A Story of Ratios
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Transcript of 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
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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.
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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
AgendaIntroduction to the ModuleConcept DevelopmentModule Review
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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
Curriculum Overview of A Story of Ratios
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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
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.
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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
G7-M2: Module Overview
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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
G7-M2: Vocabulary and Representations
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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
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
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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
Topic A: Addition and Subtraction of Integers and Rational Numbers
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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
AgendaIntroduction to the ModuleConcept Development – Topic AModule Review
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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
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
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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
Opposite Quantities Combine to Make Zero
Lesson 1/Example 3
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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
Opposite Quantities Combine to Make Zero
The Integer Game
Lesson 1, Concept Development
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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
Opposite Quantities Combine to Make Zero
Let’s Play the Integer Game!
Lesson 1, Concept Development
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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
Using the Number Line to Model the Addition of Integers
Lesson 2/Example 1
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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
Using the Number Line to Model the Addition of Integers
Lesson 2/Example 3
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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
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.
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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
Understanding Addition of Integers
Lesson 3/Exit Ticket
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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
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.
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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
Understanding Subtraction of Integers and Other Rational Numbers
Lesson 5/Example
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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
The Distance Between Two Rational Numbers
Lesson 6/Closing
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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
The Distance Between Two Rational Numbers
Lesson 6/Exit Ticket
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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
The Distance Between Two Rational Numbers
Lesson 6/Problem Set
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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
Addition and Subtraction of Rational Numbers
Lesson 7
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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
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
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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
Applying the Properties of Operations to Add and Subtract Rational Numbers
Lessons 8 and 9
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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
AgendaIntroduction to the ModuleConcept Development – Topic BModule Review
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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
Topic B: Multiplication and Division of Integers and Rational Numbers
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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
Understanding the Multiplication of Integers
Lesson 10/Example 3
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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
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.
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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
Develop Rules for Multiplying Signed Numbers
Lesson 11/Example 1
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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
Develop Rules for Multiplying Signed Numbers
Lesson 11
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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
Converting Rational Numbers to Decimals Using Long Division
Lesson 14/Example 2
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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
Mid-Module Assessment
Question 6
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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
AgendaIntroduction to the ModuleConcept Development – Topic CModule Review
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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
Topic C: Applying Operations with Rational Numbers to Expressions and Equations
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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
Comparing Tape Diagram Solutions to Algebraic Solutions
Lesson 17
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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
Comparing Tape Diagram Solutions to Algebraic Solutions
Lesson 17/Exit Ticket
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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
If-Then Moves with Integer Cards
Lesson 21/ Exercises 1
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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
If-Then Moves with Integer Cards
Lesson 21/ Exercises 1
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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
Solving Equations Using Algebra
Lesson 23/Exercise 2
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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
Solving Equations Using Algebra
Lesson 23/Exit Ticket
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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
End-of-Module Assessment
Questions 2 and 3
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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
AgendaIntroduction to the ModuleConcept DevelopmentModule Review
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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
Biggest TakeawayTurn and Talk:• What questions were answered for you?• What new questions have surfaced?
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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
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.