But first… a quick Math review:

When solving equations:Step One: Write down what you know/What you

are given in the questionStep Two: Pick and write down the correct formula

(one where you know everything but one variable)

Step Three: Using the same format of the equation, fill in the proper numbers in the right places

Step Four: Solve, and include proper units

Time Intervals

• ∆t = tfinal – tinitial • ∆ = delta or “change in”• “final” means the time you ended• “initial” means the time you started• You can find the tfinal or tinitial using these

formulas:• tfinal = ∆t + tinitial

• tinitial = tfinal – ∆t

Example One:

If you left home at 7:33 a.m. and arrived to school at 8:05 a.m., how long did it take you?

∆t = tfinal – tinitial tfinal = 8:05 a.m.

∆t = 8:05 – 7:33 tinitial = 7:33 a.m

∆t = 32 min. .

Example Two:

If a runner on a track passed you at 39 seconds, and it takes him 54 seconds to run the length of the track, at what time will he pass you by a second time?

tfinal = ∆t + tinitial ∆t = 54 sec.

Tfinal = 54 sec. + 39 sec. Tinitial = 39 sec.

Tfinal = 93 sec. or 1 min. 33 sec.

Distance Intervals

• ∆d = dfinal – dinitial • “final” means the distance at which you

ended• “initial” means the distance at what you

started• You can find the dfinal or dinitial using these

formulas:• dfinal = ∆d + dinitial

• dinitial = dfinal – ∆d

Example One:

If you left home and walked for 30 minutes and traveled a distance of 2500 metres, how far did you walk?

∆d = dfinal – dinitial dfinal = 2500 m.

∆d = 2500 – 0 dinitial = 0 m.

∆d = 2500 m .

Example Two:

What was your initial position if you ran for 850 m and ended at a position of 1387 m?

dinitial = dfinal – ∆d dfinal = 1387 m

dinitial = 1387 – 850 ∆d = 850 m

dinitial = 537 m

The rate of change of a linear relationship is the steepness of the line.

rise

run

Rate of Change =

rise

run

Engineers refer to the steepness of the roof of a house as the pitch

Engineers refer to the rate of change of a road as the grade

Engineers often represent the rate of change as a percentage.

A grade of 8% would mean for every rise of 8 units there is a run of 100 units.

8100

= 8%

Rate of change = 8

100

The steepness of wheelchair ramps is of great importance for safety.

Rate of change of wheelchair ramp =

112

If the rise is 1.5 m, what is the run?

Answer: 18 m because

1

1 2

1

12

15

18

.X 1.5

3 m

5 m

Determine the rate of change (pitch) of the roof.

3

5change of rate

2

3

3

3

3

2

change of rate

3

3

change of rate

1

Determine the rate of change of each staircase.

Determine the rate of change.Which points will you use to determine rise and run?

= $5/hr

run

rise

change of rate

hr 4

20 $20

4

Earnings

Number of Hours Worked

What does this rate of change represent?

The hourly wage

Rate of Change Calculations

Rate of change (r) = ∆ x ÷ ∆t

∆ x = xfinal – xinitial

∆t = tfinal - tinitial

Example One:

At 6:00 a.m. the temperature is 2 degrees. At 10:00 a.m. the temperature is 14 degrees. What is the average rate of change in the temperature?

r = ∆ x ÷ ∆t

r = (14 – 2) ÷ (10:00 – 6:00)

r = 12 ÷ 4

r = 3 degrees/hour

Example Two:

Ron weighed 64 kg in 2008. He weighed 52 kg in 2012. What was his rate of change in his weight?

r = ∆ x ÷ ∆t

r = (64 – 52) ÷ (2012 – 2008)

r = 12 ÷ 4

r = 3 kg/year