Lesson 9

The purpose of this lesson is to develop the rules for multiplying two negative numbers. Students use the familiar fact that \(\displaystyle \mbox{distance} = \mbox{velocity}\times\mbox{time}\) to make sense of this rule. They interpret negative time as time before a chosen starting time and then figure out what the position is of an object moving with a negative velocity at a negative time. An object moving with a negative velocity is moving from right to left along the number line. At a negative time it has not yet reached its starting point of zero, so it is to the right of zero, and therefore its position is positive. So a negative velocity times a negative time gives a positive position. When students connect reasoning about quantities with abstract properties of numbers, they engage in MP2.

There is also an optional activity where students can use another approach to understanding why the product of two negative numbers is positive, by examining patterns in an extended multiplication table that includes both positive and negative numbers (MP7).

 It is optional to provide 1 copy of the Rational Numbers Multiplication Grid blackline master to each student.