A Story of Ratios

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosGrade 7 – Module 2


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU 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.

• Identify the big idea within each topic in order to support instructional choices that achieve the lesson objectives while maintaining rigor within the curriculum.



AgendaIntroduction to the ModuleConcept DevelopmentModule Review



Curriculum Overview of A Story of Ratios



Module’s Foundation• Standards: 7.NS.A.1, 7.NS.A.2, 7.NS.A.3, 7.EE.A.2, 7.EE.B.4

• Pages 9 – 13 in the Progressions Document (The Number System, 6-8) serves as a foundation.

• Using vectors to represent the addition of rational numbers on the number line; arriving at the additive inverse: p + (-p) = 0.

• Representing subtraction as distance on the number line; and |p – q | is the distance between p and q on the number line.

• Students use the properties of operations to justify multiplication and division of rational numbers.



G7-M2: Module Overview



G7-M2: Vocabulary and Representations



G7-M2 Rational Numbers – Topic Overview

Topic A: Addition and Subtraction of Integers and Rational Numbers

Topic B: Multiplication and Division of Integers and Rational Numbers

Topic C: Applying Operations with Rational Numbers to Expressions and Equations



Topic A: Addition and Subtraction of Integers and Rational Numbers



AgendaIntroduction to the ModuleConcept Development – Topic AModule Review



Opposite Quantities Combine to Make ZeroOutcomes:• Students add positive integers by counting up and negative

integers by counting down (using curved arrows on the number line).

• Students play the Integer Game to combine integers, justifying that an integer plus its opposite add to zero.

• Students know the opposite of a number is called the additive inverse because the sum of the two numbers is zero.

Lesson 1



Opposite Quantities Combine to Make Zero

Lesson 1/Example 3




The Integer Game

Lesson 1, Concept Development




Let’s Play the Integer Game!

Lesson 1, Concept Development



Using the Number Line to Model the Addition of Integers

Lesson 2/Example 1



Using the Number Line to Model the Addition of Integers

Lesson 2/Example 3



Understanding Addition of Integers

Lesson 3

Outcomes:• Students understand addition of integers as putting together or

counting up, where counting up a negative number of times is counting down.

• Students use arrows to show the sum of two integers, , on a number line and to show that the sum is distance from to the right if is positive and to the left if is negative.

• Students refer back to the Integer Game to reinforce their understanding of addition.



Understanding Addition of Integers

Lesson 3/Exit Ticket



Understanding Subtraction of Integers and Other Rational Numbers

Lesson 5

Outcomes:• Students justify the Rule for Subtraction: Subtracting a

number is the same as adding its opposite.

• Students relate the subtraction to the Integer Game.

• Students justify the Rule for Subtraction for all rational numbers, using the inverse relationship between addition and subtraction.



Understanding Subtraction of Integers and Other Rational Numbers

Lesson 5/Example



The Distance Between Two Rational Numbers

Lesson 6/Closing








Lesson 6/Problem Set



Addition and Subtraction of Rational Numbers

Lesson 7



Applying the Properties of Operations to Add and Subtract Rational NumbersOutcomes:• Students use properties of operations to add and subtract rational numbers

without the use of a calculator.• Students recognize that any problem involving addition and subtraction of

rational numbers can be written as a problem using addition and subtraction of only positive numbers.

• Students use the commutative and associative properties of addition to rewrite numerical expressions in different forms. They know the opposite of a sum is the sum of its opposites: -(3 + (-4)) = -3 + 4.

the opposite

Lessons 8 and 9



Applying the Properties of Operations to Add and Subtract Rational Numbers

Lessons 8 and 9



AgendaIntroduction to the ModuleConcept Development – Topic BModule Review



Topic B: Multiplication and Division of Integers and Rational Numbers



Understanding the Multiplication of Integers

Lesson 10/Example 3



Develop Rules for Multiplying Signed Numbers

Lesson 11

Outcomes:• Students understand the rules for multiplication of integers and that

multiplying the absolute values of integers results in the absolute value of the product. The sign, or absolute value, of the product is positive if the factors have the same sign and negative if they have opposite signs.

• Students realize that and see that it can be proven to be true mathematically through the use of the distributive property and the additive inverse.

• Students use the rules for multiplication of signed numbers and give real-world examples.




Lesson 11/Example 1




Lesson 11



Converting Rational Numbers to Decimals Using Long Division

Lesson 14/Example 2



Mid-Module Assessment

Question 6



AgendaIntroduction to the ModuleConcept Development – Topic CModule Review



Topic C: Applying Operations with Rational Numbers to Expressions and Equations



Comparing Tape Diagram Solutions to Algebraic Solutions

Lesson 17



Comparing Tape Diagram Solutions to Algebraic Solutions




If-Then Moves with Integer Cards

Lesson 21/ Exercises 1



If-Then Moves with Integer Cards

Lesson 21/ Exercises 1



Solving Equations Using Algebra

Lesson 23/Exercise 2



Solving Equations Using Algebra




End-of-Module Assessment

Questions 2 and 3



AgendaIntroduction to the ModuleConcept DevelopmentModule Review



Biggest TakeawayTurn and Talk:• What questions were answered for you?• What new questions have surfaced?



Key Points• The additive inverse is the opposite of a number because, when added to

a number, the sum is zero.• We can use arrows on a number line to show the sum, p + q, on a number

line. The sum is the distance|q|from p to the right of p if p is positive, and to the left of p if p is negative.

• Subtracting a number is the same as adding its opposite.• The properties of operations justify the rules for multiplication and

division of integers. (-1)(-1) = 1 can be justified using the distributive property and additive inverse.

• The opposite of a sum is the sum of its opposites. Ex.: –(-3 + 4) = 3 + (-4).• Distance between 2 rational numbers p and q on a number line is |p - q|• Every rational number can be written as a decimal that either terminate s

in zeros or repeat.

A Story of Ratios

Documents

module overview

module focus sessionnovember

common core institutegrade

positive rational numbers

negative rational numbers

division of rational

story of ratiosgrade

number line