In this lesson, students explore multiplying a negative number by a positive number. To make sense of why such a product is negative, they use the context of constant velocity. Students also 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.

Scientists use the term "velocity" to describe the speed of an object in a specified direction. If one object is moving with a positive velocity, then any object moving in the opposite direction will have a negative velocity. Building on their previous work with constant speed, students calculate the final position for several different combinations of velocities and times. They see that the product of a negative velocity and a positive travel time results in a negative position relative to the starting point. As students reason through multiple examples to develop these generalizations about multiplication, they are making use of repeated reasoning (MP8).

Next, 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).