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The purpose of this discussion is to get students thinking about constant speeds and negative times. Begin by displaying the image from the Task Statement for all to see. Invite students to share where they think the person was 5 seconds earlier and will be 5 seconds later.
If necessary, point out that if we assume the person is walking at a constant speed, their before and after locations should be equally far from their current position in the image. Ask students how they might represent time in this situation. (5 seconds before the picture was taken could be -5, while 5 seconds after the picture was taken would be +5. The picture was taken when the time was 0.)
In this activity, students consider a context of vehicles traveling past a traffic camera at varied constant velocities and in different directions. Students reason about what it means to have a negative time (the vehicle has not yet passed the traffic camera) or a negative velocity (the vehicle is traveling from east to west) in this context. They engage in reasoning abstractly and quantitatively as they represent velocity, distance, and time relationships using an equation, then interpret the results in terms of the situation (MP2).
Students also use repeated reasoning to describe a pattern for identifying the sign of the product of two negative numbers (MP8).
Keep students in the same groups. Remind the students of movement east or west as positive or negative velocity.
This activity is the same context as one in the previous lesson, and the questions are related. So students should be able to get to work rather quickly. However, each question requires some careful thought, and one question builds on the other. Consider suggesting that students check in with their partner frequently and explain their thinking. Additionally, you might consider asking students to pause after each question for a quick whole-class discussion before continuing to the next question.
A traffic safety engineer was studying traffic patterns. She set up a camera to record the speed and direction of cars and trucks that passed by. She decided to represent positions to the east of the camera with positive numbers and positions to the west of the camera with negative numbers.
Number line. To the left is the word west. To the right is the word east. The numbers negative 50 through 50, in increments of 10, are indicated. There are tick marks midway between.
A car was traveling east at 12 meters per second. Where was the car 10 seconds before it passed the camera?
A car was traveling west at -14 meters per second. Where was the car 10 seconds before it passed the camera?
Complete the table to show the position of each vehicle after traveling at a constant velocity for the given amount of time.
| velocity (meters per second) |
time after passing the camera (seconds) |
position (meters) |
equation | |
|---|---|---|---|---|
| car A | +12 | -10 | -120 | |
| car B | -14 | -10 | ||
| car C | +9 | -6 | ||
| car D | -11 | -9 | ||
| car E | -15 | -4 | ||
| car F | +8 | -13 |
Complete the sentences. Be prepared to explain your reasoning.
A positive number times a negative number equals a _______________________.
A negative number times a negative number equals a _______________________.
The key thing for students to understand from this discussion is that a negative number multiplied by another negative number results in a positive number. Begin by displaying the number line and table from the Task Statement for all to see. Invite students to share their responses, and record them in the table.
Then select a car whose equation has a positive and a negative factor, and demonstrate its path on the number line. For example, explain that Car A is traveling from west to east since its velocity is positive. Since its time is represented by a negative number, the car has not yet passed the camera and, therefore, must be located on the left side of the camera, making its position a negative number.
Then select a car whose equation has two negative factors. For example, Car D has a negative velocity, which means that it is traveling from east to west. It also has a negative time, which means it has not yet passed the camera. Car D must be located to the right of the camera, making its position a positive number.
Ask students to complete the sentences in the last problem. Consider posting these in the classroom for all to see for the next few lessons.
We can use signed numbers to represent time relative to a chosen point in time. We can think of this as starting a stopwatch. The positive times are after the watch starts, and negative times are times before the watch starts.
Three points are labeled on a number line. The numbers negative 10 through 10, in increments of 1, are indicated. A dot at negative 4 is labeled "4 seconds before the start time". A dot at 0 is labeled "start time". A dot at 7 is labeled "7 seconds after the start time".
If a car is at position 0 and is moving in a positive direction, then for times after that (positive times), it will have a positive position.
Number line. 7 evenly spaced tick marks. Scale negative 15 to 15, by 5's. Three equal sized arrows pointing to the right, one from 0 to 5, 5 to 10, and 10 to 15.
For times before that (negative times), it must have had a negative position.
A blank horizontal number line from negative 15 to 15 by 5’s. Above the number line with three arrows pointing right and a dot are plotted. The first arrow points from negative 15 to negative 10. The second arrow points from negative 10 to negative 5. The third arrow points from negative 5 to 0. A dot is above 0.
If a car is at position 0 and is moving in a negative direction, then for times after that (positive times), it will have a negative position.
A blank horizontal number line from negative 15 to 15 by 5’s. Above the number line with three arrows pointing left and a dot are plotted. The first arrow points from 0 to negative 5. The second arrow points from negative 5 to negative 10. The third arrow points from negative 10 to negative 15. A dot is above 0.
For times before that (negative times), it must have had a positive position.
A blank horizontal number line from negative 15 to 15 by 5’s. Above the number line with three arrows pointing left and a dot are plotted. The first arrow points from 15 to 10. The second arrow points from 10 to 5. The third arrow points from 5 to 0. A dot is above 0.
Here is another way of seeing this:
Arrange students in groups of 2. Give them 6 minutes of quiet work time followed by time for partner discussion. Then follow with a whole-class discussion.
Around noon, a car was traveling -25 meters per second down a highway. Use the position of the car at exactly noon as the reference point.
Complete the table.
| time () | -8 | -6 | -4 | -2 | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| position () | 0 |
Arrange students in groups of 3. If desired, distribute 1 copy of the blackline master to every student, and instruct students to ignore the chart printed in their books or devices. (Also if desired, instruct students to fold their papers according to the directions on the top and right sides of the chart so that the decimal rows and columns are temporarily hidden.) Give students 30 seconds of quiet think time. Have them share what patterns they notice about the numbers that are already filled in. Give the groups 5 minutes of work time, and follow with a whole-class discussion.
A blank multiplication chart. The horizontal axis is labeled negative 5 through 5. The vertical axis is labeled negative 5 through 5. The top right corner, from 0 through positive 5 on the horizontal and vertical axes, is shaded. Starting with 0 on the horizontal axis, each row is as follows: Row for 5: 0, 5, 10, 15, 20, blank Row for 4: 0, 4, 8, 12, 16, blank Row for 3: 0, 3, 6, 9, 12, blank Row for 2: 0, 2, 4, 6, 8, blank Row for 1: 0, 1, 2, 3, 4, blank Row for 0: 0, 0, 0, 0, blank Row 2, column negative 1 is filled in with the number negative 2.