© 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 Participant Poll Classroom teacher Math trainer Principal or school leader District representative / leader Other

© 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 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 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 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 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 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 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? 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 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 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 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 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.

© 2012 Common Core, Inc. All rights reserved. commoncore.org 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 input- output values. What type of formula could be used to model this data? What is the formula?

© 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 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 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.

© 2012 Common Core, Inc. All rights reserved. commoncore.org 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 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

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios G9-M3 Module Overview  Foundational Standards 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 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.1A-REI.B.3A-REI.C.6A-REI.D.10 Grade 8 Grade 9

© 2012 Common Core, Inc. All rights reserved. commoncore.org 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 The topic of Transformation is not commonly covered in current Grade 8 material Lesson 17 Lesson Notes

© 2012 Common Core, Inc. All rights reserved. commoncore.org 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) TransformationReflectionImage of a Point Basic Rigid MotionsTranslationDilation Distance-preservingRotationScale factor

© 2012 Common Core, Inc. All rights reserved. commoncore.org 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. Examples: 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.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Making Instructional Decisions Key Ideas Honor the objective of the lesson Maintain the rigor of the lesson

© 2012 Common Core, Inc. All rights reserved. commoncore.org 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

© 2012 Common Core, Inc. All rights reserved. commoncore.org 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?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Biggest Takeaway Turn and Talk: What questions were answered for you? What new questions have surfaced?

© 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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