NYS COMMON CORE MATHEMATICS CURRICULUM A Story of

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions A Close Look at Grade 9 Module 4 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Opening Exercise Answer the following and discuss your responses with a neighbor: • Why should students spend so much time studying quadratics? • Why are quadratics (polynomials of degree 2) called quadratics anyway? • Can any u-shaped graph be represented by a quadratic function? © 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 4 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Participant Poll • • • Classroom teacher Math trainer Principal or school leader District representative / leader Other © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

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 4 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 (if needed) A Foundation for the Study of Quadratics Examination and exploration of: • Topic A • Fluency Exercises – The Rapid White Board Exchange • Mid-Module Assessment • Topic B • Topic C • End of Module Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM 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 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 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 Agenda Orientation to Materials (if needed) A Foundation for the Study of Quadratics Examination and exploration of: • Topic A • Fluency Exercises – The Rapid White Board Exchange • Mid-Module Assessment • Topic B • Topic C • End of Module Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions A Foundation for the Study of Quadratics: Part 1 – A look back at sequences What is the next number in the sequence? • 4, 7, 10, 13, 16, … • 4, 5, 8, 13, 20, 29, … • 2, 4, 9, 22, 48, 102, … What does the leading diagonal look like for each of the following: • 1: 1, 1, 1, …. • n: 1, 2, 3, 4, 5, 6, … • n 2: 1, 4, 9, 16, 25, 36, … • n 3: 1, 8, 27, 64, 125, 216, … • n 2 +n: © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions A Foundation for the Study of Quadratics: Part 2 – Why such fascination? © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions A Foundation for the Study of Quadratics: Part 3 – A closer look at u-shaped curves 1. Tape the ends of the chain so that the lowest part of the chain falls right at the origin. 2. Identify several other points that the chain goes through. 3. Create a quadratic equation that goes through the points you identified. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Orientation to Materials (if needed) A Foundation for the Study of Quadratics Examination and exploration of: • Topic A • Fluency Exercises – The Rapid White Board Exchange • Mid-Module Assessment • Topic B • Topic C • End of Module Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Flow of Module 4 • Topic A: Quadratic Expressions, Equations, Functions, and their Connection to Rectangles • Reversing multiplication yields factored expressions (recall geometric models); practice factoring of quadratics. • When combined with the Zero Product Property we have a new power to solve quadratic equations. • What does the graph of a quadratic equation look like? It’s symmetric (and u-shaped). • Factored form + symmetry makes graphing simple. • Relating quadratic equations and their graphs to real-world context, giving contextual interpretations of key features. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Flow of Module 4 • Topic B: Using Different Forms for Quadratic Functions • Other ways to see structure in quadratics – solving by completing the square; the quadratic formula. • Why does completing the square yield something we call “vertex form”? The relationship between vertex form and transformations; the helpfulness of vertex form in graphing. • Further examination of quadratic functions and their graphs in context. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Flow of Module 4 • Topic C: Function Transformations and Modeling • The square root function and its relationship to the basic quadratic function; the cube root and cubic functions. • Transformations of all of these types of functions. • Analyzing and comparing functions represented in different forms, all done in context. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What are students coming in with? • Experience multiplying with polynomials using the distributive property (G 9 -M 1) • Experience relating the distributive property to an area model or an modified area model (the tabular method) (G 9 -M 1) • Experience writing a sum as a product of two factors (G 7 -M 3) and factoring out a greatest common factor (G 6 -M 2) • Experience transforming graphs, transforming functions and relating the transformed function to the transformed graph (G 9 -M 3) © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A – Lesson 1 • Opening Exercise • Example 1 • Extension: Is there another option? How many possible answers are there? • The language of p. 19 may prove difficult; scaffolding suggestions: • Prime numbers can be related to ‘counting by’ instead of factors • (Before presenting the given description) How can we describe what we mean by a factor being prime? How could I describe what it means when you can’t factor it any more than you already have? • Write a simple binomial. Now write that binomial as a product of two other polynomials. (Remember, even a simple integer is a polynomial) © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Topic A – Lesson 2 • Why are they called quadratics anyway? • Note the scaffold box at the top of page 31 • Exercises 7 -8 © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A – Lessons 3 -4 • Lesson 3 Opening Exercise • Continue to use the tabular model as needed. • Encourage students to verbalize their process of finding factors that work. • Lesson 4 Problem Set #3, an example of MP. 1 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A – Lessons 5 -7 • Lesson 5 Opening Exercise, Exercises 1 -4 lead students to know and apply the zero product property. • Example 1 provides context for its application. • Reasoning through a problem is still a valid approach. Factoring is only one means to the end. • Lesson 7 calls upon students to build their own equations from context. (Work through exercises 5 -7. ) © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A – Lessons 8 • Scaffold: If needed, begin this lesson with an opportunity to graph a selection of relatively simple quadratic functions, allowing students to work in pairs on a problem of appropriate complexity. • What do you notice about these graphs? Allow students to notice the symmetry, and begin with an informal description of the vertex. • This lesson brings up the question, are all u-shaped curves represented by quadratics. • Work the Extension question after Exploratory Challenge 2. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A – Lessons 9 -10 • Lesson 9 Opening Exercise • Lesson 9 Example 2. Pose the questions to students, how on earth did they come up with this formula. • Scaffold in formal terms by using contextual everyday language and then repeating with more formal words. • Lesson 10 Example 1. Ask the students to think critically about the reasonableness of this graph for this situation. • Lesson 10 Example 2. Spend ample time challenging the students with the ‘How do you know’ question. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Points – Topic A • Consider having students come up with their own summaries for how they approach factoring /solving / graphing a quadratic. • It’s better to study deeply a given application problem and the analysis of its graph’s features than to do multiple problems. • Introduce concepts like domain, range, increasing, decreasing, average rate of change, etc. by using words that feel natural in the context, and then repeat the statement or question using the more formal words. • Scaffolds are a critical tool for successful implementation. In addition to those given in the module, consider the ones we explored in this session. (Take time now to reflect and take note of them. ) © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Orientation to Materials (if needed) A Foundation for the Study of Quadratics Examination and exploration of: • Topic A • Fluency Exercises – The Rapid White Board Exchange • Mid-Module Assessment • Topic B • Topic C • End of Module Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Rapid White Board Exchange Factoring trinomials © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Orientation to Materials (if needed) A Foundation for the Study of Quadratics Examination and exploration of: • Topic A • Fluency Exercises – The Rapid White Board Exchange • Mid-Module Assessment • Topic B • Topic C • End of Module Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Mid-Module Assessment Work with a partner on this assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM Scoring the Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Orientation to Materials (if needed) A Foundation for the Study of Quadratics Examination and exploration of: • Topic A • Fluency Exercises – The Rapid White Board Exchange • Mid-Module Assessment • Topic B • Topic C • End of Module Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Topic B – Lesson 11 © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM Topic B – Lessons 12 -13 © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM Topic B – Lesson 14 • Deriving the quadratic formula • Algebraic approach. • Using geometric square model scaffold. © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM Topic B – Lessons 15 -16 © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B – Lesson 17 • Lesson 17: • Work the Opening Exercise, then challenge students to develop their own ‘general strategy’ for graphing a quadratic function before reviewing what is provided before Example 1. • Example 1 provides another opportunity to ask, ‘How do you suppose the math class was able to determine this formula? ’ • It is not explicitly asked or stated, but is suggested, ‘How can I put a function into vertex form? ’ • Have students to come up with their general approach to graphing on their own before considering the approach provided. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Key Points – Topic B © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Orientation to Materials (if needed) A Foundation for the Study of Quadratics Examination and exploration of: • Topic A • Fluency Exercises – The Rapid White Board Exchange • Mid-Module Assessment • Topic B • Topic C • End of Module Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C – Lesson 18 • • Exercise 1 Exercise 2 Exercise 3 Suggestion: Don’t give away the relationship between the graphs of these inverse functions. Ask the question, then spend ample time letting students contemplate and articulate to the best of their ability what they notice. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C – Lessons 19 -20 • Make use of technology to demonstrate and apply previous understanding of transformations of functions. • Completing the square when working with a function or an equation in two variables. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Topic C – Lessons 22 -24 © 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 C • Comparing features of functions provided in different forms deepens and consolidates student understanding of the relationship between the structure of expressions and equations, the graphs of equations and functions, and the contexts they model. • Students should walk away from quadratics understanding that a primary use of these functions is in modeling height over time of projectile objects, that they are naturally related to rectangular area problems, and that there also used in an early study of business applications. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Opening Exercise Answer the following and discuss your responses with a neighbor: • Why should students spend so much time studying quadratics? • Why are quadratics (polynomials of degree 2) called quadratics anyway? • Can any u-shaped graph be represented by a quadratic function? © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Points – Module 4 Lessons • Students are called upon to Look for and make use of structure (MP. 7) as they choose equivalent forms of quadratics to gain insight into the function’s behavior and its graph. • Students are called upon to reason abstractly and quantitatively (MP. 2) as they decontextualize and work with quadratic equations representing real-world contexts and then re-contextualize as they analyze and interpret the key features of the function and its graph in the context of the problem. • Note that the physics contexts have the same coefficients due to the mathematics of objects in motion. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Orientation to Materials (if needed) A Foundation for the Study of Quadratics Examination and exploration of: • Topic A • Fluency Exercises – The Rapid White Board Exchange • Mid-Module Assessment • Topic B • Topic C • End of Module Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM End-of-Module Assessment Work with a partner on this assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM Scoring the Assessment © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

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 4? 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