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 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 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 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 Examples of Recursive Definitions © 2012 Common Core, Inc. All rights reserved. commoncore.org 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: = − © 2012 Common Core, Inc. All rights reserved. commoncore.org – − + ( − ) + − + 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 = 1. Let’s agree to do that for now. • If we start with = 1, which formula should we use for finding the ℎ term? © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 © 2012 Common Core, Inc. All rights reserved. commoncore.org 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. © 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: ( + 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? © 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: + 1 = 5 − 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 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 © 2012 Common Core, Inc. All rights reserved. commoncore.org 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. © 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? © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 misleading © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 • Exponential Decay • Understanding of when b > 1 and b < 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 © 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 = − 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, 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. © 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 = 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. © 2012 Common Core, Inc. All rights reserved. commoncore.org 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. © 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 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 © 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 real Let () = + Initialize G as {} For all such that ≤ ≤ Append (, ()) to G Next 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 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 , () | ∈ © 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 = () 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 © 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 : 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 = (). © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 13 • Increasing vs. decreasing is different that positive and negative © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lessons 14 • Review of linear vs. exponential • Fractional Exponents not needed • Compare and contrasting growth rates © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 . © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 © 2012 Common Core, Inc. All rights reserved. commoncore.org 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! © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Explain why… © 2012 Common Core, Inc. All rights reserved. commoncore.org 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.” © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 = © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 © 2012 Common Core, Inc. All rights reserved. commoncore.org 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. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 21: © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 © 2012 Common Core, Inc. All rights reserved. commoncore.org 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. © 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. = 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) © 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? © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 • Step and piecewise functions • Utilizes ceiling function notation • Practice © 2012 Common Core, Inc. All rights reserved. commoncore.org 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. © 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

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