Download Common Core Algebra 2A Critical Area 3: Quadratic Functions

Common Core Algebra 2A
Critical Area 3: Quadratic Functions
Standards for Mathematical Practice
Pacing: Weeks 18 – 36
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Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics
Use appropriate tools strategically.
Attend to precision.
Look for and make use structure.
Look for express regularity in repeated reasoning.
Quadratic Functions
Building Functions
F-BF
A. Build a function that models a relationship between two quantities.
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. For example, build a function that models the temperature of a cooling body by
adding a constant function to a decaying exponential, and relate these functions to the model.
2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two
forms★
B. Build new functions from existing functions.
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.
Interpreting Functions
F-IF
B. Interpret functions that arise in applications in terms of the context.
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.★
Instructional Guide for Common Core Algebra 2A
Property of MPS
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Common Core Algebra 2A
Critical Area 3: Quadratic Functions
Interpreting Functions
F-IF
C. Analyze functions using different representations.
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.
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.
9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Seeing Structure in Expressions
A-SSE
B. Write expressions in equivalent forms to solve problems.
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.
Reasoning with Equations and Inequalities
A-REI
B. Solve equations and inequalities in one variable.
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 x2 = 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.
D. Represent and solve equations and inequalities graphically.
10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could
be a line).
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find
the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x)
and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★
Expressing Geometric Properties with Equations
G-GPE
A. Translate between the geometric description and the equation for a conic section
2. Derive the equation of a parabola given a focus and directrix.
Instructional Guide for Common Core Algebra 2A
Property of MPS
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Common Core Algebra 2A
Critical Area 3: Quadratic Functions
The Complex Number System
N-CN
A. Perform arithmetic operations with complex numbers.
1. Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
C. Use complex numbers in polynomial identities and equations.
7. Solve quadratic equations with real coefficients that have complex solutions.
Standards for Mathematical Practice
SMP
2. Reason abstractly and quantitatively
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on
problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed
during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing
and flexibly using different properties of operations and objects.
7. Look for and make use of structure
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the
same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the
14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for
solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single
objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize
that its value cannot be more than 5 for any real numbers x and y.
Disciplinary Literacy
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Disciplinary Literacy Framework
Disciplinary Literacy Framework Math Companion Document
Disciplinary Literacy in Mathematics
Resources on Using Literacy Skills in Mathematics
Instructional Guide for Common Core Algebra 2A
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Common Core Algebra 2A
Critical Area 3: Quadratic Functions
Key Student Understandings
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Assessment
Students will expand the number system to include the complex
numbers allowing for the solution of any quadratic equation.
Students will understand algebraic expressions can be written in
infinitely many equivalent forms, showing key features of the
relationship.
Students will understand the relationship between zeros and the
vertex.
Formative Assessment Strategies
Evidence for Standards-based Grading
Rubric
Common Misconceptions/Challenges
Algebraic expressions can be written in infinitely many equivalent forms; different forms may be useful for answering different questions about the context
of the problem
 Students have a difficult time predicting the characteristics of the graph by looking at the equation
 Students have a difficult time translating between multiple representations of an algebraic equation
 Students struggle to write quadratic expressions as a product of its factors due to weakness in number sense. In particular, students have difficulty with
expressions where a ≠ 1 and/or where c < 0.
 Students misapply formal algebraic rules without understanding the meaning of the operation that it carries out. For example, when a binomial is squared,
students may omit the middle term.
 Students who are not strong at symbolic manipulation might benefit from a geometric representation (the area model) that can be used for finding
equivalent forms
 Watch for the ability to solve an equation with graphing and with a calculator table
 Have students visualizing the graph before graphing the equation
 When students are solving algebraic equations, continue to ask: “what does this value mean?”, “does this make sense?”, “why are you doing this next
step?”
 Plan for the Math Practice Standard – Reason Abstractly and Quantitatively for students to have a better conceptual understanding of applying algebraic
rules to expressions
Expanding the number system to include complex numbers allows for the solution of quadratic (and higher degree) equations
There are deep connections between geometry and complex numbers
 Students have a hard time visualizing the imaginary numbers within the number system (where is i?)
Instructional Guide for Common Core Algebra 2A
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Common Core Algebra 2A
Critical Area 3: Quadratic Functions
Instructional Practices
Suggested
Timeline
Suggested Learning Experiences
Textbook Resources
Introduction to
Quadratics
Develop and use quadratic functions to
model real-world situations.
The High School standards are not fully
developed in the Board Adopted programs.
Recommended Lessons/Tasks
**Common SBG Evidence Items
Illustrative Mathematics:
Springboard Dive (TE)
(Weeks 18 – 22) Solve quadratic equations using graphs, *Teachers will need to incorporate some of the
Recommended Lessons/Tasks to fully meet the
tables, and symbolic methods.
standards.
You Tube: Rocket Science Describe real-world meaning related to
https://youtu.be/I91f9nDSnWI
quadratic models.
Discovering Algebra:
Lesson 9.1 - Solving Quadratic Equations
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EngageNY: Algebra I Module 4
One Step Investigation, in blue matter
from the DA teacher edition
Example B
Exercises #4, 5, 6, 8
Lesson 9.2 - Finding the Roots and the Vertex
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“Making the Most of It”, Investigation or
One Step, in the blue matter from the DA
teacher edition
Example B, read the blue matter from the
DA teacher edition for questions to
promote student thinking
Exercises #7, 8, 9, 11
Lesson 9.3 - From Vertex to General Form
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Instructional Guide for Common Core Algebra 2A
“Sneaky Squares” Investigation, this
investigation is highly recommended for
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Common Core Algebra 2A
Critical Area 3: Quadratic Functions
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students to visualize and make
connections to area models from prior
years
Exercises #9, 10, 11
Lesson 9.4 - Factored Form
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Key Features
(Weeks 22 – 32)
Utilize appropriate forms for the
situation.
Manipulate expressions into desired
forms.
Derive the quadratic formula.
Explore real-world situations modeled
by quadratics.
“Getting to the Root of the Matter”
Investigation or One Step, in the blue
matter from the DA teacher edition
Example A, connect to family of functions
Example B, foreshadows rational
functions
Exercises #9, 12, 13
Discovering Advanced Algebra:
Illustrative Mathematics:
Section 7.2 – Equivalent Quadratic Forms
Which Function? (TE)
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“Rolling Along” Investigation
Example C
Exercises #6, 8, 12
Section 7.3 – Completing the Square
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“Completing the Square” Investigation
Example B, C
Exercises #7 – 11
Graphs of Quadratic Functions (TE)
Springboard Dive (TE)
Building an Explicit Quadratic Function by
Composition (TE)
Medieval Archer
(TE)
Zero Product Property 4
(TE)
Section 7.4 – The Quadratic Formula
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Instructional Guide for Common Core Algebra 2A
Example A, continue after Example A into EngageNY: Algebra I Module 4
the derivation of the Quadratic Formula.
“How High Can You Go?” Investigation
Exercises #6, 8 - 11,
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Common Core Algebra 2A
Critical Area 3: Quadratic Functions
Complex
Numbers
Recognize when the quadratic formula
gives complex solutions.
(Weeks 32 – 36) Perform arithmetic operations with
complex numbers.
Explore situations that result in
complex solutions.
Discovering Advanced Algebra:
EngageNY:
Section 7.5 – Complex Numbers
Algebra 2 Module 1: Lessons 37 – 40
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Example, pg. 410
“Complex Arithmetic” Investigation
(Parts 1 – 3)
Exercises #11, 14
Differentiation
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Literacy Connections
Consider number of roots, vertex, and number of transformations
when choosing examples for students.
Consider multiple representations used to solve quadratic
equations. For example, students who understand graphing as a
method to solve equations will have alternate access to solving
quadratics with any real root(s).
Consider linking factoring to the area model so that students can be
successful is writing quadratic expressions as a product of its
factors. This model may provide students with representations
where they can see the relationships between the terms of a
quadratic expression.
Academic Vocabulary Terms
Vocabulary Strategies
Literacy Strategies
Additional Resources
Discovering Mathematics Online Textbook
Desmos Online Graphing Calculator
Instructional Guide for Common Core Algebra 2A
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