NYS COMMON CORE MATHEMATICS CURRICULUM A Story of

NYS COMMON CORE MATHEMATICS CURRICULUM Agenda Introduction to the Module Concept Development Module Review

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Progressions for the Common Core

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Curriculum Overview of A Story

NYS COMMON CORE MATHEMATICS CURRICULUM Grade 7 Module 3 Overview • • Lessons and

NYS COMMON CORE MATHEMATICS CURRICULUM Topic A A Story of Ratios Use Properties of

NYS COMMON CORE MATHEMATICS CURRICULUM Important Terminology Equivalent Expressions An Expression in Expanded Form

NYS COMMON CORE MATHEMATICS CURRICULUM Equivalent Expressions Lesson 1 / Example 1(a) © 2012

NYS COMMON CORE MATHEMATICS CURRICULUM Equivalent Expressions Lesson 1 / Example 1 (a) ©

NYS COMMON CORE MATHEMATICS CURRICULUM Equivalent Expressions Lesson 1 / Example 1 (b) ©

NYS COMMON CORE MATHEMATICS CURRICULUM Equivalent Expressions © 2012 Common Core, Inc. All rights

NYS COMMON CORE MATHEMATICS CURRICULUM Equivalent Expressions Lesson 1 / Example 4 (f) ©

A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Writing Products as Sums Lesson

NYS COMMON CORE MATHEMATICS CURRICULUM Writing Products as Sums Lesson 3 / Example 6

NYS COMMON CORE MATHEMATICS CURRICULUM Writing Products as Sums 4(s + 1) = 4

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Writing Sums as Products 2

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Using the Identity and Inverse

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Collecting Rational Number Like Terms

NYS COMMON CORE MATHEMATICS CURRICULUM Topic B A Story of Ratios Solve Problems Using

NYS COMMON CORE MATHEMATICS CURRICULUM Understanding Equations Where does an equation come from? Expressions

A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Understanding Equations Youngest Sister Middle

NYS COMMON CORE MATHEMATICS CURRICULUM Understanding Equations Lesson 7 / Example 1 © 2012

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Using If-Then Moves in Solving

A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Using If-Then Moves in Solving

NYS COMMON CORE MATHEMATICS CURRICULUM Lessons 1 through 9 Lessons 1 - 9/ Concepts

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Angle Problems and Solving Equations

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Fluency Sprint Opening Exercise: Round

NYS COMMON CORE MATHEMATICS CURRICULUM Properties of Inequalities Lesson 12 / Example 1 ©

A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Inequalities Lesson 13 / Opening

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Solving Inequalities The camp may

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Graphing Solutions to Inequalities Lesson

NYS COMMON CORE MATHEMATICS CURRICULUM Mid-Module Assessment Topics A and B Mid Module Assessment

NYS COMMON CORE MATHEMATICS CURRICULUM Topic C Use Equations and Inequalities to Solve Geometry

NYS COMMON CORE MATHEMATICS CURRICULUM The Most Famous Ratio of All Lesson 16 /

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios The Most Famous Ratio of

NYS COMMON CORE MATHEMATICS CURRICULUM The Area of a Circle Discussion: Cut a circle

NYS COMMON CORE MATHEMATICS CURRICULUM The Area of a Circle Lesson 17 / Discussion

NYS COMMON CORE MATHEMATICS CURRICULUM The Area of a Circle Lesson 17 / Example

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios The Area of a Circle

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios More Problems on Area and

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Unknown Area Problems in the

NYS COMMON CORE MATHEMATICS CURRICULUM Composite Area Problems Lesson 20 / Example 3 ©

NYS COMMON CORE MATHEMATICS CURRICULUM Composite Area Problems Lesson 20 / Problem Set #5

NYS COMMON CORE MATHEMATICS CURRICULUM Lessons 10 through 20 Lessons 10 - 20/ Concepts

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Surface Area Outcomes: • Students

NYS COMMON CORE MATHEMATICS CURRICULUM Surface Area Lesson 21/Example 1 © 2012 Common Core,

NYS COMMON CORE MATHEMATICS CURRICULUM Surface Area – Day 2 Lesson 22 © 2012

NYS COMMON CORE MATHEMATICS CURRICULUM Surface Area Lesson 22/Exit Ticket © 2012 Common Core,

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios The Volume of a Right

NYS COMMON CORE MATHEMATICS CURRICULUM The Volume of a Right Prism Lesson 23/Exit Ticket

NYS COMMON CORE MATHEMATICS CURRICULUM The Volume of a Right Prism Lesson 24/Example 3

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Volume and Surface Area Outcomes:

NYS COMMON CORE MATHEMATICS CURRICULUM Volume and Surface Area Lesson 25/Exercise 1 © 2012

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Volume and Surface Area –

NYS COMMON CORE MATHEMATICS CURRICULUM Volume and Surface Area – Day 2 Lesson 26/Exercise

NYS COMMON CORE MATHEMATICS CURRICULUM End-of-Module Assessment Topics A, B, and C © 2012

NYS COMMON CORE MATHEMATICS CURRICULUM End-of-Module Assessment Questions 3 © 2012 Common Core, Inc.

NYS COMMON CORE MATHEMATICS CURRICULUM Biggest Takeaway Turn and Talk: • What questions were

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Key Points • • Properties

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NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Session Objectives • Examine the development of mathematical understanding across the module using a focus on concept development within the lessons. • Identify the big idea within each topic in order to support instructional choices that achieve the lesson objectives while maintaining rigor within the curriculum. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Progressions for the Common Core State Standards in Mathematics What to expect in Grade 7: • More complex expressions; containing signed numbers • Various forms of equivalent expressions to model contextual relationships • Relating numeric solution process to algebraic steps • Using mental math and estimation to check answers to complex problems • Using variables and equation models to represent real-world problems • Extending the process of solving equations to inequalities, while realizing that multiplying or dividing both sides of an inequality by a negative number reverses the comparison order. Expressions and Equations Grades 6 -8 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Grade 7 Module 3 Overview • • Lessons and topics Narrative Focus Standards Builds upon Foundations Mathematical Practices Terminology (old and new) Tools and Representations © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Ratios

NYS COMMON CORE MATHEMATICS CURRICULUM Topic A A Story of Ratios Use Properties of Operations to Generate Equivalent Expressions • Take a moment to read through: • • • Module Overview paragraphs 2 -3 Topic A Opener What do you expect to see conceptually as Topic A unfolds? © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Important Terminology Equivalent Expressions An Expression in Expanded Form An Expression in Standard Form An Expression in Factored Form © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Ratios

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Using the Identity and Inverse to Write Equivalent Expressions Lesson Highlights: • Students come to recognize the identity properties of 0 and 1. • Students come to recognize the existence of inverses (opposites and reciprocals) that together make 0’s and 1’s in expressions. • Why do we emphasize student awareness of these properties? © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Topic B A Story of Ratios Solve Problems Using Expressions, Equations, and Inequalities • Take a moment to read through: • • • Module Overview paragraph 4 Topic B Opener What do you expect to see conceptually as Topic B unfolds? © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Understanding Equations Where does an equation come from? Expressions and Equations Progression p. 4 How will we approach equations? Lots and lots of wonderful word problems! © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Ratios

A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Understanding Equations Youngest Sister Middle Sister 1 Oldest Sister 1 Lesson 7 / Example 1 © 2012 Common Core, Inc. All rights reserved. commoncore. org 45 1

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Using If-Then Moves in Solving Equations Example 1: Julia, Keller, and Israel are volunteer firefighters. On Saturday the volunteer fire department held its annual coin drop fundraiser at a streetlight. After one hour, Keller had collected $42. 50 more than Julia and Israel had collected $15 less than Keller. Altogether, the three firefighters collected $125. 95. How much did each person collect? Find the solution using a tape diagram. Lesson 8 / Example 1 © 2012 Common Core, Inc. All rights reserved. commoncore. org

A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Using If-Then Moves in Solving Equations In terms of Julia Collected Isreal Collected i - 27. 50 k - 27. 50 Lesson 8 / Example 1 © 2012 Common Core, Inc. All rights reserved. commoncore. org Keller Collected i + 15 k - 15

A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Using If-Then Moves in Solving Equations Present Age (years) Future Age (years) Bonnie Shelby x+5 7 x + 5 Lesson 9 / Example 2 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Angle Problems and Solving Equations Example 4: Two line intersect in the following figure. The ratio of the measurements of the obtuse angle to the acute angle in any adjacent angle pair is 2: 1. 2 xᵒ Lesson 10 / Example 4 © 2012 Common Core, Inc. All rights reserved. commoncore. org xᵒ

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Angle Problems and Solving Equations Exercise 2 In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for x and y. Confirm your answers by measuring with a protractor. Lesson 11 / Exercise 2 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Fluency Sprint Opening Exercise: Round 1: You have 1 minute to complete as many as possible…do not skip around, go in order! You only need to provide the answer. • If you have the answer correct, clap once! • Finish the remaining problems. Lesson 12 / Opening Exercise © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Topic C Use Equations and Inequalities to Solve Geometry Problems • Take a moment to read through: • • • Module Overview paragraphs 5 -7 Topic C Opener What do you expect to see conceptually as Topic C unfolds? © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Ratios

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios The Most Famous Ratio of All Example 1: 1. Mark a point on the wheel with a piece of masking tape or chalk. 2. Mark a starting point on the floor, align it with the mark on the wheel, and carefully roll the wheel so that is rolls one complete revolution. 3. Mark the endpoint on the floor with a piece of masking tape or chalk. Lesson 16 / Example 1 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios The Most Famous Ratio of All From the Educator Guide to the 2013 Grade 7 Common Core Mathematics Test: © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM The Area of a Circle Discussion: Cut a circle into 16 equal size sectors. Arrange the “triangles” by alternating their directions and sliding them together to form a parallelogram. Lesson 17 / Discussion © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Ratios

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios The Area of a Circle Problem Set #10 If the diameter of a circle is double the diameter of a second circle, what is the ratio of area of the first circle to the area of the second? Lesson 17 / Problem Set #10 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios More Problems on Area and Circumference Opening Exercise: Draw a circle of diameter 12 cm and a square of side length 12 cm on grid paper. Determine the area of the square and the circle. Lesson 18 / Opening Exercise © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Surface Area Outcomes: • Students find the surface area of three-dimensional objects whose surface area is composed of triangles and quadrilaterals. They use polyhedron nets to understand that surface area is simply the sum of the area of the lateral faces and the area of the base(s). Lessons 21 -22 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios The Volume of a Right Prism Outcomes: • Students use the known formula for the volume of a right rectangular prism (length x width x height). • Students understand the volume of a right prism to be the area of the base times the height. • Students compute volumes of right prisms involving fractional values for length. Lesson 23 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios The Volume of a Right Prism Question: How could we create a right triangular prism with five times the volume of the right triangular prism pictured, without changing its base? What would its volume be? Lesson 23 / Discussion and Concept © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Volume and Surface Area Outcomes: • Students solve real-world and mathematical problems involving volume and surface areas of three-dimensional objects composed of cubes and right prisms. Lesson 25 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Volume and Surface Area – Day 2 Outcomes: • Students solve real-world and mathematical problems involving volume and surface areas of three-dimensional objects composed of cubes and right prisms. Lesson 26 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM 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 A Story of Ratios

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Key Points • • Properties of operations are used to generate equivalent expressions; and different forms of expressions serve different purposes. “Any order, any grouping” and the distributive property are important for generating equivalent expressions (Grade 6); in Grade 7 an understanding of operations involving negative numbers is also necessary. The steps in solving an equation mirror the numeric process; if-then statements justify the making of zeros and ones. Visual models relate algebraic steps to numeric steps. As equations get more complex, estimation skills and mental math verify that an answer is reasonable. The solution process for inequalities is similar to that of equations, with the exception of multiplying or dividing by negative numbers. Conceptual understanding of pi and the use of equations, allow us to solve real-life problems related to circles. Facts about angle relationships can be represented with equations and used to find unknown angles. Nets serve as visual connectors between the area of 2 -dimensional objects and the surface area of 3 -dimensional prisms. © 2012 Common Core, Inc. All rights reserved. commoncore. org