9.1: Before and After
Where was the girl
- 5 seconds after this picture was taken? Mark her approximate location on the picture.
- 5 seconds before this picture was taken? Mark her approximate location on the picture.
Let's multiply signed numbers.
Where was the girl
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.
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 |
What does it mean when the time is zero?
What could it mean to have a negative time?
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 |
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 |
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.
Complete the table.
time (s) | -10 | -7 | -4 | -1 | 2 | 5 | 8 | 11 |
---|---|---|---|---|---|---|---|---|
position (m) |
Find the value of these expressions without using a calculator.
$(\text-1)^2$
$(\text-1)^3$
$(\text-1)^4$
$(\text-1)^{99}$
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?
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.
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:
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!