© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions A Close Look at Grade 9 Module.

© 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 A Story of Functions 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 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 Let’s clarify - what do I mean by “the n th 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? = 2 0 = 2 1 = 2 2 = 2 3 … 100 n 1 2 4 8 = 2 99 = 2 n-1

© 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 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: = 5 + 3 x ? = 5 + 3 x 1 = 5 + 3 x 2 = 5 + 3 x 3 = 5 + 3 x 4 = 5 + 3 x ?

© 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 2 nd 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 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 NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 9-10: Definition of Function FunctionRule {1,2,3}  {0,1,2}f(x)=x-1 {1,2,3}  {0,2,1} {1,2,3}  {1,0,2} {1,2,3}  {1,2,0} {1,2,3}  {2,0,1} {1,2,3}  {2,1,0} {1,2,3}  {0,0,0}f(x)=0 {1,2,3}  {1,1,1}f(x)=1 {1,2,3}  {2,2,2}f(x)=2 {1,2,3}  {0,0,1} FunctionRule {1,2,3}  {0,1,0} {1,2,3}  {1,0,0} {1,2,3}  {0,0,2} {1,2,3}  {0,2,0} {1,2,3}  {2,0,0} {1,2,3}  {1,1,0} {1,2,3}  {1,0,1} {1,2,3}  {0,1,1} {1,2,3}  {1,1,2} {1,2,3}  {1,2,1} FunctionRule {1,2,3}  {2,1,1} {1,2,3}  {2,2,0} {1,2,3}  {2,0,2} {1,2,3}  {0,2,2} {1,2,3}  {2,2,1} {1,2,3}  {2,1,2} {1,2,3}  {1,2,2}

© 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 6 th & 7 th Grade: Proportional Relationships 7 th Grade: Scale drawings 8 th 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 The Graph of a Function vs. The Graph of an Equation in 2 Variables 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)

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions 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 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 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 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 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 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 A Story of Functions Lesson 24 Step and piecewise functions Utilizes ceiling function notation Practice

© 2012 Common Core, Inc. All rights reserved. commoncore.org 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 NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions A Close Look at Grade 9 Module.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions A Close Look at Grade 9 Module.

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