Common Core Algebra I Unit 3 Starting Points
Unit 3: Quadratic Functions and Modeling
Part 1: Graphical Analysis and Modeling of Quadratic Functions
Essential Questions:
o When linear and quadratic expressions are added, subtracted, or multiplied, what is the
resulting function?
o What are the differences between the numeric patterns and graphical representations of
linear, exponential and quadratic functions? How might these differences appear in a
real-world problem?
o How can quadratic models be represented recursively? How might they be represented
explicitly?
o How can quadratic models be used to represent real-world problems?
o What do the key features of the graph of a quadratic function reveal about the problem?
o How do parameters introduced in the context of the problem affect the symbolic, numeric
and graphical representations of a quadratic function?
o How do changes to the symbolic representation of a quadratic function affect the
graphical representation of the function?
Curriculum Standards:
Perform arithmetic operations on polynomials.
A.APR.A.1 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. Focus on polynomial expressions that are linear or quadratic in a positive
integer power of x.
Build a function that models a relationship between two quantities.
F.BF.A.1 Write a quadratic 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.
Analyze functions using different representations.
F.IF.C.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.
Interpret functions that arise in applications in terms of a context.
F.IF.B.4 For a quadratic 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, and end behavior.
F.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative
relationships it describes.
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Build new functions from existing functions.
F.BF.B.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.
Approximate Length of Unit Part 1 (Graphical Analysis and Modeling of Quadratic
Functions): 12 days
Standard(s)
Days
Notes
Big Ideas:
A.APR.A.1
3-4
Add, subtract, and multiply polynomials. (focus on linear and
quadratic expressions with positive integer powers)
Resources:
 Teacher Resource:
Cooney, T. J., Beckmann, S., & Lloyd, G. M. (2010).
Developing Essential Understanding of Functions for
Teaching Mathematics in Grades 9-12. P. S. Wilson (Ed.).
Reston, VA: The National Council of Teachers of
Mathematics, Inc. "Function Sorting Activity" p. 80-81.
"Comparison of Linear and Quadratic Patterns Through
Tables" p. 100.
Assessment Limit/Clarification:
F.BF.A.1a
2-3
Big Ideas:
Express quadratic patterns recursively and explicitly.
Distinguish between linear, exponential, and quadratic functions
given verbal, numeric, or graphical representations.
Resources:
 Lesson: The Bears Problem
Assessment Limit/Clarification:
Tasks have a real-world context. Tasks are limited to linear,
quadratic, and exponential functions with domains in the integers.
F.IF.C.7
F.IF.B.4
F.IF.B.5
2-3
Big Ideas:
Graph linear and quadratic functions by hand, and in some cases
using technology.
Identify and interpret key features (intercepts, intervals of
increasing/decreasing, intervals of positive/negative, relative
maxima/minima, symmetries, and end behavior) in order to sketch
graphs and in the context of a problem.
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Identify domain restrictions within the context of a problem.
Resources:
 Lesson: Angry Birds
 Lesson: Real-world Quadratics Graphing
 Task: Flying T-Shirts
 Task: Planning for Prom
 What is your weather?
Assessment Limit/Clarification:
Tasks have a real-world context. Tasks are limited to linear
functions, quadratic functions square root functions, cube root
functions, piecewise-defined functions (including step and absolute
value functions), and exponential functions with domains in the
integers.
F.BF.B.3
1-2
Big Ideas:
Explore transformations of quadratic functions.
(experimentation should lead to the development of rules).
Resources:
 Lesson: Transformations of Functions
 Web Lesson: Up, Down, Right, Left-Function Families
 Web Resource: Function Transformation applet
Assessment Limit/Clarification:
Identifying 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) is limited to linear and quadratic functions.
Experimenting with cases and illustrating an explanation of the
effects on the graph using technology is limited to linear functions,
quadratic functions, square root functions, cube root functions,
piecewise-defined functions (including step functions and absolute
value functions), and exponential functions with domains in the
integers. Tasks do not involve recognizing even and odd functions.
UTA Dana Center Prototype Assessment Item:
Quadratic Transformation
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Part 2: Algebraic Analysis of Quadratic Functions
Essential Questions:
o What are the parts of the expression and what information do they reveal about the
problem?
o What do different forms of an expression reveal about the graph of a quadratic function?
o Can the methods of completing the square or factoring be used to gain more information
about the problem or the graph of the function?
o How can an expression be rewritten to reveal the necessary information?
o What method is most efficient for solving a quadratic equation given in specific forms?
o When does a quadratic equation have a complex solution?
Curriculum Standard(s):
Interpret the structure of expressions.
A.SSE.A.1 Interpret quadratic 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.
A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
Write expressions in equivalent forms to solve problems.
A.SSE.B.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.
A.REI.B.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.
Analyze functions using different representations.
F.IF.C.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
zeroes, extreme values, and symmetry of the graph, and interpret these in terms of a
context.
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Approximate Length of Part 2(Algebraic Analysis of Quadratic Functions): 30 days
Standard(s)
A.SSE.A.1.a
A.SSE.A.1.b
Days
2
Notes
Big Ideas:
Interpret parts of a quadratic expression in terms of a problem
situation (terms, coefficients, and factors).
Understand what different forms of quadratic expressions might
reveal about a problem or graph.
Connect to key features of graphical representations.
Resources:
 Lesson: Exploring 3 forms of quadratic functions
Assessment Limit/Clarifications:
PARCC Prototype Assessment Item:
Seeing Structure in an Equation
A.SSE.A.2
A.SSE.B.3.b
F.IF.C.8.a
10-13
Big Ideas:
Factor quadratic functions to find zeros.
Complete the square to determine maxima/minima of function.
Re-write expressions through factoring and completing the square.
Resources:
 Lesson: Rewriting quadratic equations
 Lesson: Factoring trinomials using algebra tiles
 Web Lesson: Using algebra tiles to factor and find intercepts
 Web Resource: Factoring with Algebra Tiles
 Web Resource: Complete the Square interactive tool
 Tutorial Video: Completing the Square
Assessment Limit/Clarification:
Tasks are limited to numerical expressions and polynomial
expressions in one variable. Examples may include identifying
quadratic difference of squares and rewriting quadratics in factored
form.
PARCC Prototype Assessment Item:
Seeing Structure in an Equation
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
A.SSE.B.3.a
A.SSE.B.3.b
A.REI.B.4.a
A.REI.B.4.b
15-18
Big Ideas:
Solve quadratic equations by taking the square root, completing the
square, factoring, or quadratic formula.
Determine which method is appropriate based on the form given.
Recognize when a quadratic function would have a complex
solution. (not required to solve complex solutions until Algebra II)
- Determine the number of solutions a quadratic function
will have based on the discriminant.
Resources:
 Task: Building a Playground
 Lesson: Completing the Square
 Lesson: Solving Quadratic Equations
 Wiki Document: TI-Nspire Resources
 Pencast: Deriving the Quadratic Formula
 Web Resource: Online quiz/practice for solving quadratic
equations and writing in different forms
 Web Resource: Completing the Square illuminations
activity
Assessment Limit/Clarification:
Tasks do not require students to write solutions for quadratic
equations that have roots with nonzero imaginary parts. However,
tasks can require the student to recognize cases in which a quadratic
equation has no real solutions.
PARCC Prototype Assessment Item:
Seeing Structure in Quadratic Equation
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
This document represents one sample Starting Points for the Unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.