NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
A Story of Ratios
Grade 6 – Module 3
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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.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development
Module Review
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Curriculum Overview of A Story of Ratios
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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.
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NYS COMMON CORE MATHEMATICS CURRICULUM
G6-M3: Module Overview
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
G6-M3: Vocabulary and Representations
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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
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topic A: Understanding Positive and Negative Numbers on
the Number Line
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NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development – Topic A
Module Review
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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
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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
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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
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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
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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
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Real World Positive and Negative Numbers and
Zero
Lesson 3 /Exit Ticket
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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., −(−𝑎) = 𝑎.
• Students locate and position opposite numbers on a
number line.
Lesson 5
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NYS COMMON CORE MATHEMATICS CURRICULUM
The Opposite of a Number’s Opposite
Lesson 5/Example
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
The Opposite of a Number’s Opposite
Lesson 5/Activity
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Rational Numbers on the Number Line
Lesson 6
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Rational Numbers on the Number Line
Lesson 6/Exercise 1
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Rational Numbers on the Number Line
Lesson 6/Exercise 1
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Rational Numbers on the Number Line
Lesson 6/Exit Ticket
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NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development – Topic B
Module Review
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Topic B: Order and Absolute Value
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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
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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
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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 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 8
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Writing and Interpreting Inequality Statements Involving
Rational Numbers
Inequalities Fluency Builder
Lesson 10
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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
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NYS COMMON CORE MATHEMATICS CURRICULUM
Mid-Module Assessment
Question 3
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development – Topic C
Module Review
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topic C: Rational Numbers and the Coordinate Plane
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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
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment
Questions 1 and 5
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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.
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