# Lesson 9

Multiplying Rational Numbers

### Lesson Narrative

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).

### Learning Goals

Teacher Facing

- Generalize (orally) that the product of two negative numbers is positive.
- Interpret signed numbers used to represent elapsed time before or after a chosen reference point.
- Use patterns to find the product of signed numbers, and explain (orally and using other representations) the reasoning.

### Student Facing

Let's multiply signed numbers.

### Required Materials

### Required Preparation

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

### Learning Targets

### Student Facing

- I can explain what it means when time is represented with a negative number in a situation about speed and direction.
- I can multiply two negative numbers.