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NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios A Story of Ratios Grade 6 – Module 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org 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 Agenda Introduction to the Module Concept Development Module Review © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Curriculum Overview of A Story of Ratios © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Module’s Foundation • Standards: 6.NS.C.5, 6.NS.C.6, 6.NS.C.7, 6.NS.C.8 • Pages 7 – 8 in the Progressions Document (The Number System, 6-8) serves as a foundation. • Directed measurement --- a rational number’s position on the number line is found using length and direction. • The opposite of a number a, is –a. Both a and –a are located an equal distance from zero, in opposite directions. • Rational numbers represent real-world situations. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM G6-M3: Module Overview © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios G6-M3: Vocabulary and Representations © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios G6-M3 Rational Numbers – Topic Overview Topic A: Understanding Positive and Negative Numbers on the Number Line Topic B: Order and Absolute Value Topic C: Rational Numbers and The Coordinate Plane © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic A: Understanding Positive and Negative Numbers on the Number Line © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Agenda Introduction to the Module Concept Development – Topic A Module Review © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Positive and Negative Numbers on the Number Line: Opposite Direction and Value Outcomes: • Students extend their understanding of the number line, which includes zero and numbers to the right, that are above zero, and numbers to the left, that are below zero. • Students use positive integers to locate negative integers, moving in the opposite direction from zero. • Students understand that the set of integers is the set of whole numbers and their opposites, and understand that zero is its own opposite. Lesson 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Positive and Negative Numbers on the Number Line: Opposite Direction and Value • The number line extends to include negative numbers. • Lengths on the right-side and left-side of zero are the same for a number and its opposite. (Use a compass for the construction.) • The set of whole numbers and their opposites (zero is its own opposite) are called integers. • The order of the set of integers is: …-5,-4,-3,-2,-1,0,1,2,3,4,5… Lesson 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Positive and Negative Numbers on the Number Line: Opposite Direction and Value • Draw a horizontal line. Place a point on the line and label it 0. • Use a compass to locate and label the next point 1, thus creating a scale. (Continue to locate other whole numbers to the right of zero using the same unit measure.) • Using the same process, locate the opposite of each number on the left side of zero. Label the first point to the left of zero, -1. Lesson 1/Activity © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Real World Positive and Negative Numbers and Zero Outcomes: • Students use positive and negative numbers to indicate a change (gain or loss) in elevation with a fixed reference point, temperature, and the balance in a bank account. • Students use vocabulary precisely when describing and representing situations involving integers; for instance, an elevation of −10 feet is the same as 10 feet below the fixed reference point. • Students will choose an appropriate scale for the number line when given a set of positive and negative numbers to graph. Lessons 2-3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Real World Positive and Negative Numbers and Zero • Use positive and negative numbers and zero to represent situations. • Graph integers on a number line, using an appropriate scale and relating points to real-world situations. • Explain the meaning of zero in the context of a situation. Lessons 2-3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Real World Positive and Negative Numbers and Zero Lesson 3 /Exit Ticket © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios The Opposite of a Number’s Opposite Outcomes: • Students understand that, for instance, the opposite of −5 is denoted −(−5) and is equal to 5. In general, they know that the opposite of the opposite is the original number; e.g., −(−) = . • Students locate and position opposite numbers on a number line. Lesson 5 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM The Opposite of a Number’s Opposite Lesson 5/Example © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM The Opposite of a Number’s Opposite Lesson 5/Activity © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Rational Numbers on the Number Line Lesson 6 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Rational Numbers on the Number Line Lesson 6/Exercise 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Rational Numbers on the Number Line Lesson 6/Exercise 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Rational Numbers on the Number Line Lesson 6/Exit Ticket © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Agenda Introduction to the Module Concept Development – Topic B Module Review © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Topic B: Order and Absolute Value © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Ordering Integers and Other Rational Numbers Outcomes: • Students write, interpret, and explain statements of order for rational numbers in real world contexts. • Students recognize that if a < b, then -a > -b, because a number and its opposite are equal distances from zero; and moving along the horizontal number line to the right means the numbers are increasing. Lesson 7 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Ordering Integers and Other Rational Numbers What is the Value of Each Number and Which is Larger? 1) The first number is 8 ½ units to the right of -5. The second number is 3 units to the right of 0. First Number: 3.5; Second Number: 3; 3.5 is larger than 3. 2) The first number is ¼ unit to the left of -7. The second number is 8 units to the left of 1. First Number: −7.25; Second Number: −7; −7 is larger than −7.25. 3) The opposite of the first number is 2 units to the right of 3. The opposite of the second number is 2 units to the left of -3. First Number: −5; Second Number: 5; 5 is larger than −5. Lesson 7 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Ordering Integers and Other Rational Numbers Outcomes: • Students write, interpret, and explain statements of order for rational numbers in the real-world. • Students recognize that if < , then − > −, because a number and its opposite are equal distances from zero; and moving along the horizontal number line to the right means the numbers are increasing. Lesson 8 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Writing and Interpreting Inequality Statements Involving Rational Numbers Inequalities Fluency Builder Lesson 10 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Absolute Value – Magnitude and Distance • Complete Exercises 1- 3 • How do we want Grade 6 students to conceptualize absolute value? • Complete Exercises 6 – 19 • Discuss how students display an understanding of absolute value as magnitude and distance. Lesson 11/Exercises © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Mid-Module Assessment Question 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Agenda Introduction to the Module Concept Development – Topic C Module Review © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic C: Rational Numbers and the Coordinate Plane © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios You’re the Expert! • Go to the appropriate table as designated by the number on your card. • You will have ten minutes to discuss with your group the following: • Lesson Study • Modeling of an Essential Portion of the Lesson • Concerns/Scaffolding Beyond Teacher’s Edition • Each group will have ten minutes to model essential portion • During the ten minutes please address: • Lesson progression • Prerequisite/Foundation Skills Lessons 14 - 19 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM End-of-Module Assessment Questions 1 and 5 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Agenda Introduction to the Module Concept Development Module Review © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios 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 • Directed measurement --- a rational number’s position on the number line is found using length and direction. • The opposite of a number a, is –a. Both a and –a are located an equal distance from zero, in opposite directions. • Rational numbers represent real-world situations. We can write and explain statements of order for rational numbers in real-world contexts. • The absolute value of a number is its distance from zero; and can be used in the context of a situation to show magnitude. We can use absolute value and the symmetry of the coordinate plane to solve problems related to distance. © 2012 Common Core, Inc. All rights reserved. commoncore.org