Rational Numbers
Students are introduced to integers and rational numbers, extending the number line to include negative values, understanding the order of rational numbers, and interpreting them in context.
Unit Summary
In Unit 4, sixth-grade students extend their understanding of numbers to include rational numbers. Prior to this unit, students have worked only with positive values, and their concepts of number lines and coordinate planes have been limited by these positive values. Students explore real-world situations that naturally connect to negative values, such as temperature, money, and elevation. The number line is a valuable tool that is referred to and used throughout the unit. Students use the number line to develop understanding of negatives, opposites, absolute value, and comparisons and inequalities (MP.5). They also discover the four-quadrant coordinate plane by intersecting two number lines at a 90-degree angle and representing locations using ordered pairs.
In elementary grades, students build and develop their sense of number with positive values. They use the number line as a tool to better understand whole numbers, fractions, and decimals. In fifth grade, students look at the first quadrant of the coordinate plane and represent locations using ordered pairs of positive numbers. In sixth grade, students build on and extend these concepts to include negative values.
In seventh grade, students will discover how to compute with rational numbers and what happens when the properties of operations are applied to negative values. The work they do in this sixth-grade unit is foundational of these seventh-grade concepts.
Pacing: 16 instructional days (13 lessons, 2 flex days, 1 assessment day)
For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 6th Grade Scope and Sequence Recommended Adjustments.
Assessment
This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.
Unit Prep
Intellectual Prep
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Internalization of Standards via the Post-Unit Assessment
- Take Post-Unit Assessment. Annotate for:
- Standards that each question aligns to
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that Assessment points to
Internalization of Trajectory of Unit
- Read and annotate the Unit Summary.
- Notice the progression of concepts through the unit using the Lesson Map.
- Do all Target Tasks. Annotate the Target Tasks for:
- Essential Understandings
- Connection to Post-Unit Assessment questions
- Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.
Unit-Specific Intellectual Prep
- Read the Progressions for the Common Core State Standards in Mathematics, The Number System, 6-8 for relevant standards in this Number System domain.
Essential Understandings
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- The number line can be extended to the left or downward to include negative values; smaller or lesser numbers are always located to the left (or downward) on the number line, and greater numbers are located to the right (or upward) on the number line (using the convention for a number line that is ordered with the smaller numbers to the left or to the bottom).
- Opposite numbers are the same distance from 0 but on opposite sides of 0 on the number line; opposite numbers have the same absolute value since they are the same distance from 0.
- Integers and other rational numbers can be used to represent and model real-world values, including situations with negatives.
- Magnitude and distance refer to a positive amount and can be represented using absolute value; ordering values involves listing values either from least to greatest or greatest to least as they would be shown on a number line.
- When two perpendicular number lines intersect, they create a four-quadrant coordinate plane. The coordinate plane can be used to describe location in two dimensions, defined by an ordered pair.
Vocabulary
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rational number
charge
integer
deposit
withdrawal
credit
elevation
inequality
absolute value
opposite
coordinate plane
quadrant
reflection (of a coordinate point)
ordered pair
To see all the vocabulary for this course, view our 6th Grade Vocabulary Glossary.
Materials
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- number line, vertical and horizontal (laminated optional)
- dry erase markers (optional)
- Coordinate plane
- graph paper
- patty (transparency) paper
Lesson Map
Topic A: Understanding Positive and Negative Rational Numbers
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6.NS.C.6
6.NS.C.6.C
Extend the number line to include negative numbers. Define integers.
2
6.NS.C.5
Use positive and negative numbers to represent real-world contexts, including money and temperature.
3
6.NS.C.5
Use positive and negative numbers to represent real-world contexts, including elevation.
4
6.NS.C.6.A
6.NS.C.6.B
Define opposites and label opposites on a number line. Recognize that zero is its own opposite.
5
6.NS.C.6.C
Find and position integers and rational numbers on the number line.
Topic B: Order and Absolute Value
6
6.NS.C.6.C
6.NS.C.7.A
Order integers and rational numbers. Explain reasoning behind order using a number line.
7
6.NS.C.6.C
6.NS.C.7.A
Compare and interpret the order of rational number for real-word contexts.
8
6.NS.C.7.A
6.NS.C.7.B
Write and interpret inequalities to compare rational numbers in real-world and mathematical problems.
9
6.NS.C.7.C
Define absolute value as the distance from zero on a number line.
10
6.NS.C.7.C
6.NS.C.7.D
Model magnitude and distance in real-life situations using order and absolute value.
Topic C: Rational Numbers in the Coordinate Plane
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6.NS.C.6.B
6.NS.C.6.C
Use ordered pairs to name locations on a coordinate plane. Understand the structure of the coordinate plane.
12
6.NS.C.6.B
Reflect points across axes and determine the impact of reflections on the signs of ordered pairs.
13
6.NS.C.7.C
6.NS.C.8
Calculate vertical and horizontal distances on a coordinate plane using absolute value in real-world and mathematical problems.
Common Core Standards
Key: Major Cluster Supporting Cluster Additional Cluster
Core Standards
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The Number System
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6.NS.C.5 — Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
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6.NS.C.6 — Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
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6.NS.C.6.A — Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.
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6.NS.C.6.B — Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
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6.NS.C.6.C — Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
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6.NS.C.7 — Understand ordering and absolute value of rational numbers.
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6.NS.C.7.A — Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right.
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6.NS.C.7.B — Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write -3 °C > -7 °C to express the fact that -3 °C is warmer than -7 °C.
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6.NS.C.7.C — Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars.
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6.NS.C.7.D — Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars.
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6.NS.C.8 — Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Foundational Standards
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Geometry
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4.G.A.3
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5.G.A.1
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5.G.A.2
Number and Operations—Fractions
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3.NF.A.2
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4.NF.A.2
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4.NF.C.6
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4.NF.C.7
Future Standards
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Expressions and Equations
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6.EE.B.8
Geometry
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6.G.A.3
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8.G.B.8
The Number System
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7.NS.A.1
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7.NS.A.2
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7.NS.A.3
Standards for Mathematical Practice
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CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
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CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
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CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
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CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
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CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
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CCSS.MATH.PRACTICE.MP6 — Attend to precision.
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CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
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CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.