In this lesson, students explore multiplying a negative number by a negative number. To make sense of why such a product is positive, they expand the context of constant velocity to include times before a chosen reference point.

First, students learn to interpret negative values for time as times before a chosen starting point. Assuming that the object was already moving at a constant velocity, they calculate its position at various negative times. An object that is moving with a constant negative velocity will be at a negative position after crossing the reference point, but before crossing this point its position was positive. That is, . As students make sense of situations that involve negative velocity or negative time and represent these situations with equations, they are reasoning abstractly and quantitatively (MP2).

The following two activities are optional, so teachers can choose what best fits the needs of their students. Both of the activities are designed to reveal patterns that reinforce what students have learned about multiplying signed numbers. The first optional activity uses the structure of the coordinate grid. The second optional activity expands the standard multiplication table to include negative factors. Teachers may select one activity for the whole class to work on or they may let each student choose.

Some of the activities in this lesson work best when each student has access to a device that can run the applet because they will benefit from seeing the relationship in a dynamic way.