominator with binomial
How are complex fractions simplified?
and polynomial denominators.
How are complex denominators and numerators
Use of a conjugate to rationalize.
rationalized?
How are fractional expressions simplified?
How to solve a linear equation?
The properties of equality.
How to solve an equation in combinations with the
The order of operations.
order of operations?
Quadratic equation.
How to solve a quadratic by factoring?
Guidelines for completing the square.
How to solve a quadratic by completing the square?
Guidelines for applying the discriminant
How to solve a quadratic by the quadratic equation?
How to solve a formula for a specific variable?
How to use the discriminant to find the number of
zeros of a quadratic?
How to solve rational equations?
How to solve an inequality?
Properties of inequalities.
How to solve ‘or’ and ‘and’ inequalities?
Properties of absolute values.
Solving a rational inequality?
How to solve an absolute value inequality?
105
Standards Addressed
California: CA A2:1.O, 2.0, 4.0, 5.0, 8.0, 12.0
Common Core: N-RN 3, A-SSE1
California: CA A2:1.O, 2.0, 4.0, 5.0, 8.0, 12.0
Common Core: N-RN 1,2, A-REI 2,3
California: CA A2:1.O, 2.0, 4.0, 5.0, 8.0, 12.0
Common Core: A-SSE 2,3 A-APR 1
California: CA A2:1.O, 2.0, 4.0, 5.0, 8.0, 12.0
Common Core: N-RN 3
California: CA A2:1.O, 2.0, 4.0, 5.0, 8.0, 12.0
Common Core: A-SSE 3, A-CED 1, A-REI 3, 4, 11
California: CA A2:1.O, 2.0, 4.0, 5.0, 8.0, 12.0
Common Core: A-REI 2, 3
Content Outcomes
Students will understand the concept of a function.
Students will be able to graph of functions.
Students will understand the transformations of
functions.
Students will understand the application of
quadratic functions: maxima and minima.
Students will understand how to combining
functions.
Students will learn about one-to-one functions and
their Inverses.
Pre-Calculus
Unit 2 – What are Functions?
Essential Questions
Key Concepts
How to evaluate a function?
Definition of a function.
How to determine the domain of a function?
Definition of a graph of a function.
How to sketch a piece-wise function?
Vertical line test.
How to apply the vertical line test?
Definition of linear and piece-wise functions.
Standards Addressed
California: CA A2: 15.0, 17.0; CA T: 4.0 - 8.0, 10.0
Common Core: F-IF 1, 2
How to recognize the shape of common functions
based on the equation?
How to sketch the graph of a function?
How to find the values of a function from a graph?
How to find domain and range of a function?
Vertical line test.
Basic functions and their graph.
California: CA A2: 15.0, 17.0; CA T: 4.0 - 8.0, 10.0
How to distinguish even and odd functions?
How to vertically and horizontally shift a graph?
How to horizontally and vertically stretch and
compress a graph?
How to graph a piecewise function?
Definition of even and odd functions.
Transformations on graphs.
How to sketch the graph of a quadratic function?
How to use the standard from of a parabola?
How to find the vertex of a parabola?
How to find the maximum and minimum of a
parabola?
Definition of a quadratic function.
Standard equation of a parabola with a vertical axis.
Theorem for locating the vertex of a parabola.
Theorem on the minimum or maximum value of a
quadratic function.
California: CA A2: 15.0, 17.0; CA T: 4.0 - 8.0, 10.0
How to perform basic operations on functions?
How to find composite functions?
How to find the domain of a composite function?
How to find the composite form of a function?
Sum, difference, produce and quotient of functions.
Definition of a composite function.
California: CA A2: 15.0, 17.0; CA T: 4.0 - 8.0, 10.0
How to determine if a function is one to one?
How to apply the horizontal line test to see is a
function is one to one?
How to find an inverse function?
Theorem on increasing or decreasing functions are
one to one.
Definition of an inverse function.
Theorem on inverse functions.
The relationship between the domain and range of
inverse functions.
Guidelines for finding inverse functions.
Common Core: F-IF 5, 6, G-CO 5, A-REI 10
California: CA A2: 15.0, 17.0; CA T: 4.0 - 8.0, 10.0
Common Core: F-IF 3
Common Core: F-IF 4, 7, Modeling
Common Core: F-IF 1, 3
106
California: CA A2: 15.0, 17.0; CA T: 4.0 - 8.0, 10.0
Common Core: F-IF 4
Content Outcomes
Students will understand polynomial functions and
their graphs.
Students will be able to divide polynomials.
Students will understand the application of real
zeros of polynomials.
Students will understand how to solve equations
relating to complex numbers.
Students will understand the application of complex
zeros and the fundamental theorem of algebra.
Students will understand the application of rational
functions.
Pre-Calculus
Unit 3 – What are Polynomial and Rational Functions?
Essential Questions
Key Concepts
How to sketch a polynomial function of degree Intermediate value theorem.
larger than two?
Guidelines for finding end behavior.
How to determine the end behavior of a function?
Guidelines for finding real zeros of a polynomial.
How to find the real zeros of a polynomial?
Shape of a graph near a zero with a given
How does multiplicity of its zeros affect the graph
multiplicity.
of a function?
Standards Addressed
California: CA A2: 3.0, 6.0, 7.0
Common Core: A-APR 1, 2, 3, 4
How to use polynomial long division?
How to apply the remainder theorem?
How the factor theorem applies to factoring a
polynomial?
How to synthetically divide a polynomial by x-c?
Polynomial long division.
Remainder theorem.
Factor theorem.
Guidelines for synthetic division.
California: CA A2: 3.0, 6.0, 7.0
How to find a rational zero?
How to use the rational zeros theorem to factor a
polynomial?
How to use the rational zeros theorem and the
quadratic equation?
How to apply Descartes rule of sign?
How to factor higher degree polynomials?
How to find the upper and lower bound for zeros of
a polynomial?
Rational zeros theorem.
Guidelines for finding a rational zero of a
polynomial.
Descartes’ rule of sign.
The upper and lower bounds theorem.
California: CA A2: 3.0, 6.0, 7.0
How to add, subtract, and multiply complex
numbers?
How to divide complex numbers?
How to find and apply the square root of a negative
number?
How to solve quadratic equations with complex
solutions?
Definition of a complex number.
Formula for adding, subtracting, and multiplying
complex numbers.
Formula for dividing complex numbers.
Formula for finding square roots of negative
numbers.
California: CA A2: 3.0, 6.0, 7.0
How to factor a polynomial completely?
How to factor a polynomial with complex zeros?
How to find a polynomial with specified zeros?
How to find all zeros of a polynomial?
How to find a polynomial with a specified complex
zero?
How to factor a polynomial into linear and
quadratic factors?
Fundamental theorem of algebra.
Complete factorization theorem.
Zeros theorem.
Conjugate zeros theorem.
Linear and quadratic factors theorem.
California: CA A2: 3.0, 6.0, 7.0
How to find the domain of a rational function?
How to find vertical and horizontal asymptotes of a
rational function?
How to sketch a rational function?
How to find an oblique asymptote of a rational
function?
Definition of vertical asymptotes.
Definition of horizontal asymptotes.
Theorem on horizontal asymptotes.
Guidelines for sketching the graph of a rational
function
California: CA A2: 3.0, 6.0, 7.0
107
Common Core: A-APR 1, 2, 3, 4
Common Core: A-APR 2, 3
Common Core: N-CNI 1, 2, 3
Common Core: N-CNI 7, 8, 9
Common Core: A-APR 6, 7
Content Outcomes
Students will understand exponential functions.
Students will understand logarithmic functions.
Students will understand the application of the laws
of logarithms.
Students will be able to solve exponential and
logarithmic equations.
Students will be able to model with exponential and
logarithmic functions.
Pre-Calculus
Unit 4 – What are Exponential and Logarithmic Functions?
Essential Questions
Key Concepts
How to solve an exponential equation?
Theorem on exponential functions being one to one.
How to sketch an exponential function?
Compound interest formula.
How to shift an exponential function using The number e.
transformations?
Definition of the natural exponential function.
How to find an exponential function that satisfies Continuously compounded interest formula
conditions?
How to apply the compound interest function?
How to use the continuously compounded interest
formula?
How to graph the natural exponential functions?
How to change an equation from exponential to
Definition of log base a.
logarithmic form?
Logarithmic and exponential forms of an equation.
How to find basic logarithms using the properties of
Properties of logs.
logarithms?
Definition of a common logarithm.
How to solve logarithmic equations?
How to graph a logarithmic equation?
How to shift a logarithmic equation with
transformations?
How to graph and solve natural logarithms?
How to apply the laws of logarithms?
Laws of logarithms.
How to solve a logarithmic equation?
The change of base formula.
How to evaluate logarithms using the change of
base formula?
Standards Addressed
California: CA A2: 11.0 - 11.2, 12.0, 14.0
Common Core: F-LE 1, 3
California: CA A2: 11.0 - 11.2, 12.0, 14.0
Common Core: F-LE 5
California: CA A2: 11.0 - 11.2, 12.0, 14.0
Common Core: N-RN 1
How to solve an exponential equation?
How to solve an exponential equation of quadratic
type?
How to use the definition of a logarithm to solve an
exponential equation?
How to solve an equation involving logarithms?
Special change of base formulas.
Guidelines for solving exponential equations.
Guidelines for solving logarithmic equations.
California: CA A2: 11.0 - 11.2, 12.0, 14.0
How to predict the values based on an exponential
or logarithmic model?
Exponential growth model.
Radioactive decay model.
California: CA A2: 11.0 - 11.2, 12.0, 14.0
Common Core: F-LE 4
Common Core: Modeling
108
Content Outcomes
Students will understand the application of the Unit
Circle.
Students will learn about trigonometric functions of
real numbers.
Students will learn how to make trigonometric
graphs.
Pre-Calculus
Unit 5 – What are the Trigonometric Functions of Real Numbers?
Essential Questions
Key Concepts
How to locate a point on the unit circle?
Definition and equation of a unit circle.
How to find a terminal point on the unit circle?
Definition of a terminal point.
How to find a reference number for each angle?
Definition of a reference angle.
How to use reference angles to find terminal points?
How to evaluate trigonometric functions?
How to determine the sign of a trigonometric
function?
How to use a calculator to evaluate trigonometric
functions?
How to apply the even odd properties to determine
trigonometric functions?
How to apply the fundamental identities to find all
trigonometric functions?
How to sketch a transformed sine and cosine graph?
Standards Addressed
California: CA T: 4.0 - 6.0, 9.0
Common Core: F-TF 1, 2
Definition of the trigonometric functions.
Domains of the trigonometric functions.
Signs of the trigonometric functions.
Even-odd properties.
Fundamental identities.
California: CA T: 4.0 - 6.0, 9.0
Theorem on amplitudes and periods.
Theorem on amplitudes, periods, and phase shifts.
California: CA T: 4.0 - 6.0, 9.0
Common Core: F-TF 3
Common Core: F-TF 4, 5, Modeling
Students will learn how to make tangent, cotangent
and cosecant trigonometric graphs.
How to sketch tangent, cotangent, secant and
cosecant?
Guidelines for sketching tangent and cotangent
curves.
Guidelines for sketching cosecant and secant
curves.
109
California: CA T: 4.0 - 6.0, 9.0
Common Core: F-TF 6
Content Outcomes
Students will learn about angle measures.
Students will understand the trigonometry of right
triangles.
Students will learn about the application of
trigonometric functions of angles.
Students will understand the application of the Law
of Sines.
Pre-Calculus
Unit 6 – What are the Trigonometric Functions of Angles?
Essential Questions
Key Concepts
How to convert between radians and degrees?
Definition of a radian measure.
How to find coterminal angles?
Relationship between degrees and radians.
How to find the length of a circular arc?
Definition of coterminal angles.
How to find the area of a circular sector?
Equation for length of circular arc and area of
How to find linear and angular speed?
circular area.
Equation for linear speed and angular speed.
How to find the six trigonometric ratios of an angle?
How to solve a right triangle?
How to use a calculator to find trigonometric ratios?
How to apply trigonometric ratios to solve a word
problem?
The trigonometric ratios of a right triangle.
How to use the reference angle to evaluate a
trigonometric ratio?
How to express a trigonometric function in terms of
another?
How to use trigonometry to find the area of a
triangle?
Fundamental identities.
Area of a triangle using trigonometry.
How to solve a triangle in ASA form?
How to solve a triangle in SSA form?
The law of sines.
Standards Addressed
California: CA T: 1.0, 3.0, 12.0 - 14.0
Common Core: G-SRT 6
California: CA T: 1.0, 3.0, 12.0 - 14.0
Common Core: G-SRT 7
California: CA T: 1.0, 3.0, 12.0 - 14.0
Common Core: G-SRT 9
California: CA T: 1.0, 3.0, 12.0 - 14.0
Common Core: G-SRT 10
Students will understand the application of the Law
of Cosines.
How to solve a triangle in SAS form?
How to solve a triangle in SSS form?
How to find the area of a triangle using Heron’s
formula?
The law of cosines.
Area of a triangle.
Heron’s formula.
110
California: CA T: 1.0, 3.0, 12.0 - 14.0
Common Core: G-SRT 11
Content Outcomes
Students will understand trigonometric identities.
Essential Questions
How to verify an identity?
Pre-Calculus
Unit 7 – What is Analytical Trigonometry?
Key Concepts
No new concepts just application of prior work.
Standards Addressed
California: CA T: 3.0, 8.0 - 11.0
Common Core: F-TF 8
Students will understand the application of addition
and subtraction formulas.
Students will understand the application of DoubleAngle, Half-Angle, and Product-Sum Formulas.
Students will understand the Inverse trigonometric
functions.
Students will understand how to solve trigonometric
equations.
How to use the subtraction formulas?
How to use the addition formulas to find the
quadrant containing an angle?
Subtraction formula for cosines.
Addition formula for cosines.
Addition and subtraction formulas for sine and
tangent.
California: CA T: 3.0, 8.0 - 11.0
How to use the double angle formulas?
How to change the form of a multiple angle
trigonometric function?
How to use the half-angle formulas and double
angle formulas to verify an identity?
How to express a product as a sum?
How to express a sum or a difference as a product?
How to verify a trigonometric identity using the
sum to product or product to sum formulas?
How to use the sum to product formulas to solve an
equation?
Double angle formulas.
Half-Angle identities.
Half-Angle formulas.
Half-Angle formulas for the tangent.
Product-to-sum formulas.
Sum-to-product formulas.
California: CA T: 3.0, 8.0 - 11.0
How to find the inverses of the six trigonometric
functions?
How to find the values of inverse trigonometric
functions?
Relationships between a function and its inverse.
Definition of the inverse sine function.
Properties of inverse sine functions.
Definition of the inverse cosine function.
Properties of the inverse cosine function.
Definition of the inverse tangent function.
Properties of the inverse tangent function.
California: CA T: 3.0, 8.0 - 11.0
How to solve a trigonometric equation?
How to solve a trigonometric equation involving
multiple angles?
How to solve a trigonometric equation by factoring?
Unit circle.
Co-terminal angles.
California: CA T: 3.0, 8.0 - 11.0
111
Common Core: F-TF 9
Common Core: F-TF 9
Common Core: F-TF 6, 7, Modeling
Common Core: F-TF 7
Content Outcomes
Students will understand how to find polar
coordinates.
Students will understand how to create graphs of
polar coordinates.
Students will rules and applications related to
vectors.
Students will understand the properties and
applications of the Dot Product.
Pre-Calculus
Unit 8 – What ways can Polar Coordinates and Vectors used?
Essential Questions
Key Concepts
How to change polar coordinates into rectangular Relationship between rectangular
coordinates?
coordinates.
How to change rectangular coordinates into polar
coordinates?
How to change a rectangular equation into a polar
equation?
and
polar
Standards Addressed
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
Common Core:
How to sketch the graph of a polar equation?
How to use the symmetry of a polar coordinate to
sketch a polar graph?
Tests for symmetry of polar graphs.
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
How to write a vector in component form?
How to sketch a vector?
How to sketch scalar vectors?
How to express vectors in i and j format?
How to perform operations on vectors?
Component form of a vector.
Definition of the magnitude of a vector.
Definition of addition of vectors.
Definition of a scalar multiple of a vector.
Definition of 0 and –a.
Properties of addition and scalar multiples of
vectors.
Definition of subtraction of vectors.
Definition of i and j.
i and j form of vectors.
Formulas for horizontal and vertical components of
a vector.
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
How to find the dot product of two vectors?
How to find the angle between two vectors?
How to show that two vectors are parallel?
How to show a pair of vectors is orthogonal?
How to find the component of a vector?
Definition of the dot product.
Properties of the dot product.
Definition of parallel and orthogonal vectors.
Theorem on the dot product.
Theorem on the cosine of the angle between
vectors.
Theorem on orthogonal vectors.
Calculating components.
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
Common Core:
112
Common Core: N-VM 1, 2, 4, 5
Common Core:
Content Outcomes
Students will be able to solve systems of Equations.
Students will be able to solve systems of Linear
Equation with Two Variables.
Students will be able to solve systems of Linear
Equations with Several Variables.
Students will understand system of Linear
Equations: Matrices.
Students will understand the application of the
Algebra of Matrices.
Students will understand the inverse Matrices and
Matrix Equations.
Students will be able to graph systems of
Inequalities.
Pre-Calculus
Unit 9 – What are Systems of Equalities and Inequalities determined and utilized?
Essential Questions
Key Concepts
How to solve a system of any two equations?
Guidelines for substitution method.
How to solve a system of two equations using
Guidelines for elimination method.
substitution, elimination, and graphing?
Guidelines for graphing method.
Standards Addressed
California: CA A2: 2.0; CA LA: 1.0, 2.0, 10.0, 11.0
Common Core: A-REI 5, 6, 7
How to solve a system of linear equations with one
solution?
How to solve a system of linear equations with
infinite solutions?
How to solve a system of linear equations with no
solutions?
Guidelines for modeling with systems of equations.
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
How to solve a three equation three variable
system?
How to solve a three equation three variable system
with no solution and many solutions?
Triangular form of a linear system.
How to use a matrix to solve a three equation three
variable system?
Definition of a matrix.
Elementary row operations.
Row echelon form of a matrix.
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
Sum, difference, and scalar product of matrices.
Properties of addition and scalar multiplication of
matrices.
Matrix multiplication.
Properties of matrix multiplication.
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
What is an identity matrix?
How to find an inverse of a 2 by 2 matrix?
How to solve a system of equations using matrices?
Definition of an identity matrix.
Guidelines for finding a 2 by 2 inverse matrix.
Solving a matrix equation.
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
How to sketch the graph of an inequality?
How to sketch the graph of a linear inequality?
How to sketch the graph of a system?
How to solve a system of linear inequalities?
Graphing inequalities.
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
Common Core: A-CED 2, A-REI 5, 6, 7
How to add, subtract and solve scalar matrices?
How to multiply matrices?
California: CA T: 15.0, 16.0; CA LA: 4.0, 5.0, 12.0
Common Core:
Common Core: A-REI 8
Common Core: A-VM 6, 7, 8, 9, 10
Common Core: A-REI 9
Common Core: A-CED 3, 12
113
Content Outcomes
Students will understand how to sketch and apply
ellipses.
Students will understand how to sketch and apply
hyperbolas.
Pre-Calculus
Unit 10 – What are the key characteristics of Analytic Geometry?
Essential Questions
Key Concepts
How to sketch an ellipse?
Definition of an ellipse.
How to find an equation of an ellipse given vertices Standard equations of an ellipse with center at the
and foci?
origin.
How to find equations for half-ellipses?
Definition of eccentricity.
How to find the eccentricity of an ellipse?
How to sketch a hyperbola?
How to find an equation of a hyperbola satisfying
prescribed condition?
How to find the equation of portions of hyperbolas?
Definition of a hyperbola.
Standard equations of a hyperbola with center at the
origin.
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Standards Addressed
California: CA A2: 16.0, 17.0
Common Core: G-GPE 2, 3
California: CA A2: 16.0, 17.0
Common Core: G-GPE 2, 3
Damien High School
Mathematics and Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Statistics
“C” in Algebra 2
Yes – Category D
Year
Brief Course Description
The course will cover primarily descriptive statistics with minimal focus on inferential statistics. The topics include conducting studies in order to collect,
organize, summarize, and analyze data, frequency distributions, correlation, regression, probability and counting, probability and normal distributions.
Assigned Textbook(s)
Supplemental Material(s)
Bluman, Allan G. Elementary Statistics: A Step by Step Approach, 6th edition : McGraw Hill, 2007.
Statistics Made Clear; Professor Michael Starbird (a video series).
Common Assessments Utilized
Common Final each semester
Homework
In-Class Problems
Semester Projects
Tests
ISOs Addressed
115
Major Content Outcomes
I. Introduction of Statistics
A. Descriptive and Inferential Statistics
B. Sample vs. Population
C. Discrete vs. Continuous Variables
D. Nominal, Ordinal, Interval, Ratio Scales
E. Data collection
1) Random Sampling
2) Systematic Sampling
3) Stratified Sampling
4) Cluster Sampling
II. Descriptive Analysis and Presentation of Single Variable Data
A. Frequency Distributions
B. Histograms, Frequency Polygons and Cumulative Frequency Polygons
C. Stem and Leaf plots
D. Time Plots
E. Mean, Median and Mode for lists of data
F. Variance and Standard Deviation
1. Biased and unbiased estimate
2. Formula and shortcut formula
3. Calculator
G. Mean and Standard Deviation of Grouped Data
H. Measure of Position
1. Quartiles
2. 5-Number Summary
3. Box Plots
4. Outliers
5. Modified Box Plots
6. Z-Scores
I. Chebyshev's Theorem
III. Descriptive Analysis and Bivariate Data
A. Scatterplots
B. Definition of Correlation and Regression
C. Correlation
1. Independent vs. Dependent variable
2. Positive vs. Negative correlation
3. Scatterplots
4. Formula for correlation
5. Correlation and Causation
D. Regression
IV. Probability
A. Formula for classical probability
B. Formula for empirical probability
C. Subjective probability
D. Counting Rules
1. Fundamental Counting Principle
2. Factorial, Permutations, and Combinations
E. Tree Diagrams
F. Addition Rules
G. Multiplication Rules and Conditional Probability
H. Bayes’ Theorem
V. Discrete Probability Distributions
A. Random Variable
B. Probability distributions
Major Skill Outcomes (include Labs???)
Student will be able to explore data through the analysis of graphical and numerical techniques to discover
patterns and deviations from pattern.
Student will be able to collect data through a well-developed plan. In addition, numerous methods of data
collection will be analyzed for strengths and weaknesses.
Student will be able to use probability as a tool for anticipating the distribution of data under different
models.
Unit 1
Students should be able to tell the difference between a descriptive and inferential statistic.
Students should be able to classify a variable by its level of measurement.
Students should be able to classify a variable as qualitative or quantitative, and classify a quantitative
variable as discrete or continuous.
Students should be able to classify a sample by the method of collection.
Students should be able to define a population from a given statement and determine how a sample might
be obtained.
Students should know the various ways that statistics could be misused and how to recognize misleading
statistics.
Unit 2
Students should be able to find class boundaries, midpoints and class widths in a frequency distribution.
Students should know the criteria for a frequency distribution.
Students should be able to construct a grouped, ungrouped, categorical and cumulative frequency
distribution given raw data.
Students should be able to construct a histogram, frequency polygon, and ogive given a frequency
distribution.
Students should be able to describe a distribution by its shape.
Students should be able to construct a bar graph, Pareto chart, time series graph, pie chart, and stem and leaf
plot.
Unit 3
Students should be able to find the mean, median, mode, and midrange given raw data and know which one
gives the best description of center based on the skew.
Students should be able to determine the mean of a frequency distribution and the mean of weighted data.
Students should be able to find the range, variance and standard deviation given raw data and frequency
distributions.
Students should be able to compare sets of data to determine how one set of data is more variable than
another.
Students should be able to apply Chebyshev’s Theorem and know when to use it.
Students should be able to find z scores, quartiles, and percentiles to determine relative position of a data
value.
Students should be able to check a data set for outliers.
Unit 4
Students should be able to describe the difference between classical, empirical, and subjective probability.
Students should be able to determine the sample space in a probability experiment.
Students should be able to determine the probabilities of outcomes and events, as well as their complements
in a probability experiment.
Students should be able to explain the difference between “and” & “or” and apply them correctly to
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1. Requirements to be a distribution
2. Discrete distributions
3. Continuous distributions
C. Mean, Variance, Standard Deviation, and Expectation.
D. Binomial distribution
VI. Normal Probability Distributions
A. Normal Probability Distribution
1. Mean and Standard Deviation
2. Bell Curve
B. Standard Normal Distribution
1. Probabilities
2. Cutoff values
C. Applications of the Normal Distribution
D. Normal Approximation to the Binomial
E. Tests for Normality
probability problems.
Students should be able to calculate empirical probabilities after running multiple trials in an experiment.
Students should be able to determine if two events are mutually exclusive and find probabilities of mutually
exclusive and non-mutually exclusive events using the addition rules of probability.
Students should be able to construct Venn diagrams for multiple events.
Students should be able to determine if two events are independent and find probabilities of dependent and
independent events using the multiplication rules of probability.
Students should be able to determine when and how to calculate a conditional probability.
Students should be able to calculate the number of all possible outcomes for a sequence of events using the
fundamental counting principle and the permutation and combination formulas.
Students should be able to calculate probabilities of events with numerous outcomes using counting rules.
Unit 5
Students should be able to describe the rankings of poker hands and calculate their probabilities and odds.
Students should be able to describe the rules of Roulette, Craps, and Texas Hold ‘Em and the various bets
allowed in each game.
Students should be able to determine the true odds and payout odds of each bet in Roulette.
Students should be able to calculate the probability of rolling a specified number with two dice, describe
how one could win and lose a bet in craps, and calculate the odds against winning each bet.
Students should be able to determine the “house edge” in Roulette and Craps.
Students should be able to calculate the probabilities and odds against being dealt each 2 card hand, and
determine the number of “outs” after the flop and turn for each scenario.
Unit 6
Students should be able to determine if a distribution represents a probability distribution and if so,
construct and graph the probability distribution for a given set of data.
Students should be able to calculate the mean, variance, and standard deviation of a probability distribution.
Students should be able to calculate the expected value of a discrete random variable and determine the
fairness of a game.
Students should be able to describe the criteria for a binomial distribution and find probabilities of the
outcomes.
Students should be able to calculate the mean, variance, and standard deviation of a binomial distribution
using the shortcut formulas.
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Content Outcomes
Students will demonstrate an understanding of the
difference between Descriptive and Inferential
Statistics
Students will demonstrate an understanding of
different types of variables and types of data
Statistics
Unit 1 – What is the Purpose of Probability and Statistics?
Essential Questions
Key Concepts
What is a variable?
Definition of descriptive and inferential Statistics
Types of variables (qualitative, quantitative,
What is a data set?
discrete, continuous)
Difference between population and sample
What defines the differences between descriptive
and inferential statistics?
What is the difference between a population and a
sample?
What is the difference between qualitative and
quantitative variables?
Standards Addressed
California:
Common Core: S-IC.1 , S-1C.3 , S-IC.5
Levels of measurement (nominal, ordinal, ratio,
interval)
Boundaries for continuous data
California:
Sampling techniques (random, systematic, stratified,
cluster, convenience
California:
What makes a quantitative variable discrete or
continuous?
Common Core: S-IC.1 , S-1C.3 , S-IC.5
How are boundaries for continuous variables
determined?
How can variables be categorized, counted, or
measured?
Students will demonstrate an understanding of data
collection and sampling techniques
What are the most common methods of collecting
data?
Common Core: S-IC.1 , S-1C.3 , S-IC.5
How should a sample be chosen to maintain the
randomness of the subjects?
Students will demonstrate an understanding of the
difference between observational and experimental
studies
What is an observational study?
How is an experimental study conducted?
Observational and experimental studies
Independent, dependent and confounding variables
Hawthorne effect
California:
Misuses of Statistics (suspect samples, ambiguous
averages, changing the subject, detached statistic,
implied connections, misleading graphs, faulty
survey questions
California:
Common Core: S-IC.1 , S-1C.3 , S-IC.5
What is the difference between the treatment and
control group?
Students will become aware of the uses and misuses
of Statistics
What are independent, dependent, and confounding
variables?
What are the various ways that a person could
misuse statistics for their own benefit?
What is a suspect sample?
How can a graph be misleading?
What is a detached statistic?
What are some examples of faulty survey
questions?
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Common Core: S-IC.1 , S-1C.3 , S-IC.5
Content Outcomes
Students will understand the process of organizing
data in frequency distributions
Statistics
Unit 2 – How do Frequency Distributions and their Graphs help us to Organize Data?
Essential Questions
Key Concepts
What is a frequency distribution?
Understand raw data, find range, and determine
class width
What are the components of a frequency
Components of a frequency distribution (class, class
distribution?
boundary, frequency, frequency percentage,
midpoint, cumulative frequency)
Why do we organize raw data in a frequency
Types of frequency distributions (categorical,
distribution?
grouped, ungrouped)
Rules to construct frequency distributions
What are the 3 types of distributions (categorical,
grouped, and cumulative) and how do they differ?
Standards Addressed
California: CA PS: 8.0 , CA APS: 14.0
Common Core: S-ID.1 , S-ID.5
What are the rules for creating a frequency
distribution?
Students will learn to construct Histograms,
Frequency Polygons, and Ogives
What is a histogram and how do you create one
using a frequency distribution?
Construct a histogram, frequency polygon, and an
ogive using raw data or frequency distribution
California: CA PS: 8.0 , CA APS: 14.0
Common Core: S-ID.1 , S-ID.5
What is a frequency polygon and how do you create
one using a frequency distribution?
What is an ogive and how do you create one using a
cumulative frequency distribution?
Students will learn how to construct other types of
graphs to describe data
What is a bar graph and how do you create one
using a categorical frequency distribution?
Construct a bar graph, Pareto chart, pie graph, time
series, and stem and leaf plot using raw data or
frequency distribution
Distribution shapes
What is a Pareto chart and how do you create one
using a categorical frequency distribution?
What is a time series graph and how do you create
one?
What is a pie chart and how do you create one using
a frequency distribution?
What is a stem and leaf plot and how do you create
one using raw data? Why is it unlike other graphs?
What are the advantages and disadvantages of each
and when do you use one instead of another?
119
California: CA PS: 8.0 , CA APS: 14.0
Common Core: S-ID.1 , S-ID.5
Content Outcomes
Students will be able to calculate the Measures of
Central Tendency
Students will be able to calculate the Measures of
Variation
Students will be able to calculate the Measures of
Position
Students will be able to calculate the Exploratory
Data Analysis
Statistics
Unit 3 – How can Data be Effectively Described?
Essential Questions
Key Concepts
What is the difference between a statistic and a
Formula for population and sample mean
parameter?
Definition of mean, median, mode, and midrange.
How do we find the mean of a data set given raw
Procedure for finding mean for group data
data and grouped data?
Formula for weighted mean
How do we find the median, mode and midrange of
Properties and advantages of each measure of
a data set?
central tendency
How do we find a weight’s mean?
When was one measure suggested over another?
Standards Addressed
California: CA PS: 5.0 – 8.0, 10.0, 11.0, 16.0
Common Core: S-ID.2 , S-1D.3 , S-IC.4
How do we determine how spread out the data:
How can we compare 2 different sets of data using
the range, variance, and standard deviation?
What steps are taken to find the standard deviation?
What shortcut can be taken to find the variance and
standard deviation?
How do we find the standard deviation for grouped
data?
What is the purpose of the Coefficient of Variation?
What is the purpose of Chebyshev’s Theorem?
What does the Empirical Rule state?
Formulas for variance and standard deviation
Formulas (and shortcuts) for variance and standard
deviation for data obtained from samples
Procedure for finding sample variance and standard
deviation for grouped data.
Formula for coefficient of variation
Uses of variance and standard deviation
Range rule of thumb to approximate standard
deviation
Chebyshev’s theorem
The empirical rule
California: CA PS: 5.0 – 8.0, 10.0, 11.0, 16.0
How do you find z-scores and what is their
purpose?
How do you find the percentile of a specific data
value?
Why are percentiles and quartiles effective
measures of position?
What is the Interquartile Range (IQR) and how is it
used to find outliers?
Formula for z-score
Percentiles and quartiles
Procedure for finding a data value corresponding to
a given percentile
Procedure for finding data values corresponding to
Q1 , Q2 , and Q3
Interquartile range
Procedure for identifying outliers
California: CA PS: 5.0 – 8.0, 10.0, 11.0, 16.0
What is the 5 number summary and why are these
values important to find in describing a data set?
How do you create a box plot using the 5 number
summary?
Five number summary
Procedure for constructing boxplots
Information obtained from a boxplot
Traditional Stats vs. Exploratory Data Analysis
California: CA PS: 5.0 – 8.0, 10.0, 11.0, 16.0
120
Common Core: S-ID.2 , S-1D.3 , S-IC.4
Common Core: S-ID.2 , S-1D.3 , S-IC.4
Common Core: S-ID.2 , S-1D.3 , S-IC.4
Content Outcomes
Students will understand sample spaces and
Probability
Statistics
Unit 4 – What is Probability and how are Counting Rules used?
Essential Questions
Key Concepts
What are Probability and a Probability experiment?
Def. of Probability, probability experiment,
outcome, sample space, simple event, compound
What are a sample space, outcome, and event?
event, and complement of an event
Formula for classical probability
What is the difference between Classical, Empirical,
Inclusive and exclusive “or” statements
and Subjective Probability?
Basic Probability rules
Formula for empirical probability
What is the difference between “and” & “or”
Law of large numbers
statements?
Standards Addressed
California: CA A2: 18.0 , CA A2: 19.0 , CA PS:
1.0 , CA PS: 2.0 , CA PS: 3.0
Common Core: S-IC.2 , S-CP.1 , S-CP.2 , S-CP.3 ,
S-CP.4 , S-CP.5 , S-CP.6 , S-CP.7 , S-CP.8 , SCP.9
What are the rules for probability?
What is the complement of an event?
Given a scenario, what is the sample space and
corresponding probabilities of each outcome or
event?
What does the law of large numbers state?
What is the difference between probability and
odds?
Students will know how and when to apply addition
rules for Probability
When are two events mutually exclusive?
How do Venn diagrams help to visualize
probabilities?
Students will know how and when to apply
multiplication rules and Conditional Probability
What are the addition rules of Probability (“or”
statements)?
When are two events independent?
What are the multiplication rules of probability
(‘and” statements)?
Addition rule for events that are mutually exclusive
Addition rule for events that are not mutually
exclusive
Venn diagrams
California: CA A2: 18.0 , CA A2: 19.0 , CA PS:
1.0 , CA PS: 2.0 , CA PS: 3.0
Independent and dependent events
Multiplication rule for 2 independent events
Conditional probability and multiplication rule for
dependent events.
Formula for conditional probability
California: CA A2: 18.0 , CA A2: 19.0 , CA PS:
1.0 , CA PS: 2.0 , CA PS: 3.0
Fundamental counting rule
Factorial formula
Def. of permutation and combination
Formula for permutation and combination
Counting rules combined with probability rules
California: CA A2: 18.0 , CA A2: 19.0 , CA PS:
1.0 , CA PS: 2.0 , CA PS: 3.0
Why are the rules different for finding probability
with and without replacement?
Common Core: S-IC.2 , S-CP.1 , S-CP.2 , S-CP.3 ,
S-CP.4 , S-CP.5 , S-CP.6 , S-CP.7 , S-CP.8 , SCP.9
Common Core: S-IC.2 , S-CP.1 , S-CP.2 , S-CP.3 ,
S-CP.4 , S-CP.5 , S-CP.6 , S-CP.7 , S-CP.8 , SCP.9
What is conditional probability? When and how is
it used?
Students will know how and when to apply
counting rules to calculate the number of ways
something can occur and how to apply counting
rules to finding Probabilities
What is the Fundamental Counting Principle and
how is it used?
How do we find the number of permutations and
combinations of a certain number of objects for a
given event?
How do we calculate probabilities of events with
numerous outcomes using counting rules?
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Common Core: S-IC.2 , S-CP.1 , S-CP.2 , S-CP.3 ,
S-CP.4 , S-CP.5 , S-CP.6 , S-CP.7 , S-CP.8 , SCP.9
Content Outcomes
Students will learn the rules, odds, and probabilities
associated with Poker
Statistics
Unit 5 – How is Probability applied to games of chance?
Essential Questions
Key Concepts
What is the ranking of 5-card poker hands?
Probability rules and counting principles
Ranking of poker hands
How do we find the number of hands of each rank?
Standards Addressed
California: CA A2: 18.0, 19.0; CA PS: 1.0 - 3.0
Common Core: S-IC.2 ; S-CP. 1 – 9
What are the probabilities of getting each poker
hand?
What are the odds against getting each poker hand?
Students will learn the rules, odds, and probabilities
associated with Roulette
What are the rules and game play of roulette?
Probability rules and counting principles
Rules and various bets in Roulette
Common Core: S-IC.2 ; S-CP. 1 – 9
What are the different bets in roulette?
Students will learn the rules, odds, and probabilities
associated with Craps
California: CA A2: 18.0, 19.0; CA PS: 1.0 - 3.0
How do you determine the true odds and payout
odds of each scenario?
What are the rules and possible bets in craps?
National Discipline
Probability rules and counting principles
Rules and various bets in Craps
California: CA A2: 18.0, 19.0; CA PS: 1.0 - 3.0
Common Core: S-IC.2 ; S-CP. 1 – 9
What is the probability of rolling a specified
number with two dice?
National Discipline
What is a pass line, come line, field, and place bet
and what are the corresponding odds against
winning?
How can a person win in various scenarios and what
are the probabilities of each?
What is an “odds” bet and why does it give the
player a bigger advantage than any other game?
How is the house edge determined on each bet?
Students will learn the rules, odds, and probabilities
associated with Texas Hold ‘Em
What are the rules and game play of Texas Hold
‘Em?
Probability rules and counting principles
Rules and various bets in Texas Hold ‘Em
California: CA A2: 18.0, 19.0; CA PS: 1.0 - 3.0
Common Core: S-IC.2 ; S-CP. 1 – 9
What are the probabilities and odds of being dealt
each 2 card hand?
National Discipline
How are the numbers of “outs” after the flop and
turn determined?
What is the total number of different types of
hands?
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Content Outcomes
Students will understand and create Probability
distributions
Statistics
Unit 6 – What is the purpose of a Discrete Probability Distribution?
Essential Questions
Key Concepts
What is a random variable?
Def. of random variable and discrete probability
distribution
How do we create a probability distribution?
Requirements for a probability distribution
Standards Addressed
California: CA PS: 4.0 – 7.0
Common Core: S-MD.1 – 6
What are the requirements for a probability
distribution?
Students will be able to compute Mean, Variance,
Standard Deviation, and Expected Value of various
types of distributions
How is the mean of a probability distribution
found?
How is the variance and standard deviation of a
probability distribution found?
Formulas for the mean, variance and standard
deviation of a probability distribution
Def. of expected value
Formula for expected value
California: CA PS: 4.0 – 7.0
Requirements for a binomial experiment
Notation for a binomial distribution
Binomial probability formula
Formulas for mean, variance, and standard
deviation for a binomial distribution
California: CA PS: 4.0 – 7.0
Common Core: S-MD.1 – 6
What is the expected value for a discrete random
variable and how is it found?
How is the fairness of a game determined?
Students will understand requirements and apply
formulas for Binomial Distributions
What are the criteria for a binomial experiment?
What is a binomial distribution?
How do we find probabilities of certain outcomes in
a binomial experiment?
How do we find the mean, variance, standard
deviation, and expected value of a binomial
distribution?
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Common Core: S-MD.1 – 6
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Brief Course Description
AP Statistics
“C” or higher in Pre-Calculus or Honors Algebra 2 with Trigonometry
Yes
Year
An introduction to the modern methods of analyzing numerical data, as dictated by the Advanced Placement syllabus. Topics include frequency distribution, measures
of central tendency, measures of dispersion, probability theory, binomial and normal distribution, hypothesis testing and liner regression.
Assigned Textbook(s)
Supplemental Material(s)
The Practice of Statistics
TI-89 Calculator
Common Assessments Utilized
Common Final each semester
Homework
In-Class Quizzes
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
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Major Content Outcomes
I. Exploring Data
A. The interpretation of graphical displays of distributions of data including dotplots, stemplots,
histograms, and frequency plots.
1. Center and spread.
2. Clusters and gaps.
3. Outliers and other unusual features.
4. Shape
B. The summarization of distributions including quantifying the graphical distributions.
1. Measuring the two measures of center: median and mean.
2. Measuring spread: range, interquartile range, and standard deviation.
3. Measuring position: quartiles, percentiles, standardized scores (z-scores).
4. Using and interpreting boxplots
5. The effect of changing units on the above summary measures.
C. Comparing distributions of data.
1. Comparing center and spread within groups and between groups.
2. Comparing clusters and gaps.
3. Comparing outliers and other unusual features.
4. Comparing shapes.
D. Exploring data.
1. Analyzing patterns in scatterplots.
2. Correlation and linearity.
3. Least square regression.
4. Residual plots, outliers, and influential points.
5. Transformations to achieve linearity: non algebraic transformations.
E. Exploring categorical data: frequency tables.
1. Marginal and joint frequencies of two way tables.
2. Conditional relative frequencies and association.
II. Planning a Study
A.Overview of methods of data collection.
1. Census.
2. Sample survey.
3. Experiment.
4. Observational study.
B. Planning and conducting surveys.
1. Characteristics of a well-designed and conducted survey.
2. Populations, samples, and random selection.
3. Sources of bias in surveys.
4. Simple random sampling.
5. Stratified random sampling.
C. Planning and conducting experiments.
1. Characteristics of a well-designed and conducted experiment
2. Treatments, control groups, experimental units, random assignments and replication.
3. Sources of bias and confounding including the placebo effect and blinding.
4. Completely randomized sample design.
5. Randomized block design including matched pairs design.
D. Generalizing results of studies, surveys and experiments.
III. Anticipating patterns using probability models, theory and simulation.
A. Probability as relative frequency.
Major Skill Outcomes
I. Explore data through the analysis of graphical and numerical techniques to discover patterns and
deviations from pattern.
II. Collect data through a well-developed plan. In addition, numerous methods of data collection will be
analyzed for strengths and weaknesses.
III. Use probability as a tool for anticipating the distribution of data under different models.
IV. Understand that statistical inference needs to be properly applied to draw conclusions from data. This
includes selecting a proper model, including a statement in probability language, and stating confidence
about the conclusion.
Unit 1
Students should be able to give a numerical summary of a distribution.
Students should be able to contrast mean and median as measures of center.
Students should be able to give a five number summary and graphically represent it using a box plot.
Students should be able to state what measures to describe a distribution are resistant.
Students should be able to construct stemplots, histograms, and bar graphs.
Students should be able to determine the effects of a linear transformation on the measures of center and
spread.
Students should be able to describe a distribution using shape, outliers, center, and spread.
Unit 2
Students should know the properties of density curves.
Students should be able to standardize an observation using a z-score.
Students should be able to use a z table to find the probability an event or events will occur.
Students should be able to apply the empirical rule to normal distributions.
Students should be able to assess normality.
Unit 3
Students should be able to recognize both quantitative and categorical variable.
Students should be able to plot two quantitative variables on a scatterplot and identify the response and
explanatory variable.
Students should be able to explain correlation and calculate it using a calculator.
Students should be able to find a regression line using a calculator and by hand.
Students should be able to describe r squared.
Students should be able to recognize outliers in a scatterplot and identify if they are influential.
Students should be able to find residuals and construct a residual plot.
Unit 4
Students should be able to perform a non-algebraic transformation to data to make it more linear.
Students should be able to identify a causation model as either confounding or common response and state
the lurking variable.
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1. Law of large numbers.
2. Addition rule, multiplication rule, conditional probability and independence.
3. Discrete random variables and their distributions.
4. Simulation of probability distributions, including binomial and geometric.
5. Mean and standard deviation of a random variable.
B. Combining independent random variables.
1. Notion of independence versus dependence.
2. Mean and standard deviation for sums and differences of random variables.
C. The normal distribution.
1. Properties of the normal distribution.
2. Using tables of the normal distribution.
3. The normal distribution as a model for measurements.
D. Sampling distributions.
1. Sampling distributions of a sample proportion.
2. Sampling distribution of a sample mean.
3. Central Limit Theorem.
4. Sampling distribution of a difference between two sample proportions.
5. Sampling distribution of a difference between two sample means.
6. Simulation of sampling distributions.
IV. Statistical Inference.
A. Confidence Intervals.
1. The meaning of a confidence interval.
2. Large sample confidence interval for a proportion.
3. Large sample confidence interval for a mean.
4. Large sample confidence interval for a difference of two proportions.
5. Large sample confidence interval for a difference of two means.
B. Tests of significance.
1. Logic of significance testing, Ho and Ha, p-values, Type I and II errors and power.
2. Large sample test for a proportion.
3. Large sample test for a mean.
4. Large sample test for a difference between two proportions.
5. Large sample test for a difference between two means.
6. Chi square test for goodness of fit and independence.
C. Special case of normally distributed data.
1. t-distribution.
2. Single sample t procedures.
3. Two sample t procedures.
4. Inference for the slope of least-square regression line.
Students should be able to compute row and column comparisons of two-way tables.
Students should be able to identify and explain Simpson’s paradox.
Unit 5
Students should be able to identify the population, experimental units, and treatment of an experiment.
Students should be able to recognize bias in sampling designs.
Students should be able to use a random method such as a random number chart to select a sample.
Students should be able to distinguish between an experiment and an observational study.
Students should be able to define and apply double blind, placebo effect, and matched pair.
Students should be able to identify different sampling models such as stratified, block and multi-stage.
Students should be able to carry out a simulation.
Unit 6
Students should be able to know and apply the probability rules of defined events.
Students should be able to determine if events are disjoint, complementary, or independent.
Students should be able to know how the multiplication rules are affected by independence.
Students should be able to understand conditional probability and reverse conditional probability.
Students should be able to make tree diagrams to organize information in a multiple stage problem.
Unit 7
Students should be able to define, identify, and know the properties of discrete and continuous random
variables.
Students should be able to calculate the mean and variance of discrete random events involving sums,
differences, and linear combinations.
Students should be able to find the expected value of a random variable.
Unit 8
Students should be able to identify both binomial and geometric situations by verifying the conditions.
Students should be able to use a calculator to find binomial and geometric probabilities.
Students should be able to find mean and standard deviation of binomial and geometric settings.
Students should be able to use normal approximation for binomial situations.
Unit 9
Students should be able to identify parameters and statistics in a sample or experiment.
Students should be able to recognize sample variability and that variability is controlled by sample size.
Students should be able to find the mean and standard deviation for both sample proportions and sample
means.
Students should be able to use a normal calculation to find probabilities that use proportions and means.
Unit 10
Students should be able to calculate and explain a confidence interval.
Students should be able to change margin of error by manipulating n and confidence level.
Students should be able to find sample size based on desired criteria.
Students should be able to define and state a null and alternate hypothesis when testing a parameter.
Students should be able to explain a p-value.
Students should be able to calculate one and two sided p-values and assess the significance level of those pvalues.
126
Students should be able to recognize the difference between statistical significance and practical
significance.
Students should be able to recognize when to use a z-test.
Students should be able to explain, define, and apply Type I error, Type II error, and power.
Unit 11
Students should be able to determine when a one sample, two sample, or matched pair mean procedure is
appropriate.
Students should be able to recognize when a t procedure is appropriate instead of a z procedure.
Students should be able to carry out a t-test and calculate a t interval.
Students should be able to recognize the limitation of a t procedure.
Students should be able to carry out a two sample t procedure and two sample t interval.
Unit 12
Students should be able to recognize when a one-sample, two-sample, or matched pair design is appropriate.
Students should be able to carry out a one proportion z test and a one proportion z interval.
Students should be able to understand when it is appropriate to use a z procedure.
Students should be able to carry out a two proportion z test and a two proportion z interval.
Unit 13
Students should be able to recognize when to use a goodness of fit test and a test of independence.
Students should be able to perform both goodness of fit and test of independence chi square tests.
Students should be able to find expected value and the chi square component of any cell.
Students should be able to either use a chart or a calculator to find the p-value of a chi square test.
Students should be able to evaluate the p-value of a chi square test to determine its significance.
Unit 14
Students should be able to recognize the type of inference for a regression model.
Students should be able to explain slope, y-intercept, and standard errors from a computer output or minitab printout.
Students should be able to use a computer output or mini-tab printout to carry out tests of significance and
compute confidence intervals for the slope.
127
Content Outcomes
Students will understand how to interpret data.
AP Statistics
Unit 1 – How To Explore Data.
Essential Questions
Key Concepts
How to identify the individuals and variables in a
Definition of individuals, variables, categorical
data set?
variables and quantitative variables.
How to distinguish categorical and quantitative
variables?
Standards Addressed
California: CA APS: 10.0
Common Core:
National Discipline:
Students will understand how to display
distributions of data.
Students will understand how to inspect
distributions.
Students will understand how to construct and
interpret time plots.
How to make and interpret a bar graph and a pie
chart?
How to make a dotplot?
How to make a stemplot?
How to make a histogram?
Definition of bar graph, pie chart, and dotplots.
Definition of a stem and a leaf.
Guidelines for making a histogram.
How to identify an overall pattern and major
deviations of a pattern from data?
How to determine shape of data from its graphical
distribution?
How to describe a distribution by giving numerical
measure of center and spread?
Definition skewness and symmetry.
Definition of mean and median.
Definition of outliers.
Strengths and weaknesses of mean and median.
How to decide the appropriate measure of center?
How to identify outliers?
How to make a time plot?
How to recognize patterns or trends in a time plot?
Definition of a time plot.
California:
Common Core: S-ID-1
National Discipline:
California:
Common Core: S-ID-2
National Discipline:
California:
Common Core: S-ID-3
National Discipline:
Students will understand how to measure center and
spread.
Students will understand how to change units of
measurement and their effect on data.
How to find the mean and median of a set of data?
How to find the five number summaries and draw a
boxplot?
How to use the calculator to find the numerical
summary of data?
Equation of mean and median.
Definition of resistance.
Properties of standard deviation.
Properties of boxplots.
Stats/Vars program on TI-89 calculator.
What is the effect of a linear transformation on
measures of center and spread?
Linear graphing models.
California:
Common Core: S-ID-2
National Discipline:
California: CA APS: 14.0
Common Core:
National Discipline:
Students will understand how to compare
distributions.
How to use side by side graphs to compare
distributions?
How to make back to back stemplots and side by
side boxplots?
Box plots.
Stem plots.
California:
Common Core: S-ID-3
National Discipline:
128
AP Statistics
Unit 2 – The Normal Distributions
Content Outcomes
Students will understand the properties of a density
curve.
Essential Questions
What are the basic properties of a density curve?
How to approximately locate the median and mean
of a density curve based on shape?
The properties of a symmetric density curve?
Key Concepts
Definition and properties of a density curve.
Standards Addressed
California: CA APS: 4.0
Common Core: S-ID-4
National Discipline:
Students will understand normal distributions.
How to use the empirical rule?
How to find a z score?
How to find a probability based on a z-score?
How to find a value to achieve a certain percentile?
Empirical rule.
Standard normal probability chart.
California: CA APS: 5.0, 7.0
Common Core:
National Discipline:
Students will understand how to assess normality.
How to determine if a distribution in normal or
relatively normal?
Normal probability charts.
Empirical rule.
California: CA APS: 11.0, 18.0
Common Core:
National Discipline:
129
Content Outcomes
Students will understand statistical data.
AP Statistics
Unit 3 – Examining Relationships
Essential Questions
Key Concepts
How to recognize an explanatory variable and a
Definition of explanatory variable and response
response variable?
variable.
Standards Addressed
California: CA APS: 12.0
Common Core:
National Discipline:
Students will understand how to construct and
interpret scatterplots.
Students will understand correlation.
Students will understand how to construct straight
lines regression lines.
How to plot the relationship between two
quantitative variables?
How to plot a categorical variable using a different
color or symbol?
How to describe the form, direction and strength of
a relationship?
How to recognize an outlier in a scatterplot?
How to find correlation using a calculator?
How to quantify the strength of a relationship using
correlation?
How to identify if data has no relationship using
correlation?
Scatterplots.
Definition of an outlier.
How to interpret slope and y-intercept in context of
a problem?
How to graph a line?
Slope-intercept form of a line.
California: CA APS: 13.0
Common Core: S-ID-6
National Discipline:
Stats/Vars program on a TI-89.
Definition of correlation.
Properties of correlation.
California:
Common Core: S-ID-6,8,9
National Discipline:
California:
Common Core: S-ID-7
National Discipline:
Students will understand how to construct and
interpret regression models.
How to use a calculator to find the least square
regression line?
How to find the equation of a least square
regression line without the calculator?
How to use the regression line to find predicted y
values for x?
What is r squared?
How to recognize outliers and influential points
from a scatterplot?
How to find residuals and complete a residual plot?
Slope and y-intercept equations.
Properties of r squared.
Definition of a residual.
Properties of residual plots.
130
California:
Common Core: S-ID-6
National Discipline:
AP Statistics
Unit 4 – Two-Variable Data
Content Outcomes
Students will understand how to model
nonlinear data.
Essential Questions
How to recognize exponential growth or power
functions?
How does a pattern in the residual plot apply back
to the original data?
Key Concepts
Exponential and power functions.
Standards Addressed
California: CA APS: 13.0
Common Core:
National Discipline:
Students will understand how to interpret
correlation and regression.
Students will understand how to describe relations
in categorical data.
How to recognize lurking variables that would
affect the response variable?
Are correlation and least square regressions
resistant?
How to identify the three causation models?
Properties of direct causation, common response,
and confounding.
How to describe the relationship between
categorical variables with row and column analysis?
How to recognize and explain Simpson’s Paradox?
Two-way tables.
Simpson’s paradox.
California:
Common Core: S-ID-9
National Discipline:
California:
Common Core: S-ID-5
National Discipline:
131
Content Outcomes
Students will understand how to construct a
sampling model.
Students will understand how to construct and
conduct a proper experiment.
Students will understand how to perform
simulations.
AP Statistics
Unit 5 – Producing Data
Essential Questions
Key Concepts
How to identify the population of interest for a
Definition of population of interest.
sample?
Definition of voluntary response and convenience
How to recognize the bias in voluntary response and sampling.
convenience sampling?
Random number charts.
How to use a random number chart to select a
Various forms of bias in sampling.
sample?
How to recognize undercoverage in a sample?
How to recognize the various forms of potential
bias in a sample?
How to recognize the various forms of sampling
including blocking, matched pair, stratified, and
multistage?
How to differentiate between an experiment and an
Definition of an experiment.
observational study?
Definition of an observational study.
How to identify the treatments, response variables,
The placebo effect.
factors and experimental units of an experiment?
Definition of a treatment.
How to diagram a randomized experiment?
Definition of double blind.
How to use a random number chart to select a
random sample?
What the placebo effect is and how it affects
experiments?
How to carry out a simulation?
How to correctly assign numbers to reflect the
probability of an event occurring?
Guidelines for carrying out a simulation.
Standards Addressed
California:
Common Core: S-IC-3
National Discipline:
California:
Common Core: S-IC-3
National Discipline:
California:
Common Core: S-ID-4
National Discipline:
132
Content Outcomes
Students will understand the basic ideas of
probability.
Essential Questions
What is randomness?
What is a probability?
AP Statistics
Unit 6 – Probability
Key Concepts
Definition of randomness.
Definition of probability.
Standards Addressed
California: CA APS: 1.0, 2.0
Common Core: S-IC-2; S-CP1,2,
National Discipline:
Students will understand the probability models.
What is a probability model?
What is sample space?
What is a Venn diagram?
How to apply and use the probability rules?
Venn diagrams.
Probability rules.
California:
Common Core: S-IC-2; S-CP-3,4,5,6,7
National Discipline:
Students will understand how to use the general
probability rules.
What are disjoint events?
What are independent events?
What conditional probability is and how to apply it?
What is a reverse conditional problem?
Definition of disjoint.
Definition of independent.
Definition of conditional probability.
Baye’s rule.
California:
Common Core: S-CP-3,4,5,6,7,8,9
National Discipline:
133
Content Outcomes
Students will understand how to recognize the
differences between discrete and continuous random
variables.
Students will understand how to find the means and
variances of random variables.
AP Statistics
Unit 7 – Random Variables
Essential Questions
Key Concepts
How to recognize and define a discrete random
Definition of a discrete random variable.
variable?
Properties of discrete random variables.
How to construct a probability distribution table and Probability tables and histograms.
a probability histogram for a discrete random
Definition of a continuous random variable.
variable?
How to recognize and define a continuous random
variable?
How to determine probabilities of events as area
under density curves?
How to find probabilities of events as areas under
the standard normal curve?
How to find the mean and variance of a discrete
Mean of a discrete random variable.
random variable?
Variance of a discrete random variable.
How to find expected value?
Law of large numbers.
How to use the law of large numbers to approximate Rules for means.
the mean of a distribution?
Rules for variances.
How to use the rules for means and rules for
variances to solve problems with combinations of
random variables?
134
Standards Addressed
California: CA APS: 3.0,4.0,5.0
Common Core: S-MD-1,2
National Discipline:
California: CA APS: 6.0
Common Core: S-MD-1,2
National Discipline:
Content Outcomes
Students will understand the binomial distribution.
Students will understand the geometric distribution.
AP Statistics
Unit 8 – Binomial and Geometric Distributions
Essential Questions
Key Concepts
How to identify a random variable as binomial
The binomial setting.
using B.I.N.S.?
Definition of a binomial distribution.
How to use the Stats/Vars program or formulas to
Binomial PDF and binomial CDF.
calculate binomial probabilities?
The binomial equation.
How to construct probability tables and histograms
Mean and standard deviation of a binomial random
for binomial distributions?
variable.
How to find cumulative functions for binomial
Normal approximation of a binomial distribution.
variables?
How to find the mean and standard deviation of
binomial variables?
How to use normal approximation of a binomial
distribution to find probabilities?
How to identify a random variable as geometric
The geometric setting.
using B.I.S.?
Rules for calculating geometric probabilities.
How to use the Stats/Vars program or formulas to
The mean and standard deviation of a geometric
calculate geometric probabilities?
variable.
How to find the cumulative functions for geometric
The probability it takes more than n trials for a
variables?
success.
How to construct probability functions for
geometric variables?
How to find the expected values and standard
deviations of geometric variables?
135
Standards Addressed
California: CA APS: 3.0, 4.0, 5.0,7.0
Common Core: S-ID-4
National Discipline:
California: CA APS: 10.0, 11.0
Common Core: S-ID-4
National Discipline:
Content Outcomes
Students will understand how to interpret a
sampling distribution.
Students will understand how to interpret sample
proportions.
Students will understand how to interpret sample
means.
AP Statistics
Unit 9 – Sampling Distributions
Essential Questions
Key Concepts
How to identify parameters and statistics in a
Definition of a parameter and a statistic.
sample or experiment?
Concept of sample variability.
How to recognize sampling variability?
Guidelines for bias and variability of a statistic.
How to interpret a sampling distribution?
How to describe the bias and variability of a
statistic?
What is the only determinant of variability in a
sample?
How to recognize a problem that involves sample
Mean and standard deviation of a proportion
proportions?
distribution.
How to find the mean and standard deviation of a
Rule of thumb for normal approximation of a
sampling distribution of a proportion?
proportion.
When to use normal approximation to estimate a
distribution of a proportion?
What happens to spread when the sample size
increases?
How to recognize a problem that involves a sample
mean?
How to find the mean and standard deviation of a
sampling distribution of a sample mean?
What happens to spread when sample size
increases?
How is the distribution of a sample mean
approximately normal even if the population is not?
Mean and standard deviation of a sample mean.
Central limit theorem.
136
Standards Addressed
California: CA APS: 5.0,6.0,7.0,10.0
Common Core: S-IC-1
National Discipline:
California: CA APS: 11.0,15.0,16.0
Common Core: S-IC-1,2
National Discipline:
California: CA APS: 18.0
Common Core: S-IC-1,2
National Discipline:
:
Content Outcomes
Students will understand how to perform confidence
intervals.
Students will understand how to perform
significance tests, type I and type II error, and
power.
AP Statistics
Unit 10 – Introduction to Inference
Essential Questions
Key Concepts
How to state what a confidence interval is in
Equation of a confidence intervals for a parameter.
nonstatistical language?
Conditions for constructing a confidence interval
How to calculate a confidence interval for the
for a mean.
mean?
Confidence intervals for a population mean.
How to recognize when it is appropriate to use the
Equation for sample size for a desired margin of
confidence interval?
error.
How the margin of error of a confidence interval
changes with sample size and level of confidence?
How to find the sample size required to obtain a
confidence interval of specified margin of error and
confidence level?
How to state the null and alternate hypotheses in a
Definition of a null hypothesis.
testing situation when the parameter is a mean?
Definition of a p-value.
How to explain a p-value in nonstatistical language? Definition of statistical significance.
How to calculate the one sample z statistic and the
Relationship between confidence intervals and two
p-value for both one-sided and two-sided tests of a
sided tests of significance.
mean?
Definition of type I and type II errors.
How to assess statistical significance at standard
Relationship between significance and type I errors.
levels alpha?
Definition of power.
How to recognize that significance testing does not
measure the size or importance of an effect?
How to explain type I and type II error, and power
in a significance test?
137
Standards Addressed
California: CA APS: 5.0, 6.0,7.0,11.017.0,18.0
Common Core: S-IC-1,2
National Discipline:
California: CA APS:5.0, 6.0,7.0,11.017.0,18.0
Common Core: S-IC-1,2
National Discipline:
Content Outcomes
Students will recognize how to recognize different
inference problems.
Students will understand how to perform a onesample t procedure.
Students will understand how to perform a twosample t procedure.
AP Statistics
Unit 11- Inference for Distributions
Essential Questions
Key Concepts
How to recognize when a problem requires
No new material.
inference about a mean or the comparison of two
means?
How to recognize whether a one-sample, matched
pair, or two sample procedure is appropriate?
Standards Addressed
California: CA APS:5.0, 6.0,7.0,11.017.0,18.0
Common Core: S-IC-1,2
National Discipline:
How to use the t procedure to obtain a confidence
interval?
How to carry out a t test for the hypothesis that a
population mean has a specific value?
How to use the table of t critical values to find the
p-value?
Why t procedures are more appropriate in practice
than normal distributions?
What are the limitations of t procedures against
outliers and skewness?
When matched pairs t procedures are appropriate?
Conditions for inference about a mean.
Definition of standard error.
The one sample t statistic and the t distributions.
The one sample t procedures.
Matched pair t procedures.
The robustness of a t procedure.
California: CA APS:5.0, 6.0,7.0,11.017.0,18.0
How to give a confidence interval for the
difference of two means?
Hot to test the hypothesis that two populations
have equal means against either a one sided or two
sided alternative?
How to recognize when two sample t procedures
are appropriate?
Conditions for comparing two means.
The two sample t procedures.
The two sample t confidence interval
California: CA APS:5.0, 6.0,7.0,11.017.0,18.0
Common Core: S-IC-1,2
National Discipline:
Common Core: S-IC-1,2
National Discipline:
138
Content Outcomes
Students will understand how to recognize different
sampling procedures.
AP Statistics
Unit 12-Inference for Proportions
Essential Questions
Key Concepts
How to recognize whether one sample, matched
Equation for counts and proportions.
pair or two sample procedures are needed?
How to recognize the parameter?
How to calculate counts from sample proportions?
Standards Addressed
California: CA APS: 5.0,6.0,7.0,11.0,17.0,18.0
Common Core: S-IC-1,2
National Discipline:
Students will understand how to perform inference
about one proportion.
Students will understand how to compare two
proportions.
How to use the z procedure to give a confidence
interval for a proportion?
How to use the z statistic to perform a test of
significance for either a one sided or two sided
procedure?
How to check if the z procedure is appropriate for a
one proportion problem?
Conditions for inference about a proportion.
Confidence intervals for one proportion.
Significance tests for one proportion.
Sample size for desired margin of error for a one
sample proportion.
California: CA APS: 5.0,6.0,7.0,11.0,17.0,18.0
How to use the two sample z procedure to find a
confidence interval?
How to use the z statistic to test a two proportion
confidence interval?
How to check if the z procedure is appropriate for a
two proportion problem?
Confidence intervals for comparing two
proportions.
Significance tests for comparing two proportions.
California: CA APS: 5.0,6.0,7.0,11.0,17.0,18.0
139
Common Core: S-IC-1,2
National Discipline:
Common Core: S-IC-1,2
National Discipline:
AP Statistics
Unit 13- Inference for Tables: Chi-Square Procedures
Key Concepts
Content Outcomes
Essential Questions
Students will understand how to choose the
appropriate chi-square procedure.
How to distinguish between tests of homogeneity
and tests of association?
How to organize data into a two way table?
Definition and conditions of a chi-squared test of
goodness of fit.
Definition and conditions for a chi-squared test of
independence.
Standards Addressed
California: CA APS: 19.0
Common Core: S-IC-1,2
National Discipline:
Students will understand how to perform chi-square
tests.
Students will understand how to interpret chi-square
tests.
How to explain the null hypothesis being tested?
How to calculate expected counts?
How to calculate chi-squared for an individual cell
and overall?
How to find the appropriate degree of freedom?
How to use the chart to approximate the p-value?
Definition of an expected count.
How to locate expected cell counts, the chi-square
statistic, and its p-value with a calculator?
How to determine the cell that contributed most to
the chi-square statistic?
Knowledge of the Stats/Vars program on a TI-89.
California: CA APS: 19.0
Common Core: S-IC-1,2
National Discipline:
California: CA APS: 19.0
Common Core: S-IC-1,2
National Discipline:
140
Content Outcomes
Students will understand how to recognize the
inference for a regression model.
Students will understand how to infer using
software and calculator output.
AP Statistics
Unit 14-Inference for Regression
Essential Questions
Key Concepts
How to recognize a regression model?
Conditions for regression inference.
How to recognize which type of inference is needed
in a regression model?
How to inspect data to recognize when inference is
not appropriate?
How to explain in a regression model the meaning
of slope of the population regression line?
How to understand a mini-tab printout for a
regression model?
How to apply the information from a mini-tab
output to perform tests and find confidence intervals
for slope?
Standard error about the least square regression line.
Confidence intervals for regression slope.
Significance tests for regression slope.
Summary of a mini-tab output.
141
Standards Addressed
California: CA APS: 5.0,6.0,7.0,11.0,17.0,18.0
Common Core: S-IC-1,2
National Discipline:
California: CA APS: 5.0,6.0,7.0,11.0,17.0,18.0
Common Core: S-IC-1,2
National Discipline:
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
AP Calculus AB
“B” or better in Precalculus or “A” in Honors Algebra 2
Yes – Category D
Year
Brief Course Description
This course will include a review of functions, an introduction to limits and continuity, derivatives and their applications, integrals and their applications,
and an introduction to differential equations. There is an emphasis on conceptual understanding and working with functions represented graphically,
numerically, analytically, and verbally.
Assigned Textbook(s)
Supplemental Material(s)
Single Variable Calculus, Early Transcendentals, 6th ed., Stewart
Change and Motion : Calculus Made Easy (DVD Lecture Series)
TI-89 Graphing Calculator
Common Assessments Utilized
Common Final each semester
Homework
Quizzes
Group Work
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
142
Major Content Outcomes
Major Skill Outcomes (include Labs???)
I. Functions, Graphs, and Limits
A. Analysis of graphs: With the aid of technology, graphs of functions are often easy to produce.
The emphasis is on the interplay between the geometric and analytic information and on the use of
calculus both to predict and to explain the observed local and global behavior of a function.
B. Limits of functions
1.
An intuitive understanding of the limiting process
2.
Calculating limits using algebra
3.
Estimating limits from graphs or tables of data
C. Asymptotic and unbounded behavior
1.
Understanding asymptotes in terms of graphical behavior
2.
Describing asymptotic behavior in terms of limits involving infinity
3.
Comparing relative magnitude of functions and their rates of change
D. Continuity as a property of functions
1.
An intuitive understanding of continuity.
2.
Understanding continuity in terms of limits
3.
Geometric understanding of graphs of continuous functions
• Students should be able to work with functions represented in a variety of ways:
graphical, numerical, analytical, or verbal. They should understand the connections among these
representations.
• Students should understand the meaning of the derivative in terms of a rate of change and local linear
approximation, and should be able to use derivatives to solve a variety of problems.
• Students should understand the meaning of the definite integral both as a limit of Riemann sums and as
the net accumulation of change, and should be able to use integrals to solve a variety of problems.
• Students should understand the relationship between the derivative and the definite integral as expressed
in both parts of the Fundamental Theorem of Calculus.
• Students should be able to communicate mathematics and explain solutions to problems both verbally and
in written sentences.
• Students should be able to model a written description of a physical situation with a function, a differential
equation, or an integral.
• Students should be able to use technology to help solve problems, experiment, interpret results, and
support conclusions.
• Students should be able to determine the reasonableness of solutions, including
sign, size, relative accuracy, and units of measurement.
• Students should develop an appreciation of calculus as a coherent body of knowledge and as a human
accomplishment.
Unit 1
Students will understand and be able to determine the limit of a function both numerically and graphically
Students will understand and be able to calculate limits using the limit laws
Students will understand and be able to determine the continuity of a function
Students will understand and be able to determine limits at infinity.
II. Derivatives
A. Concept of the derivative
Unit 2
1. Derivative presented graphically, numerically, and analytically
Students will understand and be able to determine average rates of change on an interval
Students will understand the tangent line problem and be able to calculate rates of change using the
difference quotient
Students will understand and be able to determine the derivative of a function and view the result as a slope.
2. Derivative interpreted as an instantaneous rate of change
3. Derivative defined as the limit of the difference quotient
4. Relationship between differentiability and continuity
B. Derivative at a point
Unit 3
1. Slope of a curve at a point.
Students will be able to determine derivatives of polynomials
Students will be able to use the product rule and quotient rule to determine the derivative of a function
Students will be able to determine derivatives of trigonometric functions
Students will be able to us the chain rule to determine the derivative of a function
Students will understand and be able to apply implicit differentiation to find the derivative of functions
defined implicitly for a given variable.
Students will be able to determine derivatives of inverse functions, including exponential, logarithmic, and
inverse trigonometric functions
2. Tangent line to a curve at a point and local linear approximation.
3. Instantaneous rate of change as the limit of average rate of change
4. Approximate rate of change from graphs and tables of values
C. Derivative as a function
1. Corresponding characteristics of graphs of f and its derivative
2. Relationship between the increasing and decreasing behavior of f and the sign of the derivative
Unit 4
3. The Mean Value Theorem and its geometric interpretation
4. Equations involving derivatives. Verbal descriptions are translated into equations involving
Students will be able to demonstrate an understanding of differentiation to solve application problems
involving rates of change in the sciences
Students will be able to demonstrate an understanding of differentiation to solve application problems
143
derivatives and vice versa.
involving related rates
Students will be able to find the linear approximation of a function and use differentials to approximate
function values
Students will be able to determine maximum and minimum values of a function
Students will be able to explain how the Mean Value Theorem applies to various situations
Students will be able to determine how derivatives affect the shape of a graph
Students will be able to demonstrate an understanding of differentiation to solve application problems
involving optimization
D. Second derivatives
1. Corresponding characteristics of the graphs of f, the 1 st derivative, and the 2nd derivative
2. Relationship between the concavity of f and the sign of the 2nd derivative
3. Points of inflection as places where concavity changes
E. Applications of derivatives
1. Analysis of curves, including the notions of monotonicity and concavity
2. Analysis of planar curves given in parametric form, polar form, and vector form, including
velocity and acceleration
3. Optimization, both absolute and relative extrema
4. Modeling rates of change, including related rate problems
5. Use of implicit differentiation to find the derivative of an inverse function
6. Interpretation of the derivative as a rate of change in varied applied contexts, including velocity,
speed, and acceleration
7. Geometric interpretation of differential equations via slope fields and the relationship between
slope fields and solution curves for differential equations
8. L’Hopital’s Rule, including its use in determining limits and convergence of improper integrals
and series
Unit 5
Students will be able to find an antiderivative of a function and use it to find the position of an object using
its velocity.
Students will be able to find an approximation of the area under a curve by using the left endpoint, right
endpoint, midpoint and trapezoidal rules.
Students will be able to describe the integral of a function as the exact area under a curve between two xvalues. Students will then be able to find both indefinite and definite integrals using the properties and
describe the difference between them.
Students will be able to use the Fundamental Theorem of Calculus to find the area under a curve and apply
it to application problems involving net change. Students will then be able to describe the relationship
between a derivative and an integral.
Students will be able to apply various integration techniques including pattern recognition & u-substitution.
Students will be able to find the general solution of a separable differential equation and a particular
solution given an initial condition.
Students will be able to solve exponential differential equations and use them in modeling
Students will be able to construct a slope field for a differential equation and interpret the significance of
various aspects of slope fields.
F. Computation of derivatives
1. Knowledge of derivatives of basic functions, including power, exponential, logarithmic,
Unit 6
trigonometric, and inverse trigonometric functions
Students will be able to calculate the area between two curves and describe the significance
Students will be able to calculate the volumes of solids of revolution and describe the significance
Students will be able to calculate the volumes of solids using cross sections perpendicular to an axis and
describe the significance
2. Derivative rules for sums, products, and quotients of functions
3. Chain rule and implicit differentiation
III. Integrals
A. Interpretations and properties of definite integrals
1. Definite integral as a limit of Riemann sums
2. Definite integral of the rate of change of a quantity over an interval interpreted as the change of
the quantity over the interval
3. Basic properties of definite integrals
B. Applications of integrals: Appropriate integrals are used in a variety of applications to model
physical, biological, or economic situations. Although only a sampling of applications can be
included in any specific course, students should be able to adapt their knowledge and techniques to
solve other similar application problems. Whatever applications are chosen, the emphasis is on
using the method of setting up an approximating Riemann sum and representing its limit as a
144
definite integral. To provide a common foundation, specific applications should include the area of
a region, the volume of a solid with known cross sections, the average value of a function, the
distance traveled by a particle along a line, the length of a curve, and accumulated change from a
rate of change.
C. Fundamental Theorem of Calculus
1. Use of the Fundamental Theorem to evaluate definite integrals
2. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and
graphical analysis of functions so defined
D. Techniques of integration
1. Antiderivatives following directly from derivatives of basic functions
2. Antiderivatives by pattern recognition and u-substitution
E. Applications of antidifferentiation
1. Finding specific antiderivatives using initial conditions, including applications to motion along a
line
2. Solving separable differential equations and using them in modeling
F. Numerical approximations to definite integrals: Use of Riemann sums and trapezoidal sums
to approximate definite integrals of functions represented algebraically, graphically, and by tables
of values
145
AP Calculus AB
Unit 1 – What is the importance of a limit?
Content Outcomes
Students will understand and be able to determine
the limit of a function both numerically and
graphically
Essential Questions
What is a limit?
How is a limit found numerically and graphically?
Key Concepts
Notation of a limit
Definition of one sided limit
Definition of infinite limits
Standards Addressed
California: CA C: 1.0 , CA C: 2.0
Properties of limits
Direct substitution property
One sided limit theorem
Definition of greatest integer function
California: CA C: 1.0 , CA C: 2.0
Definition of continuity at x = c
Properties of continuity
Definition of continuity on an interval
Types of functions that are continuous in their
domains
Intermediate Value Theorem
Types of discontinuities
California: CA C: 1.0 , CA C: 2.0
Definition of limit at infinity
Definition of horizontal asymptote
Meaning of infinite limit at infinity
California: CA C: 1.0 , CA C: 2.0
Common Core:
When does a limit fail to exist?
What is an infinite limit and when do they occur?
Students will understand and be able to calculate
limits using the limit laws
What are the properties of infinite limits?
What are the properties of limits?
What are one sided limits?
Common Core:
What does indeterminate form mean?
Students will understand and be able to determine
the continuity of a function
What techniques can be used to find limits of
functions analytically?
What does it mean for a function to be continuous at
x = c?
How do we prove continuity at a single point?
What are the properties of continuity and why are
they true?
Common Core:
What functions are continuous everywhere in their
domain?
What is the purpose of the intermediate value
theorem?
Students will understand and be able to determine
limits at infinity
How is a limit at infinity found?
What does a limit at infinity describe?
146
Common Core:
AP Calculus AB
Unit 2 – What is a tangent line and how does it relate to a function?
Content Outcomes
Students will understand and be able to determine
average rates of change on an interval
Essential Questions
What is a rate of change?
Key Concepts
Formula for average rate of change.
How do you find the average rate of change on a
given interval?
Students will understand the tangent line problem
and be able to calculate rates of change using the
difference quotient
Standards Addressed
California: CA C: 4.0 , CA C: 7.0
Common Core:
What does the slope of a tangent line tell us?
Graphically, how can we make a secant line
between two points become a tangent line at a given
point?
Definition of a secant line
Definition of a tangent line
Definition of derivative at a number x = c
Alternative definition of derivative at x = c
Definition of velocity and acceleration
California: CA C: 4.0 , CA C: 7.0
Formula for the derivative of f, f’(x)
Definition of differentiability
Differentiation notation
Ways a function can be non-differentiable
California: CA C: 4.0 , CA C: 7.0
Common Core:
What does a derivative describe?
How can we find the slope of a tangent line at a
given point using the formal definition of the
derivative?
What is the alternative form of the derivative?
Students will understand and be able to determine
the derivative of a function and view the result as a
slope.
What is the interpretation of a tangent line with
respect to various applications in the sciences?
How is differentiability related to continuity?
Where is a function not differentiable?
How is a higher ordered derivative found and what
are their meanings with respect to position?
147
Common Core:
AP Calculus AB
Unit 3 – How can derivatives be found for any function?
Content Outcomes
Students will be able to determine derivatives of
polynomials
Essential Questions
What is the power rule?
Key Concepts
Power Rule, Constant Multiple Rule, Sum and
Difference Rule, Derivative of e^x
What is the derivative of a sum, difference, and
constant multiple of functions?
Standards Addressed
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
What is the derivative of an exponential function?
How do you write the equation of a tangent line?
Where does a tangent line have a given slope?
Where does a function have a horizontal tangent
line?
Students will be able to use the product rule and
quotient rule to determine the derivative of a
function
What is the product rule and when is it used?
Students will be able to determine derivatives of
trigonometric functions
How are the derivatives of sin x and cos x derived?
Product Rule, Quotient Rule
What is the quotient rule and when is it used?
Derivatives of Trigonometric functions
How can the derivatives of the remaining trig
functions be found using the quotient rule?
Students will be able to us the chain rule to
determine the derivative of a function
What is the chain rule and when should it be used?
Chain Rule
Chain Rule using u-substitution
Derivative of au
How can u-substitution be used to differentiate a
composite function?
Students will understand and be able to apply
implicit differentiation to find the derivative of
functions defined implicitly for a given variable.
Students will be able to determine derivatives of
inverse functions, including exponential,
logarithmic, and inverse trigonometric functions
When is an equation defined implicitly and
explicitly?
Steps for implicit differentiation
Derivative of the inverse of f(x)
Steps for logarithmic differentiation
What steps are used to implicitly differentiate a
function?
How do you find the derivative of an inverse
function?
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
Derivative of the inverse of f(x)
Steps for logarithmic differentiation
Derivatives of inverse trigonometric functions
Derivatives of logau, and lnu, where u is a function
of x
What is the relationship between the slope of the
original function and the slope of the inverse?
What is the derivative of au , logau, and lnu, where u
is a function of x?
How do we use logarithms to separate a function so
differentiation is easier (logarithmic
differentiation)?
How do we find the derivative of inverse
trigonometric functions using implicit
differentiation?
148
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
AP Calculus AB
Unit 4 – What are derivatives used for and how are they applied to real situations?
Content Outcomes
Students will be able to demonstrate an
understanding of differentiation to solve application
problems involving rates of change in the sciences
Essential Questions
What is the relationship between position, velocity
and acceleration?
What other rates of change can be found using
derivatives?
Students will be able to demonstrate an
understanding of differentiation to solve application
problems involving related rates
What do we mean by related rates?
Students will be able to find the linear
approximation of a function and use differentials to
approximate function values
How can a linearization of f help to approximate a
function value?
Key Concepts
Formula for average rate of change
Relationship between position, velocity and
acceleration
Formula for the law of natural growth and decay
Standards Addressed
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Definition of related rates
Strategy for solving related rate problems
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
How do you solve related rate problems?
Common Core:
Common Core:
Formula for linear approximation of f(x) at x = c
Definition and use of differentials
What is the meaning of the differentials dx and dy?
Students will be able to determine maximum and
minimum values of a function
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
What is a critical point?
What is the difference between absolute and relative
extrema?
Definition of absolute maximum, absolute
minimum, relative maximum, relative minimum,
critical number
Extreme Value Theorem
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
Where do extrema occur?
Steps to find absolute extrema on [a,b]
Students will be able to explain how the Mean
Value Theorem applies to various situations
What do the Mean Value Theorem and Rolle’s
Theorem say?
Rolle’s Theorem
Mean Value Theorem
Under what conditions do they apply?
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
What applications does this have in the real world?
Students will be able to determine how derivatives
effect the shape of a graph
When is a function increasing, decreasing, concave
up, and concave down?
Increasing/Decreasing Test, First Derivative Test,
Concavity Test, Second Derivative Test
Definition of concavity and inflection point
Guidelines for sketching a curve
What are the first and second derivative tests?
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
What is concavity and how does it relate to the first
derivative?
How does the slope of the graph give the graph of
the derivative?
How can one interpret f(x) or f’(x) from a graph?
Students will be able to demonstrate an
understanding of differentiation to solve application
problems involving optimization
How do we create a function from a word problem?
How do we maximize or minimize a quantity of a
real application using calculus techniques?
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
149
AP Calculus AB
Unit 5 – How are antiderivatives found and what is their purpose?
Content Outcomes
Students will be able to find an antiderivative of a
function and use it to find the position of an object
using its velocity.
Essential Questions
What is an antiderivative?
Key Concepts
Definition of antiderivative and differential equation
Antidifferentiation formulas
What is a differential equation?
Standards Addressed
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
How is a general solution and particular solution to
a differential equation found?
Common Core:
What are the rules for integration of common
functions?
Students will be able to find an approximation of
the area under a curve by using the left endpoint,
right endpoint, midpoint and trapezoidal rules.
What is the graphical meaning of the integral of a
function?
How can the area under a curve, bounded by the xaxis be found using left and right endpoints and
midpoints?
Graphical understanding of finding the sum of areas
of multiple rectanges
Definition of area of a region under a curve
Area formula using left endpoints, right endpoints,
midpoints, and trapezoidal rule
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Definition of definite integral
Integration notation
Interpretation of the definite integral as a net area
Integrability Theorems
Properties of integrals
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Fundamental Theorem of Calculus
Inverse nature of differentiation and integration
Definition of indefinite integral
Table of common indefinite integrals
Net Change Theorem
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Substitution Rule for integration
Integrals of trigonometric functions
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
Common Core:
What is the trapezoidal rule?
What happens if we use an infinite number of
rectangles to approximate the area under a curve?
Students will be able to describe the integral of a
function as the exact area under a curve between
two x-values. Students will then be able to find
both indefinite and definite integrals using the
properties and describe the difference between
them.
What is a definite integral?
What is the purpose of finding the norm of a
partition and how is it used to prove that the definite
integral from x = a to x = b describes the exact area
under the curve?
Common Core:
What are the properties of definite integrals?
Students will be able to use the Fundamental
Theorem of Calculus to find the area under a curve
and apply it to application problems involving net
change. Students will then be able to describe the
relationship between a derivative and an integral.
What is the Fundamental Theorem of Calculus and
how is it used to evaluate a definite integral?
What is the mean value theorem for integrals and
what is the meaning of the average value of a
function?
Common Core:
What is the purpose of the 2nd Fundamental
Theorem of Calculus?
What is the Net Change Theorem and what real
world applications does it allow us to solve?
How are total distance and displacement found?
Students will be able to apply various integration
techniques including pattern recognition and u-
What is the substitution rule for integration?
150
substitution.
What situations require integration to be done by usubstitution?
Substitution rule for definitie integrals
Rules for integrals of symmetric functions
, CA C: 27.0
Common Core:
What steps are used to integrate by u-substitution?
How is a definite integral found by u-substitution?
Students will be able to find the general solution of
a separable differential equation and a particular
solution given an initial condition.
How are separable differential equations solved and
what does their solution mean?
How do we verify that a general solution is a
solution to a differential equation?
Students will be able to construct a slope field for a
differential equation and interpret the significance
of various aspects of slope fields.
What is a slope field and what does the graph
signify?
How do we graph a slope field and it’s
corresponding solution through a particular point?
Definition of general solution, particular solution
and initial condition
Form of a separable differential equation
Steps for solving differential equations
Solution for the law of natural growth
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Definition of a slope field
Construct slope field given differential equation
Draw a solution to a differential equation, given an
initial condition
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Common Core:
Common Core:
151
AP Calculus AB
Unit 6 – What are the applications of antiderivatives?
Content Outcomes
Students will be able to calculate the area between
two curves and describe the significance
Essential Questions
How do we find the area between 2 non intersecting
or 2 intersecting curves?
Key Concepts
Rule for finding area between curves
Common Core:
How do we determine whether to integrate with
respect to x or y?
Students will be able to calculate the volumes of
solids of revolution and describe the significance
How do we find the volume of a solid generated by
rotating a region about a vertical or horizontal line?
Definition of solids of revolution
Formulas for disk and washer methods of finding
volumes of rotated solids
California: CA C: 16.0
Volume formulas of common solids
Definition of volume using the integral of area
Interpretation of volume as a sum of infinite areas
California: CA C: 16.0
How is area under a curve related to volume of
rotated solids?
Students will be able to calculate the volumes of
solids using cross sections perpendicular to an axis
and describe the significance
When do we use the disk method as opposed to the
washer method for finding volumes of rotated
solids?
What are the functions for area of common shapes
with respect to their base, i.e. equilateral triangle,
isosceles triangle, semicircle, rectangle, square, etc.
Standards Addressed
California: CA C: 16.0
How do we use these formulas to find the volume of
a solid where cross sections perpendicular to an
axis, for a given region, are a specified shape?
152
Common Core:
Common Core:
Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
AP Calculus BC
“B” or better in AP Calculus AB and a score of 3 or higher on the AP Calculus AB exam
Yes – Category D
Year
Brief Course Description
This course will include a review of functions, an introduction to limits and continuity, derivatives and their applications, integrals and their applications,
and an introduction to differential equations. There is an emphasis on conceptual understanding and working with functions represented graphically,
numerically, analytically, and verbally.
Assigned Textbook(s)
Supplemental Material(s)
Single Variable Calculus, Early Transcendentals, 6th ed., Stewart
Change and Motion : Calculus Made Easy (DVD Lecture Series)
TI-89 Graphing Calculator
Common Assessments Utilized
Common Final each semester
Homework
Quizzes
Group Work
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
153
Major Content Outcomes
Major Skill Outcomes
I. Functions, Graphs, and Limits
A. Analysis of graphs: With the aid of technology, graphs of functions are often easy to produce.
The emphasis is on the interplay between the geometric and analytic information and on the use of
calculus both to predict and to explain the observed local and global behavior of a function.
B. Limits of functions
4.
An intuitive understanding of the limiting process
5.
Calculating limits using algebra
6.
Estimating limits from graphs or tables of data
C. Asymptotic and unbounded behavior
4.
Understanding asymptotes in terms of graphical behavior
5.
Describing asymptotic behavior in terms of limits involving infinity
6.
Comparing relative magnitude of functions and their rates of change
D. Continuity as a property of functions
4.
An intuitive understanding of continuity.
5.
Understanding continuity in terms of limits
6.
Geometric understanding of graphs of continuous functions
• Students should be able to work with functions represented in a variety of ways:
graphical, numerical, analytical, or verbal. They should understand the connections among these
representations.
• Students should understand the meaning of the derivative in terms of a rate of change and local linear
approximation, and should be able to use derivatives to solve a variety of problems.
• Students should understand the meaning of the definite integral both as a limit of Riemann sums and as
the net accumulation of change, and should be able to use integrals to solve a variety of problems.
• Students should understand the relationship between the derivative and the definite integral as expressed
in both parts of the Fundamental Theorem of Calculus.
• Students should be able to communicate mathematics and explain solutions to problems both verbally and
in written sentences.
• Students should be able to model a written description of a physical situation with a function, a differential
equation, or an integral.
• Students should be able to use technology to help solve problems, experiment, interpret results, and
support conclusions.
• Students should be able to determine the reasonableness of solutions, including
sign, size, relative accuracy, and units of measurement.
• Students should develop an appreciation of calculus as a coherent body of knowledge and as a human
accomplishment.
Unit 1
Students will understand and be able to determine the limit of a function both numerically and graphically
Students will understand and be able to calculate limits using the limit laws
Students will understand and be able to determine the continuity of a function
Students will understand and be able to determine limits at infinity.
II. Derivatives
G. Concept of the derivative
Unit 2
5. Derivative presented graphically, numerically, and analytically
Students will understand and be able to determine average rates of change on an interval
Students will understand the tangent line problem and be able to calculate rates of change using the
difference quotient
Students will understand and be able to determine the derivative of a function and view the result as a slope.
6. Derivative interpreted as an instantaneous rate of change
7. Derivative defined as the limit of the difference quotient
8. Relationship between differentiability and continuity
Unit 3
H. Derivative at a point
Students will be able to determine derivatives of polynomials
Students will be able to use the product rule and quotient rule to determine the derivative of a function
Students will be able to determine derivatives of trigonometric functions
Students will be able to us the chain rule to determine the derivative of a function
Students will understand and be able to apply implicit differentiation to find the derivative of functions
defined implicitly for a given variable.
Students will be able to determine derivatives of inverse functions, including exponential, logarithmic, and
inverse trigonometric functions
5. Slope of a curve at a point.
6. Tangent line to a curve at a point and local linear approximation.
7. Instantaneous rate of change as the limit of average rate of change
8. Approximate rate of change from graphs and tables of values
I. Derivative as a function
5. Corresponding characteristics of graphs of f and its derivative
Unit 4
6. Relationship between the increasing and decreasing behavior of f and the sign of the derivative
7. The Mean Value Theorem and its geometric interpretation
8. Equations involving derivatives. Verbal descriptions are translated into equations involving
Students will be able to demonstrate an understanding of differentiation to solve application problems
involving rates of change in the sciences
Students will be able to demonstrate an understanding of differentiation to solve application problems
involving related rates
154
derivatives and vice versa.
Students will be able to find the linear approximation of a function and use differentials to approximate
function values
Students will be able to determine maximum and minimum values of a function
Students will be able to explain how the Mean Value Theorem applies to various situations
Students will be able to determine how derivatives affect the shape of a graph
Students will be able to demonstrate an understanding of differentiation to solve application problems
involving optimization
J. Second derivatives
4. Corresponding characteristics of the graphs of f, the 1 st derivative, and the 2nd derivative
5. Relationship between the concavity of f and the sign of the 2nd derivative
6. Points of inflection as places where concavity changes
K. Applications of derivatives
Unit 5
9. Analysis of curves, including the notions of monotonicity and concavity
10.
Analysis of planar curves given in parametric form, polar form, and vector form,
including velocity and acceleration
11.
Optimization, both absolute and relative extrema
12.
Modeling rates of change, including related rate problems
13.
Use of implicit differentiation to find the derivative of an inverse function
14.
Interpretation of the derivative as a rate of change in varied applied contexts,
including velocity, speed, and acceleration
15.
Geometric interpretation of differential equations via slope fields and the relationship
between slope fields and solution curves for differential equations
L’Hopital’s Rule, including its use in determining limits and convergence of improper
16.
integrals and series
L. Computation of derivatives
4. Knowledge of derivatives of basic functions, including power, exponential, logarithmic,
trigonometric, and inverse trigonometric functions
Students will be able to find an antiderivative of a function and use it to find the position of an object using
its velocity.
Students will be able to find an approximation of the area under a curve by using the left endpoint, right
endpoint, midpoint and trapezoidal rules.
Students will be able to describe the integral of a function as the exact area under a curve between two xvalues. Students will then be able to find both indefinite and definite integrals using the properties and
describe the difference between them.
Students will be able to use the Fundamental Theorem of Calculus to find the area under a curve and apply
it to application problems involving net change. Students will then be able to describe the relationship
between a derivative and an integral.
Students will be able to apply various integration techniques including pattern recognition, u-substitution,
integration by parts, and partial fraction decomposition
Students will be able to evaluate the two types of an improper integral: Infinite interval and discontinuous
integrand.
Students will be able to use L’Hospital’s Rule to calculate limits of functions that yield the various types of
indeterminate forms
Students will be able to find the general solution of a separable differential equation and a particular
solution given an initial condition.
Students will be able to solve exponential and logistic differential equations and use them in modeling
Students will be able to construct a slope field for a differential equation and interpret the significance of
various aspects of slope fields.
5. Derivative rules for sums, products, and quotients of functions
Unit 6
6. Chain rule and implicit differentiation
Students will be able to calculate the area between two curves and describe the significance
Students will be able to calculate the volumes of solids of revolution and describe the significance
Students will be able to calculate the volumes of solids using cross sections perpendicular to an axis and
describe the significance
Students will be able to find the length of a curve on a given interval
IV. Integrals
F.
Interpretations and properties of definite integrals
Unit 7
4. Definite integral as a limit of Riemann sums
5. Definite integral of the rate of change of a quantity over an interval interpreted as the change of
the quantity over the interval
Students will be able to apply calculus techniques to parametric curves
Students will be able to demonstrate he relationship between rectangular and polar coordinates and
equations
Students will be able to determine the area bounded inside a polar curve
6. Basic properties of definite integrals
G. Applications of integrals: Appropriate integrals are used in a variety of applications to model
physical, biological, or economic situations. Although only a sampling of applications can be
included in any specific course, students should be able to adapt their knowledge and techniques to
solve other similar application problems. Whatever applications are chosen, the emphasis is on
using the method of setting up an approximating Riemann sum and representing its limit as a
Unit 8
Students will be able to identify the different types of infinite series.
Students will be able to apply the Integral Test to test for convergence and divergence
Students will be able to apply the Comparison Tests to test for convergence and divergence
Students will be able to apply the Alternating Series Test to test for convergence and divergence
Students will be able to interpret when a series is Absolutely Convergent using the Ratio and Root Tests
Students will be able to represent a function as a power series and recognize and interpret when a Power
Series is convergent or divergent
Students will be able to find and approximate the Taylor and Maclaurin Series for certain functions and use
155
definite integral. To provide a common foundation, specific applications should include the area of
a region, the volume of a solid with known cross sections, the average value of a function, the
distance traveled by a particle along a line, the length of a curve, and accumulated change from a
rate of change.
H. Fundamental Theorem of Calculus
3. Use of the Fundamental Theorem to evaluate definite integrals
4. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and
graphical analysis of functions so defined
I.
Techniques of integration
3. Antiderivatives following directly from derivatives of basic functions
4. Antiderivatives by pattern recognition and u-substitution
J.
Applications of antidifferentiation
3. Finding specific antiderivatives using initial conditions, including applications to motion along a
line
4. Solving separable differential equations and using them in modeling
F. Numerical approximations to definite integrals: Use of Riemann sums and trapezoidal sums
to approximate definite integrals of functions represented algebraically, graphically, and by tables
of values
156
them in various applications
AP Calculus BC
Unit 1 – What is the importance of a limit?
Content Outcomes
Students will understand and be able to determine
the limit of a function both numerically and
graphically
Essential Questions
What is a limit?
How is a limit found numerically and graphically?
Key Concepts
Notation of a limit
Definition of one sided limit
Definition of infinite limits
Standards Addressed
California: CA C: 1.0 , CA C: 2.0
Properties of limits
Direct substitution property
One sided limit theorem
Definition of greatest integer function
California: CA C: 1.0 , CA C: 2.0
Definition of continuity at x = c
Properties of continuity
Definition of continuity on an interval
Types of functions that are continuous in their
domains
Intermediate Value Theorem
Types of discontinuities
California: CA C: 1.0 , CA C: 2.0
Definition of limit at infinity
Definition of horizontal asymptote
Meaning of infinite limit at infinity
California: CA C: 1.0 , CA C: 2.0
Common Core:
When does a limit fail to exist?
What is an infinite limit and when do they occur?
Students will understand and be able to calculate
limits using the limit laws
What are the properties of infinite limits?
What are the properties of limits?
What are one sided limits?
Common Core:
What does indeterminate form mean?
Students will understand and be able to determine
the continuity of a function
What techniques can be used to find limits of
functions analytically?
What does it mean for a function to be continuous at
x = c?
How do we prove continuity at a single point?
What are the properties of continuity and why are
they true?
Common Core:
What functions are continuous everywhere in their
domain?
What is the purpose of the intermediate value
theorem?
Students will understand and be able to determine
limits at infinity-
How is a limit at infinity found?
What does a limit at infinity describe?
157
Common Core:
AP Calculus BC
Unit 2 – What is a tangent line and how does it relate to a function?
Content Outcomes
Students will understand and be able to determine
average rates of change on an interval
Essential Questions
What is a rate of change?
Key Concepts
Formula for average rate of change.
How do you find the average rate of change on a
given interval?
Students will understand the tangent line problem
and be able to calculate rates of change using the
difference quotient
Standards Addressed
California: CA C: 4.0 , CA C: 7.0
Common Core:
What does the slope of a tangent line tell us?
Graphically, how can we make a secant line
between two points become a tangent line at a given
point?
Definition of a secant line
Definition of a tangent line
Definition of derivative at a number x = c
Alternative definition of derivative at x = c
Definition of velocity and acceleration
California: CA C: 4.0 , CA C: 7.0
Formula for the derivative of f, f’(x)
Definition of differentiability
Differentiation notation
Ways a function can be non-differentiable
California: CA C: 4.0 , CA C: 7.0
Common Core:
What does a derivative describe?
How can we find the slope of a tangent line at a
given point using the formal definition of the
derivative?
What is the alternative form of the derivative?
Students will understand and be able to determine
the derivative of a function and view the result as a
slope.
What is the interpretation of a tangent line with
respect to various applications in the sciences?
How is differentiability related to continuity?
Where is a function not differentiable?
How is a higher ordered derivative found and what
are their meanings with respect to position?
158
Common Core:
AP Calculus BC
Unit 3 – How can derivatives be found for any function?
Content Outcomes
Students will be able to determine derivatives of
polynomials
Essential Questions
What is the power rule?
Key Concepts
Power Rule, Constant Multiple Rule, Sum and
Difference Rule, Derivative of e^x
What is the derivative of a sum, difference, and
constant multiple of functions?
Standards Addressed
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
What is the derivative of an exponential function?
How do you write the equation of a tangent line?
Where does a tangent line have a given slope?
Where does a function have a horizontal tangent
line?
Students will be able to use the product rule and
quotient rule to determine the derivative of a
function
What is the product rule and when is it used?
Students will be able to determine derivatives of
trigonometric functions
How are the derivatives of sin x and cos x derived?
Product Rule, Quotient Rule
What is the quotient rule and when is it used?
Derivatives of Trigonometric functions
How can the derivatives of the remaining trig
functions be found using the quotient rule?
Students will be able to us the chain rule to
determine the derivative of a function
What is the chain rule and when should it be used?
Chain Rule
Chain Rule using u-substitution
Derivative of au
How can u-substitution be used to differentiate a
composite function?
Students will understand and be able to apply
implicit differentiation to find the derivative of
functions defined implicitly for a given variable.
Students will be able to determine derivatives of
inverse functions, including exponential,
logarithmic, and inverse trigonometric functions
When is an equation defined implicitly and
explicitly?
Steps for implicit differentiation
Derivative of the inverse of f(x)
Steps for logarithmic differentiation
What steps are used to implicitly differentiate a
function?
How do you find the derivative of an inverse
function?
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
Derivative of the inverse of f(x)
Steps for logarithmic differentiation
Derivatives of inverse trigonometric functions
Derivatives of logau, and lnu, where u is a function
of x
What is the relationship between the slope of the
original function and the slope of the inverse?
What is the derivative of au , logau, and lnu, where u
is a function of x?
How do we use logarithms to separate a function so
differentiation is easier (logarithmic
differentiation)?
How do we find the derivative of inverse
trigonometric functions using implicit
differentiation?
159
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0
, CA C: 7.0
Common Core:
AP Calculus BC
Unit 4 – What are derivatives used for and how are they applied to real situations?
Content Outcomes
Students will be able to demonstrate an
understanding of differentiation to solve application
problems involving rates of change in the sciences
Essential Questions
What is the relationship between position, velocity
and acceleration?
What other rates of change can be found using
derivatives?
Students will be able to demonstrate an
understanding of differentiation to solve application
problems involving related rates
What do we mean by related rates?
Students will be able to find the linear
approximation of a function and use differentials to
approximate function values
How can a linearization of f help to approximate a
function value?
Key Concepts
Formula for average rate of change
Relationship between position, velocity and
acceleration
Formula for the law of natural growth and decay
Standards Addressed
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Definition of related rates
Strategy for solving related rate problems
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
How do you solve related rate problems?
Common Core:
Common Core:
Formula for linear approximation of f(x) at x = c
Definition and use of differentials
What is the meaning of the differentials dx and dy?
Students will be able to determine maximum and
minimum values of a function
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
What is a critical point?
What is the difference between absolute and relative
extrema?
Definition of absolute maximum, absolute
minimum, relative maximum, relative minimum,
critical number
Extreme Value Theorem
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
Where do extrema occur?
Steps to find absolute extrema on [a,b]
Students will be able to explain how the Mean
Value Theorem applies to various situations
What do the Mean Value Theorem and Rolle’s
Theorem say?
Rolle’s Theorem
Mean Value Theorem
Under what conditions do they apply?
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
What applications does this have in the real world?
Students will be able to determine how derivatives
affect the shape of a graph
When does a function increase? Decrease?
Increasing/Decreasing Test, First Derivative Test,
Concavity Test, Second Derivative Test
Definition of concavity and inflection point
Guidelines for sketching a curve
What is the first derivative test?
What is concavity and how does it relate to the first
derivative?
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
What is the second derivative test?
How does the slope of the graph give the graph of
the derivative?
Students will be able to demonstrate an
understanding of differentiation to solve application
problems involving optimization
How can one interpret f(x) or f’(x) from a graph?
How do we create a function from a word problem?
How do we maximize or minimize a quantity of a
real application using calculus techniques?
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0
, CA C: 11.0 , CA C: 12.0
Common Core:
160
AP Calculus BC
Unit 5 – How are antiderivatives found and what is their purpose?
Content Outcomes
Students will be able to find an antiderivative of a
function and use it to find the position of an object
using its velocity.
Essential Questions
What is an antiderivative?
Key Concepts
Definition of antiderivative and differential equation
Antidifferentiation formulas
What is a differential equation?
Standards Addressed
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
How is a general solution and particular solution to
a differential equation found?
Common Core:
What are the rules for integration of common
functions?
Students will be able to find an approximation of
the area under a curve by using the left endpoint,
right endpoint, midpoint and trapezoidal rules.
What is the graphical meaning of the integral of a
function?
How can the area under a curve, bounded by the xaxis be found using left and right endpoints and
midpoints?
Graphical understanding of finding the sum of areas
of multiple rectanges
Definition of area of a region under a curve
Area formula using left endpoints, right endpoints,
midpoints, and trapezoidal rule
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Definition of definite integral
Integration notation
Interpretation of the definite integral as a net area
Integrability Theorems
Properties of integrals
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Fundamental Theorem of Calculus
Inverse nature of differentiation and integration
Definition of indefinite integral
Table of common indefinite integrals
Net Change Theorem
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Substitution Rule for integration
Integrals of trigonometric functions
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
Common Core:
What is the trapezoidal rule?
What happens if we use an infinite number of
rectangles to approximate the area under a curve?
Students will be able to describe the integral of a
function as the exact area under a curve between
two x-values. Students will then be able to find
both indefinite and definite integrals using the
properties and describe the difference between
them.
What is a definite integral?
What is the purpose of finding the norm of a
partition and how is it used to prove that the definite
integral from x = a to x = b describes the exact area
under the curve?
Common Core:
What are the properties of definite integrals?
Students will be able to use the Fundamental
Theorem of Calculus to find the area under a curve
and apply it to application problems involving net
change. Students will then be able to describe the
relationship between a derivative and an integral.
What is the Fundamental Theorem of Calculus and
how is it used to evaluate a definite integral?
What is the mean value theorem for integrals and
what is the meaning of the average value of a
function?
Common Core:
What is the purpose of the 2nd Fundamental
Theorem of Calculus?
What is the Net Change Theorem and what real
world applications does it allow us to solve?
How are total distance and displacement found?
Students will be able to apply various integration
techniques including pattern recognition, u-
What is the substitution rule for integration?
161
substitution, integration by parts, and partial
fraction decomposition
What situations require integration to be done by usubstitution?
What situations require integration by parts to be
done?
Substitution rule for definitie integrals
Rules for integrals of symmetric functions
Rule for integration by parts
Technique of decomposing a fraction
, CA C: 27.0
Definition of improper integrals with infinite
intervals
Definition of improper integrals with discontinuous
integrands
Definitions of convergence and divergence
California: CA C: 22.0 , CA C : 23.0
Types of Indeterminate Form
L’Hospital’s Rule
California: CA C: 22.0 , CA C : 23.0
Common Core:
What steps are used to integrate by u-substitution
and integration by parts?
How is a definite integral found by u-substitution
and integration by parts?
Students will be able to evaluate the two types of an
improper integral: Infinite interval and
discontinuous integrand.
How do we find an integral of a rational function by
partial fraction decomposition (linear factors only)?
What makes an integral improper and what are the
two types of improper integrals?
How do we evaluate an improper integrals of both
types?
Common Core:
How do we determine whether an improper integral
is convergent or divergent?
Students will be able to use L’Hospital’s Rule to
calculate limits of functions that yield the various
types of indeterminate forms
Students will be able to find the general solution of
a separable differential equation and a particular
solution given an initial condition.
What is the comparison theorem and how is it used
to determine convergence or divergence?
What are the various types of indeterminate forms?
How can we find limits of functions using
L’Hospital’s Rule?
Common Core:
How are separable differential equations solved and
what does their solution mean?
How do we verify that a general solution is a
solution to a differential equation?
Definition of general solution, particular solution
and initial condition
Form of a separable differential equation
Steps for solving differential equations
Euler’s Method
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
The Logistic Model
The Law of Natural Growth
California: : CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Common Core:
How do we find a numerical solution of a
differential equation using Euler’s method?
Students will be able to solve exponential and
logistic differential equations and use them in
modeling
How do we solve a logistic differential equation and
what is its purpose?
How can we derive the law of natural growth and
solve natural growth and decay applications?
Students will be able to construct a slope field for a
differential equation and interpret the significance
of various aspects of slope fields.
What is a slope field and what does the graph
signify?
How do we graph a slope field and it’s
corresponding solution through a particular point?
Definition of a slope field
Construct slope field given differential equation
Draw a solution to a differential equation, given an
initial condition
Common Core:
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0
, CA C: 27.0
Common Core:
162
AP Calculus BC
Unit 6 – What are the applications of antiderivatives?
Content Outcomes
Students will be able to calculate the area between
two curves and describe the significance
Essential Questions
How do we find the area between 2 non intersecting
or 2 intersecting curves?
Key Concepts
Rule for finding area between curves
Common Core:
How do we determine whether to integrate with
respect to x or y?
Students will be able to calculate the volumes of
solids of revolution and describe the significance
How do we find the volume of a solid generated by
rotating a region about a vertical or horizontal line?
Definition of solids of revolution
Formulas for disk and washer methods of finding
volumes of rotated solids
California: CA C: 16.0
Volume formulas of common solids
Definition of volume using the integral of area
Interpretation of volume as a sum of infinite areas
California: CA C: 16.0
Formula for Arc Length
California: CA C: 16.0
How is area under a curve related to volume of
rotated solids?
Students will be able to calculate the volumes of
solids using cross sections perpendicular to an axis
and describe the significance
When do we use the disk method as opposed to the
washer method for finding volumes of rotated
solids?
What are the functions for area of common shapes
with respect to their base, i.e. equilateral triangle,
isosceles triangle, semicircle, rectangle, square, etc?
Students will be able to find the length of a curve on
a given interval
How do we use these formulas to find the volume of
a solid where cross sections perpendicular to an
axis, for a given region, are a specified shape?
How do we find the length of the arc of a function
on [a , b]?
Standards Addressed
California: CA C: 16.0
Common Core:
Common Core:
Common Core:
How do we determine a function s(x) for the length
of a curve on [a , x]?
163
AP Calculus BC
Unit 7 – What is the calculus of parametric and polar equations?
Content Outcomes
Students will be able to apply calculus techniques to
parametric curves
Essential Questions
What is a parametric equation and how do we graph
them?
Key Concepts
The slope of the tangent line of a parametric curve
The concavity of a parametric curve
Arc length formula for parametric equations
Standards Addressed
California: CA C 6.0
Definition of Polar Coordinates
Conversion formulas between rectangular and polar
coordinates
Slope of tangent line formula for polar coordinates
California:
Formula for area inside polar curves
California:
How do you find the slope of the tangent line for a
set of parametric equations?
Common Core:
How do you use derivatives of parametric equations
to find the graph’s concavity?
Students will be able to demonstrate the relationship
between rectangular and polar coordinates and
equations
How do you find the arc length of a parametric
curve?
What are polar coordinates and how are they related
to rectangular coordinates?
What are the conversion formulas between
rectangular and polar coordinates?
Common Core:
How do you find the slope of the tangent line for
polar graphs?
Students will be able to determine the area bounded
inside a polar curve
How do you find the area inside a polar curve?
Common Core:
164
AP Calculus BC
Unit 8 – What are infinite series?
Content Outcomes
Students will be able to identify the different types
of infinite series.
Essential Questions
What are infinite series and how are they considered
convergent or divergent?
What is the difference between a geometric,
telescoping, harmonic and alternating series?
Students will be able to apply the Integral Test to
test for convergence and divergence
How do we recognize the terms of a series as areas
of rectangles & their relationship to improper
integrals?
How and when do we test a series for convergence
and divergence using the integral test?
Key Concepts
Definition of Infinite Series
Test for Convergence and Divergence
Definition of Geometric Series
Definition of Telescoping Series
Definition of Alternating Series
Standards Addressed
California: CA C 22.0 – 26.0
Integral Test
California: CA C 22.0 – 26.0
Common Core:
Common Core:
Students will be able to apply the Comparison
Tests to test for convergence and divergence
How and when do we test a series for convergence
and divergence using the direct comparison test?
Direct Comparison Test
Limit Comparison Test
California: CA C 22.0 – 26.0
Common Core:
How and when do we test a series for convergence
and divergence using the limit comparison test?
California: CA C 22.0 – 26.0
Students will be able to apply the Alternating
Series Test to test for convergence and divergence
How and when do we test a series for convergence
and divergence using the alternate series test?
Definition of an alternating series
Alternating Series Test
Students will be able to interpret when a series is
Absolutely Convergent using the Ratio and Root
Tests
What is the definition of absolute convergence?
Definition of absolute convergence
The Ratio Test
The Root Test
California: CA C 22.0 – 26.0
Definition of a Power Series
Definition of the interval of convergence of a power
series
Differentiation and Integration Theorem of Power
Series
California: CA C 22.0 – 26.0
Definition of Taylor Series
Definition of Maclaurin Series
Maclaurin series of ex , sin x, and cos x
California: CA C 22.0 – 26.0
Common Core:
Students will be able to represent a function as a
power series and recognize and interpret when a
Power Series is convergent or divergent
What are the Ratio and Root Tests and when do we
use them?
What is a power series?
How do you find the interval of convergence for a
power series?
Common Core:
Common Core:
How do we find the radius and interval of
convergence of power series?
Students will be able to find and approximate the
Taylor and Maclaurin Series for certain functions
and use them in various applications
How can we represent a function as a power series?
How do we find the Taylor polynomial
approximation with a graphical demonstration of
convergence?
How do we determine the Maclaurin series and the
general Taylor series centered at x = a?
How can we formally manipulate Taylor series and
apply shortcuts to computing Taylor series,
including substitution, differentiation,
antidifferentiation, and the formation of
new series from known series?
How do we determine the Lagrange error bound for
Taylor polynomials?
165
Common Core:
Damien High School
Mathematics and Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Differential Equations
“A” in AP Calculus BC and a score of “5” on the AP Calculus BC exam
Yes – Category C
Year
Brief Course Description
This course provides an introduction to ordinary differential equations with an emphasis on applications. Topics include first-order, linear higher-order,
and systems of differential equations; numerical methods; series solutions; eigenvalues and eigenvectors; Laplace transforms; and Fourier series. Upon
completion, students should be able to use differential equations to model physical phenomena, solve the equations, and use the solutions to analyze the
phenomena.
Assigned Textbook(s)
Supplemental Material(s)
A First Course in Differential Equations with Modeling Applications, 9th ed., by Dennis Zill
Graphing Calculator
Common Assessments Utilized
Common Final each semester
Homework
Tests
ISOs Addressed
166
Major Content Outcomes
Students will be able to solve first-order differential equations.
Students will be able to solve higher-order differential equations.
Students will be able to use matrices.
Students will be able to use the properties of the Laplace Transform and Inverse Laplace.
Students will be able to analyze, interpret, and solve applications of ODE’s
Students will be able to solve systems of linear first order differential equations.
I.
First Order Differential Equations
A. Initial Value Problems
B. Differential Equations as Mathematical Models
C. Solution Curves Without a Solution
D. Direction Fields and Autonomous First Order D.E.
E. Separable Variables
F. Linear Equations
G. Exact Equations
II. Higher Order Differential Equations
A. Preliminary Theory for Linear Equations
1. Initial Value and Boundary Value Problems
2. Homogeneous Equations
3. Nonhomogeneous Equations
B. Reduction of Order
C. Homogeneous Linear Equations with Constant Coefficients
D. Variation of Parameters
E. Cauchy Euler Equations
III. The Laplace Transform
A. Definition of Laplace Transforms
B. Inverse Transforms and Transforms of Derivatives
C. Operational Properties
1. Translation on the s-axis
2. Translation on the t-axis
3. Derivatives of a Transform
4. Transforms of Integrals
5. Transform of a Periodic Function
D. The Dirac-Delta Function
IV. Matrices
A. Operations with Matrices
B. Derivatives and Integrals of Matrices of Functions
C. Row Echelon Form and Gauss-Jordan Elimination
D. Eigenvalues and Eigenvectors
V. Systems of Linear First-Order Differential Equations
Major Skill Outcomes (include Labs???)
Students will learn what an Ordinary Differential Equation (ODE) is, how to classify them,
what initial value problems are, & what constitutes a solution.
Students will learn to visualize and manipulate ODEs in graphical, numerical, and symbolic
form.
Students will understand the concepts of the existence and uniqueness of solutions.
Students will learn to work with matrices and apply them when dealing with determinants,
Cramer’s Rule, and Gauss Jordan Elimination.
Students will recognize certain basic types of first order and higher order ODEs for which exact
solutions may be obtained and will solve them using the corresponding methods.
Students will be introduced to the concept of the Laplace transform and will apply the
properties to solve linear ODE’s.
Students will be introduced to system of linear first order ODE’s and discuss graphical,
numerical, and analytical solution methods
Students will work with a variety of applications, using appropriate models, and will analyze the
validity of the solutions obtained.
Unit 1
Students will be able to identify differential equations by various criteria.
Students will be able to solve separable differential equations.
Students will be able to solve first order linear differential equations.
Students will be able to solve exact equations.
Students will be able to solve homogeneous differential equations.
Unit 2
Students will recognize and solve initial value problems, boundary value problems,
homogeneous, & non-homogeneous differential equations.
Students will be able to find a second solution of a differential equation from a known solution,
using reduction of order.
Students will be able to solve homogeneous and non-homogeneous linear equations with
constant coefficients.
Students will learn how to solve Cauchy Euler equations.
Students will be able to solve certain types of nonlinear differential equations.
Unit 3
Students will be able to find the Laplace (and inverse Laplace) transform of functions and
derivatives by using the definition and formulas.
Students will be able to use Laplace transforms to solve differential equations with initial
conditions.
Students will be able to translate on the s-axis and t-axis.
Students will be able to find derivatives of transforms and transforms of integrals.
167
A.
B.
C.
D.
Preliminary Theory for Linear Systems
Homogeneous Linear Systems
1. Distinct Real, Repeated, and Complex Eigenvalues
Nonhomogeneous Linear Systems
1. Undetermined Coefficients
2. Variation of Parameters
Matrix Exponential
Students will be able to find the transform of a periodic function.
Students will be able to solve differential equations having a Dirac Delta function.
Unit 4
Students will be able to find when two matrices are equal and apply operations involving
matrices.
Students will be able to find the derivative and integral of a matrix of functions.
Students will be able to solve a system of equations by Gaussian and Gauss-Jordan elimination.
Students will be able to find eigenvalues and eigenvectors of a matrix.
Unit 5
Students will be able to solve a system of homogeneous linear systems.
Students will be able to solve a system of nonhomogeneous linear systems.
Students will be able to use the matrix exponential.
168
Content Outcomes
Students will be able to identify differential
equations by various criteria.
Differential Equations
Unit 1 – What are First-order Differential Equations?
Essential Questions
Key Concepts
How can we identify the order of an ordinary
Definition of separable equations
differential equation and determine whether it is
linear or nonlinear?
Standards Addressed
California:
Common Core:
How can we sketch a slope field for a first-order
differential equation as well as solution curves on
the slope field?
How do we apply the Existence-Uniqueness
Theorem for first-order differential equations.
Students will be able to solve separable differential
equations.
How do we identify a separable first-order equation
and find a family of solutions or a particular
solution?
Students will be able to solve first order linear
differential equations.
How do we identify a first-order linear equation and
find the general solution using an integrating factor?
Definition of linear equations
California:
Common Core:
Students will be able to solve exact equations.
What is an exact equation?
Definition of exact equations
How do we identify an exact differential equation
and find a family of solutions?
California:
Common Core:
How do we solve initial-value problems involving
first-order separable, linear, and exact equations?
Students will be able to solve homogeneous
differential equations.
What is a homogeneous equation?
Definition of homogeneous equations
How do you solve a homogeneous equation?
California:
Common Core:
169
Content Outcomes
Students will recognize and solve initial value
problems, boundary value problems, homogeneous,
& non-homogeneous differential equations.
Students will be able to find a second solution of a
differential equation from a known solution, using
reduction of order.
Students will be able to solve homogeneous and
non-homogeneous linear equations with constant
coefficients.
.
Differential Equations
Unit 2 – What are Higher Order Equations?
Essential Questions
Key Concepts
What is an initial value problem?
Definition of linear dependence and independence
Definition of Wronskian
What is a boundary value problem?
Definition of a fundamental set of solutions
Definition of complementary and particular
What is a higher order homogeneous equation?
solutions.
What is a nonhomogeneous equation?
How do we find a second solution if we know one
solution of a differential equation?
Formula for finding a second solution
Standards Addressed
California:
Common Core:
California:
Common Core:
What is an auxiliary equation?
Variation of Parameters technique
How do we solve homogeneous linear equations
with constant coefficients?
California:
Common Core:
When do we use the method of Variation of
Parameters and how do we apply the process?
Students will learn how to solve Cauchy Euler
equations.
What is the form of Cauchy Euler Equations?
Definition of Cauchy Euler equations
How do we solve a Cauchy Euler Equation?
Students will be able to solve certain types of
nonlinear differential equations.
How do we solve nonlinear differential equations
where the dependent variable x or independent
variable y is missing?
California:
Common Core:
Reduction of order for nonlinear differential
equations
California:
Common Core:
170
Content Outcomes
Students will be able to find the Laplace (and
inverse Laplace) transform of functions and
derivatives by using the definition and formulas.
Differential Equations
Unit 3 – What is a Laplace Transform?
Essential Questions
Key Concepts
What is the definition of the Laplace transform?
Definition of the Laplace transform
Formulas of Laplace transforms
How do we find an inverse Laplace transform?
Formulas of inverse Laplace transforms
Formula for Laplace transforms of derivatives
How do we find the Laplace transform of
derivatives?
Students will be able to use Laplace transforms to
solve differential equations with initial conditions.
How do we use the Laplace transform to solve
differential equations and initial value problems?
Procedure for applying the Laplace transform and
inverse Laplace transform to solve an ODE
Standards Addressed
California:
Common Core:
California:
Common Core:
Students will be able to translate on the s-axis and
t-axis.
How do we translate on the s-axis?
First Translation Theorem
Second Translation Theorem
How do you convert a piecewise function into unit
step functions?
California:
Common Core:
How do we translate on the t-axis?
Students will be able to find derivatives of
transforms and transforms of integrals.
How do we find the derivative of transforms?
How do we find the transforms of integrals?
Students will be able to find the transform of a
periodic function.
How do we find the transform of a periodic
function?
Derivatives of transforms
Convolution Theorem
Transforms of integrals
California:
Transform of a periodic functions
California:
Common Core:
Common Core:
Students will be able to solve differential equations
having a Dirac Delta function.
What is the Dirac Delta function and how do we
solve differential equations involving them?
Transform of the Dirac Delta function
California:
Common Core:
171
Content Outcomes
Students will be able to find when two
matrices are equal and apply operations
involving matrices.
Students will be able to find the derivative and
integral of a matrix of functions.
Students will be able to solve a system of
equations by Gaussian and Gauss-Jordan
elimination.
Differential Equations
Unit 4 – What are Matrices?
Essential Questions
Key Concepts
When are 2 matrices equal?
Dimensions of a matrix
Equality of matrices
When and how can we find the sum, difference, Sums and products of matrices
and product of two matrices?
Inverse of a matrix
Determinant of a square matrix
How do we find the transpose and inverse of a Transpose of a matrix
matrix?
How do we differentiate a matrix of functions?
Formula for derivative of a matrix
Formula for integral of a matrix
How do we integrate a matrix of functions?
What row operations can be applied to a matrix to
maintain its equivalency?
Definition of augmented matrix
Definition of row echelon/ reduced row echelon
form
How do we solve a system of linear equations by
applying elementary row operations?
What are eigenvalues and eigenvectors?
How can we find the eigenvectors and eigenvalues
of a matrix?
Common Core:
California:
Common Core:
How do we find an inverse matrix by using
Gaussian elimination?
Students will be able to find eigenvalues and
eigenvectors of a matrix.
Standards Addressed
California:
Definition of eigenvalue
Definition of eigenvector
172
California:
Common Core:
Content Outcomes
Students will be able to solve a system of
homogeneous linear systems.
Students will be able to solve a system of
nonhomogeneous linear systems.
Differential Equations
Unit 5 – How do you Solve Systems of Linear First-order Differential Equations?
Essential Questions
Key Concepts
What is the definition of a system of linear first- Definition of a system of linear first-order
order differential equations?
differential equations
How do you solve a system with distinct real
eigenvalues?
Process of solving systems of homogeneous linear
systems
Standards Addressed
California:
Common Core:
How do you solve a system with repeated
eigenvalues?
Students will be able to solve a system of
nonhomogeneous linear systems.
Students will be able to use the matrix
exponential.
How do you solve a system with complex
eigenvalues?
How do you solve nonhomogeneous linear systems?
Solving non homogeneous linear systems by
Variation of Parameters
What is Variation of Parameters?
What is the matrix exponential?
California:
Common Core:
Definition of the matrix exponential
California:
Common Core:
173
Damien High School
Mathematics and Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Introduction to Computer Science
“B” or better in Algebra 1
Yes
Semester
Brief Course Description
An introductory look at the design and implementation of software engineering.
Assigned Textbook(s)
Supplemental Material(s)
Charles E. Cook, Blue Pelican Java, Virtualbookworm.com Publishing Inc. http://www.bluepelicanjava.com/

Introduction to Computer Science using Java, Bradley Kjell, Central Connecticut State University, http://programmedlessons.org/java5/index.html#12

Java’s Application Programming Interface (API), Sun Microsystems, www.java.sun.com
Common Assessments Utilized
Common Semester Final
Homework
In-Class Projects
Quizzes
Exams
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
174
Major Content Outcomes
I.
Understanding of variables
A. Their types
B. Use in software engineering
C. Assignment and declaration
D. Passing and receiving variables as arguments to methods.
II.
Understanding of conditional programming
A. Use of if
B. Use of if, else
C. Use of if, else if, else
D. Counter controlled while loops
E. Sentinel controlled while loops
F. Traditional for loops
III.
Class Design
A. Fields, reference variables
B. Methods
C. Variable scope
D. Constructors
E. Accessor methods
F. Mutator methods
G. Instantiation
IV.
Random number generation
A. Creating random numbers
B. Psuedo-random number generation
C. Applications in simulations and game design
V.
Strings
A. Formatting output
B. Escape sequences
C. Substrings
D. Iteration over string variables
VI.
Web Programming
A. Graphical User Interfaces
B. Software engineering for online applications
C. HTML
VII.
Arrays
A. One-dimensional arrays of primitive data
B. The use of subscripts to access and modify data
C. Iteration of arrays
Major Skill Outcomes (include Labs???)
By the end of the year students should be able to:
I. Develop and implement algorithms to solve common problems in computer science.
II. Make use of compilers and Interactive Development Environments for the use of creating
software.
III. Evaluate potential ethical dilemmas related to the use and creation of computer programs.
IV. Apply the cycle of software creation (coding, debugging, testing, documenting).
Unit 1
Students should be able to understand computer programs, how they are created, maintained and designed.
Students should be able to construct a simple one-class program.
Students should be able to produce properly formatted output.
Students should be able to properly document a computer program.
Students should be able to understand and discuss the ethical dilemmas in computer programming.
Students should be able to properly use a interactive development environment and compiler.
Unit 2
Students should be able to use primitive variable types.
Students should be able to use basic mathematical operations on variable types.
Students should be able to declare and assign values to variables.
Students should be able to output a variables value.
Students should be able to input a variables value from the keyboard.
Unit 3
Students should be able to use if statements.
Students should be able to use if-else statements.
Students should be able to use if-else if-else statements.
Students should be able to use and analyze sentinel controlled while loops.
Students should be able to use and analyze counter controlled while loops.
Students should be able to use and analyze standard for loops.
Students should be able to create if statements to change program execution based on variable input.
Students should be able to repeat a set of instructions with the proper type of loop.
Unit 4
Students should be able to critically think about a problem and decompose it into parts.
Students should be able to create methods to solve specific tasks.
Students should be able to write methods given pre and post conditions.
Students should be able to understand variable scope.
Students should be able to understand a programs flow of control.
Students should be able to collaborate on a programming project by created separate methods.
Students should be able to create a web based program.
Students should be able to use simple Graphical User Interface (GUI) elements.
Unit 5
Students should be able to create arrays of primitive data.
Students should be able to store user input in arrays.
Students should be able to iterate over arrays of primitive data.
Students should be able to implement simple sorting algorithms over arrays of primitive data.
175
Introduction to Computer Science
Unit 1 – Introduction to Programming
Key Concepts
Content Outcomes
Essential Questions
Students will learn about software engineering.
What is software engineering?
What skills are involved in computer programming?
What is a programming language?
What is a compiler?
What is an Interactive Development Environment?
What ethical issues concern technology today?
Java, C++, C#, PHP, HTML, Binary
Interpreted vs Compiled language
IDE
End-User-License-Agreement
Digital Millennium Copyright Act
Students will learn how to create a simple program.
What is a computer program?
How do we create computer programs?
How does an IDE facilitate this process?
What language should we use?
Main class
Method main
Programming libraries
Compiling
Use of an IDE
ISTE-CSE: B: v, vi
Students will learn how to produce output.
What is output?
Why is it important for a program to produce
output?
How many rules do we have in the English
language for producing output?
Output methods and functions
Using escape sequences
Arguments to a method
Variable type String
ISTE-CSE: A: i
C: i
Students will learn how to properly document a
program.
Why would we want to document our program?
Typically, how many people work on a software
program and why would it be useful to use
documentation?
How can documenting your own program help you?
Single-line comments
Multiple-line comments
Design documentation
ISTE-CSE: B: vi
176
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: D: i, ii
Content Outcomes
Students will understand the different types of
variables.
Students will be able to declare and assign variables
a value.
Students will learn how to use variables in
mathematical expressions.
Students will learn how to output and input a
variables value.
Introduction to Computer Science
Unit 2 – Variables
Essential Questions
Key Concepts
What kind of data to computer programs need to
Integers
store?
Decimals
Why would we store data by type?
Strings
How many different types of data can you list?
Boolean
How much memory do you think different types of
Memory locations
data consumes?
How would we tell the computer to create a variable Variable declaration
for our program?
Using the assignment operator
What value do variables initially hold?
Valid data types
How would we change the value of a variable?
Common errors in variables
What kind of data would cause an error in a
variable?
What can we do to numbers mathematically and
Order of operations
how will that translate to programming?
Addition
How do we store the result of an expression?
Subtraction
What practical use can you apply using a
Multiplication
mathematical expression?
Division, integer division vs decimal division
Modulus
How would we output the value of a variable?
How would we input the value of a variable?
How can a programming library help us input data
from the keyboard?
How can we output more than one piece of data?
Output methods and functions
String concatenation
Input from the keyboard
Output a variable
177
Standards Addressed
ISTE-CSE: A: i
C: i
ISTE-CSE: A: i
C: i
ISTE-CSE: A: i
C: i
ISTE-CSE: A: i
C: i
Content Outcomes
Students will be able to program if, if-else and ifelse if-else statements.
Students will be able to program sentinel controlled
while loops.
Students will be able to program counter controlled
while loops.
Students will be able to program a standard for
loop.
Introduction to Computer Science
Unit 3 – Conditional Programming
Essential Questions
Key Concepts
How can we use a computer to make simple
Conditional statements
decisions?
Boolean types
How can we change our program output based upon
Equality and relational operators
program input?
The if statement
How would a program know what instruction to
The else statement
execute and what instructions to skip?
The else if statement
How does a program repeat itself?
The while loop
How do programs continually run without a definite
Sentinel value
stopping point?
Infinite loops
Why applications exist for a program that continues
Loop continuation condition
forever until a certain condition is met?
Are there times when we know how many times we
The while loop
want to repeat a set of instructions?
Counter variable
How could we modify the sentinel while loop to
Loop continuation condition
loop a set number of times instead?
Infinite loops
Increment/decrement of counter variable
How might we place all the instructions for a
counter controlled while loop into a one-line
instruction?
What benefit would this serve?
The for loop
For loop initialization
For loop continuation condition
For loop increment/decrement
178
Standards Addressed
ISTE-CSE: A: i
B: i
C: i
ISTE-CSE: A: i
B: i
C: i
ISTE-CSE: A: i
B: i
C: i
ISTE-CSE: A: i
B: i
C: i
Content Outcomes
Students will understand the purpose of methods.
Introduction to Computer Science
Unit 4 – Methods
Essential Questions
Key Concepts
Why would we not want to write our program all
Functional decomposition
from within method main?
Method signatures
What does it mean to decompose a problem into
Returning data from a method
parts?
Passing arguments to a method
What is an argument/parameter?
Using arguments from within a method
Standards Addressed
ISTE-CSE: A: i
A: iv
B: i
B: vi
C: i
Students will be able to program methods.
How can we write a separate piece of code in order
to solve a smaller portion of a larger problem?
How to we link our methods together in order to
solve the larger problem?
Method signatures
Passing arguments to a method
Using arguments from within a method
Returning data from a method
Variable scope, local variables vs global variables
The programs flow of control
Invoking a method
ISTE-CSE: A: i
A: iv
B: i
B: vi
C: i
Students will understand how to implement
programming collaboration.
How could we use methods to work together on a
programming project?
How can we make sure that I will know how to run
your methods and vice versa?
What real word applications exist?
Functional Decomposition
Programming by contract
Pre and post conditions
ISTE-CSE: A: i
A: iv
B: i
B: vi
C: i
Students will be able to write a web based program.
What kinds of programs run online vs offline?
What are some real world applications of online
programs?
What different elements might we concern
ourselves with when programming for an online
environment?
HTML
Graphical User Interfaces
Adding a program into a web page
Web based programming concerns and ethical
issues.
ISTE-CSE: A: i
A: iv
B: i
B: vi
C: i
179
Content Outcomes
Students will understand the importance and need
for data structures.
Introduction to Computer Science
Unit 5 – Data Structures: Arrays
Essential Questions
Key Concepts
How would we store a 100 pieces of data? 1000?
Data Structures
10000? 100000? 1000000?
Arrays
What real world applications exist that would need
Memory concepts
to store this much data?
What shortcuts could you conceive to help us create
mass amounts of storage for data?
Standards Addressed
ISTE-CSE: A: i
A: ii
A: iv
B: i
B: ii
C: i
Students will be able to create one-dimensional
arrays of primitive data.
How would we create a bunch of memory
locations?
What limitations might be places on these memory
locations?
What would these memory locations hold initially?
Declaration of arrays.
Initialization of arrays.
Memory concepts.
Homogenous data concepts
ISTE-CSE: A: i
A: ii
A: iv
B: i
B: ii
C: i
Students will be able to access, change and iterate
of one-dimensional arrays of primitive data.
How would we access a single location in an array?
How would we change a single location in an array?
What does iterate mean? How does this apply to
arrays?
Access array elements using subscripts.
Modifying array elements.
Iterating over arrays to access and/or change each
element.
ISTE-CSE: A: i
A: ii
A: iv
B: i
B: ii
C: i
Students will learn how to sort a one-dimensional
array of primitive data.
Why might we want to sort the data in an array?
How would you go about sorting an array?
Iterating over arrays.
Bubble sort algorithm.
Selection sort algorithm.
ISTE-CSE: A: i
A: ii
A: iv
B: i
B: ii
C: i
180
Damien High School
Mathematics and Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
Introduction to Game Design
“B” or better in Algebra 1
Yes
Semester
Brief Course Description
An introductory look at the design and implementation of game development.
Assigned Textbook(s)
Supplemental Material(s)
Charles E. Cook, Blue Pelican Java, Virtualbookworm.com Publishing Inc. http://www.bluepelicanjava.com/

Introduction to Computer Science using Java, Bradley Kjell, Central Connecticut State University, http://programmedlessons.org/java5/index.html#12

Java’s Application Programming Interface (API), Sun Microsystems, www.java.sun.com

Slick2D Wiki, Slick2D Development Team, http://www.slick2d.org/wiki/index.php/Main_Page
Common Assessments Utilized
Common Semester Final
Homework
In-Class Projects
Quizzes
Exams
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
181
Major Content Outcomes
VIII.
Understanding of variables
A. Their types
B. Use in game programming
C. How complex data types are stored
D. Variables types used to display graphics
IX.
Understanding of conditional programming
A. If, else-if, else structure
B. While loops
C. For loops
D. Using conditional structures to make decisions
E. Using conditional structures for the main game loop
i. User input
ii. Artificial agents act
iii. Render new screen
X.
User input
A. Retrieve input from the keyboard
B. Retrieve input from the mouse
C. Track the mouse location and event
D. Understanding the basics of user interaction using the keyboard, mouse and
interactive graphics.
i.
XI.
Random number generation
A. Generating random numbers within given ranges
B. Understanding random number generation for games
C. Using random numbers to make artificial agents act within probability parameters.
i. What decision should be made by artificial agents given the game state?
XII.
Artificial Intelligence
A. Chasing algorithms
B. Evading algorithms
C. Spawning algorithms
D. Shooting algorithms
XIII.
Fundamentals of game design and theory
A. Designing reward systems
B. Designing reward schedules
C. The tidiness theory of game design
D. Creating the design document
Major Skill Outcomes (include Labs???)
By the end of the year students should be able to:
V. Effectively use compilers and Interactive Development Environments for the use of creating
interactive games.
VI. Program sprites, including: movement, rotation and collision.
VII. Program basic Artificial Intelligence routines including, chasing, evading, attacking and
random decision making.
VIII. Develop video game software both individually and collaboratively.
Unit 1
Students should be able to open a game window.
Students should be able to draw graphic primitives onto the game window using the correct parameters.
Students should be able to effectively use the RGB color system.
Students should be able to effectively use the screen coordinate system to place graphic primitives.
Unit 2
Students should be able to prepare the mouse and keyboard for input.
Students should be able to use if statements to poll the mouse and keyboard devices.
Students should be able to read in key presses from the keyboard.
Students should be able to read in mouse clicks.
Students should be able to track the mouse’s cursor position.
Students should be able to make decisions based upon user input.
Unit 3
Students should be able to create and load a bitmap image.
Students should be able to place a bitmap image onto the screen.
Students should be able to add movement to a bitmap based upon user input.
Students should be able to create and modify bitmap images.
Students should be able to transform bitmap images.
Unit 4
Students should be able to add sprite movement and rotation based upon input.
Students should be able to add sprite movement based upon velocity.
Students should be able to add sprite collision between sprites and the game window.
Students should be able to add sprite collision between two sprites.
Unit 5
Students should be able to create random numbers and use simple probability to make decisions.
Students should be able to employ simple chasing algorithms.
Students should be able to employ simple evading algorithms.
Students should be able to employ simple shooting algorithms.
Students should be able to employ simple spawning algorithms.
Unit 6
Students should be able to describe reward systems and their strength and weaknesses.
Students should be able to describe reward ratios and how they can be used.
Students should be able to effectively apply reward systems and ratios into a game.
Students should be able to describe the tidiness theory of game design.
Students should be able to effectively use the tidiness theory of game design.
182
Introduction to Game Design
Unit 1 – Introduction to Game Programming
Key Concepts
Content Outcomes
Essential Questions
Students will learn how games are created.
What does game programming entail?
What is a programming language and game engine?
What languages are used for the creation of games?
What are the ethical concerns in game design?
What is a pixel?
What is screen resolution?
What is a game window?
What parameters would we need to supply to open a
game window?
What is a graphic primitive?
What parameters would we need to supply in order
to draw different graphic primitives?
How does resolution play a factor in where and how
we draw?
Game engines
Game programming libraries
Java, C++, C#
Ethical concerns in terms of violence
Pixel
Screen resolution
Image resolution
Parameter
Methods
Parameters
Graphic primitives
Invoking a method
How resolution affects graphics
ISTE-CSE: A: i, iii
D: vi
How do monitors display color?
How can we change the color of our graphic
primitives?
How do we control where we want to display
graphics?
The RGB color system.
Declaring and creating colors.
The screen coordinate system.
Properly placing graphics onto the game window.
ISTE-CSE: A: i, iii
C: i
Students will learn how to open a game window.
Students will learn how to draw graphic primitives
onto the game window.
Students will learn how to use the RGB color and
screen coordinate system.
183
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: D: i, ii
ISTE-CSE: A: i, iii
Introduction to Game Design
Unit 2 – Input
Key Concepts
Content Outcomes
Essential Questions
Students will be able to obtain input from the
keyboard and mouse.
What kind of input do we need to obtain for games?
What would we use these input devices for?
How do you suppose we obtain this input?
Class Input org.newdawn.slick
Obtaining key presses from keyboard
Obtaining mouse clicks
Students will learn how to use if statements to react
to input from the keyboard and mouse.
How can we use this input?
What events might we create based upon user
input?
How would we code such events?
If statements
If-else statements
If-else if-else statements
Boolean conditions
ISTE-CSE: A: iii
B: i
Students will understand how keyboard polling and
mouse input is handled by the computer.
How does class Input obtain and store the user
input?
Why often do you think it attempts to read the input
devices?
What do you think the term ‘polling’ means?
Why would we need to know where the mouse
cursor is at?
What events might depend upon the position of the
mouse cursor?
Data storage
Programming threads
Accessing data in a data structure
ISTE-CSE: A: i, iii
B: i
C: i
Obtaining (x,y) position of mouse cursor
Responding to cursor position
Combining cursor position with mouse clicks
ISTE-CSE: A: i
B: i
Students will learn how to track the mouse’s cursor
position.
184
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: iii
B: i
Introduction to Game Design
Unit 3 – Bitmaps
Key Concepts
Content Outcomes
Essential Questions
Students will be able to load bitmaps images (jpeg,
bmp, png) onto the screen.
What is a bitmap image?
What are the common file extensions for bitmap
images?
What do you see as the limitations of using bitmap
images in games?
How would we move a bitmap image?
Why would we want to move a bitmap image?
How can we use user input to move an image?
What key bindings and events would you check for
in order to move a bitmap image?
Bitmap images
Memory concepts
Loading a bitmap image into a game
Rendering a bitmap image onto the screen
The (x,y) screen coordinate system
Using variables to define position
Using user input to move an image
Applying variable delta to ensure proper speed
across user experience.
ISTE-CSE: A: i, iii
B: i
C: i
Students will learn how to use a common image
editing software program.
What attributes might we want to change in an
image?
How would we change an image?
How would we create an image?
Bitmap creation software
Saving to the proper file type
Modifying color by use of filters and selections
Modifying size and rotating bitmap files.
ISTE-CSE: A: i, iii
B: i
C: i
Students will be able to apply common
transformations with bitmap images.
What is a transformation?
What transformations might we want to apply to
bitmap images in our code? Why?
How might we apply these transformations?
Scaling bitmap images
Rotating bitmap images
Flipping bitmap images
Applying color filters to bitmap images
ISTE-CSE: A: i, iii
B: i,ii
C: i
Students will be able to add movement to bitmap
images bases upon user input.
185
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: i, iii
B: i
C: i
Introduction to Game Design
Unit 4 – Sprite Programming
Key Concepts
Content Outcomes
Essential Questions
Students will create a Sprite class for reusability.
Why would we want to create a separate class with
all the image programming code in it?
What does making a separate class allow us to do?
What algorithms and data should this Sprite class
contain?
Object Oriented Programming
Code re-use
Class declaration
Fields
Constructors
Access modifiers
Methods
Students will be able to move a sprite based upon
input and Sprite state.
What is the state of a Sprite?
How would the data values affect the movement of
a Sprite?
What types of different movements may we want to
program into our Sprite class?
Obtaining user input
Using and storing Sprite angle
Calculating x and y velocities based upon Sprite
angle
Moving Sprite according to Sprite angle
ISTE-CSE: A: i, ii, iii
B: i, ii, vi
C: i
Students will understand and be able to write simple
Sprite collision.
What is Sprite collision?
What might a Sprite collide with?
What would happen if we allow a Sprite to move
off the screen?
Screen resolution
Image resolution
The (x,y) coordinate system of the screen
Sprite collision with edges of visible window
Calculating boundaries of a bitmap image
ISTE-CSE: A: i, ii, iii
B: i, ii, vi
C: i
Students will understand and be able to write Sprite
to Sprite collision.
What happens when one Sprite collides with
another?
How would we know when two Sprites are
occupying the same space?
What decision might we make when two Sprites
collide?
Calculating boundaries of a bitmap image
Calculating bounding boxes of Sprites
Using ifs to determine if two Sprites are colliding
Common practices in Sprite Collision
ISTE-CSE: A: i, ii, iii
B: i, ii, vi
C: i
186
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: i,ii, iii
B: i, vi
Introduction to Game Design
Unit 5 – Artificial Intelligence
Key Concepts
Content Outcomes
Essential Questions
Students will understand the importance and usage
of random number generators.
Why would we need a random number generator?
What decision could be artificially made by using a
random number generator?
How would we generate a random number in code?
Psuedo-random numbers
Class Random from java.util
Creating random numbers within a specified range
Making decision based upon probability and
random number generation
Students will understand and learn to program
simple chasing and evading algorithms.
How would we make an artificial agent chase/evade
us?
Why would we want to write different types of
chasing/evading algorithms?
What mathematical principles will we need to be
versed in, in order to write these algorithms.
How would we make artificial agents shoot?
How would we represent bullet data?
How many bullets can an artificial agent shoot?
Would it become more complicated with more
agents/bullets? How?
Geometry and using right triangles to determine the
angle from one Sprite to another.
Simple evading algorithms.
Simple chasing algorithms.
ISTE-CSE: A: i, iv
B: i, ii, vi
C: i
Memory concepts
Composition of classes
Simple shooting algorithms
Constructing shooting patterns
Class ArrayList from java.util
ISTE-CSE: A: i, iv
B: i, ii, vi
C: i
How would we re-spawn or spawn enemies?
How could we use random number generation in
spawning algorithms?
Why would we use random number generation in
spawning algorithms?
How could we hold a series of enemies data-wise?
Class ArrayList from java.util
Re-spawning enemies after a time-out delay
Creating enemy spawners
Using random number generation to spawn enemies
ISTE-CSE: A: i, ii, iv
B: i, ii, vi
C: i
Students will understand and learn to program
simple shooting algorithms.
Students will understand and learn to program
simple spawning algorithms.
187
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: i, iv
Introduction to Game Design
Unit 6 – Game Design
Key Concepts
Content Outcomes
Essential Questions
Students will understand different reward systems
employed in games.
What makes a game fun?
What makes a game addicting?
Can you identify key elements in popular games?
Currency rewards
Rand rewards
Mechanical rewards
Narrative rewards
Emotional rewards
New toys
New places
Completeness
Victory
Students will understand different rewards ratios
employed in games.
How often do you reward a player?
What does the frequency of rewards do to the fun
factor and addiction of games?
Can you identify some popular games and how
often you are rewarded in them?
What do you suppose is the tidiness theory of game
design?
What about game mechanics is a reference to
tidiness?
Fixed ratio schedules
Variable ratio schedules
Fixed interval schedules
Variable interval schedules
ISTE-CSE: A: iv
B: i, vi
The tidiness theory
Applications of the tidiness theory
ISTE-CSE: A: iv
B: i, vi
Think of a popular game and list out its rewards
systems, reward ratios and its applications to the
tidiness theory.
What rewards do you think are most popular and
well received?
What rewards systems to different genres of games
use?
Applying reward systems and ratios
Applying the tidiness theory
Creating a design doc
The importance of fun and addiction in games
ISTE-CSE: A: iv
B: i, vi
Students will understand the tidiness theory of game
design.
Students will understand how to effectively use
rewards systems and the tidiness theory.
188
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: iv
B: i, vi
Damien High School
Mathematics and Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
AP Computer Science
3.5 Cumulative GPA and “B” or better in Algebra 1
Yes
Year
Brief Course Description
To learn the basics of JAVA programming from an object oriented approach. Including but not limited to: data types, unary and binary operators, logical operators, string
manipulation, output, input, conditional statements, loops, classes, methods, inheritance, polymorphism, interfaces, one dimensional arrays, and array lists.
Assigned Textbook(s)
Supplemental Material(s)
Java How to Program: Early Objects Version (9th Edition), Deitel and Deitel, Prentice Hall, 2012

Barron’s AP Computer Science A (6th Edition), Barron’s Educational Series, 2013

Java’s Application Programming Interface (API), Oracle, http://docs.oracle.com/javase/7/docs/api/

GridWorld Case Study, College Board, http://apstudent.collegeboard.org/apcourse/ap-computer-science-a/gridworld-case-study
Common Assessments Utilized
Common Semester Final
Homework
In-Class Projects
Quizzes
Exams
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
189
Major Content Outcomes
XIV.
Understanding of variables
A. Their types
B. Use in game programming
C. How complex data types are stored
D. Variables types used to display graphics
XV.
Understanding of conditional programming
A. If, else-if, else structure
B. While loops
C. For loops
D. For-each loops
E. Using conditional structures to make decisions
F. Using conditional structures for the main game loop
i. User input
ii. Artificial agents act
iii. Render new screen
XVI.
Understanding of classes, objects and methods
A. Class declaration and creating objects
B. Implementation of class behaviors and methods
C. Implementation of class attributes and instance variables
D. Invoking methods of a class
E. Constructing new objects
F. Differences between primitive and reference types
G. Encapsulation and data hiding
H. Keyword this
I.
Static variables and methods
XVII.
Understanding of arrays and ArrayLists
A. Declaring arrays and ArrayLists
B. Accessing data in arrays and ArrayLists
C. Modifying data in arrays and ArrayLists
D. Iterating over arrays and ArrayLists
E. Searching and sorting algorithms
F. Differences and uses of arrays versus ArrayLists
XVIII.
Understanding inheritance and polymorphism
A. Promoting reusability with inheritance
B. The relationship between superclasses and subclasses
C. Keyword extends
D. Access modifier protected
E. Keyword super
F. Using constructors in inheritance hierarchies
G. The Object class
H. Overridden methods
I.
Abstract classes
J.
Interfaces
K. Determining an objects type at execution time
XIX.
Recursion
A. Writing an using recursive methods
B. Determining base case and recursive step
C. The system stack
D. Differences between recursion and iteration
Major Skill Outcomes (include Labs???)
By the end of the year students should be able to:
IX. Develop and implement algorithms to solve common problems in computer science.
X. Make use of compilers and Interactive Development Environments for the use of creating
software.
XI. Evaluate potential ethical dilemmas related to the use and creation of computer programs.
XII. Apply the cycle of software creation (coding, debugging, testing, documenting).
XIII. Create, analyze and modify large programs of multiple classes that include inheritance and
polymorphism.
XIV. Create and utilize large amount of data with the use of complex data structures.
Unit 1
Students should be able to understand computer programs, how they are created, maintained and designed.
Students should be able to construct a simple one-class program.
Students should be able to produce properly formatted output.
Students should be able to properly document a computer program.
Students should be able to understand and discuss the ethical dilemmas in computer programming.
Students should be able to properly use an interactive development environment and compiler.
Students should be able to use primitive variable types.
Students should be able to use basic mathematical operations on variable types.
Students should be able to declare and assign values to variables.
Students should be able to output a variables value.
Unit 2
Students should be able to use if statements.
Students should be able to use if-else statements.
Students should be able to use if-else if-else statements.
Students should be able to use and analyze sentinel controlled while loops.
Students should be able to use and analyze counter controlled while loops.
Students should be able to use and analyze standard for loops and for-each loops.
Students should be able to create if statements to change program execution based on variable input.
Students should be able to repeat a set of instructions with the proper type of loop.
Unit 3
Students should be able to critically think about a problem and decompose it into parts.
Students should be able to create methods to solve specific tasks.
Students should be able to write methods given pre and post conditions.
Students should be able to understand variable scope.
Students should be able to understand a programs flow of control.
Students should be able to collaborate on a programming project by created separate methods.
Students should be able to construct objects and create instances of objects.
Unit 4
Students should be able to declare and create an array.
Students should be able to declare and create an ArrayList.
Students should be able to access and modify elements of an array or ArrayList.
Students should be able to iterate over the elements of an array or ArrayList.
Students should be able to analyze and implement common searching and sorting algorithms for array and
ArrayLists.
Students should be able to discuss and analyze the differences between an array and ArrayList
Unit 5
Students should be able to create and implement abstract classes.
190
Students should be able to create and implement interfaces.
Students should be able to implement an inheritance hierarchy.
Students should be able to iterate over polymorphic data and determine object type at runtime.
Unit 6
Students should be able to write recursive methods.
Students should be able to analyze recursive methods.
Students should be able to describe the system stack in reference to method calls and priorities.
Students should be able to describe the differences between recursion and iteration.
191
AP Computer Science
Unit 1 – Introduction to Software Engineering
Key Concepts
Content Outcomes
Essential Questions
Students will learn about software engineering.
What is software engineering?
What skills are involved in computer programming?
What is a programming language?
What is a compiler?
What is an Interactive Development Environment?
What ethical issues concern technology today?
What is a computer program?
How do we create computer programs?
How does an IDE facilitate this process?
What language should we use?
Students will learn how to create a simple program.
Students will understand the different types of
variables.
What kind of data to computer programs need to
store?
Why would we store data by type?
How many different types of data can you list?
How much memory do you think different types of
data consumes?
Students will learn how to use variables in
mathematical expressions.
What can we do to numbers mathematically and
how will that translate to programming?
How do we store the result of an expression?
What practical use can you apply using a
mathematical expression?
Java, C++, C#, PHP, HTML, Binary
Interpreted vs Compiled language
IDE
End-User-License-Agreement
Digital Millennium Copyright Act
Main class
Method main
Programming libraries
Compiling
Use of an IDE
Using System.out for output
Integers
Doubles
Strings
Boolean
Memory locations
Using the assignment operator
Common errors in variables
Order of operations
Addition
Subtraction
Multiplication
Division, integer division vs decimal division
Modulus
192
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: D: i, ii
ISTE-CSE: B: v, vi
ISTE-CSE: A: i
C: i
ISTE-CSE: A: i
C: i
AP Computer Science
Unit 2 – Control Statements
Key Concepts
Content Outcomes
Essential Questions
Students will be able to program if, if-else and ifelse if-else statements.
How can we use a computer to make simple
decisions?
How can we change our program output based upon
program input?
How would a program know what instruction to
execute and what instructions to skip?
How does a program repeat itself?
How do programs continually run without a definite
stopping point?
Why applications exist for a program that continues
forever until a certain condition is met?
Conditional statements
Boolean types
Equality and relational operators
The if statement
The else statement
The else if statement
The while loop
Sentinel value
Infinite loops
Loop continuation condition
Students will be able to program counter controlled
while loops.
Are there times when we know how many times we
want to repeat a set of instructions?
How could we modify the sentinel while loop to
loop a set number of times instead?
Students will be able to program a standard for
loop.
How might we place all the instructions for a
counter controlled while loop into a one-line
instruction?
What benefit would this serve?
The while loop
Counter variable
Loop continuation condition
Infinite loops
Increment/decrement of counter variable
The for loop
For loop initialization
For loop continuation condition
For loop increment/decrement
Students will be able to program sentinel controlled
while loops.
193
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: i
B: i
C: i
ISTE-CSE: A: i
B: i
C: i
ISTE-CSE: A: i
B: i
C: i
ISTE-CSE: A: i
B: i
C: i
Content Outcomes
Students will understand the purpose of methods.
Students will be able to program methods.
AP Computer Science
Unit 3 – Classes. Methods and Object Oriented Programming
Essential Questions
Key Concepts
Why would we not want to write our program all
from within method main?
What does it mean to decompose a problem into
parts?
What is an argument/parameter?
How can we write a separate piece of code in order
to solve a smaller portion of a larger problem?
How to we link our methods together in order to
solve the larger problem?
Students will be able to construct and instantiate
classes.
Why might we want to place code into a separate
file?
What would creating separate classes allow us to
do?
How might we mimic a real world object through
class creation?
Students will understand the memory concepts of
instance variables.
How does a computer store objects in memory?
What is a memory leak?
Would it be possible to run out of memory? How?
How do we access different fields of a memory
location?
Functional decomposition
Method signatures
Returning data from a method
Passing arguments to a method
Using arguments from within a method
Method signatures
Passing arguments to a method
Using arguments from within a method
Returning data from a method
Variable scope, local variables vs global variables
The programs flow of control
Invoking a method
Class declaration
Instance variables
Constructors
Accessor and mutator methods
Instantiating an object of a class
Reference variables versus primitive data types
Invoking methods on objects
Reference variables as pointers to memory locations
The dot separator
Comparing objects
Passing objects as arguments to a method
The garbage collector and removing unused
memory
194
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: i, iv
B: i, vi
C: i
ISTE-CSE: A: i, iv
B: i, vi
C: i
ISTE-CSE: A: i, iv
B: i, vi
C: i
ISTE-CSE: A: i, iv
B: i, vi
C: i
AP Computer Science
Unit 4 – Arrays and ArrayLists
Key Concepts
Content Outcomes
Essential Questions
Students will be able to create, access and modify
arrays of data.
How would we store a bunch of objects?
How would the computer allocate memory for a
bunch of objects?
Array declaration
Accessing elements of an array
Modifying elements of an array
Memory allocation
Students will be to iterate over elements of an array.
How would we access every single element of an
array?
Why would we want to access every single element
of an array?
How long would it take and how complex would it
be to access every element of an array.
What if we don’t know how much data our program
will be required to store?
It what circumstances would an array
implementation fail?
How can we fix the size limitations of arrays?
How can we access every single element of an
ArrayList? How will this differ from an array?
Is an ArrayList more or less efficient than an array?
Why?
In what circumstances is an ArrayList better? An
array?
Iteration over an array
For loops
For-each loops
Complexity analysis
Time analysis
ISTE-CSE: A: i, ii, iv
B: i, iii, vi
C: i, ii
ArrayList declaration
Accessing elements of an ArrayList
Modifying Elements of an ArrayList
Memory allocation
ISTE-CSE: A: i, ii, iv
B: i, iii, vi
C: i, ii
Iteration over an ArrayList
For loops
For-each loops
Complexity analysis
Time analysis
Array versus ArrayList implementations
ISTE-CSE: A: i, ii, iv
B: i, iii, vi
C: i, ii
Students will be able to create, access and modify
ArrayLists of data.
Students will be to iterate over elements of an
ArrayList.
195
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: i, ii, iv
B: i, vi
C: i, ii
AP Computer Science
Unit 5 – Inheritance and Polymorphism
Key Concepts
Content Outcomes
Essential Questions
Students will be able to implement abstract classes.
Why would we want to write a class that is
incomplete?
How might we tag the class as incomplete?
What restrictions would be placed upon such a
class?
Keyword abstract
Method signatures
Abstract methods
Collaborative programming
Students will be able to implement interfaces.
Why would we want to write a completely empty
class?
What would exist inside such a class?
What restrictions would be placed upon such a
class?
How can we utilize abstract classes?
How can we utilize interfaces?
What benefit do these two programming structures
serve?
How do they relate to code reusability?
Keyword interface
Method signatures
Collaborative programming
Abstract classes versus interfaces
ISTE-CSE: A: i, ii, iv
B: i, vi
C: i
Interfaces
Abstract classes
Concrete classes
Keyword extends
Keyword implements
Superclass
Subclass
Overridden methods
Keyword super
ISTE-CSE: A: i, ii, iv
B: i, vi
C: i
How can we store different objects, all of the same
inheritance hierarchy, in a single data structure?
What problems does this pose?
What is dynamic binding?
Dynamic binding
Polymorphism
Overridden methods
ISTE-CSE: A: i, ii, iv
B: i, vi
C: i
Students will be able to implement an inheritance
hierarchy.
Students will be able to use polymorphism and
dynamic binding to make the proper method call at
runtime.
196
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: i, ii, iv
B: i, vi
C: i
AP Computer Science
Unit 6 – Recursion
Key Concepts
Content Outcomes
Essential Questions
Students will be able to analyze recursive methods.
What would happen if a method calls itself?
What would happen in memory?
Can we use this ability somehow?
Iteration
The base case
The recursive step
Divide and conquer algorithms
Complexity analysis
Stacks
Students will be able to write recursive methods.
What kind of problems does recursion lend itself to
solving?
Can recursion simplify an algorithm?
Iteration
The base case
The recursive step
Divide and conquer algorithms
ISTE-CSE: A: i, ii, iv
B: i, iii, vi
C: i, ii
Students will understand the system stack and how
programs and memory are managed.
How does a computer keep track of what instruction
it is on?
What does the computer do when a method is called
and flow of control is passed?
Do we ever need recursion? Why or why not?
Are recursive algorithms better than iterative
algorithms?
What is the difference in terms of memory storage
and runtime complexity of recursive algorithms
versus iterative algorithms?
The System Stack
Method calls
Memory concepts
ISTE-CSE: A: i, ii, iv
B: i, iii, vi
C: i, ii
Out of memory exceptions
Complexity analysis
Time analysis
ISTE-CSE: A: i, ii, iv
B: i, iii, vi
C: i, ii
Students will understand the differences and
similarities between recursive and iterative
algorithms.
197
International Society for Technology in
Education, Computer Science Standards
Addressed
ISTE-CSE: A: i, ii, iv
B: i, iii, vi
C: i, ii