Lesson 9Multiplying Rational Numbers

Let's multiply signed numbers.

Learning Targets:

  • 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.

9.1 Before and After

Where was the girl

  1. 5 seconds after this picture was taken? Mark her approximate location on the picture.
  2. 5 seconds before this picture was taken? Mark her approximate location on the picture.

9.2 Backwards in Time

A traffic safety engineer was studying travel patterns along a highway. She set up a camera and recorded the speed and direction of cars and trucks that passed by the camera. Positions to the east of the camera are positive, and to the west are negative.

  1. Here are some positions and times for one car:

    position (feet) -180 -120 -60 0   60  120
    time (seconds) -3 -2 -1 0 1 2
    1. In what direction is this car traveling?
    2. What is its velocity?
    1. What does it mean when the time is zero?

    2. What could it mean to have a negative time?

  2. Here are the positions and times for a different car whose velocity is -50 feet per second:

    position (feet) 0   -50 -100
    time (seconds) -3   -2   -1   0 1 2
    1. Complete the table with the rest of the positions.
    2. In what direction is this car traveling? Explain how you know.
  3. Complete the table for several different cars passing the camera.

    velocity
    (meters per
    second)
    time after
    passing
    camera
    (seconds)
    ending
    position
    (meters)
    equation  
    car C +25 +10 +250 25\boldcdot 10 = 250
    car D -20 +30
    car E +32 -40
    car F -35 -20
    car G -15 -8
    1. If a car is traveling east when it passes the camera, will its position be positive or negative 60 seconds before it passes the camera?
    2. If we multiply a postive number and a negative number, is the result positive or negative?
    1. If a car is traveling west when it passes the camera, will its position be positive or negative 60 seconds before it passes the camera?
    2. If we multiply two negative numbers, is the result positive or negative?

9.3 Cruising

Around noon, a car was traveling -32 meters per second down a highway. At exactly noon (when time was 0), the position of the car was 0 meters.

  1. Complete the table.

    time (s) -10  -7  -4  -1  11 
    position (m)
  2. Graph the relationship between the time and the car's position.
    1. What was the position of the car at -3 seconds?
    2. What was the position of the car at 6.5 seconds?

Are you ready for more?

Find the value of these expressions without using a calculator.

(\text-1)^2

(\text-1)^3

(\text-1)^4

(\text-1)^{99}

9.4 Rational Numbers Multiplication Grid

Look at the patterns along the rows and columns and continue those patterns to complete the table. When you have filled in all the boxes you can see, click on the "More Boxes" button.

What does this tell you about multiplication by a negative?

Lesson 9 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.

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. A positive times a positive is positive.

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 negative times a positive is negative.

If a car is at position 0 and is moving in a positive direction, then for times before that (negative times), it must have had a negative position. A positive times a negative is negative.

If a car is at position 0 and is moving in a negative direction, then for times before that (negative times), it must have had a positive position. A negative times a negative is positive.

Here is another way of seeing this:

We can think of 3\boldcdot 5 as 5 + 5 + 5 , which has a value of 15 .

We can think of 3\boldcdot (\text-5) as (\text-5) + (\text-5) + (\text-5) , which has a value of \text-15 .

We can multiply positive numbers in any order: 3\boldcdot  5=5\boldcdot  3

If we can multiply signed numbers in any order, then \text-5\boldcdot 3 = \text-15 .

We can find \text-5\boldcdot (3+(\text-3)) two ways:

  • \text-5\boldcdot 0 = 0
  • \text-5\boldcdot  3 + (\text-5)\boldcdot  (\text-3) (this is the distributive property)

That means that

\text-5\boldcdot  3 + (\text-5)\boldcdot (\text-3) = 0

Which is the same as

\text-15 + (\text-5)\boldcdot  (\text-3) = 0

So

\text-5\boldcdot  (\text-3) = 15

There was nothing special about these particular numbers. This always works!

  • A positive times a positive is always positive
  • A negative times a positive or a positive times a negative is always negative
  • A negative times a negative is always positive

Lesson 9 Practice Problems

  1. Fill in the missing numbers in these equations

    1. (\text-2) \boldcdot (\text-4.5) = {?}
    2. (\text-8.7) \boldcdot (\text-10) = {?}
    3. (\text-7) \boldcdot {?} = 14
    4. {?} \boldcdot (\text-10) = 90
  2. A weather station on the top of a mountain reports that the temperature is currently 0^\circ\text{C} and has been falling at a constant rate of 3^\circ\text{C} per hour. Find each temperature. Explain or show your reasoning.

    1. If it continues fall at this rate, what will the temperature be:

      1. in 2 hours?
      2. in 5 hours?
      3. in half an hour?
    1. What was the temperature:

      1. 1 hour ago?
      2. 3 hours ago?
      3. 4.5 hours ago?
  3. Find the value of each expression.

    1. \frac14 \boldcdot (\text-12)
    2. (\text{-}\frac {1}{3}) \boldcdot 39
    3. (\text{-}\frac {4}{5}) \boldcdot (\text-75)
    4. (\text{-}\frac {2}{5}) \boldcdot (\text{-}\frac {3}{4})
    5. \frac83 \boldcdot (\text-42)
  4. To make a specific hair dye, a hair stylist uses a ratio of 1\frac18 oz of red tone, \frac 34 oz of gray tone, and \frac58 oz of brown tone.

    1. If the stylist needs to make 20 oz of dye, how much of each dye color is needed?

    2. If the stylist needs to make 100 oz of dye, how much of each dye color is needed?

    1. Here are the vertices of rectangle FROG : (\text-2,5), (\text-2,1), (6,5), (6,1) . Find the perimeter of this rectangle. If you get stuck, try plotting the points on a coordinate plane.
    2. Find the area of the rectangle FROG .
    3. Here are the coordinates of rectangle PLAY : (\text-11, 20), (\text-11, \text-3), (\text-1, 20), (\text-1, \text-3) . Find the perimeter and area of this rectangle. See if you can figure out its side lengths without plotting the points.