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A Story of Functions

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A Story of Functions

A Close Look at Grade 9 Module 3

Opening Exercise

- Discuss with a neighbor – which of the following phrases contain incorrect usage of language or symbols?
- The graph of has an average rate of change of on the interval .
- The f is increasing on the interval (0, ∞).
- The function defined above is the function shifted to the left units.
- The terms of the arithmetic sequence, , is a straight line.

A Story of Functions

A Close Look at Grade 9 Module 3

Participant Poll

- Classroom teacher
- Math trainer
- Principal or school leader
- District representative / leader
- Other

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.

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

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

Types of Lessons

- Problem Set
Students and teachers work through examples and complete exercises to develop or reinforce a concept.

- Socratic
Teacher leads students in a conversation to develop a specific concept or proof.

- Exploration
Independent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge.

- Modeling
Students practice all or part of the modeling cycle with real-world or mathematical problems that are ill-defined.

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

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

Mathematical Themes of Module 3

- Functional relationships
- Graphs and transformational geometry
- Linear functions versus exponential 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)

Prior Experience with Sequences

G9-M1 Lesson 26-27: Recursive Challenge ProblemThe 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 the result of the first round and the result of the second round, what should we call our starting number?

Write a recursive formula for .

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:

Lesson 1 – Example 1Start 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 . Let’s agree to do that for now.
- If we start with , which formula should we use for finding the term?

Terms, Term Numbers, and “the term”

- 1
- 2
- 4
- 8
- = 299
- = 2n-1

- = 20
- = 21
- = 22
- = 23
- …
- 100
- n

- 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
- 2
- 3
- 4
- What would be an appropriate heading for each of our columns?

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 term, where I pick what is.
- In this case:
- A formula for the term
- Would it be ok if I wrote to stand for “a formula for the term”?

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.

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.

Lesson 2: (Explicit and) Recursive Formulas for Sequences

- = 5 + 3 x ?
- = 5 + 3 x 1
- = 5 + 3 x 2
- = 5 + 3 x 3
- = 5 + 3 x 4
- = 5 + 3 x ?

- Example 1
- Term 1: 5
- Term 2: 8 = 5 + 3
- Term 3: 11 = 5 + 3 + 3
- Term 4: 14 = 5 + 3 + 3 + 3
- Term 5: 17 = 5 + 3 + 3 + 3 + 3
- ...
- Term n:

Lesson 2: Recursive Formulas for Sequences

- When Johnny saw Akeila’s sequence he wrote the following:
- for and
- Why do you suppose he would write that? Can you make sense of what he is trying to convey?
- What does the part mean?

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: .
- Lesson Summary:
- Provides a description of a what a recursively defined sequence is.
- See page 32 of the teacher materials

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 geometricif there is a real number such that each term in the sequence is a product of the previous term and .

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.

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

Key Points – Topic A

- Function notation is introduced simply as a shorthand for ‘the formula for the 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.

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

Lesson 8Why Stay With Whole Numbers?

- Why are square numbers called square numbers?
- If denotes the square number, what is a formula for ?
- In this context what would be the meaning of
- Exercises 5-8:
- Suppose we extend our thinking to consider squares of side-length cm… Create a formula for the area, cm2 of a square of side length cm.
- Review Exercises 9-12, taking time to do #10 and #12
- Do Exercises 13-14

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.

Lesson 9-10: Definition of Function

- A rule, like, “Let ,” 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.

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

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.

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 is a function and is an element of its domain, then denotes the output of corresponding to the input .
- I, 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.

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 ,” 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,is equivalent to, “Let , 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.

Lesson 11-12: Graphs

- Let for any real number. Discuss the meaning of
- Now discuss the meaning of
- 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.

Lesson 11The Graph of a Function

2

4

8

16

32

- 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

- What would be printed out if this code were executed?
- Work through Exercise 1

Lesson 11The Graph of a Function

- 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

- Consider this pseudo code:
Declare real

Let

Initialize G as {}

For all such that

Append to G

Next

Plot G

The Graph of a Function vs. The Graph of an Equation in 2 Variables

- The Graph of :
- Given a function 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

The Graph of an equation in two variables:

The set of all its solutions, plotted in the coordinate plane, often forming a curve (which could be a line)

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

Lessons 11-12The Graph of a Function

- The Graph of :
- Given a function 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

- The Graph of is the same as the graph of the equation .

The Graph of :

Given a function 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

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 .

- 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

Mid-Module Assessment

Work with a partner on this assessment

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.

- 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

Lesson 16Graphs Can Solve Equations Too

Solve for x in the following equation:

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!

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.”

Lessons 17-20: Transformations

- Lesson 18:
- Complete Example 1
- Continue working through the Exercises and Examples.
- Lesson 19:
- Horizontal scaling with a scale factor of the graph of corresponds to changing the equation from to

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

- 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

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.

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.

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 input-output values.
- What type of formula could be used to model this data?
- What is the formula?

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.

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.

˚F (the temperature of the room)

˚F (the initial temperature of the body)

( per hour - calculated by the investigator from the data collected)

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?

Math Modeling Exercise

- Work through the modeling exercise.
- If it is ˚F outside, can a cup of coffee ever cool to below ˚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 ˚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?

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.

Opening Exercise

- Discuss with a neighbor – which of the following phrases contain incorrect usage of language or symbols?
- The graph of has an average rate of change of on the interval .
- The f is increasing on the interval (0, ∞).
- The function defined above is the function shifted to the left units.
- The terms of the arithmetic sequence, ,is a straight line.

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.

- 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

End-of-Module Assessment

Work with a partner on this assessment

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.

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?

Session Objectives

- To find ways to bridge gaps in student understanding for Grade 9, Module 3.
- Determine strategies to support teaching decisionswhile maintaining the rigor of the instruction

Agenda

Foundational Standards

Topics to Consider during Transition Years

Making Instructional Decisions

Agenda

Foundational Standards

Topics to Consider during Transition Years

Making Instructional Decisions

Topics to Consider during Transition Years

Lesson 17

Lesson Notes

- 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
- The topic of Transformation is not commonly covered in current Grade 8 material

Topics to Consider during Transition Years

- Grade 8 Module 2 Overview:

Topics to Consider during Transition Years

- Grade 8 Module 3 Overview:

- Strategies to efficiently deliver the key information in Grade 8 regarding Transformations:
- Present key vocabulary to students (complete list in Module Overviews)

- 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.

Agenda Grade 8 regarding Transformations:

Foundational Standards

Topics to Consider during Transition Years

Making Instructional Decisions

Making Instructional Decisions Grade 8 regarding Transformations:

- Key Ideas
- Honor the objective of the lesson
- Maintain the rigor of the lesson

Bridging the Gaps Grade 8 regarding Transformations:

- Student Outcomes and Assessments
- Focus Standards linked to Foundational Standards from previous material
- Lessons Grouped within Topics
- Terminology, Tools and Representations

Turn and Talk Grade 8 regarding Transformations:

- 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?

Biggest Takeaway Grade 8 regarding Transformations:

- Turn and Talk:
- What questions were answered for you?
- What new questions have surfaced?