Rate of Change

Earnings

Hours Worked

Jack’s Babysitting Earnings

Amount

Time

Amount of Interest Earned

We will attach the informal idea of rate of change with the mathematical structure of slope

Engineers have created a mathematical measure for the

steepness of a line

Slope: The measure of a line’s steepness

Rise

Run

Slope = Rise

Run

8cm

16cm

Slope = Rise

Run8cm

16cm

= 0.5

=

Step Count Technique

Count the number of steps going from left to right.

5

2

m = 52

-5

2

m = -52

Find the slope of the line on a plain

X

Y

(-2,-3)

-4 -3 -2 -1 1 2 3 4

3 2

-2 -3

(4,1)

4

6

m =

46

= 23

Another way to get 2/3, is to use the following formula…

m = y2 – y1

x2 – x1

Where (x1,y1) and (x2,y2) simply represent any points on the plane.

X

Y

(-2,-3)

-4 -3 -2 -1 1 2 3 4

3 2

-2 -3

(4,1)

4

6

m =

-3 - 1-2 - 4

= -4

3

(x1, y1)

(x2, y2) = 2

-6

Find the slope of a line that passes through the points A

(1,4) and B (4, -2)

Use the equation:

m = y2 – y1

x2 – x1

Determine the slope:

A( 1, 4 ) B( 4, -2)

x1, y1 x2, y2

m = -2 – (+4)

4 - 1

m = -6

3

m = -2

Notice the chart on pg 283.

First Differences

Be sure to copy down all the text in black.

All the colored text is just for reading and reflecting

Imagine a snake skimming across the water

(click on skimming above)

If you were told that the snake was swimming at 2 m/s, you could write an equation (d = 2t) (where d = distance and t = time)

To determine if the relationship was linear or not, we could make a graph….

There is a quicker way. Make a Table of Values

Time Distance

0

1

2

3

4

0

2

4

6

8

Add a First Differences Column to your chart

First

Differences

When the FDs are the same, the relationship is linear

2 – 0 = 2

4 – 2 = 2

6 – 4 = 2

8 – 6 = 2

Imagine jumping out of an airplane

(click on jumping above)

As the skydiver falls towards the mountain, he travels faster and faster. This is true for all objects that fall towards the surface of the Earth.

This relationship of speed vs time is easy to understand. Obviously, the skydiver speeds up as time passes.

(Can you imagine skydiving at the same slow speed the entire time? Kind of boring….

This relationship can also be modelled with a TOVs.

The acceleration due to gravity on Earth is 9.8m/s2

This leads to the following

chart

Time Distance

0

1

2

3

4

0

10

40

90

160

Add a First Differences Column to your chart

First

Differences

When the FDs are not the same,

the relationship is non linear

10 – 0 = 10

40 – 10 = 30

90 – 40 = 50

160 – 90 = 70

Use a TOVs with a FD column to determine if the following relationship is linear or not.

In summary

FDs are the same, then linear

FDs are different, then non-linear

Time

(months)

Plant Height (cm)

0

1

2

3

4

0

2

3

9

19

First

Differences

Page 284

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