NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
A Story of Functions
A Close Look at Grade 9 Module 3
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Opening Exercise
Discuss with a neighbor – which of the following phrases contain
incorrect usage of language or symbols?
• The graph of 𝑓(𝑥) has an average rate of change of 3 on the
interval (0, 1).
• The function 𝑔(𝑥) = 𝑥2 + 3 is increasing on the interval
(0, ∞).
• The function 𝑔(𝑥) defined above is the function 𝑓(𝑥) = 𝑥2
shifted to the left 3 units.
• The terms of the arithmetic sequence, 𝑓(𝑛) = 2 + 3𝑛, is a
straight line.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
A Story of Functions
A Close Look at Grade 9 Module 3
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Participant Poll
•
•
•
•
•
Classroom teacher
Math trainer
Principal or school leader
District representative / leader
Other
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
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
the grade.
• Make connections from the content of previous modules and
grade levels to the content of this module.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
What’s In a Module?
Teacher Materials
• Module Overview
• Topic Overviews
• Daily Lessons
• Assessments
Student Materials
• Daily Lessons with Problem Sets
Copy Ready Materials
• Exit Tickets
• Fluency Worksheets / Sprints
• Assessments
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
What’s In a Lesson?
Teacher Materials Lessons
• Student Outcomes and Lesson Notes (in select lessons)
• Classwork
• General directions and guidance, including timing guidance
• Bulleted discussion points with expected student responses
• Student classwork with solutions (boxed)
• Exit Ticket with Solutions
• Problem Set with Solutions
Student Materials
• Classwork
• Problem Set
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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)
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Prior Experience with Sequences
G9-M1 Lesson 26-27: Recursive Challenge Problem
The Double and Add 5 Game
Make 3 more entries into the table.
What is the smallest starting whole number that produces a result of 100 or
greater in 3 rounds or less?
If we call 𝒂𝟏 the result of the first round and 𝒂𝟐 the result of the second
round, what should we call our starting number?
Write a recursive formula for 𝒂𝒏.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Examples of Other Recursive Definitions
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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:
𝟕
𝒇 𝒏 =
𝒏−𝟏
𝟐𝟒
© 2012 Common Core, Inc. All rights reserved. commoncore.org
𝟒
𝟕
–
𝒏−𝟏
𝟒
𝟑
𝟕𝟕
𝟏
𝟐
+
(𝒏 − 𝟏) + 𝒏 − 𝟏 + 𝟐
𝟐𝟒
𝟒
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1 – Example 1
Start with n = 0 or with n = 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 𝑛 = 1.
Let’s agree to do that for now.
• If we start with 𝑛 = 1, which formula should we use
for finding the 𝑛𝑡ℎ term?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Terms, Term Numbers, and “the 𝑛𝑡ℎ 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Introducing the 𝑓(𝑛) 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 𝑛𝑡ℎ term, where I pick what 𝑛 is.
• In this case:
A formula for the 𝑛𝑡ℎ term = 2𝑛−1
• Would it be ok if I wrote 𝑓(𝑛) to stand for “a formula for
the 𝑛𝑡ℎ term”?
𝑓(𝑛) = 2𝑛−1
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Introducing the 𝑓(𝑛) 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 𝑓(𝑛) does not mean 𝑓 times 𝑛.
We agreed, we will use it to mean, “a formula for the nth term.”
• In exercises to come students will use 𝐴(𝑛) 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: Closing & Lesson Summary
Closing:
• Why is it important to have a formula to represent a
sequence?
• Can one sequence have two different formulas?
• What does f(n) represent? How is it read aloud?
Lesson Summary:
• A sequence can be thought of as an ordered list of elements.
To define the pattern of the sequence, an explicit formula is
often given, and unless specified otherwise, the first term is
found by substituting 1 into the formula.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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:
𝐴(𝑛 + 1) = 𝐴(𝑛) + 3 for 𝑛 ≥ 1 and 𝐴 1 = 5
• Why do you suppose he would write that? Can you make
sense of what he is trying to convey?
• What does the 𝐴(𝑛 + 1) part mean?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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: 𝑓 𝑛 + 1 = 5𝑓 𝑛 − 3.
Lesson Summary:
• Provides a description of a what a recursively defined sequence is.
• See page 32 of the teacher materials
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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 𝑑 such that each term in the
sequence is the sum of the previous term and 𝑑.
GEOMETRIC SEQUENCE - described as follows: A sequence is called
geometric if there is a real number 𝑟 such that each term in the
sequence is a product of the previous term and 𝑟.
Exercise: Think of a real-world example of an arithmetic
sequence. Describe it and write its formula.
Exercise: Think of a real-world example of an geometric
sequence. Describe it and write its formula.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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?
• Review the Opening Exercise and Examples 1 and 2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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 𝑛𝑡ℎ 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.
• It is not accurate to say simply that geometric sequences “grow
faster” than arithmetic sequences.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8
Why Stay With Whole Numbers?
• Why are square numbers called square numbers?
If 𝑆(𝑛) denotes the 𝑛𝑡ℎ square number, what is a formula for 𝑆(𝑛)?
• In this context what would be the meaning of
𝑆 0 , 𝑆 𝜋 , 𝑆 −1 ?
• Exercises 5-8:
Suppose we extend our thinking to consider squares of sidelength 𝑥 cm… Create a formula for the area, 𝐴(𝑥) cm2 of a
square of side length 𝑥 cm.
• Review Exercises 9-12, taking time to do #10 and #12
• Do Exercises 13-14
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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}
© 2012 Common Core, Inc. All rights reserved. commoncore.org
{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 𝑓 𝑥 = 𝑥 − 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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
• 7th Grade: Scale drawings
• 8th Grade: Transformations
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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, 𝑋 and 𝑌, in
which each element of 𝑋 is matched (assigned) to one and only one element
of 𝑌. The set 𝑋 is called the domain of the function.
• 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.
• If 𝑓: 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 → {𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠}, then 𝑓(𝑥) stands for a real number, not
the function itself. To refer to a function, we use its name: 𝑓.
• For example, “graph of 𝑓(𝑥)” doesn’t make sense, while the “graph of 𝑓” does.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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 𝑓 𝑥 = 2𝑥 ,” mean, and why is it okay to use it?
• Read the Lesson Notes to Lesson 10 and do Exercise 3.
• The exercise shows that the definition of the exponential function with base 2,
𝑓: 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 → 𝑥 | 𝑥 > 0 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥 ↦ 2𝑥
is equivalent to, “Let 𝑓 𝑥 = 2𝑥 , where 𝑥 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11-12: Graphs
• Let 𝑓 𝑥 = 𝑥 2 for 𝑥 any real number. Discuss the meaning of
𝑥, 𝑓 𝑥 | 𝑥 𝑟𝑒𝑎𝑙
• Now discuss the meaning of
𝑥, 𝑦 | 𝑥 𝑟𝑒𝑎𝑙 and 𝑦 = 𝑓(𝑥)
• 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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 𝒙 integer
For all 𝒙 from 1 to 5
Print 𝟐𝒙
Next 𝒙
2
4
8
16
32
What would be printed out if this code were executed?
Work through Exercise 1
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11
The Graph of a Function
Consider this pseudo code:
Declare 𝒙 real
Let 𝒇(𝒙) = 𝟐𝒙 + 𝟑
Initialize G as {}
For all 𝒙 such that 𝟐 ≤ 𝒙 ≤ 𝟖
Append (𝒙, 𝒇(𝒙)) to G
Next 𝒙
Plot G
© 2012 Common Core, Inc. All rights reserved. commoncore.org
The Graph of f:
Given a function f whose domain
D and range are subsets of the
real numbers, the graph of 𝑓 is
the set of ordered pairs in the
Cartesian plane given by
𝑥, 𝑓(𝑥) |𝑥 ∈ 𝐷
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 𝑓:
two variables:
Given a function 𝑓 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 𝑓 is the set of
(which could be a line)
ordered pairs in the Cartesian
plane given by
𝑥, 𝑓(𝑥) |𝑥 ∈ 𝐷
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: The graph of the equation 𝑦 = 𝑓(𝑥)
Declare 𝒙 and 𝒚 real
Let 𝒇 𝒙 = 𝒙 𝒙 − 𝟐 𝒙 + 𝟐
Initialize G as { }
For all 𝒙 in the real numbers
For all 𝒚 in the real numbers
If 𝒚 = 𝒇(𝒙) then
Append (𝒙, 𝒚) to G
else
Do NOT append (𝒙, 𝒚) to G
End If
Next 𝒚
Next 𝒙
Plot G
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 11-12
The Graph of a Function
The Graph of 𝑓:
The Graph of y = 𝑓(𝑥):
Given a function 𝑓 whose
Given a function 𝑓 whose
domain D and range are
domain D and range are
subsets of the real numbers,
subsets of the real numbers,
the graph of 𝑓 is the set of
the graph of 𝑦 = 𝑓(𝑥) is the
ordered pairs in the
set of ordered pairs (𝑥, 𝑦) in
Cartesian plane given by
the Cartesian plane given by
𝑥, 𝑓(𝑥) |𝑥 ∈ 𝐷
𝑥, 𝑦 |𝑥 ∈ 𝐷 𝑎𝑛𝑑 𝑦 = 𝑓(𝑥)
The Graph of 𝑓 is the same as the graph of the equation 𝑦 = 𝑓(𝑥).
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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 𝑓.
• The graph of 𝑓 is the same set of points as the graph of the
equation 𝑦 = 𝑓(𝑥).
• In either case, the axes are labeled as 𝑥 and 𝑦.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid-Module Assessment
Work with a partner on this assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Scoring the Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Mid-Module Assessment
• As much as possible, assessment items are designed to asses
the standards while emulating PARCC Type 2 and Type 3 tasks.
• Rubrics are designed to inform each district / school / teacher
as they make decisions about the use of assessments in the
assignment of grades.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16
Graphs Can Solve Equations Too
Solve for x in the following equation:
𝑥 + 2 − 3 = 0.5𝑥 + 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!
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NYS COMMON CORE MATHEMATICS CURRICULUM
Explain why…
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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.”
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 17-20: Transformations
Lesson 18:
• Complete Example 1
• Continue working through the Exercises and Examples.
Lesson 19:
• Horizontal scaling with a scale factor 𝑘 of the graph of 𝑦 =
𝑓(𝑥) corresponds to changing the equation from 𝑦 =
𝑓(𝑥) to 𝑦 = 𝑓
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1
𝑥
𝑘
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 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥)
provide solution set to the equation 𝑓(𝑥) = 𝑔(𝑥).
• 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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
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A Story of Functions
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.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 21: Delve more deeply
• Moral from Lesson 14:
• “Linear functions grow additively while exponential functions grow
multiplicatively.”
• Do Exercises 1 and 2 from Lesson 21.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Exercise 2
• Is Terrance correct or incorrect?
• If students claim he is correct, ask them for the formula. Examine the
formula as a class. Does this formula satisfy all of the values in the table?
• What mistake did Terrance make?
• Have students find the average rate of change between each pair of inputoutput values.
• What type of formula could be used to model this data?
• What is the formula?
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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.
−
𝒌𝒕
𝑻 𝒕 = 𝑻𝒂 + 𝑻𝟎 – 𝑻𝒂 ∙ 𝟐. 𝟕𝟏𝟖
𝑻(𝒕) is the temperature of the object after a time of t hours has elapsed,
𝑻𝒂 is the ambient temperature (the temperature of the surroundings), assumed to be
constant, not impacted by the cooling process,
𝑻𝟎 is the initial temperature of the object, and
𝒌 is the decay constant.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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.
𝑇𝑎 = 68˚F (the temperature of the room)
𝑇0 = 98.6˚F (the initial temperature of the body)
𝑘 = 0.1335 (13.35 % per hour - calculated by the investigator from
the
data collected)
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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?
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Math Modeling Exercise
•
•
•
•
Work through the modeling exercise.
If it is 42˚F outside, can a cup of coffee ever cool to below 42˚F?
How did you find the answer for part c of number 1?
Which should you do to drink the coffee sooner: walk outside in the 42˚F
temperature or pour milk into your coffee?
• How does changing the initial coffee temperature affect the graph?
• How does changing the ambient temperature of the coffee affect the
graph?
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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.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Opening Exercise
Discuss with a neighbor – which of the following phrases contain
incorrect usage of language or symbols?
• The graph of 𝑓(𝑥) has an average rate of change of 3 on the
interval (0, 1).
• The function 𝑔(𝑥) = 𝑥2 + 3 is increasing on the interval
(0, ∞).
• The function 𝑔(𝑥) defined above is the function 𝑓(𝑥) = 𝑥2
shifted to the left 3 units.
• The terms of the arithmetic sequence, 𝑓(𝑛) = 2 + 3𝑛, is a
straight line.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment
Work with a partner on this assessment
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Scoring the Assessment
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – End-of-Module Assessment
• End of Module assessment are designed to assess all standards
of the module (at least at the cluster level) with an emphasis
on assessing thoroughly those presented in the second half of
the module.
• Recall, as much as possible, assessment items are designed to
asses the standards while emulating PARCC Type 2 and Type 3
tasks.
• Recall, rubrics are designed to inform each district / school /
teacher as they make decisions about the use of assessments
in the assignment of grades.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Biggest Takeaway
What are your biggest takeaways from the study of
Module 3?
How can you support successful implementation of
these materials at your schools given your role as a
teacher, trainer, school or district leader,
administrator or other representative?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of
Functions
Making Instructional Decisions
Grade 9 – Module 3
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Session Objectives
• To find ways to bridge gaps in student understanding
for Grade 9, Module 3.
• Determine strategies to support teaching decisions
while maintaining the rigor of the instruction
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NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Foundational Standards
Topics to Consider during Transition Years
Making Instructional Decisions
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A Story of Ratios
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
G9-M3 Module Overview  Foundational Standards
Grade 8
8.EE.A.1
8.EE.A.2
8.F.A.1
8.F.A.2
8.F.A.3
8.F.B.4
8.F.B.5
Grade 9
N-Q.A.1
N-Q.A.2
N-Q.A.3
A-SSE.A.1
A-SSE.A.2
A-CED.A.2
A-CED.A.3
A-CED.A.4
A-REI.A.1
A-REI.B.3
A-REI.C.6
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A-REI.D.10
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Foundational Standards
Topics to Consider during Transition Years
Making Instructional Decisions
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topics to Consider during Transition Years
• As you look through the Grade 9 curriculum, you will discern what content
and skills that your students need to be successful with the Grade 9
material.
• Grade 9, Module 3,Topic C: Transformations of Functions
Lesson 17
Lesson Notes
• The topic of Transformation is not commonly covered in current Grade 8
material
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topics to Consider during Transition Years
• Grade 8 Module 2 Overview:
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topics to Consider during Transition Years
• Grade 8 Module 3 Overview:
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
• Strategies to efficiently deliver the key information in Grade 8 regarding
Transformations:
• Present key vocabulary to students (complete list in Module Overviews)
Transformation
Reflection
Image of a Point
Basic Rigid Motions
Translation
Dilation
Distance-preserving
Rotation
Scale factor
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
• Strategies to efficiently deliver the key information in Grade 8 regarding
Transformations:
• Select exercises from the relevant lessons that require students to articulate
vocabulary and highlights understanding needed for G9-M3 Topic C.
1.
To illustrate what makes a translation a rigid motion, have students complete
G8-M2-L2, Example 3. The hands on nature of the exercise will help cement
why the graph of the equation of y = f(x - 4) translates all the points of y = f(x)
to the right 4 units.
2.
To provide context for reflections, have students complete G8-M2-L4,
Exercises 3-5. Students know that a reflection is rigid, that there must be a
line to reflect over, and that any point on the line of reflection will be
unmoved by the reflection.
3.
To highlight how dilations are related to vertical and horizontal scaling, have
students complete G8-M3-L2, Example 1. Students will distinguish how
scaling on the coordinate plane is a limited form of dilation in the plane.
Examples:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Foundational Standards
Topics to Consider during Transition Years
Making Instructional Decisions
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Making Instructional Decisions
Key Ideas
• Honor the objective of the lesson
• Maintain the rigor of the lesson
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Bridging the Gaps
• Student Outcomes and Assessments
• Focus Standards linked to Foundational Standards from
previous material
• Lessons Grouped within Topics
• Terminology, Tools and Representations
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Turn and Talk
-
What are other gaps in understanding you foresee students having?
How can you use the days built into each module for review to address
these issues?
How will planning change for the following school year?
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NYS COMMON CORE MATHEMATICS CURRICULUM
Biggest Takeaway
Turn and Talk:
• What questions were answered for you?
• What new questions have surfaced?
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A Story of Ratios