7.5 Rational Number Arithmetic

In this unit, students interpret signed numbers in contexts (e.g., temperature, elevation, deposit and withdrawal, position, direction, speed and velocity, percent change) together with their sums, differences, products, and quotients. (“Signed numbers” include all rational numbers, written as decimals or in the form \(\frac a b.\)) Students use tables and number line diagrams to represent sums and differences of signed numbers or changes in quantities represented by signed numbers such as temperature or elevation, becoming more fluent in writing different numerical addition and subtraction equations that express the same relationship. They compute sums and differences of signed numbers. They plot points in the plane with signed number coordinates, representing and interpreting sums and differences of coordinates. They view situations in which objects are traveling at constant speed (familiar from previous units) as proportional relationships. For these situations, students use multiplication equations to represent changes in position on number line diagrams or distance traveled, and interpret positive and negative velocities in context. They become more fluent in writing different numerical multiplication and division equations for the same relationship. Students extend their use of the “next to” notation (which they used in expressions such as \(5x\) and \(6 (3 + 2)\) in grade 6) to include negative numbers and products of numbers, e.g., writing \(\text-5x\) and \((\text-5) (\text-10)\) rather than \((\text-5)\boldcdot (x)\) and \((\text-5)\boldcdot (\text-10)\). They extend their use of the fraction bar to include variables as well as numbers, writing \({\text-8.5}\div{x}\) as well as \(\frac{\text-8.5}{x}\).