# Lesson 14

Position, Speed, Direction

### Lesson Narrative

In this lesson, students are introduced to multiplying signed numbers, using the context of velocity, time, and position.

The context of elevation is an example of using signed numbers to represent the position of an object along a line relative to a reference position (sea level in the case of elevation). In the general case, zero represents the reference position, positive numbers represent positions on one side of the reference position, and negative numbers represent positions on the other side. In this lesson, students see that signed numbers can also be used to represent speed with direction. 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.

In previous units, students solved problems about moving objects, using the fact that the product of the (positive) speed and the (positive) travel time gives the (positive) distance traveled. In this lesson, students use several examples in the context of moving along a line to see that the product of a negative velocity and a positive travel time results in a negative position relative to the starting point.

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.

### Learning Goals

Teacher Facing

• 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.
• Write a multiplication equation to represent a situation involving constant speed with direction.

### Student Facing

Let's use signed numbers to represent movement.

### 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.
• I can use rational numbers to represent speed and direction.

Building On