Download Clear Learning Targets

Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
1. Explain how the definition of
the meaning of rational exponents
follows from extending the
properties of integer exponents to
those values, allowing for a
notation for radicals in terms of rational 1/3 exponents. For
example, we define 5^1/3 to be the cube root of 5 because
we want (5^1/3)3 = 5^(1/3)3 to hold, so (5^1/3)3 must
equal 5.
2. Rewrite expressions involving radicals and rational
exponents using the properties of exponents.
3. Explain why sums and products of rational numbers are
rational,that the sum of a rational number and an irrational
number is irrational, and that the product of a nonzero
rational number and an irrational number is
irrational.
N.RN.1-3
Essential Understanding
- Students use laws of exponents to
understand radicals as rational
exponents.
Extended Understanding
- Students use logarithms to solve
exponential equations.
Academic
Vocabulary/Language
- Rational
- Radicand
- Numerator
- Denominator
- Properties of exponents
Tier 2 Vocabulary
- Explain
- “in terms of”
- define
I Can Statements


can define radical notation as a convention used to represent rational exponents.
 I can explain the properties of operations of rational exponents as an extension of the properties of integer exponents.
 I can explain how radical notation, rational exponents, and properties of integer exponents relate to one another.
 I can, using the properties of exponents, rewrite a radical expression as an expression with a rational exponent.
 I can, using the properties of exponents, rewrite an expression with rational exponent as a radical expression.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Common Misconceptions and Challenges
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Properties of Exponents and Roots- HSN-RN.A.1
Chapter 0-10, Page P27; Chapter 4-3 (extend), 4-4, 4-5
Rewriting Radical and Exponential Expressions- HSN-RN.A.2
Fractions with Exponents- HSN.RN.A.1
Multiplying and Adding Rational and Irrational Numbers - HSN-RN.B.3
http://ccssmath.org/?s=rn.1
http://ccssmath.org/?s=rn.2
Adding and Subtracting Rational Numbers - HSN-RN.B.3
http://ccssmath.org/?s=rn.3
Rational and Irrational Numbers - HSN-RN.B.3
Career Connections
Computer and information systems managers
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Funeral directors
Industrial production managers
Medical and health services managers
Property, real estate, and community
association managers
Columbus City Schools
Actuaries
Computer software engineers
Mathematicians
Statisticians
Teacher assistants
Teachers-adult literacy and remedial and selfenrichment education
Teachers-postsecondary
Teachers-preschool, kindergarten, elementary,
middle, and secondary
Clear Learning Targets Integrated Math II 2016-2017
Grounds maintenance workers
Computer control programmers and operators
Budget analysts
Insurance underwriters
Biological scientists
Medical scientists
Mechanical engineers
Nuclear engineers
Petroleum engineers
3
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
Essential Understanding
N.CN.1 Know there is a
complex number i such that
i^2 = −1, and every complex
number has the form a + bi with a and b real.
N.CN.1-2
N.CN.2 Use the relation i ^2 = –1 and the
commutative, associative, and distributive
properties to add, subtract, and multiply complex
numbers.
- Students understand that i is the
square root of negative 1.
- Students can perform operations
with complex numbers.
Academic
Vocabulary/Language
Extended Understanding
- imaginary number
- complex number
- commutative property
- associative property
- distributive property
- Students solve equations with
complex solutions.
Tier 2 Vocabulary
- know
- form
- use
I Can Statements





I can define i as the square root of −1 or i^2 =-1.
I can define complex numbers.
I can write complex numbers in the form a+ bi with a and b being real numbers.
I can recognize that the commutative, associative, and distributive properties extend to the set of complex numbers over the
operations of addition and multiplication.
I can use the relation i^2 = -1 to simplify.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Common Misconceptions and Challenges
Simplifying Complex Numbers- CN.1
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Adding, Subtracting, Multiplying Complex Numbers- CN.2
Chapter 3-2





http://ccssmath.org/?page_id=2030
http://ccssmath.org/?page_id=2032
Career Connections
Computer and information systems managers
Farmers, ranchers, and agricultural managers
Funeral directors
Industrial production managers
Medical and health services managers
Property, real estate, and community association managers
Purchasing managers, buyers, and purchasing agents
Columbus City Schools
A-REI, A-SSE, N-CN Vertex of a parabola with complex roots
N-CN Complex number patterns
N-CN, A-REI Complex Square Roots
N-CN Complex Cube and Fourth Roots of 1
N-CN Powers of a complex number
Drafters and engineering technicians
Engineering technicians
Physical scientists
Physicists and astronomers
Education, training, library, and museum occupations
Teachers-adult literacy and remedial and self-enrichment education
Teachers-postsecondary
Teachers-preschool, kindergarten, elementary, middle, and secondary
Media and communications-related occupations
Writers and editors
Clear Learning Targets Integrated Math II 2016-2017
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Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
Essential Understanding
Understand that polynomials
form a system analogous to
the
integers, namely, they are closed under the
operations of addition,
subtraction, and multiplication; add, subtract, and
multiply polynomials.
A.APR.1
- Students can add, subtract, and
multiply polynomials.
Extended Understanding
- Students can divide polynomials.
- Students can factor polynomials.
Academic
Vocabulary/Language
- Polynomial
- Monomial
- Binomial
- Trinomial
- distribute
- like terms
Tier 2 Vocabulary
- understand
- analogous
I Can Statements
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

I can identify that the sum, difference, or product of two polynomials will always be a polynomial, which means that polynomials are
closed under the operations of addition, subtraction, and multiplication.
I can define “closure”.
I can apply arithmetic operations of addition, subtraction, and multiplication to polynomials.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Common Misconceptions and Challenges
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 1-1, 1-2, 1-3, 1-3 (explore), 1-4
http://ccssmath.org/?s=apr.1
Polynomial Addition and Subtraction- APR.1
Polynomial Multiplication- APR.1
Adding, Multiplying, and Subtracting Monomials- APR.1
https://www.sophia.org/ccss-math-standard-9-12aapr1-pathway


A-APR Non-Negative Polynomials
A-APR Powers of 11
https://sites.google.com/site/commoncorewarwick/home/unit-ofstudies/algebra-2/a-apr-1
Career Connections
Social scientists and related occupations
Economists
Education, training, library, and museum occupations
Teachers-adult literacy and remedial and self-enrichment education
Teachers-postsecondary
Teachers-preschool, kindergarten, elementary, middle, and secondary
Teachers-special education
Health diagnosing and treating occupations
Registered nurses
Columbus City Schools
Aerospace engineers
Chemical engineers
Civil engineers
Electrical engineers
Environmental engineers
Industrial engineers
Materials engineers
Mechanical engineers
Nuclear engineers
Petroleum engineers
Clear Learning Targets Integrated Math II 2016-2017
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Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
4. For a function that models a
relationship between two quantities,
interpret key features of graphs and
tables in terms of the quantities, and
sketch graphs showing key features given
a verbal description of the relationship.
Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and
minimums; symmetries; end behavior; and periodicity.
5. Relate the domain of a function to its graph and, where applicable, to
the quantitative relationship it describes. For example, if the function
h(n) gives the number of person-hours it takes to assemble n engines in
a factory, then the positive integers would be an appropriate domain for
the function.
6. Calculate and interpret the average rate of change of a function
(presented symbolically or as a table) over a specified interval. Estimate
the rate of change from a graph.
F.IF.4-6
I Can Statements
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
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


- Students understand how to interpret domain
and range from a graph.
-Students understand how to interpret domain
and range from a real-world context.
-Students understand how to calculate average
rate of change of a functions and understand
why it is the slope between two points.
Academic
Vocabulary/Language
- average rate of change
- intercepts
- local/relative max & min
- global max & min
- end behavior
- periodicity
Extended Understanding
Tier 2 Vocabulary
- Students can think about how they would try
to calculate instantaneous rate of change.
- calculate
- interpret
- model
Common Misconceptions and Challenges
I can define and recognize the key features in tables and
graphs of linear and exponential functions: intercepts;
intervals where the function is increasing, decreasing, positive,
or negative, and end behavior.
I can identify whether the function is linear or exponential,
given its table or graph.
I can interpret key features of graphs and tables of function in
the terms of the contextual quantities the function represents.
I can sketch graphs showing key features of a function that
models a relationship between two quantities from a given
verbal description of the relationship.
I can explain why a domain is appropriate for a given situation.
I can recognize slope as an average rate of chang.
Columbus City Schools
Essential Understanding
Students may believe it is reasonable to input any x-value into a
function, so they will need to examine multiple situations in which there
are various limitations to the domain.
Students may also believe that the slope of a linear function is merely a
number used to sketch the graph of the line. In reality, slopes have realworld meaning, and the idea of a rate of change is fundamental to
understanding major concepts from geometry to calculus.
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 2-1, 2-1 (explore), 2-2, 2-6, 2-7, 2-7 (extend); 3-1 (extend),
3-4 (explore, extend); 7-4
http://ccssmath.org/?s=if.4
http://ccssmath.org/?s=if.5
http://ccssmath.org/?s=if.6
Functions versus Relations (Solutions Included)- HSF-IF.B.5
Determining and Predicting the Rate of Change of Functions- HSF-IF.B.6
Columbus City Schools
Career Connections
Computer and mathematical occupations
Actuaries
Computer programmers
Computer software engineers
Mathematicians
Statisticians
Engineers
Aerospace engineers
Chemical engineers
Farming
Forest, conservation, and logging workers
Clear Learning Targets Integrated Math II 2016-2017
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Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
F.IF.7-9
7. Graph functions expressed
symbolically and show key features of the
graph, by hand in simple cases and using
technology for more complicated cases.
a. Graph linear and quadratic functions
and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions.
8. Write a function defined by an expression in different but equivalent
forms to reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
b. Use the properties of exponents to interpret expressions for
exponential functions.
9. Compare properties of two functions each represented in a different
way (algebraically, graphically, numerically in tables, or by verbal
descriptions).
I Can Statements
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


- Students can interpret key features of
graphs of quadratic functions in a realworld context.
-Students can complete the square to
determine the vertex of a quadratic
function.
-Students can factor a quadratic
equation to determine the x-intercepts
-Students can solve quadratics using
factoring and completing the square.
Extended Understanding
- Students can write a quadratic
equation given a table of values or a
quadratic pattern.
Academic
Vocabulary/Language
- quadratic equation
- quadratic function
- standard form
- vertex form
- factor
- complete the square
Tier 2 Vocabulary
- interpret
- compare
Common Misconceptions and Challenges
I can graph exponential functions by hand in simple cases or
using technology for more complicated cases, and show
intercepts and end behavior
I can determine the difference between simple and
complicated linear and exponential functions and know when
the use of technology is appropriate
I can differentiate between exponential and linear functions
using a variety of descriptors (graphically, verbally,
numerically, and algebraically)
I can use a variety of function representations algebraically,
graphically, numerically in tables, or by verbal descriptions) to
compare and contrast properties of two functions
Columbus City Schools
Essential Understanding
Students may believe that each family of functions (e.g., quadratic, square root,
etc.) is independent of the others, so they may not recognize commonalities
among all functions and their graphjs.
Students may also believe that skills such as factoring a trinomial or
completing the square are isolated within a unit on polynomials, and that they
will come to understand the usefulness of these skills in the context of
examining characteristics of functions.
Additionally, students may believe that the process of rewriting equations into
various forms is simply an algebra sumbol manipulation exercise, rather than
serving a purpose of allowing different features of the function to be exhibited.
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 2-1, 2-3 (all); 3-1; 3-4; 4-1
Graphing Linear and Quadratic Functions- IF.C.7a
http://ccssmath.org/?s=if.7
Graphing Polynomial Functions- IF.C.7c
Classifying Even and Odd Functions - IF.C.8
Computer and mathematical occupations
Actuaries
Computer programmers
Business and financial operations occupations
Budget analysts
Insurance underwriters
Engineers
Aerospace engineers
Chemical engineers
Expressions for Exponential Functions- IF.C.8b
Farming
Forest, conservation, and logging workers
Comparing Functions in Different Formats- -IF.C.9
Financial clerks
Bookkeeping, accounting, and auditing clerks
Graphing Square and Cube Roots- IF.C.7b
Graphing Rational Functions -IF.C.7d
http://ccssmath.org/?s=if.8
http://ccssmath.org/?s=if.9
Columbus City Schools
Career Connections
Graphing Exponential and Log Functions- IF.C.7e
Clear Learning Targets Integrated Math II 2016-2017
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Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
Essential Understanding
F.BF.1
Write a function that describes a
relationship between two
quantities.
a. Determine an explicit expression, a recursive process, or
steps for calculation from a context.
b. Combine standard function types using arithmetic
operations.
c. (+) Compose functions.
- Students understand how to write
a mathematical function to model a
given situation.
-Students understand how to
compose functions and use the
proper notation to do so.
Extended Understanding
Academic
Vocabulary/Language
- explicit function
- recursive function
- composite functions
Tier 2 Vocabulary
- determine
- Students can graph these functions - model
and use them to solve problems.
- compose
- combine
I Can Statements





I can define “explicit function” and “recursive process”.
I can write a function that describes a relationship between two quantities by determining an explicit expression, a recursive
process, or steps for calculation from a context.
I can combine two functions using the operations of addition, subtraction, multiplication, and division.
I can evaluate the domain of the combines function.
I can build standard functions to represent relevant relationships/quantities given a real-world situation or mathematical process.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Common Misconceptions and Challenges
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
15
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Not found in book.
http://ccssmath.org/?s=bf.1
http://ccssmath.org/?page_id=2189
https://learnzillion.com/resources/72403-combine-standard-function-types-using-arithmetic-operations
Khan Academy videos: https://www.khanacademy.org/commoncore/grade-HSF-F-BF
Open Ed: https://www.opened.com/search?category=building-functions&grade_group=high-schoolfunctions&standard=F.BF.1.b&standard_group=common-core-math
Career Connections
Electrical and electronic equipment mechanics,
installers, and repairers
Electronic home entertainment equipment
installers and repairers
Financial clerks
Bookkeeping, accounting, and auditing clerks
Business and financial operations occupations
Budget analysts
Insurance underwriters
Columbus City Schools
Management occupations
Computer and information systems managers
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Funeral directors
Industrial production managers
Medical and health services managers
Property, real estate, and community
association managers
Purchasing managers, buyers, and purchasing
agents
Top executives
Clear Learning Targets Integrated Math II 2016-2017
Computer and mathematical occupations
Actuaries
Computer programmers
Computer software engineers
Mathematicians
Statisticians
Metal workers and plastic workers
Computer control programmers and operators
Other production occupations
Photographic process workers and processing
machine operators
16
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
3. Identify the effect on the graph
of replacing f(x) by f(x) + k,
k f(x), f(kx), and f(x + k) for specific
values of k (both positive and
negative); find the value of k given
the graphs. Experiment with cases and illustrate an
explanation of the effects on the graph using technology.
Include recognizing even and odd functions from their
graphs and algebraic expressions for them.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function
f that has an inverse and write an expression for the inverse.
F.BF.3-4
Essential Understanding
- Students understand how to
transform graphs of parent
functions.
-Students understand how to
describe transformations of parent
functions including how to use the
proper notation.
-Students understand how to find
inverses of functions.
Extended Understanding
- Students find inverses of more
complex functions.
Academic
Vocabulary/Language
- translation/shift
- dilation/stretch
- inverse function
Tier 2 Vocabulary
- experiment
- identify
- include
- find
- understand
I Can Statements





I can identify the effect a single transformation will have on the function (symbolic or graphic).
I can use technology to identify effects of single transformations on graphs of functions.
I can graph a given function by replacing f(x) with f(x)+k, kf(x), f(kx), or f(x+k) for specific values of k (both positive and negative).
I can describe the differences and similarities between a parent function and the transformed function.
I can recognize even and odd functions from their graphs and from their equations.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Common Misconceptions and Challenges
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 2-3 (explore); 3-4, 3-4 (explore)
Manipulating the Graphs of Functions- BF.B.3
http://ccssmath.org/?s=bf.3
Graphing The Inverse of Functions- BF.B.4c
http://ccssmath.org/?s=bf.4
http://www.shmoop.com/common-core-standards/ccss-hs-f-bf-3.html
Inverses of Discrete Functions- BF.B.4a
https://www.sophia.org/ccss-math-standard-9-12fbf4pathway?standard=mathematics-functions
Career Connections
Electrical and electronic equipment mechanics,
installers, and repairers
Electronic home entertainment equipment
installers and repairers
Financial clerks
Bookkeeping, accounting, and auditing clerks
Business and financial operations occupations
Budget analysts
Insurance underwriters
Columbus City Schools
Management occupations
Computer and information systems managers
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Funeral directors
Industrial production managers
Medical and health services managers
Property, real estate, and community
association managers
Purchasing managers, buyers, and purchasing
agents
Top executives
Clear Learning Targets Integrated Math II 2016-2017
Computer and mathematical occupations
Actuaries
Computer programmers
Computer software engineers
Mathematicians
Statisticians
Metal workers and plastic workers
Computer control programmers and operators
Other production occupations
Photographic process workers and processing
machine operators
19
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
Essential Understanding
F.LE.3
Observe using graphs
and tables that a
quantity increasing
exponentially eventually
exceeds a quantity
increasing linearly,
quadratically, or (more
generally) as a
polynomial function.
- Students should know how to
make all different types of graphs.
Extended Understanding
- Students should be able to
understand beginning and end
behavior.
Academic
Vocabulary/Language
- quantity
- exponentially
- linearly
- quadratically
- polynomial function
Tier 2 Vocabulary
- observe
- exceed
I Can Statements


Common Misconceptions and Challenges
I can informally define the concept of “end behavior”.
I can compare tables and graphs of linear and exponential
functions to observe that a quantity increasing
exponentially exceeds all others to solve mathematical
and real-world problems.
Columbus City Schools
Students need to remember how to make each graph as they learn how
to make new ones. For example, when learning to graph a parabola, it is
important not to forget how to graph lines. Continually remind them of
the process so they do not get the different graph types and methods
confused.
Clear Learning Targets Integrated Math II 2016-2017
20
Instructional Strategies
Use the activities cited below found on Illustrative Mathematics to teach this standard.
F-LE, A-REI Population and Food Supply
F-LE Exponential growth versus linear growth I
F-LE Exponential growth versus linear growth II
F-LE Exponential growth versus polynomial growth
http://ccssmath.org/?s=le.3
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
This specific standard is not found in the book (as far as showing the differences between graph types).
Functions are covered in chapters 2-4, but they aren’t compared to each other.
Video using calculator: https://www.youtube.com/watch?v=DPuFGDXPGB8
https://www.opened.com/search?category=linear-quadratic-and-exponential-models&grade_group=high-schoolfunctions&standard=F.LE.3&standard_group=common-core-math
Career Connections
Management occupations
Computer and information systems managers
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Financial managers
Funeral directors
Industrial production managers
Medical and health services managers
Property, real estate, and community association managers
Purchasing managers, buyers, and purchasing agents
Columbus City Schools
Media and communications-related occupations
Writers and editors
Health diagnosing and treating occupations
Optometrists
Registered nurses
Health technologists and technicians
Medical records and health information technicians
Nuclear medicine technologists
Social scientists and related occupations
Economists
Market and survey researchers
Clear Learning Targets Integrated Math II 2016-2017
21
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
A.SSE.1-2
1. Interpret expressions that
represent a quantity in
terms of its context.
a. Interpret parts of an
expression, such as terms,
factors, and coefficients.
b. Interpret complicated
expressions by viewing one
or more of their parts as a
single entity.
Essential Understanding
- Students will become fluent in
identifying parts of an expression.
- Students will understand what is
meant by coefficient, factor, and
term.
Extended Understanding
- Students will be able to examine a
large expression and see parts of it
as single entities.
2. Use the structure of an
expression to identify ways
to rewrite it.
I Can Statements



- expression
- context
- term
- factor
- coefficient
- single entity
Tier 2 Vocabulary
- interpret
- represent
- viewing
- use
- structure
- identify
Common Misconceptions and Challenges
I can, for expressions that represent a contextual
quantity, define and recognize parts of an expression,
such as terms, factors, and coefficients.
I can, for expressions that represent a contextual
quantity, interpret parts of an expression, such as terms,
factors, and coefficients in terms of the context.
I can, for expressions that represent a contextual quantity,
interpret complicated expressions, in terms of the context,
by viewing one or more of their parts as a single entity.
Columbus City Schools
Academic
Vocabulary/Language
Students may believe that use of algebraic expressions is merely the abstract
manipulation of symbols. Use of real-world context examples to demonstrate
the meaning of the parts of algebraic expressions is needed to counter this
misconception.
Students may also believe that an expression cannot be factored because it
does fit into a form they recognize. They need help with reorganizing the terms
until structures become evident.
Clear Learning Targets Integrated Math II 2016-2017
22
Instructional Strategies
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Career Connections
Chapter 1-1, 1-5, 1-5 (explore), 1-6, 1-6 (explore); 3-1, 3-3
Interpret the Context of Expressions- A.SSE.1a
Complicated Expressions- A.SSE.1b
Rewriting Expressions – A.SSE.2
Columbus City Schools
Computer and mathematical occupations
Actuaries
Computer programmers
Computer software engineers
Computer support specialists and systems administrators
Computer systems analysts
Mathematicians
Statisticians
Construction
Carpenters
Electricians
Clear Learning Targets Integrated Math II 2016-2017
23
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
A.SSE.3
A.SSE.3 Choose and produce an
equivalent form of an expression to
reveal and explain properties of the
quantity represented by the
expression.
a. Factor a quadratic expression to
reveal the zeros of the function it
defines.
b. Complete the square in a quadratic
expression to reveal the maximum or
minimum value of the function it
defines.
Essential Understanding
- Students will know how to factor.
- After factoring, students will set
each factor to zero and find the
value of each root.
Extended Understanding
- Students will learn how to solve a
quadratic equation by completing
the square.




Tier 2 Vocabulary
Common Misconceptions and Challenges
I can factor a quadratic expression to produce an equivalent
form of the original expression.
I can explain the connection between the factored form of a
quadratic expression and the zeros of the function it defines.
I can explain the properties of the quantity represented by the
quadratic expression.
I can choose and produce an equivalent form of a quadratic
expression to reveal and explain properties of the quantity
represented by the original expression.
Columbus City Schools
- expression
- properties
- quantity
- factor
- quadratic expression
- maximum/minimum
- choose
- produce
- reveal
- explain
- represented
- use
c. Use the properties of exponents to
transform expressions for exponential
functions.
I Can Statements
Academic
Vocabulary/Language
Remind students that completing the square requires the value of a to
be 1. They may have to divide first.
Students often have difficulty when working with fractions. For
instance, if the value of b is odd, dividing by 2 will result in a fraction.
Students also will need to know how to square a fraction.
Clear Learning Targets Integrated Math II 2016-2017
24
Instructional Strategies
Go over the Illustrative Mathematics activities below with your students.
Also, the worksheets from Math Worksheets Land can be quite helpful.
When teaching completing the square it can be helpful to start with problems where the value of a is 1. Also, the value of b should be an
even number.
From Illustrative Mathematics:
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 1-5, 1-6, 1-7, 1-8, 1-9; 2-3, 2-4 (extend); 4-2 (extend)
Solving Quadratic Equations- A.SSE.3a
Completing the Square in a Quadratic Expression- HSA-SSE.B.3b
Career Connections
Information and record clerks
Human resources assistants, except payroll and timekeeping
Material recording, scheduling, dispatching, and distributing occupations
Stock clerks and order fillers
Healthcare support occupations
Nursing, psychiatric, and home health aides
Columbus City Schools
A-REI, A-SSE, N-CN Vertex of a parabola with complex roots
A-SSE Graphs of Quadratic Functions
A-SSE Ice Cream
A-SSE Increasing or Decreasing? Variation 2
A-SSE Profit of a company
A-SSE Profit of a company, assessment variation
http://ccssmath.org/?page_id=2093
Properties of Exponents- A.SSE.3c
Computer and mathematical occupations
Actuaries
Computer programmers
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Management occupations
Advertising, marketing, promotions, public relations, and sales managers
Computer and information systems managers
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Financial managers
Funeral directors
Industrial production managers
Medical and health services managers
Property, real estate, and community association managers
Clear Learning Targets Integrated Math II 2016-2017
25
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
1. Create equations and
inequalities in one variable and
use them to solve problems.
Include equations arising from
linear and quadratic functions,
and simple rational and exponential functions.
A.CED.1,2,4
2. Create equations in two or more variables to represent
relationships between quantities; graph equations on
coordinate axes with labels and scales.
4. Rearrange formulas to highlight a quantity of interest,
using the same reasoning as in solving equations. For
example, rearrange Ohm’s law V = IR to highlight resistance
R.
Essential Understanding
- Students can solve equations and
inequalities in one variable.
- Students will understand the
effects of dependent and
independent variables.
Extended Understanding
- Students will have a good grasp of
equation solving skills and be able
to solve an equation for any
variable in a problem. They will be
able to apply inverse operations to
solve literal equations.
Academic
Vocabulary/Language
- variable
- linear
- quadratic
- rational function
- exponential function
Tier 2 Vocabulary
- create
- use
- solve
- include
- rearrange
I Can Statements
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can solve linear and exponential equations in one variable.
I can solve inequalities in one variable.
I can describe the relationships between the quantities in the problem (for example, how the quantities are changing or growing with respect to
each other); express these relationships using mathematical operations to create an appropriate equation or inequality to solve.
I can create equations (linear and exponential) and inequalities in one variable and use them to solve problems.
I can create equations and inequalities in one variable to model real-world situations.
I can create at least two equations in two or more variables to represent relationships between quantities.
I can justify which quantities in a mathematical problem or real-world situation are dependent and independent of one another and which
operations represent those relationships.
I can solve for other variables in Literal Equation problem.
I can define a “quantity of interest” to mean any number or algebraic quantity.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
27
Textbook and Curriculum
Resources
McGraw-Hill: Integrated Math II
Chapter 3-5; 4-2, 4-5
http://ccssmath.org/?s=ced.4
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Creating Equations and Inequalities- A.CED.1
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A-CED Equations and Formulas
A-CED Rewriting equations
A-CED Sum of angles in a polygon
N-Q Harvesting the Fields
A-CED, G-GMD Global Positioning
System I
A-CED Silver Rectangle
A-CED Throwing a Ball
G-MG, A-CED Regular Tessellations of
the plane
Creating Equations with Two or More Variables- A.CED.2
Graphing Equations-A.CED.2
Rearranging and Understanding Formulas – A.CED.4
Common Misconceptions and Challenges
Career Connections
Drafters and engineering technicians
Engineering technicians
Life scientists
Medical scientists
Physical scientists
Chemists and materials scientists
Environmental scientists and hydrologists
Physicists and astronomers
Columbus City Schools
Engineers
Aerospace engineers
Chemical engineers
Civil engineers
Electrical engineers
Environmental engineers
Industrial engineers
Nuclear engineers
Petroleum engineers
Clear Learning Targets Integrated Math II 2016-2017
Farming
Forest, conservation, and logging workers
Metal workers and plastic workers
Computer control programmers and operators
Information and record clerks
Human resources assistants, except payroll and
timekeeping
28
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
A.REI.4
Solve quadratic equations in one
variable.
a. Use the method of completing
the square to transform any
quadratic equation in x into an equation of the form (x –
p)^2 = q that has the same solutions. Derive the quadratic
formula from this form.
b. Solve quadratic equations by inspection (e.g., for x^2 =
49), taking square roots, completing the square, the
quadratic formula and factoring, as appropriate to the
initial form of the equation. Recognize when the quadratic
formula gives complex solutions and write them as a ± bi for
real numbers a and b.
Essential Understanding
- Students will be able to complete
the square.
- Students should be able to choose
from multiple methods for solving
quadratic equations the best
method for each problem.
Academic
Vocabulary/Language
- variable
- complete the square
- quadratic equation
Extended Understanding
Tier 2 Vocabulary
- Students should be shown how
completing the square can be used
to derive the quadratic formula
itself.
- solve
- transform
- derive
- inspection
- recognize
I Can Statements
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I can use the method of completing the square to transform any quadratic equation in x into an equation of the form (x-p)2 = q that has the same
solutions.
I can solve quadratic equations in one variable.
I can derive the quadratic formula by completing the square on a quadratic equation in x.
I can solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring.
I can recognize when the quadratic formula gives complex solutions.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
30
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Quadratic Equations: Completing the Square- A.REI.4a
Chapter 1-6, 1-7, 1-8, 1-9; 2-2, 2-3, 2-4, 2-5
Quadratics: Using Square Roots and Zero Property- A.REI.4b
Below is a simple SMART Board file showing how to derive the Quadratic Formula
by completing the square:
https://columbus.blackboard.com/bbcswebdav/courses/S-093020141/Lessons%20Math%20II/deriving%20the%20quadratic%20formula%20by%20completi
ng%20the%20square.notebook
Solving Quadratic Equations By Factoring- A.REI.4b
Using the Quadratic Formula – A.REI.4b
Finding and Using the Discriminant- A.REI.4b
Common Misconceptions and Challenges
Career Connections
Drafters and engineering technicians
Engineering technicians
Life scientists
Medical scientists
Physical scientists
Chemists and materials scientists
Columbus City Schools
Engineers
Aerospace engineers
Chemical engineers
Civil engineers
Electrical engineers
Environmental engineers
Industrial engineers
Nuclear engineers
Clear Learning Targets Integrated Math II 2016-2017
Farming
Forest, conservation, and logging workers
Metal workers and plastic workers
Computer control programmers and operators
Metal workers and plastic workers
Computer control programmers and operators
31
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
N.CN.7
7. Solve quadratic equations
with real coefficients that
have complex solutions.
Essential Understanding
Academic Vocabulary/Language
- Students will understand that
there are graphs of parabolas where
solutions do not cross the x-axis.
There must be an equation to model
this situation. This is where
complex solutions come from.
- quadratic equation
- complex solution
- real
- coefficient
- polynomial
- identities
- Fundamental Theorem of Algebra
Extended Understanding
- Students will know and state and
verify the Fundamental Theorem of
Algebra.
Tier 2 Vocabulary
- solve
- extend
- know
- verify
I Can Statements
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I can solve quadratic equations that have complex solutions.
I can define identity.
I can give examples of polynomial identifies.
I can extend polynomial identities to the complex numbers.
I can state the Fundamental Theorem of Algebra.
I can verify that the Fundamental Theorem of Algebra is true for second-degree quadratic polynomials.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
33
Common Misconceptions and Challenges
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Career Connections
Chapter 3-2B (extend), 3-3
CN8 and 9 are optional standards.
Solving Quadratic Equations- CN.7
https://www.khanacademy.org/math/algebra2/polynomial_and_rationa
l/fundamental-theorem-of-algebra/v/fundamental-theorem-of-algebraintro
Columbus City Schools
Engineers
Aerospace engineers
Electrical engineers
Industrial engineers
Nuclear engineers
Drafters and engineering technicians
Engineering technicians
Physical scientists
Physicists and astronomers
Education, training, library, and museum occupations
Teachers-adult literacy and remedial and self-enrichment education
Teachers-postsecondary
Teachers-preschool, kindergarten, elementary, middle, and secondary
Media and communications-related occupations
Writers and editors
Clear Learning Targets Integrated Math II 2016-2017
34
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
Essential Understanding
A.REI.7
A.REI.7 Solve a simple
system consisting of a linear
equation and a quadratic
equation in two variables
algebraically and
graphically. For example,
find the points of
intersection between the
line y = –3x and the circle
x^2 + y^2 = 3.
I Can Statements
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Extended Understanding
- Students can use technology such
as graphing calculators or online
sites such as Desmos.com to quickly
create different types of graphs at
the same time.
- system
- linear equation
- quadratic equation
- point of intersection
Tier 2 Vocabulary
- solve
- transform
- explain
- correspondence (between)
Common Misconceptions and Challenges
I can solve simple systems consisting of linear, quadratic,
and circular equations.
I can transform a simple system consisting of a linear
equation and quadratic equation in 2 variables so that a
solution can be found algebraically and graphically.
I can explain the correspondence between the algebraic
and graphical solutions to a simple system consisting of a
linear equation and a quadratic equation in 2 variables.
Columbus City Schools
- Students should be able to solve
systems by using algebra, as well as
making graphs and looking at
intersection points.
Academic
Vocabulary/Language
Some of the algebra gets tricky for students. Watch closely to ensure
mistakes are not made, particulary when doing inverse operations and
using substitution.
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Download the worksheets from Math Worksheet Lands (below) and go over the step-by-step lesson “Equal to y”.
Complete the practice and matching worksheets.
Finish with the Illustrative Mathematics lessons below.
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
http://ccssmath.org/?s=rei.7
From Math Worksheets Land:
Solving Simultaneous Equations (Linear and Quadratics)- A.REI.7
Chapter 2-3
From Illustrative Mathematics:
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A-REI A Linear and Quadratic System
A-REI Pythagorean Triples
A-REI The Circle and The Line
Career Connections
Electrical and electronic equipment mechanics,
installers, and repairers
Electronic home entertainment equipment
installers and repairers
Business and financial operations occupations
Insurance underwriters
Columbus City Schools
Management occupations
Computer and information systems managers
Farmers, ranchers, and agricultural managers
Funeral directors
Industrial production managers
Medical and health services managers
Property, real estate, and community association
managers
Purchasing managers, buyers, and purchasing agents
Clear Learning Targets Integrated Math II 2016-2017
Computer and mathematical occupations
Actuaries
Computer software engineers
Mathematicians
Statisticians
Engineers
Aerospace engineers
Electrical engineers
Industrial engineers
Nuclear engineers
Engineering technicians
36
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
1. Describe events as subsets of a sample
space (the set of outcomes) using
characteristics (or categories) of the
outcomes, or as unions, intersections, or
complements of other events (“or,” “and,”
“not”).
2. Understand that two events A and B are independent if the probability
of A and B occurring together is the product of their probabilities, and
S.CP.1-5
use this characterization to determine if they are independent.
3. Understand the conditional probability of A given B as P(A and
B)/P(B), and interpret independence of A and B as saying that the
conditional probability of A given B is the same as the probability
of A, and the conditional probability of B given A is the same as
the probability of B.
4. Construct and interpret two-way frequency tables of data when
two categories are associated with each object being classified.
Use the two-way table as a sample space to decide if events are
independent and to approximate conditional probabilities.
5. Recognize and explain the concepts of conditional probability
and independence in everyday language and everyday situations.
Essential Understanding
- Students will multiply
probabilities to find the probability
of independent events.
- Students will know the difference
between dependent and
independent events. They will be
able to determine if the outcome of
one event has an impact on the
outcome of another event.
Academic
Vocabulary/Language
- union
- intersection
- subset
- event
- independence
- sample space
- probability
Extended Understanding
Tier 2 Vocabulary
- Students will be able to find real
world situations that model
conditional probability and
independence.
- recognize
- explain
- describe
- construct
- interpret
I Can Statements
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I can define unions, intersections and complements of events.
I can describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events.
I can determine the outcome of independent events as the product of their probabilities.
I can categorize events as independent or not using the characterization that two events A and B are independent when the probability of A and B occurring
together is the product of their probabilities.
I can recognize the conditional probability of A given B is the same as P(A and B)P(B).
I can interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional
probability of B given A is the same as probability of B.
I can use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
I can recognize the concepts of conditional probability and independence in everyday language and everyday situations.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Common Misconceptions and Challenges
Students may believe that multiplying across branches of a tree diagram has nothing to do with conditional probability.
Students may believe that “independence of events” and “mutually exclusive events” are the same thing.
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 13-5, 13-5 (extend), 13-6
Career Connections
http://ccssmath.org/?s=cp.1
http://ccssmath.org/?s=cp.2
http://ccssmath.org/?s=cp.3
http://ccssmath.org/?s=cp.4
http://ccssmath.org/?s=cp.5
Columbus City Schools
Health diagnosing and treating occupations
Chiropractors
Optometrists
Pharmacists
Physicians and surgeons
Registered nurses
Clinical laboratory technologists and technicians
Medical records and health information technicians
Nuclear medicine technologists
Clear Learning Targets Integrated Math II 2016-2017
38
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
6. Find the conditional probability of A
given B as the fraction of B’s outcomes
that also belong to A, and interpret the
answer in terms of the model.
7. Apply the Addition Rule, P(A or B) =
P(A) + P(B) – P(A and B), and interpret the answer in terms of the
model.
S.CP.6-7
Essential Understanding
- Students will know how to find
conditional probabilities.
- Students will be able to interpret
answers found in models.
Extended Understanding
- Students will become familiar with
the Addition Rule and
Multiplication Rule.
Academic
Vocabulary/Language
- permutation
- combination
- multiplication rule
- addition rule
Tier 2 Vocabulary
- find
- apply
- use
Common Misconceptions and Challenges
Students may believe that the probability of A or B is always the sum of the two events individually.
Students may believe that the probability of A and B is the product of the two events individually, not realizing that one of the probabilities may be
conditional.
Students often switch permutations and combinations. Remind them that the order of their locker combination matters; however, it’s the opposite in
math: when the order matters, it’s a permutation. Perhaps your locker should have a permutation, instead of a combination.
I Can Statements
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I can explain the concepts of conditional probability and independence in everyday language and everyday situations.
I can find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A.
I can use the Additional Rule, P(A or B) = P(A)+P(B)–P(A and B).
I can interpret the answer in terms of the model.
I can use the multiplication rule with correct notation
I can apply the general Multiplication Rule in a uniform probability model P(A and B) = P(A)P(B|A)] = P(B)P(A|B)
I can interpret the answer in terms of the model.
I can identify situations that are permutations and those that are combinations.
I can use permutations and combinations to compute probabilities of compound events and solve problems.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
39
Instructional Strategies
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 13-1, 13-2, 13-5 (extend), 13-6
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S-CP How Do You Get to School?
S-CP The Titanic 1
S-CP The Titanic 2
S-CP The Titanic 3
S-CP.7 Coffee at Mom's Diner
S-CP Rain and Lightning.
The Addition Rule
Columbus City Schools
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S-CP False Positive Test Results
S-CP Alex, Mel, and Chelsea Play a
Game
S-CP, MD Return to Fred's Fun
Factory (with 50 cents)
S-CP Random Walk III
S-CP Random Walk IV
http://ccssmath.org/?s=cp.6
http://ccssmath.org/?s=cp.7
Clear Learning Targets Integrated Math II 2016-2017
Career Connections
Physical scientists
Atmospheric scientists
Chemists and materials scientists
Physicists and astronomers
Social scientists and related occupations
Economists
Market and survey researchers
Social scientists, other
Farming
Forest, conservation, and logging workers
40
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Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
41
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
42
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
1. Verify experimentally the
properties of dilations given by a
center and a scale factor.
a. A dilation takes a line not passing
through the center of the dilation to a
parallel line, and leaves a line passing through the center
unchanged.
b. The dilation of a line segment is longer or shorter in the ratio
given by the scale factor.
2. Given two figures, use the definition of similarity in terms of
similarity transformations to decide if they are similar; explain
using similarity transformations the meaning of similarity for
triangles as the equality of all corresponding pairs of angles and
the proportionality of all corresponding pairs of sides.
3. Use the properties of similarity transformations to establish
the AA criterion for two triangles to be similar.
G.SRT.1-3
Essential Understanding
- Students can use scale factor to
determine how a pre-image and
image relate to one another.
Extended Understanding
- Students can determine if figures
are similar or not by examining
their sides and angle measures.
Academic
Vocabulary/Language
- scale factor
- dilation
- center of dilation
- similarity
- transformation
- image / pre-image
Tier 2 Vocabulary
- verify
- use the properties
I Can Statements
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I can define image, pre-image, scale factor, center, and similar figures as they relate to transformations.
I can identify a dilation stating its scale factor and center.
I can verify experimentally that a dilated image is similar to its pre-image by showing congruent corresponding angles and
proportional sides.
I can identify a dilation stating its scale factor and center.
I can explain that the scale factor represents how many times longer or shorter a dilated line segment is than its pre-image.
I can, by using similarity transformations, explain that triangles are similar if all pairs of corresponding angles are congruent and all
corresponding pairs of sides are proportional.
I can, given two figures, decide if they are similar by using the definition of similarity in terms of similarity transformations.
I can recall the properties of similarity transformations.
I can establish the AA criterion for similarity of triangles by extending the properties of similarity transformations to the general
case of any two similar triangles.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
43
Instructional Strategies
Common Misconceptions and Challenges
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
44
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
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Chapter 9-2, 9-6; 10-8, 10-8 (explore)
G-SRT Are They Similar?
G-SRT Congruent and Similar Triangles
G-SRT Similar Quadrilaterals
G-SRT Similar Triangles
G-SRT Similar triangles
G-SRT Dilating a Line
http://ccssmath.org/?s=srt.2
http://ccssmath.org/?s=srt.3
Dilations and Parallel Lines – G.SRT.1a
http://www.geometrycommoncore.com/content/unit2/gsrt1/gsrt1
.html
Dilations and Scale Factors- G.SRT.1b
http://njcore.org/ccss/ccssmathcontenthsg-srta1
Career Connections
Air transportation occupations
Aircraft pilots and flight engineers
Motor vehicle operators
Busdrivers
Construction
Brickmasons, blockmasons, and stonemasons
Construction and building inspectors
Construction laborers
Glaziers
Painters and paperhangers
Pipelayers, plumbers, pipefitters, and
steamfitters
Printing occupations
Prepress technicians and workers
Columbus City Schools
Similarity Transformations- G.SRT.2
Corresponding Angles of Similar of Triangles- G.SRT.3
Electrical and electronic equipment mechanics,
installers, and repairers
Electrical and electronics installers and
repairers
Electronic home entertainment equipment
installers and repairers
Vehicle and mobile equipment mechanics,
installers, and repairers
Automotive body and related repairers
Automotive service technicians and mechanics
Food processing occupations
Food processing occupations
Metal workers and plastic workers
Computer control programmers and operators
Machinists
Welding, soldering, and brazing workers
Semiconductor processors
Clear Learning Targets Integrated Math II 2016-2017
Financial clerks
Bookkeeping, accounting, and auditing clerks
Information and record clerks
Human resources assistants, except payroll
and timekeeping
Material recording, scheduling, dispatching, and
distributing occupations
Stock clerks and order fillers
Other office and administrative support
occupations
Data entry and information processing
workers
Desktop publishers
Sales
Counter and rental clerks
Insurance sales agents
Real estate brokers and sales agents
45
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
9. Prove theorems about lines and
angles. Theorems include: vertical
angles are congruent; when a
transversal crosses parallel lines,
alternate interior angles are
congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those
equidistant from the segment’s endpoints.
G.CO.9-11
10. Prove theorems about triangles. Theorems include: measures
of interior angles of a triangle sum to 180°; base angles of
isosceles triangles are congruent; the segment joining midpoints
of two sides of a triangle is parallel to the third side and half the
length; the medians of a triangle meet at a point.
Essential Understanding
- Students will be able to prove
theorems involving lines and angles.
- Students will be able to prove that
the sum of the angles in a triangle is
always 180 degrees.
Extended Understanding
- Students will understand the
basics of parallelograms and be able
to prove various theorems related
to them.
Academic
Vocabulary/Language
- vertical angles
- transversal
- alternate interior angles
- perpendicular bisector
- parallelograms
Tier 2 Vocabulary
- prove
11. Prove theorems about parallelograms. Theorems include:
opposite sides are congruent, opposite angles are congruent, the
diagonals of a parallelogram bisect each other, and conversely,
rectangles are parallelograms with congruent diagonals.
I Can Statements
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I can identify and use properties of perpendicular bisector.
I can identify and use properties of equidistant from endpoint.
I can identify and use properties of all angle relationships.
I can prove vertical angles are congruent.
I can identify the hypothesis and conclusion of a triangle sum theorem.
I can identify the hypothesis and conclusion of a base angle of isosceles triangles.
I can identify the hypothesis and conclusion of mid-segment theorem.
I can classify types of quadrilaterals.
I can explain theorems for various parallelograms involving opposite sides and angles and relate to figure.
I can use properties of special quadrilaterals in a proof.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
46
Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
47
Common Misconceptions and Challenges
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 5-2, 5-3, 5-4, 5-6; 6-1, 6-3, 6-4, 6-5, 6-6;
7-1, 7-2, 7-3, 7-4, 7-5, 7-6 (explore); 10-2 (explore)
http://ccssmath.org/?s=co.9
http://ccssmath.org/?s=co.10
http://ccssmath.org/?s=co.11
Career Connections
Construction
Boilermakers
Carpenters
Construction and building inspectors
Construction laborers
Electricians
Glaziers
Pipelayers, plumbers, pipefitters, and
steamfitters
Roofers
Personal and home care aides
Columbus City Schools
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G-C, G-CO Tangent Lines and the Radius of a Circle
G-CO Congruent angles made by parallel lines and a transverse
G-CO Points equidistant from two points in the plane
G-C, G-SRT Seven Circles I
G-CO Classifying Triangles
G-CO Congruent angles in isosceles triangles
G-CO Midpoints of Triangle Sides
G-CO Sum of angles in a triangle
G-SRT Finding the Area of an Equilateral Triangle
G-CO, G-SRT Congruence of parallelograms
G-CO Is this a parallelogram?
G-CO Midpoints of the Sides of a Paralellogram
G-CO Parallelograms and Translations
Protective service occupations
Firefighting occupations
Police and detectives
Food preparation and serving related
occupations
Chefs, cooks, and food preparation workers
Building and grounds cleaning and maintenance
occupations
Grounds maintenance workers
Personal care and service occupations
Animal care and service workers
Barbers, cosmetologists, and other personal
appearance workers
Clear Learning Targets Integrated Math II 2016-2017
Electrical and electronic equipment mechanics,
installers, and repairers
Electrical and electronics installers and
repairers
Electronic home entertainment equipment
installers and repairers
Vehicle and mobile equipment mechanics,
installers, and repairers
Automotive body and related repairers
Automotive service technicians and mechanics
Other installation, maintenance, and repair
occupations
Line installers and repairers
48
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
G.SRT.4-5
4. Prove theorems about
triangles. Theorems include:
a line parallel to one side of
a triangle divides the other
two proportionally, and
conversely; the Pythagorean
Theorem proved using
triangle similarity.
5. Use congruence and
similarity criteria for
triangles to solve problems
and to prove relationships in
geometric figures.
I Can Statements


- Students will continue to prove
key elements of triangles.
Extended Understanding
- Students will be able to solve
problems and prove relationships in
geometric figures by using
congruence and similarity.
Academic
Vocabulary/Language
- theorem
- triange
- parallel
- proportionally
- Pythagorean Theorem
- congruence
- similarity
Tier 2 Vocabulary
- prove
- use
- solve problems
Common Misconceptions and Challenges
I can recall postulates, theorems, and definitions to prove
theorems about triangles. 
I can prove theorems involving similarity about triangles.
(Theorems include: a line parallel to one side of a triangle
divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity).
Columbus City Schools
Essential Understanding
Some students may confuse the alternate interior angle theorem and its
converse as well as the Pythagorean Theorem and its converse.
Clear Learning Targets Integrated Math II 2016-2017
49
Instructional Strategies
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 6-2, 6-3, 6-3 (extend),
6-4, 6-4 (extend),
9-3, 9-4, 9-5, 9-6; 10-1
http://ccssmath.org/?s=srt.4
http://ccssmath.org/?s=srt.5













Columbus City Schools
G-SRT Joining two midpoints of sides of a triangle
G-SRT Pythagorean Theorem
8.G, G-GPE, G-SRT, G-CO Is this a rectangle?
8.G, G-SRT Points from Directions
G-CO, G-SRT Congruence of parallelograms
G-GPE, G-CO, G-SRT Unit Squares and Triangles
G-GPE, G-SRT Finding triangle coordinates
G-GPE, G-SRT Slope Criterion for Perpendicular Lines
G-SRT Bank Shot
G-SRT Extensions, Bisections and Dissections in a
Rectangle
G-SRT Folding a square into thirds
G-SRT, G-MG How far is the horizon?
G-SRT Tangent Line to Two Circles
Clear Learning Targets Integrated Math II 2016-2017
Career Connections
Management occupations
Computer and information systems managers
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Funeral directors
Industrial production managers
Medical and health services managers
Property, real estate, and community
association managers
Purchasing managers, buyers, and purchasing
agents
50
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
Essential Understanding
G.GPE.6
Find the point on a
directed line segment
between two given
points that partitions
the segment in a given
ratio.
- Students will be able to find the
point on a directed line segment.
- Students will able to determine a
given ratio.
Extended Understanding
- When given a ratio and a line segment
with either a positive or negative slope,
students will be able to find the point
that partitions the segment into the
given ratio.
I Can Statements



- directed line segment
- point
- partitions
- given ratio
Tier 2 Vocabulary
- recall
- given
- find
Common Misconceptions and Challenges
I can recall the definition of ratio.
I can recall previous understandings of coordinate
geometry.
I can, given a line segment (including those with positive
and negative slopes) and ratio, find the point on the
segment that partitions the segment into the given ratio.
Columbus City Schools
Academic
Vocabulary/Language
When finding the length of a line segment in order to determine its
midpoint, students may perform simple arithmetic incorrectly. For
example, you are given the coordinates (4, -3) and (12, -8). Students
sometimes want to remove the negative sign on the answer to -3 + -8.
Clear Learning Targets Integrated Math II 2016-2017
51
Instructional Strategies
This standard is not specifically covered in the textbook.
Show the 12 minute video at https://www.educreations.com/lesson/view/g-gpe-6-partitioning-a-segment/9730636/
Textbook and Curriculum Resources
http://ccssmath.org/?page_id=2315
Here’s a quick video explanation: https://www.youtube.com/watch?v=vhO6NA2Vwp4
For an explanation of midpoint specifically:
https://partiallyderivative.wordpress.com/2013/10/20/partitioning-a-line-segment-and-finding-the-midpoint/
Career Connections
Business and financial operations occupations
Accountants and auditors
Budget analysts
Claims adjusters, appraisers, examiners, and
investigators
Financial analysts and personal financial
advisors
Insurance underwriters
Healthcare support occupations
Dental assistants
Medical assistants
Columbus City Schools
Education, training, library, and museum
occupations
Archivists, curators, and museum technicians
Librarians
Teacher assistants
Teachers-adult literacy and remedial and selfenrichment education
Teachers-preschool, kindergarten, elementary,
middle, and secondary
Teachers-special education
Art and design occupations
Artists and related workers
Fashion designers
Clear Learning Targets Integrated Math II 2016-2017
Social scientists and related occupations
Economists
Market and survey researchers
Psychologists
Social scientists, other
Urban and regional planners
Community and social services occupations
Counselors
Legal occupations
Judges, magistrates, and other judicial workers
Sales
Counter and rental clerks
Insurance sales agents
52
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
G.SRT.6-8
6. Understand that by similarity,
side ratios in right triangles are
properties of the angles in the
triangle, leading to definitions of
trigonometric ratios for acute
angles.
7. Explain and use the relationship
between the sine and cosine of
complementary angles.
8. Use trigonometric ratios and the
Pythagorean Theorem to solve
right triangles in applied problems.
Essential Understanding
- Students will understand the
foundation of trigonometric ratios
for acute angles comes from the
similarity between sides of right
triangles.
Extended Understanding
- There is a defined relationship
between sine and cosine of
complementary angles.
Academic
Vocabulary/Language
- acute angle
- similarity
- trigonometric ratio
- sine
- cosine
- complementary angles
Tier 2 Vocabulary
- understand
- explain and use
- solve
- applied problems
I Can Statements



I can name the sides of right triangles as related to an acute angle.
I can recognize that if two right triangles have a pair of acute, congruent angles that the triangles are similar.
I can compare common ratios for similar right triangles and develop a relationship between the ratio and the acute angle leading to
the trigonometry ratios.
 I can use the relationship between the sine and cosine of complementary angles.
 I can identify sine and cosine of acute angles in right triangles.
 I can recognize which methods could be used to solve right triangles in applied problems.
 I can solve for an unknown angle or side of a right triangle using sine, cosine, and tangent.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
53
Instructional Strategies
Common Misconceptions and Challenges
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
54
Textbook and Curriculum Resources
Illustrative Mathematics G.SRT.6:
Not found in textbook.


http://ccssmath.org/?s=srt.6
Illustrative Mathematics G.SRT.7:
http://ccssmath.org/?s=srt.7


http://ccssmath.org/?s=srt.8
G-SRT Defining Trigonometric Ratios
G-SRT Tangent of Acute Angles
G-SRT Sine and Cosine of Complementary Angles
G-SRT Trigonometric Function Values
Illustrative Mathematics G.SRT.8:
Tangent Ratios- G.SRT.6
 G-C, G-SRT Neglecting the Curvature of the Earth
Cos and Sin Trigonometric Ratios – G.SRT.7
 G-C, G-SRT Setting Up Sprinklers
 G-SRT Ask the Pilot
Trigonometric Ratios and the Pythagorean Theorem- G.SRT.8
 G-SRT Constructing Special Angles
Working with Right Triangles – G.SRT.8
 G-SRT, G-MG Coins in a circular pattern
 G-SRT, G-MG Seven Circles III
 G.SRT Shortest line segment from a point P to a line L
Career Connections
Management occupations
Advertising, marketing, promotions, public relations, and sales managers
Computer and information systems managers
Construction managers
Education administrators
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Funeral directors
Industrial production managers
Medical and health services managers
Farming
Agricultural workers
Columbus City Schools
Computer and mathematical occupations
Actuaries
Computer software engineers
Mathematicians
Statisticians
Architects, surveyors, and cartographers
Architects, except landscape and naval
Surveyors, cartographers, photogrammetrists, and surveying technicians
Education, training, library, and museum occupations
Teacher assistants
Teachers-adult literacy and remedial and self-enrichment education
Teachers-preschool, kindergarten, elementary, middle, and secondary
Teachers-special education
Clear Learning Targets Integrated Math II 2016-2017
55
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
F.TF.8
Prove the Pythagorean
identity sin2 θ + cos2 θ =
1 and use it to find sin θ,
cos θ, or tan θ, given sin
θ, cos θ, or tan θ, and the
quadrant of the angle.
Essential Understanding
- Students will learn the
Pythagorean identity and be able to
use it to find the values of the
different trig functions.
Extended Understanding
- Students will begin to see how the
trigonometric functions are actually
related; they are not some random
occurrence.
I Can Statements



Academic
Vocabulary/Language
- Pythagorean Identity
- sin
- cos
- tan
- quadrant
Tier 2 Vocabulary
- prove
- use it to find
Common Misconceptions and Challenges
I can define trigonometric ratios as related to the unit circle.
I can prove the Pythagorean identity sin2 (Ө) + cos2 (Ө) =1.
I can use the Pythagorean identity, sin2 (Ө) + cos2 (Ө) =1, to
find sin (Ө), cos (Ө), or tan (Ө), given sin(Ө), cos(Ө), or tan(Ө),
and the quadrant of the angle.
Students may believe that there is no connection between the Pythagorean
Theorem and the study of trigonometry.
Students may also believe that there is no relationship between the sine and
cosine values for a particular angles. The fact that the sum of the squares of
these values always equals 1 provides a unique way to view trigonometry
through the lens of geometry.
Additionally, students may believe that sind(A + B) = sinA + sinB and need
specific examples to disprove this assumption.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
56
Instructional Strategies
Textbook and Curriculum Resources
Career Connections
Not found in textbook.
http://ccssmath.org/?s=tf.8
Illustrative Mathematics:



F-TF Calculations with sine and cosine
F-TF Finding Trig Values
F-TF Trigonometric Ratios and the Pythagorean Theorem
http://www.orglib.com/hs.ftf.8-standard-formative-assessmentviewTestQuestions_0d1520c2bb_ef8ccc04bf3d41e5b9247158ac6bb4b7_4f902e5b9a244
98c975f1804c55d2f8a.html
Columbus City Schools
Life scientists
Medical scientists
Physical scientists
Chemists and materials scientists
Physicists and astronomers
Social scientists and related occupations
Economists
Management occupations
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Medical and health services managers
Property, real estate, and community association managers
Purchasing managers, buyers, and purchasing agents
Construction
Electricians
Farming
Forest, conservation, and logging workers
Clear Learning Targets Integrated Math II 2016-2017
57
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
1. Prove that all circles are similar.
G.C.1-4
2. Identify and describe
relationships among inscribed
angles, radii, and chords. Include
the relationship between central, inscribed, and
circumscribed angles; inscribed angles on a diameter are
right angles; the radius of a circle is perpendicular to the
tangent where the radius intersects the circle.
3. Construct the inscribed and circumscribed circles of a
triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
Essential Understanding
- Students will realize that all circles
are similar.
- Students can describe
relationships between different key
terms regarding circles.
Extended Understanding
- Students will construct a tangent
line from a point outside a circle to
the circle.
4. (+) Construct a tangent line from a point outside a given
circle to the circle.
Academic
Vocabulary/Language
- similar
- inscribed angle
- radii (radius)
- chords
- central angle
- diameter
- tangent
Tier 2 Vocabulary
- prove
- identify
- construct
I Can Statements











I can compare the ratio of the circumference of a circle to the diameter of the circle.
I can discuss, develop and justify this ratio for several circles.
I can determine this ratio is constant for all circles.
I can identify inscribed angles, radii, chords, central angles, circumscribed angles, diameter, tangent.
I can recognize that inscribed angles on a diameter are right angles.
I can define inscribed and circumscribed circles of a triangle.
I can recall midpoint and bisector definitions.
I can define a point of concurrency.
I can construct the tangent line.
I can construct the perpendicular bisector of the line segment between the center C to the outside point P.
I can construct arcs on circle C from the midpoint Q having length CQ.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
58
Instructional Strategies
Common Misconceptions and Challenges
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
59
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
From Illustrative Mathematics:

G-C Similar circles

G-C, G-CO Tangent Lines and the Radius of a Circle

G-C, G-SRT Neglecting the Curvature of the Earth
http://ccssmath.org/?s=g-c.2

G-C Right triangles inscribed in circles I
http://ccssmath.org/?s=g-c.3

G-C Right triangles inscribed in circles II
http://ccssmath.org/?s=g-c.4

G-C Circumcenter of a triangle

G-C Circumscribed Triangles

G.C Inscribing a circle in a triangle I
Circles: Inscribed Angles, Arcs and Chords- G.C.2

G.C Inscribing a circle in a triangle II
Perimeter of Polygons with Inscribed Circles- G.C.3

G-C Inscribing a triangle in a circle

G-C Locating Warehouse

G-C Opposite Angles in a Cyclic Quadrilateral

G-C Placing a Fire Hydrant

G-C Tangent to a circle from a point
Chapter 11-1, 11-2, 11-3, 11-4, 11-5, 11-5 (extend), 11-7
http://ccssmath.org/?s=g-c.1
Similarity of Circles – G.C.1
Angles in Inscribed Right Triangles and Quadrilaterals- G.C.3
Constructing and Using Tangent Lines- G.C.4
Career Connections
Drafters and engineering technicians
Drafters
Engineering technicians
Physical scientists
Chemists and materials scientists
Environmental scientists and hydrologists
Physicists and astronomers
Computer and mathematical occupations
Actuaries
Computer software engineers
Mathematicians
Statisticians
Columbus City Schools
Education, training, library, and museum
occupations
Teacher assistants
Teachers-adult literacy and remedial and selfenrichment education
Teachers-preschool, kindergarten, elementary,
middle, and secondary
Teachers-special education
Art and design occupations
Fashion designers
Entertainers and performers, sports and related
occupations
Athletes, coaches, umpires, and related workers
Clear Learning Targets Integrated Math II 2016-2017
Management occupations
Computer and information systems managers
Construction managers
Education administrators
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Industrial production managers
Medical and health services managers
Property, real estate, and community
association managers
Purchasing managers, buyers, and purchasing
agents
60
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
G.C.5
Derive using similarity the fact
that the length of the arc
intercepted by an angle is
proportional to the radius, and
define the radian measure of
the angle as the constant of
proportionality; derive the
formula for the area of a sector.
Essential Understanding
- Students are introduced to the
concept of radian measure.
- Students should recall that all
circles are similar.
Extended Understanding
- Students will derive the formula
for the area of a sector.
Academic
Vocabulary/Language
- similiarity
- intercepted arc
- proportional
- radian measure
- constant of proportionality
- area of a sector
Tier 2 Vocabulary
- derive
- define
I Can Statements






Common Misconceptions and Challenges
I can recall how to find the area and circumference of a
circle.
I can explain that 1° = 180 radians.
I can recall from G.C.1, that all circles are similar.
I can determine the constant of proportionality.
I can justify the radii of any two circles (r1 and r2) and the
arc lengths (s1 and s2) determined by congruent central
angles are proportional, such that r1/ s1 = r2/ s2.
I can verify that the constant of a proportion is the same as
the radian measure of the given central angle.
Columbus City Schools
Sectors and segments are often used in everyday conversation. Care
should be taken to distinguish these two geometric concepts.
The forumals for converting radians to degrees and vice versa are easily
confused. Knowing that the degree measure of a given angle is always a
number larger than the radian measure can heko students use the
correct unit.
Clear Learning Targets Integrated Math II 2016-2017
61
Instructional Strategies
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 11-2, 11-9
Career Connections
Other installation, maintenance, and repair occupations
Millwrights
Area of Sectors of A Circle
Construction
Boilermakers
Construction and building inspectors
Construction laborers
Glaziers
Roofers
Arc Length and Radian Measure
Metal workers and plastic workers
Machinists
G-C Mutually Tangent Circles
Protective service occupations
Police and detectives
http://ccssmath.org/?s=g-c.5
Measurements of Arcs
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
62
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
G.GPE.12
1. Derive the equation of a circle of
given center and radius using the
Pythagorean Theorem; complete the
square to find the center and radius
of a circle given by an equation.
2. Derive the equation of a parabola
given a focus and directrix.
Essential Understanding
- Students will derive the equation
of a circle and understand the
horizontal and vertical shifts (h, k).
- Students will understand how to
determine the radius of a circle
when given its equation.
Academic
Vocabulary/Language
- focus
- directrix
- parabola
- complete the square
- circle
Extended Understanding
- Students will be able to derive the
equation of a parabola when given
its focus and directrix.
Tier 2 Vocabulary
- derive
I Can Statements







I can define a circle.
I can use Pythagorean Theorem and distance formula.
I can complete the square of a quadratic equation.
I can derive the equation of a circle using the Pythagorean Theorem – given coordinates of the center and length of the radius.
I can determine the center and radius by completing the square.
I can use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points
in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point
(0,2).
I can recall previous understandings of coordinate geometry (including, but not limited to: distance, midpoint and slope formula, equation of a
line, definitions of parallel and perpendicular lines, etc).
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
63
Instructional Strategies
Common Misconceptions and Challenges
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
64
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
http://ccssmath.org/?s=pe.1
http://ccssmath.org/?s=pe.2
Chapter 11-8, 11-8 (extend)
Download a free graphing circles packet at:
Finding the Equation of Circles – G.PE.1
https://columbus.blackboard.com/bbcswebdav/courses/S-093020141/Lessons%20Math%20II/CircleseverywhereGraphingthemunderstandingshiftspluspuzzl
es.pdf
Finding the Equation of a Parabola- G.PE.2
From Illustrative Mathematics:
https://www.khanacademy.org/commoncore/grade-HSG-G-GPE



G-GPE Explaining the equation for a circle
G-GPE Slopes and Circles
G-GPE Defining Parabolas Geometrically
https://www.opened.com/homework/g-c-1-prove-that-all-circlesare-similar/3689753
Career Connections
Management occupations
Computer and information systems managers
Construction managers
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Funeral directors
Industrial production managers
Medical and health services managers
Property, real estate, and community
association managers
Purchasing managers, buyers, and purchasing
agents
Business and financial operations occupations
Insurance underwriters
Columbus City Schools
http://www.openmiddle.com/tag/g-gpe-1/
https://www.sophia.org/ccss-math-standard-9-12ggpe1pathway?standard=mathematics-geometry
Engineers
Aerospace engineers
Chemical engineers
Civil engineers
Computer hardware engineers
Electrical engineers
Environmental engineers
Industrial engineers
Materials engineers
Mechanical engineers
Nuclear engineers
Petroleum engineers
Drafters and engineering technicians
Drafters
Clear Learning Targets Integrated Math II 2016-2017
Computer and mathematical occupations
Actuaries
Computer software engineers
Mathematicians
Statisticians
Architects, surveyors, and cartographers
Architects, except landscape and naval
Landscape architects
Surveyors, cartographers, photogrammetrists,
and surveying technicians
Life scientists
Biological scientists
Conservation scientists and foresters
Farming
Agricultural workers
65
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
G.GPE.4
Use coordinates to prove
simple geometric theorems
algebraically. For example,
prove or disprove that a
figure defined by four given
points in the coordinate
plane is a rectangle; prove or
disprove that the point (1,
√3) lies on the circle
centered at the origin and
containing the point (0, 2).
I Can Statements


- Students can prove simple
theorems. They will use coordinates
of points to do this in the coordinate
plane.
Academic
Vocabulary/Language
- coordinates
- geometric theorem
- origin
Extended Understanding
- Students will be comfortable with
irrational coordinates.
- Students will be able to
approximate square roots and find
them on a coordinate plane.
Tier 2 Vocabulary
- use
- prove
- disprove
Common Misconceptions and Challenges
I can recall previous understandings of coordinate geometry
(including, but not limited to: distance, midpoint and slope
formula, equations of a line, definitions of parallel and
perpendicular lines, etc.).
I can use coordinate to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure
defined by four given points in the coordinate plane is a
rectangle; prove or disprove that the point (1, sqrt(3)) lies on
the circle centered at the origin and containing the point (0,2).
Columbus City Schools
Essential Understanding
Students often have trouble estimating square roots. Remind them to
think of the nearest perfect square greater and less than the value they
are given.
Clear Learning Targets Integrated Math II 2016-2017
66
Instructional Strategies
Use the links to the tasks from Illustrative Mathematics below to explain this standard.
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
http://commoncoretools.me/forums/topic/g-gpe-4/
http://www.geometrycommoncore.com/content/unit4/ggpe4/ggpe4.html
Chapter 6-6
http://ccssmath.org/?s=pe.4
Teacher Tube: http://www.teachertube.com/video/ggpe4-use-coordinatesadn-algebra-to-prove-geometric-theorems-289842
Illustrative Mathematics:


G-GPE A Midpoint Miracle
G-GPE, G-CO, G-SRT Unit Squares and Triangles
Information and record clerks
Human resources assistants, except payroll and timekeeping
Material recording, scheduling, dispatching, and distributing occupations
Stock clerks and order fillers
Other office and administrative support occupations
Data entry and information processing workers
Career Connections
Computer and mathematical occupations
Actuaries
Computer programmers
Computer software engineers
Mathematicians
Statisticians
Business and financial operations occupations
Budget analysts
Insurance underwriters
Columbus City Schools
https://www.opened.com/homework/g-gpe-4-use-coordinates-to-provesimple-geometric-theorems/3689764
Information and record clerks
Human resources assistants, except payroll and timekeeping
Material recording, scheduling, dispatching, and distributing occupations
Stock clerks and order fillers
Clear Learning Targets Integrated Math II 2016-2017
67
Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics II
Essential Understanding
1. Give an informal argument for
the formulas for the
circumference of a circle, area of
a circle, volume of a cylinder,
pyramid, and cone. Use dissection arguments, Cavalieri’s
principle, and informal limit arguments.
- Students will be able to explain the
formula for the circumference of a
circle.
- Students will be able to explain
Cavalieri’s principle.
3. Use volume formulas for cylinders, pyramids, cones, and
spheres to solve problems.
- Students will be able to use
volume formulas to solve problems.
G.MD.1,3
Extended Understanding
Academic
Vocabulary/Language
- informal argument
- circumference
- volume
- cylinder
- cone
- dissection argument
- Cavalieri’s principle
Tier 2 Vocabulary
- give (informal argument)
- use (dissection argument)
I Can Statements
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I can recognize cross-sections of solids as two-dimensional shapes.
I can recognize formulas for area and circumference of a circle and volume of a cylinder, pyramid, and cone.
I can use the techniques of dissection and limit arguments.
I can recognize Cavalieri’s principle.
I can decompose volume formulas into area formulas using cross-sections.
I can apply dissection and limit arguments (e.g. Archimedes’ inscription and circumscription of polygons about a circle) and as a component of
the informal argument for the formulas for the circumference and area of a circle.
I can apply Cavalieri’s Principle as a component of the informal argument for the formulas for the volume of a cylinder, pyramid, and cone.
I can utilize the appropriate formula for volume depending on the figure.
I can use volume formulas for cylinders, pyramids, cones, and spheres to solve contextual problems.
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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Common Misconceptions and Challenges
From Illustrative Mathematics:
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 5-1; 12-4, 12-5, 12-6
http://ccssmath.org/?page_id=2319
http://ccssmath.org/?s=gmd.3
Cavalieri's Principle - GMD.1
Volume of Cylinders and Trigular Prisms- GMD.3
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G-GMD Area of a circle
G-GMD Circumference of a circle
G-GMD Volume formulas for cylinders and prisms
G-GMD Volume of a Special Pyramid
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G-GMD Centerpiece
G-GMD Doctor's Appointment
G-GMD The Great Egyptian Pyramids
G-GMD Volume Estimation
Volume of Cones and Spheres- GMD.3
Career Connections
Protective service occupations
Police and detectives
Building and grounds cleaning and maintenance
occupations
Grounds maintenance workers
Protective service occupations
Police and detectives
Columbus City Schools
Architects, surveyors, and cartographers
Architects, except landscape and naval
Surveyors, cartographers, photogrammetrists,
and surveying technicians
Other office and administrative support
occupations
Data entry and information processing
workers
Clear Learning Targets Integrated Math II 2016-2017
Computer and mathematical occupations
Actuaries
Computer software engineers
Mathematicians
Statisticians
Engineers
Aerospace engineers
Chemical engineers
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References
I Can Statements: http://www.etown.k12.ky.us/Instructional-Links.asp
Most Instructional Strategies sections come from Ohio Department of Education
Many great worksheets can be found at Math Worksheets Land at http://www.mathworksheetsland.com
Also used in this document is the website OpenEd at https://www.opened.com/
Career ideas came from the website http://www.xpmath.com/
Columbus City Schools
Clear Learning Targets Integrated Math II 2016-2017
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