Information for course "Unit 7 - Rational Numbers"

Description: In this unit, students are introduced to signed numbers and plot points in all four quadrants of the coordinate plane for the first time. They work with simple inequalities in one variable and learn to understand and use “common factor,” “greatest common factor,” “common multiple,” and “least common multiple.”

The first section of the unit introduces signed numbers. Students begin by considering examples of positive and negative temperatures, plotting each temperature on a vertical number line on which 0 is the only label. Next, they consider examples of positive and negative numbers used to denote height relative to sea level. In the second lesson, they plot positive and negative numbers on horizontal number lines, including “opposites”—pairs of numbers that are the same distance from zero. They use “less than,” “greater than,” and the corresponding symbols to describe the relationship of two signed numbers, noticing correspondences between the relative positions of two numbers on the number line and statements that use these symbols, e.g., 0.8>-1.3 means that 0.8 is to the right of -1.3 on the number line. Students learn that the sign of a number indicates whether the number is positive or negative, and that zero has no sign. They learn that the absolute value of a number is its distance from zero, how to use absolute value notation, and that opposites have the same absolute value because they have the same distance from zero.

Previously, when students worked only with non-negative numbers, magnitude and order were indistinguishable: if one number was greater than another, then on the number line it was always to the right of the other number and always farther from zero. In comparing two signed numbers, students distinguish between magnitude (the absolute value of a number) and order (relative position on the number line), distinguishing between “greater than” and “greater absolute value,” and “less than” and “smaller absolute value.”

Students examine opposites of numbers, noticing that the opposite of a negative number is positive.

The second section of the unit concerns inequalities. Students graph simple inequalities in one variable on the number line, using a circle or disk to indicate when a given point is, respectively, excluded or included. Student will use the following symbols: ≤ , ≥ , < and >.

Students represent situations that involve inequalities, symbolically and with the number line, understanding that there may be infinitely many solutions for an inequality. They interpret and graph solutions in contexts, understanding that some results do not make sense in some contexts, and thus the graph of a solution might be different from the graph of the related symbolic inequality. For example, the graph describing the situation “A fishing boat can hold fewer than 9 people” omits values other than the whole numbers from 0 to 8, but the graph of x<8 includes all numbers less than 8. Students encounter situations that require more than one inequality statement to describe, e.g., “It rained for more than 10 minutes but less than 30 minutes” (t>10 and t<30, where t is the amount of time that it rained in minutes) but which can be described by one number line graph.

The third section of the unit focuses on the coordinate plane. In grade 5, students learned to plot points in the coordinate plane, but they worked only with non-negative numbers, thus plotted points only in the first quadrant. In a previous unit, students again worked in the first quadrant of the coordinate plane, plotting points to represent ratio and other relationships between two quantities with positive values. In this unit, students work in all four quadrants of the coordinate plane, plotting pairs of signed number coordinates in the plane. They understand that for a given data set, there are more and less strategic choices for the scale and extent of a set of axes. They understand the correspondence between the signs of a pair of coordinates and the quadrant of the corresponding point. They interpret the meanings of plotted points in given contexts (MP2), and use coordinates to calculate horizontal and vertical distances between two points.

The last section of the unit returns to consideration of whole numbers. In the first lesson, students are introduced to “common factor” and “greatest common factor,” and solve problems that illustrate how the greatest common factor of two numbers can be used in real-world situations, e.g., determining the largest rectangular tile with whole-number dimensions that can tile a given rectangle with whole-number dimensions. The second lesson introduces “common multiple” and “least common multiple,” and students solve problems that involve listing common multiples or identifying common multiples of two or more numbers. In the third and last lesson, students solve problems that revisit situations similar to those in the first two lessons and identify which of the new concepts is involved in each problem.