# N YS CO MMO N COR E MAT N YS CO MMO N COR E MAT H E MAT I C S C U RR I C U LU M A Story of Ratios A Story of Ratios Grade 7 Module 3 Session 2 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Session Objectives Examine the development of mathematical understanding across the module using a focus on

Concept Development within the lessons. Introduce mathematical models and instructional strategies to support implementation of A Story of Ratios. 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Agenda Introduction to the Module Introduction to the Module Concept Development Concept Development Module Review

Module Review 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Curriculum Overview of A Story of Ratios 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Topic A 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 A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Ratios Equivalent Expressions Example 1 (a): Rewrite and by combining like terms. Write the original expressions and expand each term using addition. What are the new expressions equivalent to? Lesson 1 / Example 1(a) 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Equivalent Expressions

Example 1 (a): (contd) Because both terms have the common factor of , we can use the distributive property to create an equivalent expression. Expressions and Equations Progression Document (p. 6) Lesson 1 / Example 1 (a) 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Equivalent Expressions Example 1 (b): Find the sum of and . Example 1 (b): Find the sum of and .

Associative property of addition Any order, any grouping property Commutative property of addition Combined like terms (distributive property) Associative property of addition Combined like terms (distributive property) Expressions and Equations Progression Document (p. 5) Lesson 1 / Example 1 (b) 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios

Equivalent Expressions Can the any order, any grouping property be used with other operations? Subtraction? No. However a difference can be rewritten as a sum using inverse relationships: Multiplication? Yes! Multiplication is both associative and commutative. Division? No. However a quotient can be rewritten as a product using inverse relationships: 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios

Equivalent Expressions Example 4 (f): Alexander says that is equivalent to because of any order, any grouping. Is he correct? Why or why not? Sample response: If I let and let , then: Lesson 1 / Example 4 (f) 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Writing Products as Sums Example 1:

a. Draw a tape diagram representing the sum using square units. b. c. Redraw the diagram without squares, as two rectangles that are 1 unit high. Draw a rectangular array representing . Lesson 3 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org

3(3 + 2) = 9 + 6 A Story of Ratios

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Writing Products as Sums (continued) d. Draw a tape diagram representing the sum using square units. e. f. Redraw the diagram without squares, as two rectangles that are 1 unit high. Draw a rectangular array representing .

3(x + 2) = 3x + 6 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Writing Products as Sums Example 6 A square fountain area with side length is bordered by a single row of square tiles as shown. Express the total number of tiles needed in terms of three different ways. Lesson 3 / Example 6 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Writing Products as Sums 4(s + 1) = 4s + 4 = 2s + 2(s + 2) Lesson 3 / Example 6 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Writing Sums as Products

Example 2: and are positive integers. , , and stand for the number of unit squares in a rectangular array. What does the large rectangle that contains the three smaller rectangles represent? 2x + 12y + 8 square units How many rows are there in this rectangular array? 2 rows 2xdo+ you 12yknow? +8 What are the missing values and how Lesson 4 / Example 2 2012 Common Core, Inc. All rights reserved. commoncore.org

= 2(x + 6y + 4) N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 0s and 1s in expressions.

Why do we emphasize student awareness of these properties? 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Collecting Rational Number Like Terms Example 4: Model how to write the expression in standard form using rules of rational numbers. Lesson 6 / Example 4 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Topic B 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 A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Understanding Equations Example 1: The ages of three sisters are consecutive integers. The sum of their ages is 45. Find their ages. a. Use a tape diagram to find their ages. Youngest Sister Middle Sister 1

Oldest Sister 1 , and The sisters are 14, 15, and 16. Lesson 7 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org 45 1 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Ratios Understanding Equations (continued) b. If the youngest sister is years old, describe the ages of the other two years old, describe the ages of the other two sisters in terms of years old, describe the ages of the other two , write an expression for the sum of their ages in terms of years old, describe the ages of the other two , and use that expression to write an equation that can be used to find their ages Youngest sisters age in years: Middle sisters age in years: Oldest sisters age in years: Sum of their ages: Sum of their ages is 45: Lesson 7 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Understanding Equations (continued) c. Determine if your answer from part (a) is a solution to the equation you wrote in part (b). TRUE! Lesson 7 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Using If-Then Moves in Solving Equations What is an If-Then move? Any order, any grouping If , then Why if-then moves? If-then moves promote reasoning about the equality of expressions. 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios

Using If-Then Moves in Solving Equations The amount that Keller collected in dollars: The amount Isreal collected in dollars: or Write an equation in terms of j that can be used to find the amount each person collected. The sum of the amounts collected by all three people equals . Lesson 8 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Using If-Then Moves in Solving Equations

any order, any grouping If a=b, then to make 0 any order, any grouping and additive inverse additive identity property of 0 If a=b, then ; to make 1 any order, any grouping multiplicative inverse Lesson 8 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Using If-Then Moves in Solving Equations

In terms of Julia Collected Isreal Collected Keller Collected Julias Amount Isreals Amount i - 27.50 Kellers Amount k - 27.50 Lesson 8 / Example 1

2012 Common Core, Inc. All rights reserved. commoncore.org i + 15 k - 15 A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Using If-Then Moves in Solving Equations Example 2: Shelby is seven times as old as Bonnie. If in 5 years, the sum of Bonnie and Shelbys ages is 98, find Bonnies present age. Present Age (years) Future Age (years) Bonnie Shelby

x+5 7x + 5 Bonnies present age is 11 years old. Lesson 9 / Example 2 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Lessons 1 through 9 Lessons 1- 9/ Concepts and Representations 2012 Common Core, Inc. All rights reserved. commoncore.org

A Story of Ratios N YS CO MMO N COR E MAT H E MAT I C S C U RR I C U LU M A Story of Ratios A Story of Ratios Grade 7 Module 3 Session 3 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

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. 2x Acute angle: , Obtuse angle: Lesson 10 / Example 4 2012 Common Core, Inc. All rights reserved. commoncore.org x N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

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. Angles and are complimentary and sum to ; angles and are complimentary and sum to . Lesson 11 / Exercise 2 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios

Fluency Sprint Opening Exercise: Round 1: You have 1 minute to complete as many as possibledo 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Properties of Inequalities

Example 1: Using the number cubes at your tables, complete the station exercises by recording each trial of integers in the first and third columns. Station 1: If c. Station 2: If then . Station 3: If and , then . Station 4: If and , then . Lesson 12 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Inequalities

Opening Exercise: Tarik is trying to save to buy a new tablet. Right now he has and can save a week from his allowance. Write and evaluate an expression to represent the amount of money saved after: 2 weeks: 3 weeks: 4 weeks: 5 weeks: 6 weeks: 7 weeks: 8 weeks: Lesson 13 / Opening Exercise 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Ratios Solving Inequalities Example 1 A youth summer camp has budgeted \$ for the campers to attend a carnival. The cost for each camper is \$. which includes general admission to the which includes general admission to the carnival and meals. The youth summer camp must also pay \$ which includes general admission to the for the chaperones to attend and \$ which includes general admission to the for transportation to and from. What is the greatest number of campers that can attend the carnival if the camp must stay within their budgeted amount? The camp may take up to 77 campers. Lesson 14 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org

A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Graphing Solutions to Inequalities Example 1 A local car dealership is trying to sell all of the cars that are on the lot. Currently it has which includes general admission to the which includes general admission to the cars on the lot and the general manager estimates that they will consistently sell which includes general admission to the cars per week. Estimate how many weeks it will take for the number of cars on the lot to be less than which includes general admission to the ? let represent the number of complete weeks of sales It will take 10 weeks or more. Lesson 15 / Example 1

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Mid-Module Assessment Topics A and B Mid Module Assessment / Problem #6 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios The Most Famous Ratio of All Opening Exercise Draw a circle using a compass. Write your own definition for the term circle. Circle: Given a point in the plane and a number , the circle with center and radius is the set of all points in the plane that are a distance from the point . Lesson 16 / Opening Exercise

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M The Most Famous Ratio of All From the Educator Guide to the 2013 Grade 7 Common Core Mathematics Test: This curriculum utilizes the exact value of pi as well as three approximations of the value of pi, including: Calculator approximation 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios

The Area of a Circle The circumference is , where is the radius. Therefore, half of the circumference is . What is the area of the rectangle using the side lengths shown? Lesson 17 / Discussion 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios The Area of a Circle

Example 1 Michael is laying tile in the floor of a square walk-in shower. He wants a circle pattern as shown. He needs to determine the area of the circle so that he can order the special tile for the shower project. Michael drew the tile pattern on graph paper so that he could have a visual model to help guide his thinking. Lesson 17 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios More Problems on Area and Circumference Example 3 The circumference of a circle is . What is the exact area of the circle? Draw a diagram to assist you in solving the problem. If , then . Which yields . The exact area of the circle is Lesson 18 / Example 3

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Unknown Area Problems in the Coordinate Plane Example 1: The coordinate plane shown contains figure , parallelogram . c. Area of the rectangle: d. Area of each triangle: e. Area of parallelogram : The coordinate plane shown contains figure , a rectangle with the same base as the parallelogram above. f. Draw triangles and next to so that you have a rectangle that is the same size as the one you created on the first coordinate plane .

Lesson 19 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Unknown Area Problems in the Coordinate Plane (Continued) Lesson 19 / Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

Composite Area Problems Example 3: Find the area of the shaded region labeling known values. The area of the shaded region is . Lesson 20 / Example 3 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Composite Area Problems

Problem Set 5. Find the area of the region which is formed by three semicircles. Lesson 20 / Problem Set #5 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Lessons 10 through 20 Lessons 10 - 20/ Concepts and Representations 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios

N YS CO MMO N COR E MAT H E MAT I C S C U RR I C U LU M A Story of Ratios A Story of Ratios Grade 7 Module 3 Session 4 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Topic C Use Equations and Inequalities to Solve Geometry Problems 2012 Common Core, Inc. All rights reserved. commoncore.org

A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Surface Area Lesson 21/Example 1 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Surface Area Day 2 Lesson 22 2012 Common Core, Inc. All rights reserved. commoncore.org

A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Surface Area Lesson 22/Exit Ticket 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M The Volume of a Right Prism Lesson 23/Exit Ticket 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios The Volume of a Right Prism Outcomes: Students will use the known formula for the volume of a right rectangular prism (length x width x height) and find the sum (repeated use of the distributive property) to give an approximation of the volume. (Length width) height or (Area of the base) height. Students will 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 24 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M The Volume of a Right Prism Lesson 24/Example 3 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Volume and Surface Area

Lesson 25/Exercise 1 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M End-of-Module Assessment Topics A, B, and C 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M End-of-Module Assessment Questions 3 2012 Common Core, Inc. All rights reserved. commoncore.org

A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 A Story of Ratios

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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 multiplying or dividing by negative numbers. the exception of

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 N YS CO MMO N COR E MAT H E MAT I C S C U RR I C U LU M A Story of Ratios A Story of Ratios