Unit 4: 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.
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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The following assessments accompany Unit 4.
Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment after lesson 10.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment Answer Key
Post-Unit Student Self-Assessment
Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.
Suggestions for how to prepare to teach this unit
Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.
The central mathematical concepts that students will come to understand in this unit
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 6th Grade Course Material Overview.
Terms and notation that students learn or use in the unit
reflection (of a coordinate point)
To see all the vocabulary for Unit 4, view our 6th Grade Vocabulary Glossary.
Topic A: Understanding Positive and Negative Rational Numbers
Extend the number line to include negative numbers. Define integers.
Use positive and negative numbers to represent real-world contexts, including money and temperature.
Use positive and negative numbers to represent real-world contexts, including elevation.
Define opposites and label opposites on a number line. Recognize that zero is its own opposite.
Find and position integers and rational numbers on the number line.
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Topic B: Order and Absolute Value
Order integers and rational numbers. Explain reasoning behind order using a number line.
Compare and interpret the order of rational numbers for real-word contexts.
Write and interpret inequalities to compare rational numbers in real-world and mathematical problems.
Define absolute value as the distance from zero on a number line.
Model magnitude and distance in real-life situations using order and absolute value.
Topic C: Rational Numbers in the Coordinate Plane
Use ordered pairs to name locations on a coordinate plane. Understand the structure of the coordinate plane.
Reflect points across axes and determine the impact of reflections on the signs of ordered pairs.
Calculate vertical and horizontal distances on a coordinate plane using absolute value in real-world and mathematical problems.
The content standards covered in this unit
— 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.
— 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.
— 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.
— 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.
— 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.
— Understand ordering and absolute value of rational numbers.
— 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.
— 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.
— 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.
— 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.
— 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.
Standards covered in previous units or grades that are important background for the current unit
— Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
— Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
— Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
— Understand a fraction as a number on the number line; represent fractions on a number line diagram.
— Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
— Use decimal notation for fractions with denominators 10 or 100.
For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
— Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Standards in future grades or units that connect to the content in this unit
— Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
— Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
— Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
— Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
— Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
— Solve real-world and mathematical problems involving the four operations with rational numbers.
Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Multi-Digit and Fraction Computation
Numerical and Algebraic Expressions