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)

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Summer Packets

Packet of problems designed to help prepare students for the start of next school year.

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Unit Launch

Online learning modules that include short videos and readings to help teachers prepare to teach a unit.

 

Assessment

This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.

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Expanded Assessment Package

Additional assessment tools to help you monitor student learning and identify any skill or knowledge gaps.

Learn how to use these tools with our Guide to Assessments

 

Unit Prep

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

ordered pair

opposite

coordinate plane

quadrant

reflection (of a coordinate point)

Related Teacher Tools:

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Vocabulary Package

Additional vocabulary tools that help reinforce and support student vocabulary development.

Unit Materials, Representations and Tools

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  • Number line, vertical and horizontal
  • Coordinate plane

Intellectual Prep

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Internalization of Standards via the Unit Assessment

  • Take 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 "Unit Summary."
  • Notice the progression of concepts through the unit using "Unit at a Glance."
  • Do all target tasks. Annotate the target tasks for: 
    • Essential understandings
    • Connection to assessment questions
  • Identify key opportunities to engage students in academic discourse. Read through A Guide to Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

Lesson Map

Topic A: Understanding Positive and Negative Rational Numbers

Topic B: Order and Absolute Value

Topic C: Rational Numbers in the Coordinate Plane

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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The Number System
  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 6.NS.C.7 — Understand ordering and absolute value of rational numbers.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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
  • 4.G.A.3

  • 5.G.A.1

  • 5.G.A.2

Measurement and Data
  • 5.MD.B.2

Number and Operations—Fractions
  • 3.NF.A.2

  • 4.NF.A.2

Future Standards

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Expressions and Equations
  • 6.EE.B.8

Geometry
  • 6.G.A.3

  • 8.G.B.8

The Number System
  • 7.NS.A.1

  • 7.NS.A.2

  • 7.NS.A.3

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.