```NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
A Story of Functions
A Close Look at Grade 9 Module 3
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Session Objectives
• Experience and model the instructional approaches to
teaching the content of Grade 9 Module 3 lessons.
• Articulate how the lessons promote mastery of the focus
standards and how the module addresses the major work of
• Make connections from the content of previous modules and
grade levels to the content of this module.
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Orientation to Materials
Examine and experience excerpts from:
• Topic A: Lessons 1-3, 5
• Topic B: Lessons 8, 9-10, 11-12
• Mid-Module Assessment
• Topic C: Lessons 16, 18-19
• Topic D: Lessons 21, 23
• End of Module Assessment
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Types of Lessons
1. Problem Set
Students and teachers work through examples and complete
exercises to develop or reinforce a concept.
2. Socratic
Teacher leads students in a conversation to develop a specific
concept or proof.
3. Exploration
Independent or small group work on a challenging problem
followed by debrief to clarify, expand or develop math knowledge.
4. Modeling
Students practice all or part of the modeling cycle with real-world
or mathematical problems that are ill-defined.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Mathematical Themes of Module 3
• Functional relationships
• Graphs and transformational geometry
• Linear functions versus exponential functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Flow of Module 3
• Arithmetic and geometric sequences: function
notation (Topic A)
• Precise definition of function and function notation
(Topic B)
• Graphs of functions (Topic B)
• Transformations of functions (Topic C)
• Applications of functions and their graphs (Topic D)
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Examples of Recursive Definitions
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 1: Integer Sequences –
Should You Believe in Patterns?
What is the next number in the sequence?
2, 4, 6, 8, …
Is it 17?
Yes, if the formula for the sequence was:

=
−

–
−

+
( − ) +  −  +

NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1 – Example 1
• Some of you have written 2n and some have written
2n-1. Who is correct?
• Is there a way that both could be correct?
• What is the 1st term of the sequence? The 2nd?
• It feels more natural in this case to start with  = 1.
Let’s agree to do that for now.
• If we start with  = 1, which formula should we use
for finding the ℎ term?
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Terms, Term Numbers, and “the ℎ term”
• Let’s clarify - what do I mean by “the nth term”?
• Let’s create a table of the terms of the sequence.
Term Number Term or Value of the Term
1 = 20
1
2 = 21
2
4 = 22
3
8 = 23
4
…
• What would be an appropriate heading for each of our
= 299
100
columns?
= 2n-1
n
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Introducing the () Notation
• I’d like to have a formula that works like this:
I pick any term number I want and plug it into the
formula, and it will give me the value of that term.
• I’d like a formula for the ℎ term, where I pick what  is.
• In this case:
A formula for the ℎ term = 2−1
• Would it be ok if I wrote () to stand for “a formula for
the ℎ term”?
() = 2−1
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Introducing the () Notation
• AFTER using the notation with accompanying language of
“formula for the nth term”; take the chance to make very
explicit to the students that () does not mean  times .
We agreed, we will use it to mean, “a formula for the nth term.”
• In exercises to come students will use () for example to be a
formula for Akelia’s sequence.
• Beginning with Example 2, students practice writing their own
formulas for situations where the pattern is given verbally.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2:
(Explicit and) Recursive Formulas for Sequences
• Example 1
= 5+3x ?
Term 1: 5
= 5+3x1
Term 2: 8 = 5 + 3
= 5+3x2
Term 3: 11 = 5 + 3 + 3
= 5+3x3
Term 4: 14 = 5 + 3 + 3 + 3
Term 5: 17 = 5 + 3 + 3 + 3 + 3 = 5 + 3 x 4
...
= 5+3x ?
Term n:
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2:
Recursive Formulas for Sequences
• When Johnny saw Akeila’s sequence he wrote the following:
( + 1) = () + 3 for  ≥ 1 and  1 = 5
• Why do you suppose he would write that? Can you make
sense of what he is trying to convey?
• What does the ( + 1) part mean?
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Closing & Lesson Summary
Closing:
• What are two types of formulas that can be used to represent
a sequence?
• What information besides the formula equation do you need
to provide when using these types of formulas?
• List the first 5 terms of the sequence:   + 1 = 5  − 3.
Lesson Summary:
• Provides a description of a what a recursively defined sequence is.
• See page 32 of the teacher materials
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3
Two types of sequences are studied:
ARITHMETIC SEQUENCE - described as follows: A sequence is called
arithmetic if there is a real number  such that each term in the
sequence is the sum of the previous term and .
GEOMETRIC SEQUENCE - described as follows: A sequence is called
geometric if there is a real number  such that each term in the
sequence is a product of the previous term and .
Practice classifying and writing formulas for different sequences
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4
Utilized simple and compound interest to provide a context for
linear and exponential.
Lots of work here with percentages. Key understanding but often
misunderstood.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 5
Which is better?
• Getting paid \$33,333.34 every day for 30 days (for a total of
just over \$ 1 million dollars), OR
• Getting paid \$0.01 today and getting paid double the previous
day’s pay for the 29 days that follow?
• Why does the 2nd option turn out to be better?
• What if the experiment only went on for 15 days?
• Is it fair to say that the values of the geometric sequence
grow faster than the values of the arithmetic sequence?
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6
• More on Exponential Growth
• Looking at the graphs – sometimes parts of a graph can be
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7
• Exponential Decay
• Understanding of when b > 1 and b < 1
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Topic A
• Function notation is introduced simply as a shorthand for ‘the
formula for the ℎ term of a sequence’. This interpretation
will later be extended to serve as shorthand for ‘a formula for
the function value for a given input value’.
• Seeing structure in the formulas for arithmetic and geometric
sequences is a crucial part of meeting both the content
standards and the MP standards.
• How the growth of arithmetic sequences compare to
geometric sequences
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8
Why Stay With Whole Numbers?
• Why are square numbers called square numbers?
If () denotes the ℎ square number, what is a formula for ()?
• In this context what would be the meaning of
0 ,   ,  −1 ?
• Moving domain beyond subset of integers
• Triangular numbers may not be necessary
• Difference in graph when domain is integers and domain is real
numbers
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9-10: Definition of Function
• On a piece of paper, write down a definition for the word
function.
• CCSS 8.F.A.1: A function is a rule that assigns to each input
exactly one output.
• This description doesn’t cover every example of a function. To
see why:
• Write down all functions from the set {1,2,3} to the set {0,1,2}, and
write a concise linear rule if there is one.
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 9-10: Definition of Function
Function
Rule
Function
{1,2,3}{0,1,2}
f(x)=x-1
{1,2,3}{0,1,0}
{1,2,3}{2,1,1}
{1,2,3}{0,2,1}
{1,2,3}{1,0,0}
{1,2,3}{2,2,0}
{1,2,3}{1,0,2}
{1,2,3}{0,0,2}
{1,2,3}{2,0,2}
{1,2,3}{1,2,0}
{1,2,3}{0,2,0}
{1,2,3}{0,2,2}
{1,2,3}{2,0,1}
{1,2,3}{2,0,0}
{1,2,3}{2,2,1}
{1,2,3}{2,1,0}
{1,2,3}{1,1,0}
{1,2,3}{2,1,2}
{1,2,3}{1,2,2}
{1,2,3}{0,0,0}
f(x)=0
{1,2,3}{1,0,1}
{1,2,3}{1,1,1}
f(x)=1
{1,2,3}{0,1,1}
{1,2,3}{2,2,2}
f(x)=2
{1,2,3}{1,1,2}
{1,2,3}{0,0,1}
{1,2,3}{1,2,1}
Rule
Function
Rule
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9-10: Definition of Function
• A rule, like, “Let   =  − 1,” only describes a subset of the types
of all functions.
• How might you describe the distinguishing features of the examples
on the previous slide (regardless of whether there is a rule or not)?
• For every input there is one and only one output.
• They all involve correspondences.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9-10: Definition of Function
• Fortunately, students have been studying correspondences
since Kindergarten:
• Kindergarten: Matching Exercises
• 6th & 7th Grade: Proportional Relationships
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9-10: Definition of Function
• Why do correspondences matter?
• F-BF.A and MP4 (modeling): Students build functions that models relationships
between two types of quantities
• To recognize a functional relationship, students first need to be able to
recognize correspondences.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9-10: Definition of Function
• CCSS F-IF.A.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.
• FUNCTION. A function is a correspondence between two sets,  and , in
which each element of  is matched (assigned) to one and only one element
of . The set  is called the domain of the function.
• If  is a function and  is an element of its domain, then () denotes the
output of  corresponding to the input .
• If :   → { }, then () stands for a real number, not
the function itself. To refer to a function, we use its name: .
• For example, “graph of ()” doesn’t make sense, while the “graph of ” does.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9-10: Definition of Function
• Recall that an equation is a statement of equality between two expressions.
When do you explicitly link functions with equations? For example, what does,
“Let   = 2 ,” mean, and why is it okay to use it?
• Lesson 10 Exercise 3.
The exercise shows that the definition of the exponential function with base 2,
:   →  |  > 0 ℎ ℎ  ↦ 2
is equivalent to, “Let   = 2 , where  can be any real number.”
• We get the best of both worlds: The equal sign “=“ still means equal and we
can use it in a formula to define a function.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11-12: Graphs
• Let   =  2 for  any real number. Discuss the meaning of
,   |
• Now discuss the meaning of
,  |   and  = ()
• How are they the same? Different?
• Both set-builder notations describe every single element in the their respective set.
• The first “constructs each point” while the second “tests every point in the plane.”
• We need to help students develop a “conceptual image” of how these sets
can be generated.
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
The Graph of a Function
To make these lessons work, it is important that teachers spend
time getting comfortable with pseudo code.
Consider this set of pseudo-code:
Declare  integer
For all  from 1 to 5
Print
Next
2
4
8
16
32
What would be printed out if this code were executed?
Work through Exercise 1
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11
The Graph of a Function
Consider this pseudo code:
Declare  real
Let () =  +
Initialize G as {}
For all  such that  ≤  ≤
Append (, ()) to G
Next
Plot G
The Graph of f:
Given a function f whose domain
D and range are subsets of the
real numbers, the graph of  is
the set of ordered pairs in the
Cartesian plane given by
, () | ∈
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
The Graph of a Function vs.
The Graph of an Equation in 2 Variables
The Graph of an equation in
The Graph of :
two variables:
Given a function  whose
The set of all its solutions,
domain D and range are
plotted in the coordinate
subsets of the real numbers,
plane, often forming a curve
the graph of  is the set of
(which could be a line)
ordered pairs in the Cartesian
plane given by
, () | ∈
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The graph of the equation  = ()
Declare  and  real
Let   =   −   +
Initialize G as { }
For all  in the real numbers
For all  in the real numbers
If  = () then
Append (, ) to G
else
Do NOT append (, ) to G
End If
Next
Next
Plot G
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 11-12
The Graph of a Function
The Graph of :
The Graph of y = ():
Given a function  whose
Given a function  whose
domain D and range are
domain D and range are
subsets of the real numbers,
subsets of the real numbers,
the graph of  is the set of
the graph of  = () is the
ordered pairs in the
set of ordered pairs (, ) in
Cartesian plane given by
the Cartesian plane given by
, () | ∈
,  | ∈    = ()
The Graph of  is the same as the graph of the equation  = ().
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 13
• Increasing vs. decreasing is different that positive and negative
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 14
• Review of linear vs. exponential
• Fractional Exponents not needed
• Compare and contrasting growth rates
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Topic B
• When referring to a function, we use the letter of the function
only, e.g. the graph of .
• The graph of  is the same set of points as the graph of the
equation  = ().
• In either case, the axes are labeled as  and .
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 15
• Solving absolute value equations (not a focus, more for setting
up graphing)
• Exploratory challenge – setting up for transformations – good
example of something that is not a function x = abs(y).
• Way to introduce piecewise functions
• Also utilizes floor, ceiling and sawtooth functions for step
functions. Notation not necessary but floor and ceiling are
good for discussions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16
Graphs Can Solve Equations Too
Solve for x in the following equation:
+ 2 − 3 = 0.5 + 1
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.★
Use technology in this lesson!
NYS COMMON CORE MATHEMATICS CURRICULUM
Explain why…
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 17-20: Transformations
• Can you translate a function 3 units up?
• We don’t translate a function up, down, left, right, or stretch
and shrink functions.
 This language applies to graphs of functions.
• We can, however, use the transformation of the graph of a
function to give meaning to the transformation of a function.
• We can describe transformed functions using language that
refers to the values of the function inputs and outputs.
 For example: “For the same inputs, the values of the
transformed function are two times as large as the
values of the original function.”
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 17-20: Transformations
Lesson 19:
• Horizontal scaling with a scale factor  of the graph of  =
() corresponds to changing the equation from  =
() to  =
1

NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Topic C
• Use technology to facilitate understanding that the
intersection of the graphs of  = () and  = ()
provide solution set to the equation () = ().
• Relate transformations of functions to the already familiar
transformations of graphs while making a clear distinction.
• Use language accurate to the transformation being described:
•
•
Shift, stretch, reflect are used when describing transformations of
graphs
Function transformations are described by talking about the values of
the inputs and outputs
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 21: Comparing Linear and
Exponential Functions Again
Student Outcomes from Lesson 14
• Students compare linear and exponential models by focusing on
how the models change over intervals of equal length.
• Students observe from tables that a function that grows
exponentially will eventually exceed a function that grows
linearly.
Student Outcomes from Lesson 21
• Students create models and understand the differences
between linear and exponential models that are represented in
different ways.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 21:
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 22:
More modeling with linear vs. exponential
Utilize calculator – don’t need to master exponential regression
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 23: Newton’s Law of Cooling
• Students are now applying knowledge of exponential functions
and the transformations studied in Topic C to a modeling
problem.
• This formula will be addressed again in subsequent math
courses once students have learned about the number e. For
now, we are using its approximate value of 2.718.
−

=  +  –  ∙ .
() is the temperature of the object after a time of t hours has elapsed,
is the ambient temperature (the temperature of the surroundings), assumed to be
constant, not impacted by the cooling process,
is the initial temperature of the object, and
is the decay constant.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Opening Exercise
A detective is called to the scene of a crime where a dead body has just been
found. He arrives at the scene and measures the temperature of the dead
body at 9:30 p.m. After investigating the scene, he declares that the
person died 10 hours prior at approximately 11:30 a.m. A crime scene
investigator arrives a little later and declares that the detective is wrong.
She says that the person died at approximately 6:00 a.m., 15.5 hours prior
to the measurement of the body temperature. She claims she can prove it
by using Newton’s Law of Cooling.
= 68˚F (the temperature of the room)
0 = 98.6˚F (the initial temperature of the body)
= 0.1335 (13.35 % per hour - calculated by the investigator from
the
data collected)
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Math Modeling Exercise
• Students will explore the effect of each parameter in Newton’s Law of
Cooling by using a demonstration on Wolfram Alpha. This can be done as a
whole class.
• At what type of graph are we looking?
• Why is it still an exponential decay function
when the base is greater than 1?
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
• Step and piecewise functions
• Utilizes ceiling function notation
• Practice
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Topic D
• Summarizes the key ideas and concepts from Topics A – C.
• Brings together the concepts of linear and exponential growth,
transformations of functions, and using key features of a graph to
solve a problem.
• Applies the functions learned (exponential, piecewise, step) to
real world situations.
• Allows students the opportunity to go through the modeling
cycle outlined in the Standards of Mathematical Practices.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Module 3 Lessons
• Lessons emphasize a freedom to ask questions, experiment,
observe, look for structure, reason and communicate.
• Timing of lessons cannot possibly meet the needs of all
student populations. Teachers should preview the lesson and
make conscious choices about how much time to devote to
each portion.
• While many exercises support the mathematical practices in
and of themselves, the discussions and dialog points are often
critical for both their content and for enacting the
mathematical practice standards.
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