1. Slope and Rate of Change
Topic 1
Slope and Rate of Change
Unit 8 Topic 1

2. Explore
Susan is shopping for the best buy on paper towels. The same brand of paper towels comes in the four sizes shown below.
1. Which is the better deal?
2. What methods did you use to determine which is the better deal?
3. Why would someone want to buy the more expensive option?

3. Explore Which is the better deal?
The better deal is 6 rolls for $4.50.

4. Explore
2. What methods did you use to determine which is the better deal?
3. Why would someone want to buy the more expensive option?
There are a number of ways to determine which of two prices are cheaper. Two of the most common methods are
Converting each to unit price
Converting each to a common number of units or a common price
Some possible reasons why someone would buy the more expensive option:
they prefer a particular brand
quality of product
prefer shopping at a particular place
quantity they need

5. Information
In the explore, the deals presented were examples of rate. Rate is a comparison of two amounts that are measured in different units. For example, km/h, $/L and m/s.
There are two ways in which rates can be compared:
1) Convert the rates so that they have the same number of units.
2) Determine the unit rate for each rate. A unit rate is a rate where the second term is 1. We usually use the key word per or the division symbol ( / ) to indicate a unit rate. For example: If a student earns $9.50 per hour, it is the same as $9.50/hour and means he earns $9.50 for every 1 hour of work.

6. Information
For example, which is the better deal: $10.00 for 5 cupcakes or $18.00 for 10 cupcakes?
Method 1:
Convert the rates so that they have the same number of units.
Method 2:
Determine the unit rate for each rate.
$10.00 for 5 cupcakes
vs.
$9.00 for 5 cupcakes
$2.00/cupcake
$1.80/cupcake

7. Example 1 Determine which is a better buy.
Try this on your own first!!!!
Try this on your own first!!!!
Determining the better buy
Determine which is a better buy.
10 bars of soap for $6.00 or 6 bars of soap for $3.90
a 12.5 oz bag of Doritos for $5.00 or a 3 oz bag for $1.05   
c) a 5 gallon bucket of paint for $97.45 or a 1 gallon bucket of paint for $21.95

8. Example 1a: Solution
Note: There are many ways to do these questions, but you
should get the same answer…
10 bars of soap for $6.00 or 6 bars of soap for $3.90
In this example, we could determine the cost for 1 bar of soap and then compare this cost.
The better buy is 10 bars of soap for $6.00.

9. Example 1b: Solution
Note: There are many ways to do these questions, but you
should get the same answer…
a 12.5 oz bag of Doritos for $5.00 or a 3 oz bag for $1.05
In this example, we would likely want to find out the price per ounce.
The better buy is a 3 oz bag for $1.05.

10. Example 1c: Solution
Note: There are many ways to do these questions, but you
should get the same answer…
a 5 gallon bucket of paint for $97.45 or a 1 gallon bucket of paint for $21.95
Since 5 is a multiple of 1, we can find out the price for 5 gallons of paint in each option.
Since the other option is $21.95 for 1 gallon, we can multiply both of these numbers by 5 to determine how much it costs for 5 gallons.
Since one option is $97.45 for 5 gallons of paint, we can leave this one.
$97.45 for 5 gallons
The better buy is 5 gallon bucket of paint for $97.45.

11. Example 2 Solve the following questions below.
Try this on your own first!!!!
Comparing Rates
Solve the following questions below.
Natasha can buy a 26.4 lb turkey for $42.89 from the butcher shop. The supermarket has turkeys on sale for $1.49/lb. Where should Natasha buy a turkey if she wants the best deal?
b) When Amelia uses a treadmill for 2 h, she burns 1400 Cal. When she uses an elliptical trainer for 30 min she burns 390 Cal. Who burns more calories per hour?
c) 50 L of oil costs $163. What is the cost of 20 L of oil?

12. Example 2a: Solution
Natasha can buy a 26.4 lb turkey for $42.89 from the butcher shop. The supermarket has turkeys on sale for $1.49/lb. Where should Natasha buy a turkey if she wants the best deal?
Since the supermarket price has a unit price of $1.49/lb, we only need to find the unit price of the turkey from the butcher shop.
The turkey from the supermarket is cheaper so it would be the best deal.

13. Example 2b: Solution
When Amelia uses a treadmill for 2 h, she burns 1400 Cal. When she uses an elliptical trainer for 30 min she burns 390 Cal. Who burns more calories per hour?
Find out how many calories each person burns in an hour.
Remember there are in 60 minutes in 1hr.
Amelia burns more calories per hour using an elliptical trainer.

14. Example 2c: Solution
50 L of oil costs $163. What is the cost of 20 L of oil?
Now that we know how much 1 litre costs, we can easily multiply by 20 to find the cost of 20 L of oil.
The cost of 20 L of oil is $65.20.

15. Example 3
Try this on your own first!!!!
Comparing Rates Graphically
A hot tub contains 1600 L of water. Graph A represents the hot tub being filled at a constant rate. Graph B represents the hot tub being emptied at a constant rate.
a) Determine the rate of change represented in each graph.
b) Describe what the rate of change represents.

16. Example 3a: Solution
The rate of change corresponds with the slope of the graph.
Filling a Hot Tub:
Emptying a Hot Tub:
When filling the hot tub, the rate of change is 20 L/min.
When emptying the hot tub, the rate of change is -40 L/min.

17. Example 3b: Solution
The rate of change of 20 L/min indicates that the hot tub is being filled (positive rate of change) at a rate of 20 litres per minute.
The rate of change of -40 L/min indicates that the hot tub is being emptied (negative rate of change) at a rate of 40 litres per minute.

18. Information
Rate of change is the rate at which one variable changes compared to another variable, where units are included.
Slope describes how two variables change in relation to each other. (Rise compared to run.)
The rate of change can be determined using a graph and the slope formula. If two points and are on the graph of a line, then the slope of the line is
where the Greek symbol delta, , means change.

19. Information
A line that rises from left to right has a positive slope, m > 0.
A line that falls from left to right has a negative slope, m < 0.
A horizontal line has a slope of zero, m = 0.
A vertical line has an undefined slope, m undefined.   

20. Example 4
Try this on your own first!!!!
Finding Slope
Haley is conducting a science experiment. She plotted the data shown on the right.
a) Determine the rate of change, or slope, of each line segment.
From the graph, determine the rate of change or slope, m, for each line segment using the formula
When moving from point A to point B, moving
upward is a positive rise
downward is a negative rise
right is a positive run
left is a negative run.
ii. Determine the slope, m, using
the formula,

21. Example 4
Line Segment AB
Line Segment BC

22. Example 4
Line Segment CD

23. Example 4
b) Describe what the rate of change means for each line segment. Is the line horizontal, does the line rise from left to right, or does the line fall from left to right?
i. line segment AB
ii. line segment BC
iii. line segment CD
A horizontal line means no change in temperature.
The line falls from left to right. The temperature is decreasing by
The line rises from left to right. The temperature is increasing by

24. Example 5
Try this on your own first!!!!
Finding the Slope Given Points
A line passes through the point A(6, 3) and B(1, 7). Find the slope of the line.
Helpful Hint
The slope of a line passing through points A and B is

25. Example 5
Finding the Slope Given Points
A line passes through the point A(6, 3) and B(1, 7). Find the slope of the line.
Helpful Hint
The slope of a line passing through points A and B is

26. Example 6
Try this on your own first!!!!
Finding the Slope of a Vertical Line
A vertical line passes through the point A(3, 2) and B (3, 4). Find the slope of the line. 5 x y
B(3, 4)
A(3, 2)

27. Example 6
Finding the Slope of a Vertical Line
A vertical line passes through the point A(3, 2) and B (3, 4). Find the slope of the line. 5 x y
B(3, 4)
A(3, 2)

28. Example 7
Try this on your own first!!!!
Graphing a Rate of Change
Jake drives a delivery truck and is paid $16.00 an hour plus a $20.00 gas allowance at the beginning of each workday for using his own vehicle.
Complete the table below to show the relationship between hours worked and pay.
Label the graph, and then sketch Jake’s earnings.
Determine the slope of the graph.
What do you notice about the calculated slope and Jake’s rate of pay?

29. Example 7 a & b: Solution
Graphing a Rate of Change
Complete the table below to show the relationship between hours worked and pay.
Label the graph, and then sketch Jake’s earnings. 20 36 52 68 84
100
116
132
148
+16
+16
+16
+16
+16
+16
+16
Hours
Pay ($)

30. Example 7 c & d: Solution c) Determine the slope of the graph.
Graphing a Rate of Change
c) Determine the slope of the graph.
d) What do you notice about the calculated slope and Jake’s rate of pay?
The slope and his hourly rate of pay are the same, $16 / hour.

31. Need to Know:
Rate is a comparison of two amounts that are measured in different units, such as, km/h, $/L, m/s.
There are two methods you can use to compare rates.
1. Convert the rates so that they have the same number of units, or
2. Determine the unit rate for each item. A unit rate is a rate in which the second term is 1.
Rate of change is the rate at which one variable changes compared to another variable, where units are included. The rate of change can be determined using a graph and the slope formula.

32. Need to Know:
•Slope describes how two variables change in relation to one another. If two points and are on a the graph of a line, then the slope of the line is .
• Slope can be positive, negative, zero or undefined.
Lines with a positive slope rise from left to right.
Lines with a negative slope fall from left to the right.
A line with a slope of 0 is horizontal.
A line with an undefined slope is vertical.
You’re ready! Try the homework from this section.