A Story of Functions

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

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 .

Examples of Other Recursive Definitions

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.

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

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 .

Mid-Module Assessment Work with a partner on this assessment

Scoring the 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.

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!

Explain why…

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.”

A Story of Functions