Unit 5, Lesson 9
Multiplying Rational Numbers (Part 2)
Grade 7
9.1
Warm-up
- Where was the person 5 seconds after this picture was taken?
- Where was the person 5 seconds before this picture was taken?
9.2
Activity
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.
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A car was traveling east at 12 meters per second. Where was the car 10 seconds before it passed the camera?
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A car was traveling west at -14 meters per second. Where was the car 10 seconds before it passed the camera?
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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.
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A positive number times a negative number equals a _______________________.
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A negative number times a negative number equals a _______________________.
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9.4
Activity
- Complete the shaded boxes in the multiplication square.
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.
- Look at the patterns along the rows and columns. Continue those patterns into the unshaded boxes.
- Complete the whole table.
- What does this tell you about multiplication with negative numbers?
Student Lesson Summary
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:
- A positive number times a positive number always results in a positive number.
- A negative number times a positive number or a positive number times a negative number always results in a negative number.
- A negative number times a negative number always results in a positive number.
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