6.7 GeoGebra Illustrative Mathematics - Rational Numbers

In this unit, students learn about negative numbers and ways to represent them on a number line and the coordinate plane. They write and graph simple inequalities in one variable and determine the greatest common factor and least common multiple of two whole numbers. In grade 7, students will perform arithmetic operations with signed numbers and write and solve more complex inequalities.

Students begin by considering situations involving temperature or elevation and interpreting what negative numbers mean in those contexts. They also plot points to represent positive and negative values and their opposites. Previously, when students worked only with nonnegative numbers, magnitude and order were indistinguishable. In this unit, when comparing two signed numbers, students learn to distinguish between the absolute value of a number (magnitude) and a number’s relative position on the number line (order).

1. 15 feet
2.  feet
3.  feet
4. -4 feet

Next, students use the symbols  and  to compare two values. They represent an unknown value with constraints as an inequality, and they graph the solutions on a number line. Students consider the inclusion or exclusion of boundary values and interpret solutions based on the context. In these grade 6 materials, inequality symbols in are limited to  and . However, in this unit students encounter situations that are best represented by both an inequality and an equation, such as “ or .”

Then students use ordered pairs to describe pairs of numbers that include negative numbers. In grade 5, they plotted pairs of positive numbers on the coordinate grid. Here, they plot pairs of rational numbers in all four quadrants of the coordinate plane. They interpret the meanings of plotted points in given contexts and use coordinates to calculate horizontal or vertical distances between two points.

The last section of the unit returns to whole numbers. Students are introduced to common factors and common multiples. They determine the greatest common factor or the least common multiple of two numbers. They identify how these new concepts are involved in real-world situations and use their understanding to solve related problems.