Lesson 10

In this lesson, students see that the difference between two numbers can be positive or negative, but the distance between two numbers is always positive. Using the geometry of the number line (MP7), they see that if you switch the order in which you subtract two numbers, the difference becomes its opposite.

For example, to find the difference in temperature between +70\(^\circ \text{C}\) and +32\(^\circ \text{C}\) we calculate \(70 - 32 = 38\), so the difference is 38\(^\circ \text{C}\). The distance between these two is also 38\(^\circ \text{C}\). On the other hand, to find the difference in temperature between +32\(^\circ \text{C}\) and \(+70^\circ \text{C}\) we calculate \(32 - 70 = -38\), so the difference is \(-38^\circ \text{C}\). The distance is still 38\(^\circ \text{C}\). In general, if \(a - b = x\), then \(b - a = -x\). By observing the outcome of several examples, students may conjecture that this is always true (MP8).

They also work with tables that show the change, positive or negative, in quantities such as inventory or energy usage, and must make sense of these tables to answer questions about the context. As students reason about quantities using signed numbers they engage in MP2.

Use of calculators is optional. In this lesson, the important insights come from observing the outcome of evaluating expressions. Practice evaluating the expressions is of secondary importance.