1
Rigorous Curriculum Design
Unit Planning Organizer
Subject(s)
Grade/Course
Unit of Study
Unit Type(s)
Pacing
High School Mathematics
Math I Standards
Unit 6: Quadratic Functions
❑Topical ❑X Skills-based ❑ Thematic
12 days for Semester Block & A-Day/B-Day; 25 days for Middle School
Unit Abstract
An important nonlinear function category is quadratics. Understanding characteristics of
quadratic functions and connections between various representations are developed in
this unit. In the table form of a quadratic function, the change in the rate of change
distinguishes it from a linear relationship. In particular, looking at the second rates of
change or differences is where a constant value occurs. The symmetry of the function
values can be found in the table. The graphical form shows common characteristics of
quadratic functions including maximum or minimum values, symmetric shapes
(parabolas), location of the y-intercept, and the ability to determine roots of the function.
This unit explores the polynomial form [f (x) = ax2 + bx + c] and factored form
[f (x) = a (x -p ) (x - q)] of quadratic functions and the impact of changing the parameters
a, b, and c. Connections should be made between each explicit form and its graph and
table. Real-world situations that can be modeled by quadratic functions include projectile
motion, television dish antennas, revenue and profit models in business, and the shape
of suspension bridge cables. Students learn to distinguish relationships between
variables that are functions from those that are not. They use f(x) notation to represent
functions and identify domain and range of functions.
Common Core Essential State Standards
Conceptual Category: Functions
Domain: 1) The Real Number System (N-RN)
2) Seeing Structure in Expressions (A-SSE)
3) Arithmetic with Polynomials & Rational Expressions (A-APR)
4) Creating Equations (A-CED)
5) Interpreting Functions (F-IF)
6) Building Functions (F-BF)
7) Linear, Quadratic & Exponential Models (F-LE)
Clusters: 1) Reason quantitatively and use units to solve problems.
2) Interpret the structure of expressions.
3) Perform arithmetic operations on polynomials.
4) Create equations that describe numbers or relationships.
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5) Understand the concept of a function and use function notation.
Interpret functions that arise in applications in terms of the context.
Analyze functions using different representations.
6) Build a function that models a relationship between two quantities.
7) Construct and compare linear and exponential models and exponential
models and solve problems.
Standards: N-Q.1 USE units as a way to understand problems and to guide the solution
of multi-step problems; CHOOSE and INTERPRET units consistently
in formulas; CHOOSE and INTERPRET the scale and the origin in
graphs and data displays.
N-Q.2 DEFINE appropriate quantities for the purpose of descriptive
modeling.
A-SSE.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. For example, interpret
as
the product of P and a factor not depending on P.
Note: At this level, limit to linear expressions, exponential expressions with
integer exponents and quadratic 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-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 REVEALl the zeros of the
function it defines.
Note: At this level, the limit is quadratic expressions of the form
ax 2  bx  c .
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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, limit to addition and subtraction of quadratics
and multiplication of linear expressions.
A-CED.2 CREATE equations in two or more variables to represent
relationships between quantities; GRAPH equations on coordinate
axes with labels and scales.
Note: At this level, focus on linear, exponential and quadratic.
Limit to situations that involve evaluating exponential functions for
integer inputs.
F-IF.1 UNDERSTAND that a function from one set (called the domain) to
another set (called the range) ASSIGNS to each element of the
domain exactly one element of the range. If f is a function and x is an
element of its domain, then f(x) DENOTES the output of f
corresponding to the input x. The graph of f is the graph of the
equation y = f(x).
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.
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, focus on linear, exponential and quadratic functions;
no end behavior or periodicity.
F-IF.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.
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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, only factoring expressions of the form ax 2  bx  c is
expected. Completing the square is not addressed at this level.
F-IF.9 COMPARE properties of two functions each represented in a
different way (algebraically, graphically, numerically in tables, or by
verbal descriptions).
Note: At this level, focus on linear, exponential and quadratic functions.
F-BF.1 WRITE a function that describes a relationship between two
quantities.
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.
Note: At this level, limit to addition or subtraction of constant to linear,
exponential or quadratic functions or addition of linear functions to
linear or quadratic functions.
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.
Note: At this level, limit to linear, exponential, and quadratic functions;
general polynomial functions are not addressed.
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Standards for Mathematical Practice
1. Make sense of problems and
persevere in solving them.
2. Reason abstractly and
quantitatively.
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in
repeated reasoning.
“UNPACKED STANDARDS”
N-Q.1 Based on the type of quantities represented by variables in a formula,
choose the appropriate units to express the variables and interpret the
meaning of the units in the context of the relationships that the formula
describes.
Ex. When finding the area of a circle using the formula
measure would be appropriate for the radius?
a. square feet
b. inches
c. cubic yards
d. pounds
, which unit of
N-Q.1 When given a graph or data display, read and interpret the scale and
origin. When creating a graph or data display, choose a scale that is
appropriate for viewing the features of a graph or data display. Understand
that using larger values for the tick marks on the scale effectively “zooms out”
from the graph and choosing smaller values “zooms in.” Understand that the
viewing window does not necessarily show the x- or y-axis, but the apparent
axes are parallel to the x- and y-axes. Hence, the intersection of the
apparent axes in the viewing window may not be the origin. Also be aware
that apparent intercepts may not correspond to the actual x- or y-intercepts of
the graph of a function.
N-Q.2 Define the appropriate quantities to describe the characteristics of
interest for a population. For example, if you want to describe how
dangerous the roads are, you may choose to report the number of accidents
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per year on a particular stretch of interstate. Generally speaking, it would not
be appropriate to report the number of exits on that stretch of interstate to
describe the level of danger.
Ex. What quantities could you use to describe the best city in North
Carolina?
Ex. What quantities could you use to describe how good a basketball player
is?
A-SSE.1a. Students manipulate the terms, factors, and coefficients in difficult
expressions to explain the meaning of the individual parts of the expression.
Use them to make sense of the multiple factors and terms of the expression.
For example, consider the expression 10,000(1.055)5. This expression can
be viewed as the product of 10,000 and 1.055 raised to the 5 th power. 10,000
could represent the initial amount of money I have invested in an account.
The exponent tells me that I have invested this amount of money for 5 years.
The base of 1.055 can be rewritten as (1 + 0.055), revealing the growth rate
of 5.5% per year. At this level, limit to linear expressions, exponential
expressions with integer exponents, and quadratic expressions.
Ex. The expression 20(4x) + 500 represents the cost in dollars of the
materials and labor needed to build a square fence with side length x feet
around a playground. Interpret the constants and coefficients of the
expression in context.
A-SSE.1b Students group together parts of an expression to reveal
underlying structure. For example, consider the expression
that
represents income from a concert where p is the price per ticket. The
equivalent factored form,
, shows that the income can be
interpreted as the price times the number of people in attendance based on
the price charged. At this level, limit to linear expressions, exponential
expressions with integer exponents, and quadratic expressions.
Ex. Without expanding, explain how the expression
can be viewed as having the structure of a quadratic expression.
A.SSE.2 Students rewrite algebraic expressions by combining like terms or
factoring to reveal equivalent forms of the same expression.
Ex. Expand the expression
to show that it is a quadratic
expression of the form
.
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A-SSE.3a Students factor quadratic expressions and find the zeros of the
quadratic function they represent. Zeroes are the x-values that yield a yvalue of 0. Students should also explain the meaning of the zeros as they
relate to the problem. For example, if the expression x2 – 4x + 3 represents
the path of a ball that is thrown from one person to another, then the
expression (x – 1)(x – 3) represents its equivalent factored form. The zeros
of the function, (x – 1)(x – 3) = y would be x = 1 and x = 3, because an xvalue of 1 or 3 would cause the value of the function to equal 0. This also
indicates the ball was thrown after 1 second of holding the ball, and caught
by the other person 2 seconds later. At this level, limit to quadratic
expressions of the form ax2 + bx + c.
Ex. The expression
is the income gathered by promoters of a rock
concert based on the ticket price, m. For what value(s) of m would the
promoters break even?
A-APR.1 The Closure Property means that when adding, subtracting or
multiplying polynomials, the sum, difference, or product is also a polynomial.
Polynomials are not closed under division because in some cases the result
is a rational expression rather than a polynomial. At this level, limit to
addition and subtraction of quadratics and multiplication of linear
expressions.
A-APR.1 Add, subtract, and multiply polynomials. At this level, limit to
addition and subtraction of quadratics and multiplication of linear
expressions.
Ex. If the radius of a circle is
kilometers, what would the area of the
circle be?
Ex. Explain why
does not equal
.
A-CED.2 Given a contextual situation, write equations in two variables that
represent the relationship that exists between the quantities. Also graph the
equation with appropriate labels and scales. Make sure students are
exposed to a variety of equations arising from the functions they have
studied. At this level, focus on linear, exponential and quadratic equations.
Limit to situations that involve evaluating exponential functions for integer
inputs.
Ex. In a woman’s professional tennis tournament, the money a player wins
depends on her finishing place in the standings. The first-place finisher wins
half of $1,500,000 in total prize money. The second-place finisher wins half
of what is left; then the third-place finisher wins half of that, and so on.
a. Write a rule to calculate the actual prize money in dollars won by the
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player finishing in nth place, for any positive integer n.
b. Graph the relationship that exists between the first 10 finishers and
the prize money in dollars.
What pattern do you notice in the graph? What type of relationship exists
between the two variables?
F-IF.1 The domain of a function is the set of all x-values, which you control
and therefore is called the independent variable. The range of a function is
the set of all y- values and is dependent on a particular x-value, thus called
the dependent variable. Students should experience a variety of types of
situations modeled by functions. Detailed analysis of any particular class of
functions should not occur at this level. Students will apply these concepts
throughout their future mathematics courses.
Ex. When is an equation a function? Explain the notation that defines a
function.
Ex. Describe the domain and range of a function and compare the concept of
domain and range as it relates to a function.
F-IF.2 Using function notation, evaluate functions and explain values based
on the context in which they are in.
Ex. Evaluate f(2) for the function f (x) 
x 5
2x .
Ex. The function
describes the height h in feet of a

tennis ball x seconds after it is shot straight up into the air from a pitching
machine. Evaluate
and interpret the meaning of the point in the
context of the problem.
F-IF.4 When given a table or graph of a function that models a real-life
situation, explain the meaning of the characteristics of the graph in the
context of the problem. The characteristics described should include rate of
change, intercepts, maximums/minimums, symmetries, and intervals of
increase and/or decrease. At this level, focus on linear, exponential, and
quadratic functions; no end behavior or periodicity.
Ex. Below is a table that represents the relationship between daily profit, P
for an amusement park and the number of paying visitors in thousands, n.
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n
0
1
2
3
4
5
6
P
0
5
8
9
8
5
0
a. What are the x-intercepts and y-intercepts and explain them in the
context of the problem.
b. Identify any maximums or minimums and explain their meaning in the
context of the problem.
c. Determine if the graph is symmetrical and identify which shape this
pattern of change develops.
d. Describe the intervals of increase and decrease and explain them in
the context of the problem.
Ex. A rocket is launched from 180 feet above the ground at time t = 0. The
function that models this situation is given by h(t) = – 16t2 + 96t + 180, where
t is measured in seconds and h is height above the ground measured in feet.
a. What is the practical domain for t in this context? Why?
b. What is the height of the rocket two seconds after it was launched?
c. What is the maximum value of the function and what does it mean in
context?
d. When is the rocket 100 feet above the ground?
e. When is the rocket 250 feet above the ground?
f. Why are there two answers to part e but only one practical answer for part
d?
g. What are the intercepts of this function? What do they mean in the
context of this problem?
h. What are the intervals of increase and decrease on the practical domain?
What do they mean in the context of the problem?
F-IF.7a Students should graph functions given by an equation and show
characteristics such as but not limited to intercepts, maximums, minimums,
and intervals of increase or decrease. Students may use calculators or a
CAS for more difficult cases. Ex. Graph
intercepts and maximum or minimum.
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f ( x ) = -4.9t 2 + 20t
, identifying it’s
10
F-IF.8a Students should take a function and manipulate it in a different form
so that they can show and explain special properties of the function such as;
zeros, extreme values, and symmetries.
Students should factor and complete the square to find special properties
and interpret them in the context of the problem. Keep in mind when
completing the square, the coefficient on the x2 variable must always be one
and what you add in to the problem, you must also subtract from the
problem. In other words, we are adding zero to the problem in order to
manipulate it and get it in the form we want. At this level, only factoring
expressions of the form ax2 + bx + c, is expected. Completing the square is
not addressed.
Ex. Suppose you have a rectangular flower bed whose area is 24ft2. The
shortest side is (x-4)ft and the longest side is (2x)ft. Find the length of the
shortest side.
F-IF.9 Students should compare the properties of two functions represented
by verbal descriptions, tables, graphs, and equations. For example, compare
the growth of two linear functions, two exponential functions, or one of each.
At this level, limit to linear, exponential, and quadratic functions.
Ex. Compare the functions represented below. Which has the lowest
minimum?
a. f(x) = 3x2 +13x +4
b.
F-BF.1b Students should take standard function types such as constant,
linear and exponential functions and add, subtract, multiply and divide them.
Also explain how the function is effected and how it relates to the model. At
this level, limit to addition or subtraction of a constant function to linear,
exponential, or quadratic functions or addition of linear functions to linear or
quadratic functions.
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F-LE.3 When students compare graphs of various functions, such as linear,
exponential, quadratic, and polynomial they should see that any values that
increase exponentially eventually increases or grows at a faster rate than
values that increase linearly, quadratically, or any polynomial function. At
this level, limit to linear, exponential, and quadratic functions; general
polynomial functions are not addressed.
“Unpacked” Concepts
(students need to know)
“Unwrapped” Skills
(students need to be able to
do)
COGNITION
DOK
N-Q.1
Numbers can be interpreted
as quantities with appropriate
units, scales, and levels of
accuracy to effectively model
and make sense of real
world problems.
I can label units through
multiple steps of a problem.
1
I can choose appropriate units
for real world problems
involving formulas.
1
I can use and interpret units
when solving formulas.
2
I can choose an appropriate
scale and origin for graphs and
data displays.
I can interpret the scale and
origin for graphs and data
displays.
1
2
N-Q.2
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I can identify the variables or
quantities of significance from
the data provided.
3
I can identify or choose the
appropriate unit of measure for
each variable or quantity.
3
12
A-SSE.1a, b
Expressions can be written in
multiple ways using the rules
of algebra; each version of
the expression tells
something about the problem
it represents.
I can define expression, term,
factor, and coefficient.
I can interpret the real-world
meaning of the terms, factors,
and coefficients of an
expression in terms of their
units.
I can group the parts of an
expression differently in order
to better interpret their
meaning.
I can define expression, term,
factor, and coefficient.
1
2
3
1
I can interpret the real-world
meaning of the terms, factors,
and coefficients of an
expression in terms of their
units.
2
I can group the parts of an
expression differently in order
to better interpret their
meaning.
3
A-SSE.2
I can look for and identify clues
in the structure of expressions
(e.g., like terms, common
factors, difference of squares,
perfect squares) in order to
rewrite it another way.
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3
I can explain why equivalent
expressions are equivalent.
2
I can apply models for factoring
and multiplying polynomials to
rewrite expressions.
3
13
A-SSE.3a
I can factor a quadratic
expression (ax2+bx+c) to find
the zeros of the function it
represents.
3
A-APR.1
Algebraic expressions, such
as polynomials and rational
expressions, symbolize
numerical relationships and
can be manipulated in much
the same way as numbers.
I can apply the definition of an
integer to explain why adding,
subtracting, or multiplying two
integers always produces an
integer.
I can apply the definition of
polynomial to explain why
adding, subtracting, or
multiplying two polynomials
always produces a polynomial.
2
2
3
I can add and subtract
polynomials.
I can multiply polynomials.
3
A-CED.2
1
Relationships between
numbers can be represented
by equations, inequalities,
and systems.
I can identify the variables and
quantities represented in
real world problems.
I can determine the best model
for the real-world problem (e.g.
linear, quadratic).
2
3
I can write the equation that
best models the problem.
I can set up coordinate axes
using an appropriate scale
and label the axes.
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14
3
I can graph equations on
coordinate axes with
appropriate labels and scales.
F-IF.1
Equations, verbal
descriptions, graphs, and
tables provide insight into the
relationship between
quantities.
I can define relation, domain,
and range.
I can define a function as a
relation in which each input
(domain) has exactly one
output (range).
I can determine if a graph,
table or set of ordered pairs
represents a function.
I can determine if states rules
(both numeric and nonnumeric) produce ordered
pairs that represent a function.
1
2
2
2
F-IF.2
I can convert a table, graph,
set of ordered pairs, or
description into function
notation by identifying the rule
used to turn inputs into outputs
and writing the rule.
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1
I can identify the numbers that
are not in the domain of a
function.
2
I can choose inputs that make
sense based on a problem
situation.
3
I can analyze the input and
output values of a function
based on a problem situation.
3
15
F-IF.4
I can locate the information
that explains what each
quantity represents.
2
I can interpret the meaning of
an ordered pair (e.g., the
ordered pair (9,90) could mean
that a person earned $90 after
working 9 hours).
3
I can determine if negative
inputs make sense in the
problem situation.
3
I can determine if negative
outputs make sense in the
problem situations.
3
I can identify the y-intercept.
1
I can use the definition of
function to explain why there
can only be one y-intercept
2
I can use the problem situation
to explain what the y-intercept
means.
2
I can identify the x-intercept(s).
I can use the definition of
function to explain why some
functions have more than one
x-intercept.
2
2
I can use the problem situation
to explain what an x-intercept
means.
I can use the problem situation
to explain where and why the
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16
function is increasing or
decreasing.
I can use the problem situation
to explain why the function has
symmetry.
3
F-IF.7a
I can explain that the minimum
or maximum of a quadratic is
called the vertex.
3
I can identify whether the
vertex of a quadratic will be a
minimum or a maximum by
looking at the equation.
3
I can find the y-intercept of a
quadratic by substituting 0 for x
and evaluating.
3
I can estimate the vertex and
x-intercepts of a quadratic by
evaluating different values of x.
2
I can graph a quadratic using
evaluated points.
I can use technology to graph
a quadratic and to find precise
values for the x-intercept(s)
and the maximum or minimum.
1
2
F-IF.8a
I can explain that there are
three forms of quadratic
functions: standard form,
vertex form, and factored form.
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1
17
I can explain that standard
form is f (x)  ax 2  bx  c .
I can explain that factored form
is f (x)  a(x  x1)(x  x2 ) ,

where x1 and x2 are x
intercepts of the function.

II can find the x-intercepts of a
quadratic written in factored
form.
1
2
2
I can use the x-intercepts of a
quadratic to find the axis of
symmetry .
2
I can use the axis of symmetry
of a quadratic to find the vertex
of a parabola.
2
I can convert a standard for
quadratic to factored form by
factoring.
2
F-IF.9
I can compare properties of
two functions when
represented in different ways
(algebraically, graphically,
numerically in tables, or by
verbal descriptions)
3
F-LE.3
Lines, exponential functions,
and parabolas each describe
a specific pattern of change.
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I can use graphs or tables to
compare the output values of
linear, quadratic, polynomial,
and exponential functions.
2
I can estimate the intervals for
2
18
which the output of one
function is greater than the
output of another function
when given a graph or table.
I can use technology to find the
point at which the graphs of
two functions intersect.
I can use the points of
intersection to precisely list the
intervals for which the output of
one function is greater than the
output of another function.
I can use graphs or tables to
compare the rate of change of
linear, quadratic, polynomial
and exponential functions.
I can explain why exponential
functions eventually have
greater output values than
linear, quadratic , or polynomial
functions by comparing simple
functions of each type.
Essential Questions
2
2
2
2
Corresponding Big Ideas
In what ways can the choice of units,
quantities, and levels of accuracy impact a
solution?
Interpret numbers as quantities with
appropriate units, scales, and levels of
accuracy to effectively model and make
sense of real world problems.
Why do we structure expressions in
Expressions can be written in multiple
ways using the rules of algebra; each
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different ways?
version of the expression tells something
about the problem it represents.
How can the properties of the real number
system be useful when working with
polynomials and rational expressions?
Algebraic expressions, such as
polynomials and rational expressions,
symbolize numerical relationships and can
be manipulated in much the same way as
numbers.
How can I use algebra to describe the
relationship between sets of numbers?
Relationships between numbers can be
represented by equations, inequalities,
and systems.
How can the relationship between
quantities best be represented?
Equations, verbal descriptions, graphs,
and tables provide insight into the
relationship between quantities.
When does a function best model a
situation?
Lines, exponential functions, and
parabolas each describe a specific pattern
of change.
Vocabulary
Units, scale, origin, expression, term, factor, coefficient, equivalent, polynomial, closure
property, integers, linear, quadratic, coordinate axes, labels, x-intercept, y-intercept,
increase, decrease, maximum, minimum, symmetry, function, domain, range
Language Objectives
Key Vocabulary
N-Q.1
N-Q.2
A-SSE.1 a,b
A-SSE.2
A-SSE.3 a
A-APR.1
A-CED.2
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SWBAT define and give examples of vocabulary (above) specific
to the standards.
20
F-IF.1
F-IF.2
F-IF.4
F-IF.7a
F-IF.8a
F-IF.9
F-BF.1b
F-LE.3
Language Function
N-Q.1,2
SWBAT use given units and the context of a problem as a way to
determine if the solution to a multi-step problem is reasonable
(e.g. length problems dictate different units than problems dealing
with a measure such as slope)
SWBAT interpret units or scales used in formulas or represented
in graphs.
A-SSE.1
SWBAT interpret parts of an expression, such as terms, factors, and
coefficients in terms of the context.
F-IF.1, 2
SWBAT write algebraic rules as functions and interpret the
meaning of expressions involving function notation.
Language Skills
F-IF.1, 2
SWBAT to understand the meaning of domain and range and to
understand the relationship between those sets and input and
output values, respectively.
F-IF.9
SWBAT use a variety of function representations (algebraically,
graphically, numerically in tables, or by verbal descriptions) to
compare and contrast properties of two functions.
F-LE.3
SWBAT 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.
Language Structures
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N-CED.2
SWBAT justify which quantities in a mathematical problem or realworld situation are dependent and independent of one another and
which operations represent those relationships.
F-IF.4
SWBAT sketch graphs showing key features of a function that
models a relationship between two quantities from a given verbal
description of the relationship.
Lesson Tasks
N-Q.3
SWBAT choose and justify a level of accuracy and/or precision
appropriate to limitations on measurement when reporting
quantities.
F-IF.4
SWBAT sketch graphs showing key features of a function that
models a relationship between two quantities from a given verbal
description of the relationship.
Language Learning Strategies
F-BF.1b
SWBAT, given a real-world situation or mathematical problem,
build standard functions to represent relevant relationships/
quantities; determine which arithmetic operation should be
performed to build the appropriate combined function; and
relate the combined function to the context of the problem.
Information and Technology Standards
HS.TT.1.1 Use appropriate technology tools and other resources to access information.
HS.TT.1.2 Use appropriate technology tools and other resources to organize
information.
Instructional Resources and Materials
Physical
Core Plus
Contemporary
Mathematics in
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Technology-Based
CPMP-Tools Software
http://www.wmich.edu/cpmp/CPMP-Tools/
22
Context (2nd
Edition) – Unit 7
Course 1, Unit 7,
NCTM Illuminations(
http://illuminations.nctm.org/)
Egg Launch Contest: Students will represent quadratic
functions as a table, with a graph, and with an equation. They
will compare data and move between representations.
http://illuminations.nctm.org/LessonDetail.aspx?id=L738
Hanging Chains: Both ends of a small chain will be attached
to a board with a grid on it to (roughly) form a parabola.
Students will choose three points along the curve and use them
to identify an equation. Repeating the process, students will
discover how the equation changes when the chain is shifted.
http://illuminations.nctm.org/LessonDetail.aspx?id=L628
Texas Instruments (
http://education.ti.com/calculators/timath/)
Applications of Parabolas(TI-84+): In this activity, students
will look for both number patterns and visual shapes that go
along with quadratic relationships. Two applications are
introduced after some basic patterns in the first two problems.
http://education.ti.com/xchange/US/Math/AlgebraI/12236/Alg1
_AppsParabolas_TI84.pdf
Exploring the Vertex Form of the Quadratic Function(TI84+): Students explore the vertex form of the parabola and
discover how the vertex, direction, and width of the parabola
can be determined by studying the parameters. They predict
the location of the vertex of a parabola expressed in vertex
form.
http://education.ti.com/calculators/downloads/US/Activities/De
tail?id=5560
Pass the Basketball – Linear and Quadratic Activities: Many
teachers have probably seen a linear version of this activity.
Students determine the time it takes for different numbers of
students to pass a ball from one student to the next. If the students
pass the ball at a relatively constant rate, the data collected and
graphed (time versus number of students) can be modeled by a
linear function. The activity can be modified to collect data that is
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logically modeled by a quadratic function. Questions are provided
for each version of the activity. A basketball and a stopwatch are
needed for both activities.
http://educator.schools.officelive.com/Documents/Pass%20The%20
Ball%20Activity-%20linear%20and%20quadratic.pdf
Kitchen Parabolas: Students use kitchen bowls to determine the
equation of a quadratic function that closely matches a set of points.
http://www.thefutureschannel.com/pdf/algebra/kitchen_paraboloids.
pdf
Quadratic Functions: Quadratic Functions are explored through
two lessons in this unit. The first lesson requires students to explore
quadratic functions by examining the family of functions described
by y = a (x - h)2 + k. In the second, students explore quadratic
functions by using a motion detector known as a Calculator Based
Ranger (CBR) to examine the heights of the different bounces of a
ball. Students will represent each bounce with a quadratic function
of the form y = a (x - h)2 + k.
http://www.learner.org/workshops/algebra/workshop4/lessonplan.ht
ml
Toothpicks and Transformations: The lesson begins with a review
of transformations of quadratic functions, vertical and horizontal
shifts, and stretches and shrinks. First, students match the symbolic
form of the function to the appropriate graph, then given the graphs,
students analyze the various transformations and determine the
equation for the functions. This review is followed by an activity
where students explore a mathematical pattern that emerges as
they build a geometric design with toothpicks. Students examine the
recursive nature of the relationship. An explicit model for the relation
is developed, and a third model is developed by examining the
scatterplot and determining the equation from the transformations.
Finally, the class uses the graphing calculators to develop another
model and to verify that all of the models;factored form, vertex form,
and general form;are equivalent.
http://www.pbs.org/teachers/connect/resources/4453/preview/
GeoGebra (
http://www.geogebra.org/)
Quadratic Fun 1: This geogebra applet allows the user to
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explore the relationship between the value of a in
f(x)=a(x−h)2+k on the shape vertex of a parabola. Also the
relationship between and the axis of symmetry and the vertex
of the parabola is explored.
http://www.mathcasts.org/gg/student/quadratics/quad_fun1/ind
ex.html
Quadratic Fun 2: This applet explores how knowing the vertex
and an additional point on the parabola can help generate the
entire parabola. In addition, using the previous information, the
student is asked to calculate a in the equation f(x)=a(x−h)2+k.
http://www.mathcasts.org/gg/student/quadratics/quad_fun2/ind
ex.html
Vertical Motion Interactvity: The motion of a mortar shell shot
directly up from the top of a cliff is used to simulate free fall
motion. Included are some very good questions or students to
consider about the meaning of points along the path of the
object. The good questions and worksheet provide scenarios to
consider and pose questions for students to explore.
http://geogebrawiki.wikispaces.com/Vertical+Motion
Professional Resources
NCTM (www.nctm.org)
Focus in High School Mathematics: Reasoning and Sense
Making: This publication elevates reasoning and sense making
to a primary focus of secondary mathematics teaching. It shifts
the teachers’ role from acting as the main source of information
to fostering students’ reasoning to make sense of the
mathematics.
http://www.nctm.org/catalog/product.aspx?ID=13494
Focus in High School Mathematics: Reasoning and Sense
Making in Algebra: Reasoning about and making sense of
algebra are essential to students' future success. This book
examines the five key elements (meaningful use of symbols,
mindful manipulation, reasoned solving, connecting algebra
with geometry, and linking expressions and functions: identified
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in Focus in High School mathematics: Reasoning and Sense
Making in more detail and elaborates on the associated
reasoning habits.
http://www.nctm.org/catalog/product.aspx?ID=13524
Articles from National Council of Teachers of Mathematics
(www.nctm.org)
Articles available as free downloads to NCTM members, or for a fee
to non-members
Eraslan, A. and Aspinwall, L. (2007). Connecting Research to
Teaching: Quadratic Functions:Students’ Graphic and Analytic
Representations.
Mathematics Teacher, 101(3), 223. Retrieved February 18,
2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&
uri=MT2007-10-233a
Math Assessment Project
http://map.mathshell.org/materials/lessons.php
Assessment Tasks
1) Patchwork: Build a Function
http://map.mathshell.org/materials/download.php?fileid=754
2) Functions
http://map.mathshell.org/materials/download.php?fileid=762
Activities for Students: Using Graphs to Introduce Functions:
Hands-on, open-ended activities that encourage problem solving,
reasoning, communication, and mathematical connections.
http://www.nctm.org/workarea/downloadasset.aspx?id=18533
Domain and Range - Graphically!: This demo is designed to help
students use graphical representations of functions to determine the
domain and range.
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http://www.mathdemos.org/mathdemos/domainrange/domainrange.
html#Level
Professional Resources
Articles from National Council of Teachers of Mathematics
(www.nctm.org)
Articles available as free downloads to NCTM members, or for a fee
to non-members.
Hartter, B. (2009). A Function or Not a Function? That is the Question.
Mathematics Teacher, 103(3), 200. Retrieved on March 7, 2012 from
http://www.nctm.org/publications/article.aspx?id=24752
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