Chapter 6 discusses the concept of rate of change in linear relationships, emphasizing its representation through steepness of lines, such as roofs and wheelchair ramps. The document provides real-life examples, including the weight gain of a baby and population changes in Chicago, illustrating how to calculate both positive and negative rates of change. The chapter concludes with the formula for determining rate of change between two points.

1. CHAPTER 6:
Slopes and Lines

2. RATE OF
CHANGE
Lesson 6.1

3. RATE OF CHANGE OF A LINEAR
RELATIONSHIP
rise
run
Rate of Change =
The rate of change of a linear relationship is
the steepness of the line.
rise
run

4. RATES OF CHANGE ARE SEEN
EVERYWHERE.

5. The steepness of
the roof of a house
is referred to as the
pitch of the roof by
home builders.

6. Give one reason
why some homes
have roofs which
have a greater
pitch.
There is less snow buildup in the
wintertime.

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

8. They often represent the rate of
change as a percentage.

9. 8
100
A grade of 8% would mean for every
rise of 8 units there is a run of 100 units.
= 8%
Rate of change =
8
100

10. The steepness of wheelchair ramps is of
great importance for safety.
1
12
Rate of change of wheelchair ramp =
1
1 2
If the rise is 1.5 m, what is the run?
Answer: 18 m because
1
12
= .
15
18

11. 3 m
Determine the rate of
change (pitch) of the
roof.
5 m
rate of change = 5
3

12. Determine the rate of change of each
staircase.
2
3
3
3
rate of change
=
2
3
rate of change
=
3
3
=1

13. Determine the rate of change.
Which points will
you use to
determine rise and
run?
rate of change
=
rise
run
20 = $ 20
4 hr
= $5/hr
4
Ear
ni
ngs
Number of Hours Worked
What does this rate of
change represent?
The hourly wage

14. REAL LIFE
Mr. and Mrs. Overjoyed had a
healthy baby girl named Joy
who weighed 7.5lbs. at birth. At
the end of 4 months, the baby
weighed 13.5lbs.
 How fast did her weight change
from birth to 4 months?

15. REAL LIFE (PART 2)
 To answer this question, we calculate the Rate of
Change of Joy’s weight, that is, how much she
gained per month.
Change in weight 13.5lbs – 7.5lbs 6lbs
Change in age 4mo – 0mo 4mo
1.5lbs./month

16. POSITIVE RATE OF CHANGE
At age 6 months, Joy weighed 15.75lbs.
 How fast did she grow from 4 months to 6
months?

17. Line Graph: Rate of Change

18. NEGATIVE RATE OF CHANGE
Joy was gaining weight, so the rate of change was
positive. But a rate of change can be negative.
 Let’s take a look at the population of Chicago
from 1830 to 2000…

19.  The rate of change, in people per year, from 1890
to 1900. For these years, the population
increased.
Change in population 1,698,575 – 1,099,850 598,725p
Change in years 1900 – 1890 10yr
59, 872.5 people/year

20.  Between 1970 and 1980 the population decreased,
so the rate of change is negative.
Change in population 3,005,072 – 3,369,357 -364, 285 p
Change in years 1980 - 1970 10 years
- 36,428.5 people per year

22. FORMULA FOR RATE OF CHANGE
y2 – y1
x2 – x1
 The rate of change between points (x1, y1)
and (x2, y2)