Quadratics Unit
Part I: Unit Information
Unit Title: Quadratics
Grade Level: 9-10
Time Frame: 3-4 weeks
Prerequisite Knowledge (where this unit sits in a scope and sequence):
Students should be able to:
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Graphing a quadratic
Factor quadratics
Interpret max, min, intercepts
Recognize from graphs, tables, and equations
Solving quadratic equations
Unit Overview :
Students will be able to:
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Interpret the structure of quadratic expressions.
Perform arithmetic operations on quadratics.
Understand the relationship between zeros and factors of quadratics
Create quadratic equations that describe numbers or relationships.
Understand solving quadratic equations as a process of reasoning and explain the reasoning.
Solve & represent graphically quadratic equations and inequalities in one variable.
Understand the concept of a quadratic function and use function notation.
Interpret quadratic functions that arise in applications in terms of the context.
Build quadratic functions that model a relationship between two quantities.
Build new quadratic functions from existing functions.
Essential Question:
How do the characteristics of quadratic functions describe various situations?
CCSS Mathematical Content Standards Addressed:
A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
Quadratics Unit
a. Interpret parts of an expression, such as terms, factors, and coefficients.
Note: At this level include polynomial expressions
A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 –
(y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
A-APR.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.
Note: At this level, add and subtract any polynomial and extend multiplication to as many as three linear
expressions.
A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomial.
Note: At this level, limit to quadratic expressions.
A-CED.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.
Note: At this level extend to quadratic and inverse variation (the simplest rational) functions and use
common logs to solve exponential equations.
A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities,
and interpret solutions as viable or nonviable options in a modeling context. For example, represent
inequalities describing nutritional and cost constraints on combinations of different foods.
Note: Extend to linear-quadratic, and linear–inverse variation (simplest rational) systems of equations.
A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at
the previous step, starting from the assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
Note: At this level, limit to factorable quadratics.
A-REI.4 Solve quadratic equations in one variable.
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.
Note: At this level, limit solving quadratic equations by inspection, taking square roots, quadratic formula,
and factoring when lead coefficient is one. Writing complex solutions is not expected; however recognizing
when the formula generates non-real solutions is expected.
A-REI.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).
Quadratics Unit
Note: At this level, extend to quadratics.
A-REI.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.
Note: At this level, extend to quadratic functions.
F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that
use function notation in terms of a context.
Note: At this level, extend to quadratic, simple power, and inverse variation functions.
F-IF.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.
Note: At this level, limit to simple trigonometric functions (sine, cosine, and tangent in standard position)
with angle measures of 180° (𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠) or less. Periodicity not addressed.
F-IF.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.
Note: At this level, extend to quadratic, right triangle trigonometry, and inverse variation functions.
F-IF.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.
Note: At this level, completing the square is still not expected.
F-IF.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.
Note: At this level, extend to quadratic, simple power, and inverse variation functions.
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.
Note: Continue to allow informal recursive notation through this level.
F-BF.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
Quadratics Unit
their graphs and algebraic expressions for them.
Note: At this level, extend to quadratic functions and, k f(x).
CCSS Mathematical Practice Standards Addressed:
MP1. Make sense of problems and persevere in solving them.
MP2. Reason abstractly and quantitatively.
MP3. Construct viable arguments and critique the reasoning of others.
MP4. Model with mathematics.
MP5. Use appropriate tools strategically.
MP6. Attend to precision.
MP7. Look for and make use of structure.
MP8. Look for and express regularity in repeated reasoning.
*While all eight mathematical practices may not be used in the culminating task, they will all be used by
students at some point within the overall unit of instruction.
Skills/Performances
Concepts
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Solutions, zeros, roots, x-intercepts
Vertex, max/min
Symmetry
End Behavior
Coefficients
Factors
Expression, equation, function
Quadratic formula
Function notation
Domain/range
Explicit vs. recursive
Translation, reflection, dilation
Standard form
Vertex form
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Model behavior of quadratic functions
Solve quadratic equations with various
methods.
Identify and interpret key features of a
quadratic graph.
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Part II: Evidence of Understanding
Quadratics Unit
Culminating Performance Task: Quadratic
CCSS Mathematical Content Standards Addressed:
A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 –
(y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
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Is the student able to translate from vertex form to standard form?
A-APR.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.
Note: At this level, add and subtract any polynomial and extend multiplication to as many as three linear
expressions.
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Is the student able to multiply binomials?
A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomial.
Note: At this level, limit to quadratic expressions.
F-IF.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.
Note: At this level, completing the square is still not expected.
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Is the student able to identify the zeros of a quadratic and relate them to the intercepts and factors?
A-CED.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.
Note: At this level extend to quadratic and inverse variation (the simplest rational) functions and use
common logs to solve exponential equations
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.
Note: Continue to allow informal recursive notation through this level.
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Is the student able to create a quadratic equation from a flow chart (given the specific rules)?
A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at
the previous step, starting from the assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
Quadratics Unit
Note: At this level, limit to factorable quadratics.
A-REI.4 Solve quadratic equations in one variable.
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.
Note: At this level, limit solving quadratic equations by inspection, taking square roots, quadratic formula,
and factoring when lead coefficient is one. Writing complex solutions is not expected; however recognizing
when the formula generates non-real solutions is expected.
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Is the student able to solve a quadratic equation by factoring?
A-REI.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.
Note: At this level, extend to quadratic functions.
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Is the student able to interpret intersections of functions as the solutions to the equation?
F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that
use function notation in terms of a context.
Note: At this level, extend to quadratic, simple power, and inverse variation functions.
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Is the student able to show that when x=5 then y=7?
Is the student able to find y when x =0?
F-IF.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.
Note: At this level, limit to simple trigonometric functions (sine, cosine, and tangent in standard position)
with angle measures of 180° (𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠) or less. Periodicity not addressed.
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Is the student able to find intercepts and relative minimum for a graphed quadratic?
Practice Standards Addressed:
MP1. Make sense of problems and persevere in solving them.
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Is the student able to take the information from each task to answer each subsequent question?
Quadratics Unit
MP2. Reason abstractly and quantitatively.
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Is the student able to recognize the solution of the equation as the intersections of the two graphs?
MP4. Model with mathematics.
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Is the student able to map the relationship using flow charts and formulas?
MP7. Look for and make use of structure.
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Is the student able to use the order of operations to create the equations for the flowchart?
Interim/Formative Performance Tasks
Pre-Assessment:
ACTIVITY: f and g
1. Here are two functions: f (x) = (x – 5)² and g (x) = x – 5
How are the algebraic expressions of f and g different?
Express f and g as flow diagrams.
2. Here are pieces of the graphs of the two functions.
Which one is f and which one is g? How do you know? How are the graphs different?
Quadratics Unit
3. Below are pieces of the tables of the two functions. Complete it. How are the tables different?
Don’t just look at the numbers, but at how the values of f and g change as the values of x change.
Quadratics Unit
Interim Performance Task 1:
Throwing Baseballs (Illustrative Mathematics)
Suppose Brett and Andre each throw a baseball into the air. The height of Brett's baseball is given by h(t) =
−16t² + 79t + 6, where is in feet and is in seconds. The height of Andre's baseball is given by the graph below:
Brett claims that his baseball went higher than Andre's, and Andre says that his baseball went higher.
a. Who is right?
b. How long is each baseball airborne?
c. Construct a graph of the height of Brett's throw as a function of time on the same set of axes as the graph
of Andre's throw (if not done already), and explain how this can confirm your claims to parts (a) and (b).
Quadratics Unit
Interim Performance Task 2:
Medieval Archer (Illustrative Mathematics)
A computer game uses functions to simulate the paths of an archer’s arrows. The -axis represents the level
ground on which the archer stands, and the coordinate pair (2,5) represents the top of a castle wall over
which he is trying to fire an arrow.
In response to user input, the first arrow followed a path defined by the function f(x) = 6 - x², failing to clear
the castle wall.
The next arrow must be launched with the same force and trajectory, so the user must reposition the archer
in order for his next arrow to have any chance of clearing the wall.
a. How much closer to the wall must the archer stand in order for the arrow to clear the wall by the greatest
possible distance?
b. What function must the user enter in order to accomplish this?
c. If the user can only enter functions of the form f(x + k), what are all the values of k that would result in the
arrow clearing the castle wall?
Interim Performance Task 3:
Catapult
Imagine in the sketch below that you launch a stone from a catapult located at the origin
(0; 0) and you want the stone to hit the enemy in their castle on the hill at the point (10; 5).
Assume that the trajectory (path) of the stone is described by a parabola.
Quadratics Unit
1. If the path of the stone is described by the following formulae, will the stone hit or miss the target? If it
will miss, is it too high or too low?
(a) y = -x² + 10x
(b) y = -x² + 11x
Find the value of b for which y = -x² + bx will hit the target.
Describe your strategy.
2. If the path of the stone is described by the following formulae, will the stone hit or miss the target? If it
will miss, is it too high or too low?
(a) y = -0.05x2 + 2x
(b) y = -0.05x2+ 6x
Find the value of b for which y = -0.05x2 + bx , will hit the target.
Describe your strategy.
3. Find the value of a for which y = ax² + 5x will hit the target.
4. Find another formula that will hit the target. Prove that it will hit the target!
Quadratics Unit
Part III: Instructional Pathway
Learning Map
Timeframe
Learning Objectives
Evidence of Performance
Week 1:
Week 2:
Week 3:
Final Week 4: Unit in Review
Culminating performance
Part IV: Scaffolding
Students need many opportunities to state comparisons and define these using
Challenges and Barriers
Content / Concept
Barrier
Scaffold
Quadratics Unit
Universal Design for Learning
Representation
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Action and Expression
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Use pictures in situations
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Have tactile manipulatives
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available to model
comparative situations
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Use color to highlight
comparison words in situations
Have students model
comparisons
Manipulatives
Graphic organizers
Sentence starters and
language objectives
Culture of a “complete
answer”
Engagement
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Have students create their
own
Reinforce classroom norms
Ask students to draw diagrams
Do not always ask for answers
Language
Language Objectives:
Sentence Frames:
Academic Language:
Part V: Resources
Texts:
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http://www.illustrativemathematics.org
http://insidemathematics.org
Malati: Mathematics Learning and Teaching Initiative, Algebra, Module 7, The Quadratic Function,
Grade 9
Quadratics Unit
Culminating Task
Quadratic
________________________________________________
This problem gives you the chance to:
• Work with a quadratic function in various forms
This is a quadratic number machine.
1. a. Show that, if x is 5, y is 7.
_________________________________________________
b. What is y if x is 0?
_______________________________________________________
c. Use algebra to show that, for this machine, y = x² – 2x – 8.
________________________________________________________________
________________________________________________________________
The diagram on the next page shows the graph of the machine’s quadratic function y = x² – 2x – 8
and the graphs of y = 3 and y = x.
2. a. Which point on the diagram shows the minimum value of y? ____________________
b. Which point(s) on the diagram show(s) the solution(s) to the equation 3 = x² – 2x – 8? How do you
know?
________________________________________________________________________________
________________________________________________________________________________
c. Which point(s) on the diagram show(s) the solution(s) to the equation x = x² – 2x – 8? How do you
know?
________________________________________________________________________________
________________________________________________________________________________
Quadratics Unit
3. a. Use the graph to solve the equation x² – 2x – 8 = 0. Mark the solutions on the graph.
x = _________ or x = ___________
b. Use algebra to solve the same equation.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
Quadratics Unit
Quadratics Unit
Rubric for Culminating Task
Criteria of Standard
Evidence of Meeting
Standard
A-SSE.2 Use the structure of an If they correctly
expression to identify ways to
transition from vertex
4
rewrite it. For example, see x
and standard form
4
2
2
2
2
– y as (x ) – (y ) , thus
recognizing it as a difference of
squares that can be factored as
(x2 – y2)(x2 + y2).
A-APR.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.
Note: At this level, add and
subtract any polynomial and
extend multiplication to as
many as three linear
expressions
A-APR.3 Identify zeros of
polynomials when suitable
factorizations are available,
and use the zeros to construct
a rough graph of the function
defined by the polynomial.
Note: At this level, limit to
quadratic expressions.
A-CED.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.
Note: At this level extend to
quadratic and inverse variation
(the simplest rational)
functions and use common
logs to solve exponential
Student correctly
rewrites the term as the
product of 2 binomials
and multiplies them
correctly.
Evidence of
Approaching
Standard
Student correctly
transition from
vertex and standard
form with small
computation errors.
Student correctly
rewrites the term as
the product of 2
binomials but
performs small
computation errors.
Evidence of Below
Standard
Student incorrectly
transitions from vertex
and standard.
Student does not rewrite
the expression.
Quadratics Unit
equations.
A-REI.1 Explain each step in
solving a simple equation as
following from the equality of
numbers asserted at the
previous step, starting from
the assumption that the
original equation has a
solution. Construct a viable
argument to justify a solution
method.
Note: At this level, limit to
factorable quadratics.
A-REI.4 Solve quadratic
equations in one variable.
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.
Note: At this level, limit
solving quadratic equations by
inspection, taking square roots,
quadratic formula, and
factoring when lead coefficient
is one. Writing complex
solutions is not expected;
however recognizing when the
formula generates non-real
solutions is expected.
A-REI.11 Explain why the xcoordinates of the points
where the graphs of the
equations y = f(x) and
y = g(x) intersect are the
Quadratics Unit
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.
Note: At this level, extend to
quadratic functions.
F-IF.2 Use function notation,
evaluate functions for inputs in
their domains, and interpret
statements that use function
notation in terms of a context.
Note: At this level, extend to
quadratic, simple power, and
inverse variation functions.
F-IF.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.
Note: At this level, limit to
simple trigonometric functions
(sine, cosine, and tangent in
standard position) with angle
measures of 180° (𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠)
or less. Periodicity not
addressed.
F-IF.8 Write a function defined
by an expression in different
Quadratics Unit
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.
Note: At this level, completing
the square is still not expected.
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.
Note: Continue to allow
informal recursive notation
through this level.