UNIT 5 – LINEAR RELATIONS: Constant Rate of Change,
Initial Condition, Direct and Partial Variation
Lesson 1
Introduction to Rates of Change ……………………………….…………..2
Lesson 2
Graphs and Their Stories ……………………………………..…….………7
Lesson 3
Rate of Change ………………………………………....…………..……….10
Lesson 4
Models of Movement ……………………………….………………………15
Lesson 5
Mid-Chapter Review ……………………………………..…….……………21
Lesson 6
Equation of a Line ………………………………………....…………..…….23
Lesson 7
Descriptions, Tables of Values, Equations, Graphs ..…………….……..30
Lesson 8
Modelling Linear Relations with Equations .……………………..…….…34
Lesson 9
Graphing Linear Relations …………………………………....……………39
Lesson 10
Working with Equations ……………………………………………….……42
Lesson 11
Applications ………………………………………………………..…….…..48
Lesson 12
Linear or Non-Linear? ………………………………………....……………53
Unit 5 Review ………………………………………………………………….……………...60
Important Dates:
Quiz #1
Quiz #2
Unit Summary Sheet Due
Unit 5 Test
Unit 5 Thinking Task
Assignment #1 Due
Assignment #2 Due
LESSON 1 – Introduction to Rates of Change
DATE:
YOUR TASK:
To create a graph similar to each of the given graphs using the CBR.
 You may need more than one attempt to match a graph.
 Once you have matched a graph, be sure to remember the motion needed (i.e.
direction you were walking, speed, etc). This will be discussed in class
following the activity.
 Be sure to give each group member a chance to “walk”.
A few notes about the CBR






The walker should always be at least 0.5 m away.
The walker should be directly in line with the CBR.
All other group members should be behind the CBR.
You may need to experiment to see the maximum distance the CBR will detect.
Be sure to hold the CBR still while recording.
You will hear a “ticking” sound when the CBR is recording data.
How to Use the CBR:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Connect the cable from the CBR to the bottom of the calculator.
Turn the calculator ON.
Press the blue APPS button.
Select 2:CBL/CBR.
The CBL/CBR title screen will appear. Press ENTER.
Select 3:RANGER.
The Ranger title screen will appear. Press ENTER.
Select 2:SET DEFAULTS.
With the small arrow pointing to START NOW, press ENTER.
You are now ready to take your first sample!
To take a sample:
 At the same time, the walker should start walking and the calculator person
should press ENTER to begin recording data.
 After you have studied your graph, press ENTER, then
select 3:REPEAT SAMPLE to try again or to attempt
to match the next graph.
MFM 1P – Unit 5 – Linear Relations
2
LESSON 1 – Introduction to Rates of change
DATE:
OTHER GROUP MEMBERS:
Discussion: What kinds of motion were needed to create each graph?
MFM 1P – Unit 5 – Linear Relations
3
LESSON 1 – Introduction to Rates of Change
DATE:
When a line is drawn on a distance-time graph, SPEED can be determined.
Types of Graphs:
1. Walking away from an object (slowly)
2. Walking away from an object (quickly)
3. Walking towards an on object (slowly)
4. Walking towards an object (quickly)
5.
Walking towards an object while
increasing in speed.
MFM 1P – Unit 5 – Linear Relations
6. Walking away from object, stops for 3
seconds then returns at the same speed.
4
LESSON 1 – Introduction to Rates of Change
DATE:
Part One: Walk the Line
Examine the graph below.
Mrs. Armstrong’s Walk
Calculate the rate of change of the graph (speed of the walk).
Draw a large right-angled triangle under the graph and label it with the height as the rise and the
base as the run. Show the lengths of each.
Calculate the rate of change of your walk using the formula: rate of change  rise
run
Complete the following:
a) The rate of change of my walk is ________________.
b) The speed of my walk is ________________ m/s away from the CBR™.
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5
LESSON 1 – Introduction to Rates of Change
DATE:
Part One: Walk the Line
Examine the graph below.
Mrs. Deeks’ Walk
Calculate the rate of change of the graph (speed of your walk).
Hint: The rise will be a negative number!
Draw a large right-angled triangle under the graph and label it with the rise and run values.
Calculate the rate of change using the formula: rate of change = rise .
run
Complete the following:
The rate of change of my walk is ________________.
The speed of my walk is ________________ m/s toward the CBR™.
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6
LESSON 2 – Graphs and Their Stories
DATE:
What kinds of motion are represented in the different graphs below?
Sunflower Seed Graphs
Ian and his friends were sitting on a deck and eating sunflower seeds. Each person had a bowl
with the same amount of seeds. The graphs below all show the amount of sunflower seeds
remaining in the person’s bowl over a period of time.
Write sentences that describe what may have happened for each person.
a)
b)
MFM 1P – Unit 5 – Linear Relations
c)
d)
7
LESSON 2 – Graphs and Their Stories
DATE:
Graphical Stories
The graphs below are three stories about walking from your locker to your class.
Match each story to the appropriate graph:
HINT: One of these stories does not match a graph, and one of these graphs does not match a
story!
1. I started to walk to class, but I realized I had forgotten my notebook, so I went back to my
locker and then I went quickly at a constant rate to class.
2. I was rushing to get to class when I realized I wasn’t really late, so I slowed down a bit.
3. I started walking at a steady, slow, constant rate to my class, and then, realizing I was late, I
ran the rest of the way at a steady, faster rate.
4. I walked to my friend’s locker, and stopped to talk to her for a few minutes. After
she had collected all of her books, we walked (a little faster this time) to class together.
Graphs which compare DISTANCE FROM A POINT and TIME are called Distance-Time
graphs. These graphs can be used to indicate direction, speed and total length of trip (from
starting point to ending point). A story can be made from a distance-time graph.
MFM 1P – Unit 5 – Linear Relations
8
LESSON 2 – Graphs and Their Stories
DATE:
ASSIGNMENT PRACTICE GRAPH (Completed as a class)
“Lunchtime at Mr. Dumpling”
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9
LESSON 3 – Rate of Change
DATE:
Rate of Change
We have been looking at graphs which compared distance
and time. When we calculated the steepness of the line on
a distance-time graph, we were calculating the speed.
speed = distance
time
Speed is an example of a RATE OF CHANGE
Other phrases used with rate of change:
 slope - steepness of a hill
 pitch - slope/steepness of a roof
 percent grade - inclination of a
road
Rate of change can be written as:
Example 1: Given a 4% grade, find the following:
a) Fraction
b) Rise
c) Run
d) Rate of change
Example 2: Given a rise of 1 and run of 20, find the following:
a) Fraction
b) Percent Grade
c) Rate of change
Example 3: Given a rate of change of 0.065, find the following:
a) Percent Grade
MFM 1P – Unit 5 – Linear Relations
b) Fraction
c) Rise
d) Run
10
LESSON 3 – Rate of Change
DATE:
Ramps
Types of inclines and recommendations by
rehabilitation specialists
Rise
Run
Rate of
(Vertical (Horizontal
Change
Distance) Distance)
The recommended incline for wheelchair uses is 1:12.
For exterior ramps in climates where ice and snow are
common, the incline should be more gradual, at 1:20.
For unusually strong wheelchair users or for motorized
chairs, the ramp can have an incline of 1:10.
The steepest ramp should not have an incline exceeding 1:8.
Building Ramps - Which ramp could be built for each of the clients?
1
2
3
4
Clients
Choice
of Ramp
Client A lives in a split-level town house. He owns a very powerful motorized chair. He wishes to build
a ramp that leads from his sunken living room to his kitchen on the next level.
Client B requires a ramp that leads from her back deck to a patio. She is of average strength and
operates a manual wheelchair.
Client C lives in Sudbury where ice and snow are a factor. She is healthy, but not particularly strong.
Her house is a single level bungalow but the front door is above ground level.
Client D will not get approval because the design of his ramp is too dangerous.
MFM 1P – Unit 5 – Linear Relations
11
LESSON 3 – Rate of Change
DATE:
Roads
The inclination of a road is called “percent grade.” Severe grades (greater than 6%) are difficult
to drive on for extended amounts of time. The normal grade of a road is between 0% and 2%.
Warning signs are posted in all areas where the grades are severe.
Percent grade
A
Fraction
Rise
Run
1
50
1%
B
C
D
0.035
4%
E
525
10 000
3
50
F
G
0.1
H
1
2
I
0.75
J
1
3
2
5
K
L
Rate of change
(decimal form)
8.25%
Which of the roads, A–L, would require a warning sign?
Some of the values in the table are fictional. There are no roads that have grades that are that
severe. Which roads, A–L, could not exist? Explain your reasoning.
Describe a road with a 0% grade.
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12
LESSON 3 – Rate of Change
DATE:
Practicing Rate of Change
1. Determine the rate of change for each object.
(a)
The pitch of the roof is the rate of
change.
7.2 m
5.6 m
(b)
The steepness of the ramp is the rate
of change.
1.2 m
4.8 m
(c)
The steepness of the staircase is the
rate of change.
4m
4m
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13
LESSON 3 – Rate of Change
DATE:
Practicing Rate of Change (continued)
2. If a wheelchair ramp must have a rate of change of
1
, determine the horizontal distance
12
required for a ramp that has a vertical distance of 5.2𝑚.
3. The grade of a road is often given as a percent. If the road rises 15𝑚 over a horizontal
distance of 180𝑚, determine the grade as a percent.
5
4. The pitch of a roof of a house is given by a rate of change of . If the horizontal distance is
6
actually 10.5𝑚, determine the vertical distance of the roof.
MFM 1P – Unit 5 – Linear Relations
14
LESSON 4 – Models of Movement
DATE:
A Runner’s Run
Chris runs each day as part of his daily exercise. The graph shows his distance from home as
he runs his route.
200
Calculate his rate of change (speed) for each segment of the graph.
SEGMENT
RATE OF CHANGE (SPEED)
MFM 1P – Unit 5 – Linear Relations
15
LESSON 4 – Models of Movement
DATE:
Micha’s Trip to the Store
Distance vs. Time
At 11 o’clock, Micha’s mother
sends him to the corner store for
milk and tells him to be back in 30
minutes. Examine the graph.
700
D
E
600
F
Distance from Home (m)
500
C
400
300
200
B
100
G
A
4
8
12
16
20
24
28
32
36
40
44
48
Time (min)
1. Why are some line segments on the graph steeper than others?
2. Calculate the rate of change (speed) of each of the line segments:
Rate of change AB =
Rate of change BC =
Rate of change CD =
Rate of change DE =
Rate of change EF =
Rate of change FG =
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LESSON 4 – Models of Movement
DATE:
Micha’s Trip to the Store
3. Over what interval(s) of time is Micha travelling the fastest?
the slowest?
4. How long did it take Micha to reach the store? How do you know?
5. How long did Micha stay at the store?
6. How long did it take Micha to get home from the store?
7. How can you use the graph to tell which direction Micha is travelling?
8. Did Micha make it home in 30 minutes? How do you know?
9. Using the information the graph provides, write a story that describes Micha’s trip to the
store and back.
MFM 1P – Unit 5 – Linear Relations
17
LESSON 4 – Models of Movement
DATE:
The Blue Car and the Red Car
Distance from parking lot (km)
Two friends are leaving a parking lot at the same time. They agree to meet later at the home of
a friend who lives 400 km from the parking lot. One friend drives a blue car and the other a red
car. The blue car is labelled B and the red car, R. Answer the questions below using the
following graph.
400
B
300
R
200
100
1
2
3
4
5
6
Time (h)
1. At what time do the cars pass each other? How far are they from the parking lot?
2. Which car stopped and for how long? How far from the parking lot did the car stop?
3. Suggest reasons for the car stopping.
4. Which car got to the final destination first? Explain.
5. The posted speed limit was 80 km/h. If you were a police officer, could you stop either of the
cars for speeding? Explain.
MFM 1P – Unit 5 – Linear Relations
18
LESSON 4 – Models of Movement
DATE:
Practice
1. Find the slopes of the lines drawn.
F
A:
E
B:
C:
D
D:
A
E:
C
F:
B
2. Ramps: A ramp is considered safe its slope is 1/8 or lower.
a) A ramp leads to a door which is 2 m above the sidewalk. The horizontal length of the ramp is
10 m. Is the ramp safe?
2m
10 m
b) A 25 m long ramp reaches a door which is 7 m above the ground. Is the ramp safe?
7m
25 m
Adapted from PDSB
MFM 1P – Unit 5 – Linear Relations
19
LESSON 4 – Models of Movement
DATE:
Answer the questions for each graph. Include units of measurement with each answer.
1.
2.
a) How much money had been saved at time 0?
b) What was the rate of saving (dollars per
month)?
a) What was the distance from home at time 0?
b) What was the rate of speed (meters per hour)?
3.
4.
a) What was the initial height of the tree?
b) What was the rate of growth?
a) What was the speed from 0 to 3h?
b) What was the speed from 3 to 4h?
c) What was the speed from 4 to 10h?
MFM 1P – Unit 5 – Linear Relations
20
LESSON 5 – Mid Chapter Review
1.
DATE:
A bakery delivery truck spends 3 hours driving the morning run. There are two
deliveries made and then the truck returns to the bakery.
Distance (km)
90
60
30
0
0.5
1.0
1.5
2.0
2.5
3.0
Time (h)
(a)
What is the speed of the truck as it approaches its first delivery?
(b)
When does the truck leave to make the second delivery?
(c)
What is the speed of the truck as it approaches its second delivery?
(d)
How long does it take to complete the second delivery?
(e)
What is the speed of the truck as it approaches the bakery at the end of the run?
MFM 1P – Unit 5 – Linear Relations
21
LESSON 5 – Mid Chapter Review
2.
DATE:
Amy takes the bus to school. Lucky for her, she is the third last stop on the way to
school. The bus arrives to pick up Amy and it drives at a constant speed for 5 minutes to
the next stop 3 km away. It takes 1 minute for the students to get on the bus. The bus
then travels 60 km/h to a stop that is 5 km away. 2 minutes later, the bus is on its way to
the school. It takes 8 minutes to reach the school which is 6 km away.
Distance (km)
Draw a distance-time graph of Amy’s bus ride to school.
Time (min)
MFM 1P – Unit 5 – Linear Relations
22
LESSON 6 – Equation of a Line
DATE:
EQUATION OF A LINE
On a distance-time graph, the rate of change represents speed.
The Rate of Change (ROC) also represents the steepness of the line.
There is an
represent the
or
equation that is used to
line:
y = ROC(x) + initial value
y = (Rate)x + initial value
There are 2 types of lines:
Direct Variation
Partial Variation
line passes through origin
initial value is 0
You open a bank account and
deposit money
y = 5x
line does not pass through the origin
initial value is not 0
You put money into an existing bank
account
y = 10x + 50
Graph
Key Feature
Initial Value
Example
Sample Equation
Example
A pizza company charges $12 for the pizza, then $2 per topping.
a) Complete the chart below
# of toppings
b) Graph the relationship
price
0
1
2
3
c) Write an equation to model this relationship
d) Is this direct or partial variation?
MFM 1P – Unit 5 – Linear Relations
23
LESSON 6 – Equation of a Line
DATE:
Outfitters
Jaraad wants to rent a canoe for a day trip. He gathers this information from two places and
decides to make a table of values and graph each of these relationships.
 Big Pine Outfitters charges a base fee of $40 and $10 per hour of use.
 Hemlock Bluff Adventure Store does not charge a base fee, but charges $30 per hour to use
the canoe.
Jaraad’s Working Sheet
1. a) What is the cost of each canoe if Jaraad cancels his reservation?
b) Compare the rate of change of cost for Big Pine and for Hemlock Bluff to the cost per
hour for each outfitter.
c) What is the cost per hour for each outfitter? What conclusions can you draw from
comparing the answers to (b) and (c)?
2. Which graph has a initial value of zero? This is called direct variation.
MFM 1P – Unit 5 – Linear Relations
24
LESSON 6 – Equation of a Line
DATE:
3. Which graph has an initial value other than zero? This is called a partial variation.
4. Which outfitter company should Jaraad choose if he estimates he will canoe for
0.5 h?… 1.5 h?…
2.5 h?
Time (h)
Big Pine Cost ($)
Hemlock Bluff Cost ($)
0.5
1.5
2.5
Explain how you determined your answers.
5. Write an equation to model the cost for each outfitter.
Let C represent the cost in dollars and h represent the time in hours.
Big Pine
C=
Hemlock Bluff
C=
6. If Big Pine Outfitters decided to change its base fee to $50 and charge $10 per hour, what
effect would this have on the graph?
a) Draw a sketch of the original cost and show the changes on the same sketch.
b) Write an equation to model the new cost.
MFM 1P – Unit 5 – Linear Relations
25
LESSON 6 – Equation of a Line
DATE:
Outfitters (continued)
7. If Hemlock Bluff Adventure Store decided to change its hourly rate to $40, what effect would
this have on the graph?
a) Draw a sketch of the original cost and show the changes on the same sketch.
b) Write an equation to model the new cost.
8. For Big Pine Outfitters, how are the pattern in the table of values, the description, the graph,
and the equation related?
Description
Big Pine Outfitters charges a base fee of $40 to deliver the canoe to the launch site and $10 per
hour of use.
Table of Values
Graph
Time (h) Cost ($)
0
40
1
50
2
60
3
70
4
80
Equation
C = 40 + 10h
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LESSON 6 – Equation of a Line
DATE:
Outfitters (continued)
9. For Hemlock Bluff, how are the pattern in the table of values, the description, the graph, and
the equation related?
Description
Graph
Hemlock Bluff charges $30 per hour.
Table of Values
Time (h) Cost ($)
0
0
1
30
2
60
3
90
4
120
Equation
C = 30h
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27
LESSON 6 – Equation of a Line
DATE:
Practice
1. A rental car costs $50 per day plus $0.20 for each kilometre it is driven.
a) What is the dependent variable?
b) Make a table of values for the rental fee up to 500 km.
c) Graph the relationship.
Number of
Kilometres
Cost vs. Number of Kilometres
Cost ($)
0
260
240
100
220
200
200
Cost ($)
180
160
140
120
100
80
60
40
20
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
Number of Kilometres
d) Write an equation to model
the relationship. C is the cost and n is the number of kilometres.
____ = _______________
e) Does this relation represent a partial or direct variation? Explain.
f)
Determine the rental fee for 45 km. Show your work.
MFM 1P – Unit 5 – Linear Relations
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LESSON 6 – Equation of a Line
DATE:
2. There is $500 in Holly’s bank account. She takes out $50 from her account each month but
doesn’t put any back in.
a) Make a table of values for up to 6 months.
b) Graph the relationship.
1
2
3
4
5
6
7
c) Write an equation to model the relationship.
____ = ______________
d) Does this relation represent a partial or direct variation? Explain.
e) How much will Holly have in her account after 8 months? Show your work.
f)
How many months will have passed when Holly has $50 in her account? Show your work.
MFM 1P – Unit 5 – Linear Relations
29
LESSON 7 – Descriptions, Tables of Values, Equations, Graphs DATE:
1. Nisha is just learning how to snowboard. White Mountain charges $10/hour for lessons and
$40 for the lift ticket and snowboard rental.
a) Make a table of values for up to 6 hours.
b) Graph the relationship.
1
2
3
4
5
6
7
8
9
10 11
12
13
14
c) Write an equation to model the relationship.
___ = _________________
d) Does this relation represent a partial or direct variation? Explain.
e) How much will it cost in total for Nisha to take 2.5 hours of lessons?
Show your work.
f)
If Nisha paid $75, how long was she at the White Mountain getting lessons?
Show your work.
MFM 1P – Unit 5 – Linear Relations
30
LESSON 7 – Descriptions, Tables of Values, Equations, Graphs DATE:
2. Ishmal sells high-definition televisions. He is paid a weekly salary of 20%
commission of his total weekly sales.
a) Complete the table of values.
b) Graph the relationship.
Weekly
Sales ($)
Total Pay ($)
0
1000
2000
1800
1600
2000
1400
1200
3000
1000
800
600
4000
400
200
5000
2000
4000
6000
8000
10000
12000
c) Write an equation to model the relationship.
___ = _________________
d) Does this relation represent a partial or direct variation? Explain.
e) Determine Ishmal’s pay if his sales for the week were $8000. Show your work.
f)
Ishmal made $975. How much were his weekly sales? Show your work.
MFM 1P – Unit 5 – Linear Relations
31
LESSON 7 – Descriptions, Tables of Values, Equations, Graphs DATE:
Practice
Recall: Rate of Change (ROC) = rise
run
Equation of line:
y = (ROC)x + initial value
Example – if the rate of change is 10 and the initial value is 5, then y = 10x + 5
Direct Variation – initial value is 0 (line goes through origin)
Partial Variation – initial value is not 0 (line does not go through origin)
1. Write an equation for each relationship. Indicate whether the relationship is direct or partial
variation.
a) A pizza costs $12 for the pizza and $0.50 for each topping.
b) A cell phone company charges $40 to use a phone, and $1.25 per minute.
c) To rent a banquet hall, the cost is $400, plus $50 for each person who attends.
2. For each graph, find the rate of change and the initial value, then write the equation.
Rate of change =
Rate of change =
Rate of change =
Initial Value =
Initial Value =
Equation =
Equation =
Initial Value =
Equation =
MFM 1P – Unit 5 – Linear Relations
32
LESSON 7 – Descriptions, Tables of Values, Equations, Graphs DATE:
3. For each of the following tables, draw a graph, then use the graph to find the rate of change,
initial value and equation.
Number of
minutes
0
2
4
6
Cost
50
60
70
80
Initial Value:
Rate of Change:
Equation:
X
1
2
3
4
5
Y
5
10
15
20
25
Initial Value:
Rate of Change:
Equation:
X
0
1
2
3
4
Y
5
7
9
11
13
Initial Value:
Rate of Change:
Equation:
MFM 1P – Unit 5 – Linear Relations
33
LESSON 8 – Modelling Linear Relations with Equations
DATE:
Writing Equations for Relationships
To write an equation for a relationship, identify two things:
1. The Rate of Change (short form: ROC). Examples: $2 per hour or $3 per ticket
or 20 km/hr).
2. The initial amount or starting amount. Examples: $500 to rent hall even if no
guests come, $15 for a pizza with no toppings; $20 a month for cell phone
coverage even if zero minutes are used.
Once you have identified these two things, you can write the equation as:
y= ROC(x) + Initial Amount
Example: A banquet hall cost $300 to rent the room, plus $25 per person



The ROC would be 25 ($25 per person). This is how the cost changes
The initial/starting amount would be 300 ($300 to rent the room with zero guests)
The equation would be y = 25x + 300 or
C = 25n + 300, where C=total cost
n=number of guests
MFM 1P – Unit 5 – Linear Relations
34
LESSON 8 – Modelling Linear Relations with Equations
DATE:
Write the equation for each relationship in the space provided. Show any calculations you
made. Indicate if the relation is a partial or direct variation.
A coaches B
1. A family meal deal at Chicken Deluxe
costs $26, plus $1.50 for every extra
piece of chicken added to the bucket.
B coaches A
2. A Chinese food restaurant has a special
price for groups. Dinner for two costs $24
plus $11 for each additional person.
3.
4.
5.
Number of
Toppings
0
1
2
3
4
Cost of a Large
Pizza ($)
9.40
11.50
13.60
15.70
17.80
MFM 1P – Unit 5 – Linear Relations
6.
Number of
Scoops
Cost of Ice
Cream with
Sugar Cone ($)
0
1
2
3
4
1.25
2.00
2.75
3.50
4.25
35
LESSON 8 – Modelling Linear Relations with Equations
DATE:
Write the equation for each relationship in the space provided. Show any calculations you
made. Indicate if the relation is a partial or direct variation and describe why these variables are
discrete.
A coaches B
1. A banquet hall charges $100 for the hall
and $20 per person for dinner.
B coaches A
2. The country club charges a $270 for their
facilities plus $29 per guest.
3.
4.
5.
Number of
Athletes
Cost of
Attending a
Hockey
Tournament
0
1
2
3
4
0
225
450
675
900
MFM 1P – Unit 5 – Linear Relations
6.
Number of
People
Cost of
Holding an
Athletic
Banquet
0
20
40
60
80
75
275
475
675
875
36
LESSON 8 – Modelling Linear Relations with Equations
DATE:
Write the equation for each relationship in the space provided. Show any calculations you
made. Indicate if the relation is a partial or direct variation.
A coaches B
1. Rent a car for the weekend costs $50
plus $0.16/km.
B coaches A
2. A race car travels at a constant speed of
220km/h. Write an equation for the total
distance travelled over time.
3.
4.
5.
Distance
(km)
0
10
20
30
40
Cost of a Taxi
Fare ($)
3.50
6.50
9.50
12.50
15.50
MFM 1P – Unit 5 – Linear Relations
6.
Distance
(km)
0
100
200
300
400
Cost of Bus
Charter ($)
170
210
250
290
330
37
LESSON 8 – Modelling Linear Relations with Equations
DATE:
Journal Activity
A pizza costs $9.00 plus $2.00 per topping.
Discuss the effect on the graph of changing the initial cost to $10.00 and lowering the cost per
topping to $1.50.
Use tables, graphs, and equations to support your conclusions.
MFM 1P – Unit 5 – Linear Relations
38
LESSON 9 – Graphing Linear Relations
DATE:
A tennis club charges $25 initial membership fee plus $5 per day. The equation of this relation is
C = 25 + 5d, where C is the cost and d is the number of days.
Total Cost vs. Number of Day Passes
65
60
55
50
45
40
Total Cost ($)
35
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
Number of Day Passes
Indicate how to find the rate of change from the graph.
If the initial membership fee is changed to $15 and daily cost to $10, graph the new relation on
the same grid.
Indicate the procedure you followed to graph the line.
MFM 1P – Unit 5 – Linear Relations
39
LESSON 9 – Graphing Linear Relations
DATE:
Write the equation for the relationship and graph the relationship.
1. A golf club charges an annual membership 2. Repair-It charges $60 for a service call plus
fee of $1000 plus $100/day for a green fee
$25/h to repair the appliance.
to play golf.
Equation:
3. Movie House charges $5 to rent each
DVD.
Equation:
MFM 1P – Unit 5 – Linear Relations
Equation:
4. A kite is 15 m above the ground when it
descends at a steady rate of 1.5 m/s.
Equation:
40
LESSON 9 – Graphing Linear Relations
DATE:
Write the equation for the relationship and graph the relationship.
1. The Recreation Centre charges a monthly
membership fee of $20 plus $5 per class.
Show the relationship for one month.
2. Repair Window charges a $20 service fee
plus $10/h to fix the window pane.
Equation:
Equation:
3. Yum-Yum Ice Cream Shop charges $0.50
for the cone plus $1 per scoop of ice
cream.
Equation:
MFM 1P – Unit 5 – Linear Relations
5. A submarine model starts 6.5 m above
the bottom of the pool. It gradually
descends at a rate of 0.25 m/s.
Equation:
41
LESSON 10 – Working with Equations
DATE:
Jenise has inquired about the cost of renting a facility for her wedding. She used the data she
received to draw the graph below.
Cost of Holding a Wedding at a Facility
3500
3000
Cost ($)
2500
2000
1500
1000
500
20
40
60
80
100
120
140
Number of Guests
1. Jenise said the graph shows a linear relationship. Justify Jenise’s answer.
2. Does this relation represent a direct or partial variation? Explain your answer.
3. State the initial value and calculate the rate of change of this relation.
MFM 1P – Unit 5 – Linear Relations
42
LESSON 10 – Working with Equations
4.
DATE:
Use the graph to complete the table of values:
Number of
Guests
10
Cost ($)
1250
110
2500
0
3500
30
5. Determine an equation for the relationship.
6. Solve the above equation to determine the number of guests Jenise could have for $1750.
Verify your answer using the graph.
7. Solve the equation to determine the cost for 175 guests. Show your work.
MFM 1P – Unit 5 – Linear Relations
43
LESSON 10 – Working with Equations
DATE:
Mathematical Models
Each situation has a graphical model (graph), an algebraic model (equation) and a numerical
model (table of values). Choose either the graphical model or the algebraic model to complete
the table of values. Show your work and justify your choice of model.
1. Big Pine Outfitters charges a base fee of $40 and $10 per hour of use.
C represents the total cost ($) and t represents the numbers of hours the canoe is used.
Algebraic Model:
C = 40 + 10t
Cost ($)
Big Pine
Graphical Model:
90
80
70
60
50
40
30
20
10
0
0
1
2
3
4
Time (h)
t (h)
a)
Numerical Model:
Solutions:
a)
MFM 1P – Unit 5 – Linear Relations
b)
C ($)
0
b)
70
c)
230
c)
44
LESSON 10 – Working with Equations
DATE:
Mathematical Models (continued)
2. A rental car costs $50 per day plus $0.20 for each kilometre it is driven.
C represents the total cost ($) and d represents the distance (km).
Algebraic Model:
C = 50 + 0.2d
Car Rental
150
Cost ($)
125
Graphical Model:
100
75
50
25
0
0
100
200
300
400
500
600
distance (km)
d (km)
Numerical Model:
a)
250
b)
1000
c)
C ($)
300
Solutions:
a)
b)
c)
Justify your choice.
Challenge: Describe a situation that could be modelled with the given graph or equation.
MFM 1P – Unit 5 – Linear Relations
45
LESSON 10 – Working with Equations
DATE:
Cooling It!
Denis measured the temperature of a cup of hot water as it cooled. He then made the graph on
the right. Complete the scale, and then answer the following questions about the graph.
a) One of the points on the graph is (6, 35).
Explain the meaning of this point, in the
context of Denis’ measurements.
b) Independent variable:
Dependent variable:
c) Use your graph to determine the temperature after 3.5 minutes.
d) Identify the rate of change and the initial value and explain what they mean in this problem.
What do they mean in this problem?
Rate of change:
Initial value:
MFM 1P – Unit 5 – Linear Relations
46
LESSON 10 – Working with Equations
DATE:
Cooling It! (continued)
e) Write an equation to model Denis’ data. Use T for temperature and t for time.
f)
Use your equation to determine the temperature of the water after:
i) 3.5 minutes
ii) 20 minutes
g) Your results for 20 minutes may conflict with what you know about cooling water. Explain.
What does this tell you about the limitations of this linear model?
h) Use your equation to predict when the temperature will be 39°C.
MFM 1P – Unit 5 – Linear Relations
47
LESSON 11 – Applications
DATE:
Planning a Special Event
Maxwell’s Catering Company prepares and serves food for large gatherings. They charge a
base fee of $200 for renting the facility, plus a cost per person based on the menu chosen.
Menu 1 is a buffet that costs $10 per person.
Menu 2 is a three-course meal that costs $14 per person.
Menu 3 is a five-course meal that costs $18 per person.
1. Complete the table of values for each relation: [*Note: n must go up by equal increments]
Menu 1: C = 10n + 200
Menu 2: C = 14n + 200
Menu 3: C = 18n + 200
n
No. of
people
n
No. of
people
n
No. of
people
C
Cost
($)
First
Difference
C
Cost
($)
First
Difference
0
0
0
50
50
50
100
100
100
150
150
150
200
200
200
C
Cost
($)
First
Difference
2. a) Graph the 3 relations on the
same set of axes.
Use an appropriate scale,
labels, and title.
b) Explain whether to use
dashed or solid lines to draw
these graphs.
MFM 1P – Unit 5 – Linear Relations
48
LESSON 11 – Applications
DATE:
Planning a Special Event (continued)
3. a) Identify the rate of change and the initial amount of the Menu 1 line. How do these relate
to the total cost?
What does it mean in this problem?
Rate of change:
Initial amount:
b) Identify the rate of change and the initial amount of the Menu 2 and 3 lines.
Line
Rate of change
Initial amount
Menu 2
Menu 3
4. a) Examine the first differences and the increment in n.
Line
Change in n
First Differences
First Differences
Change in n
Menu 1
Menu 2
Menu 3
b) How do they relate to the graph and the equation?
5. Compare the three graphs. How are the graphs the same? different?
Same
MFM 1P – Unit 5 – Linear Relations
Different
49
LESSON 11 – Applications
DATE:
Planning a Special Event (continued)
6. a) For Menu 2, what does the ordered pair (120, 1780) mean?
b) For Menu 3, what does the ordered pair (80, 1540) mean?
7. Seventy people are expected to attend a school event. How much will it cost for each
menu?
Menu
Cost (show your work)
1
2
3
8. Vadim and Sheila are planning a celebration. They have $3000 to spend on dinner. They
would like to have Menu 3. What is the greatest number of guests they can have?
9. Logan’s Plastics employs 50 people. Each year the company plans a party for its
employees.
a) Find the cost for Menu 2 and write your answer as the ordered pair (50, C).
b) Find the cost for Menu 3 and write your answer as the ordered pair (50, C).
c) How many more dollars will Logan’s Plastics have to pay if they choose Menu 3 instead
of Menu 2?
MFM 1P – Unit 5 – Linear Relations
50
LESSON 11 – Applications
DATE:
Practice
1.
To fix a car, Joe’s Garage charges a base fee of $25 and $40/h.
a.
Make a table of values of the cost of fixing a car for each hour up to 4 hours.
Number of hours
(h)
b.
Cost to Fix the Car
(C)
Using your table from (a), calculate the first differences and the rate of change.
Number of hours
(h)
Rate of Change =
MFM 1P – Unit 5 – Linear Relations
Cost to Fix the Car
(C)
First Difference
Change in C
Change in h
51
LESSON 11 – Applications
DATE:
Practice (continued)
c.
Graph the cost of fixing a car for up to 4 hours.
d.
Identify the rate of change and the initial value. What do they mean in this problem?
e.
Determine an equation to model the graph.
f.
Determine the cost of a 2.5 hour repair job.
Show your work using the equation from part (e).
Check your answer using your graph from part (d).
g.
What does the point (6, 265) represent?
h.
If it costs $155, how long was spent working on the car?
MFM 1P – Unit 5 – Linear Relations
52
LESSON 12 – Linear or Non-Linear
DATE:
Complete the tables of values and determine if
the relationship is linear or non-linear.
Figure Number
Number of Shaded Circles
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
Figure Number
because ________________________________________
Number of Unshaded Circles
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
Figure Number
because ________________________________________
Total Number of Circles
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
MFM 1P – Unit 5 – Linear Relations
because ________________________________________
53
LESSON 12 – Linear or Non-Linear
DATE:
Complete the tables of values and determine if the
relationship is linear or non-linear.
Figure Number
Number of Shaded Squares
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
Figure Number
because ________________________________________
Number of Unshaded
Squares
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
Figure Number
because ________________________________________
Total Number of Squares
First Differences
1
2
3
4
5
This relationship is
linear or non-linear
MFM 1P – Unit 5 – Linear Relations
because ________________________________________
54
LESSON 12 – Linear or Non-Linear
DATE:
Number Patterns
Determine an algebraic expression for the nth term.
1.
Term
Number
1
2
3
4
Term
2.
1
2
3
4
n
3.
Term
Number
1
2
3
4
Term
Number
1
2
3
4
Term
4.
10
13
16
19
Term
Number
1
2
3
4
0
1
2
3
Term
Number
1
2
3
4
Term
7
12
17
22
n
Term
14
18
22
26
n
7.
Term
n
n
5.
Term
Number
1
2
3
4
6.
Term
Number
1
2
3
4
Term
26
31
36
41
n
Term
1
3
5
7
n
MFM 1P – Unit 5 – Linear Relations
55
LESSON 12 – Linear or Non-Linear
DATE:
Feeding Frenzy – Patterning to Algebraic Modelling
Part A
Frieda runs a catering business. She often has to set up table arrangements like the ones
shown below.
Help her determine the number of chairs and/or tables that she needs.
1. Start by completing the Number of Chairs column.
Term
Number
Picture
1
Number of
Chairs
6
2
3
4
5
6
2. Find an expression for the Number of Chairs if the term number is n.
(i.e. if there are n tables)
MFM 1P – Unit 5 – Linear Relations
56
LESSON 12 – Linear or Non-Linear
DATE:
Feeding Frenzy (continued)
Part B
Frieda runs a catering business. She often has to set up table arrangements like the ones
shown below. Frieda sometimes uses trapezoidal tables.
Help her determine the number of chairs and/or tables that she needs.
1. Start by completing the Number of Chairs column.
Term
Number
Picture
1
Number of
Chairs
5
2
3
4
5
2. Find an expression for the Number of Chairs if the term number is n.
(i.e. if there are n tables)
MFM 1P – Unit 5 – Linear Relations
57
LESSON 12 – Linear or Non-Linear
DATE:
Practice
1. How many toothpicks are needed for n squares?
2. The Capture-It Company makes picture frames.
Tiles are used for the border of the frames.
The light area represents the square space for the picture.
Frame
Number
(n)
Number of
Dark Tiles (d)
Determine your own equation to represent the
relationship between the frame (n) and the number of
dark tiles (d).
1
2
3
4
5
6
MFM 1P – Unit 5 – Linear Relations
58
LESSON 12 – Linear or Non-Linear
DATE:
3. The Larry’s Landscaping Company makes walkways. One walkway starts with a hexagonal
piece of concrete. To make the walkway longer, square pieces are added.
Number of
Squares
(n)
Perimeter
(P)
1
2
Sample Diagram:
This walkway begins with a hexagon
and has three square pieces added.
The length of each side is the same.
The perimeter of this walkway is 12.
3
4
5
6
a) Draw the diagrams for the first 2 terms (i.e. for 1 square and 2 squares).
b) Find the perimeter for each of your diagrams in part (a) and record it in the table.
c) Complete the remainder of the perimeter column (draw diagrams if necessary).
d) Determine your own equation to represent the relationship between the number of
squares (n) and the perimeter (P).
MFM 1P – Unit 5 – Linear Relations
59
REVIEW
DATE:
1. What is the formula for rate of change?
2. How do you find the initial value using
a) the graph
b) the table of values (chart)
c) the equation
3. How do you find the rate of change using
b) the graph
b) the table of values (chart)
c) the equation
4. Explain the difference between direct variation and partial variation. Draw an example of
each.
5. Draw an example for each of the following movements. Distance is measured from the
motion sensor.
Moving slowly
away from sensor
Moving quickly
away from sensor
Moving towards sensor
while speeding up
Moving slowly
towards sensor
Moving away from sensor
while slowing down
MFM 1P – Unit 5 – Linear Relations
Moving quickly
towards sensor
Moving away from sensor
then standing still
60
REVIEW
DATE:
6. Complete the chart:
Percent grade
A
Fraction
Rise
Run
1
40
Rate of change
(decimal form)
7%
B
C
0.09
4
25
D
7. Determine the rate of change for each object.
The pitch of the roof is the rate of
a)
change.
Rate of change =
7.2 m
The pitch is
5.6 m
b)
The steepness of the ramp is the rate
of change.
1.2 m
Rate of change =
4.8 m
The rate of change is
8. The grade of a road is often given as a percent. If the road rises 20 m over a horizontal
distance of 160 m, determine the grade as a percent.
MFM 1P – Unit 5 – Linear Relations
61
REVIEW
DATE:
9. Draw a graph matching the following story.
Bob is at home playing games on his X-Box when he realizes his buddies are coming
over and he has no food. He heads out to the grocery store to pick up some munchies
for his friends. It takes him 5 minutes to run to the store. At the grocery store, Bob sees
his high school math teacher and they chat for about his recent math mark during which
time he learns that he should be studying rather than playing video games. Bob cuts his
teacher short as he has now been at the store for 10 minutes and is running late. He
begins to quickly run back home, but after 1 minute, he realizes he has no plates. He
stops at the dollar store (which takes 8 minutes since there is a big line-up). He then
runs the rest of the way home in 3 minutes just in time to meet his friends at the front
door.
10. Write the equation for each relationship in the space provided. Show any
calculations you made. Indicate if the relation is a partial or direct variation.
A Chinese food restaurant has a special price The country club charges a $270 for their
for groups. Dinner for two costs $24 plus $11 facilities plus $29 per guest.
for each additional person.
Number of
People
0
20
40
60
80
MFM 1P – Unit 5 – Linear Relations
Cost of Holding an
Athletic Banquet
75
275
475
675
875
62
REVIEW
DATE:
11. A rental car costs $60 per day plus $0.30 for
each kilometre it is driven. Complete the table
then graph the relationship.
Number of
Kilometres
Cost vs. Number of Kilometres
260
Cost ($)
240
220
200
0
200
400
Cost ($)
180
160
140
120
100
80
600
60
40
20
Write an equation to model the relationship. C is
the cost and n is the number of kilometres.
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
Number of Kilometres
____ = _______________
Does this relation represent a partial or direct variation? Explain.
12. There is $400 in Mike’s bank account. She takes out $25 from her account each month but
doesn’t put any back in. Make a table of values for up to 5 months, then graph the
relationship.
Write an equation to model the
relationship.
____ = ______________
MFM 1P – Unit 5 – Linear Relations
63
REVIEW
DATE:
13. A pizza costs $10.00 plus $1.50 per topping.
Discuss the effect on the graph of changing the initial cost to $8.00 and raising the cost
per topping to $2.00. Use tables, graphs, and equations to support your conclusions.
14. a) Determine an equation to represent the relationship
between the term number, n, and the term, T.
b) Use your equation from part (a) to find the value of term number 20.
c) Use your equation from part (a) to find the term number if the value of
the term is 41.
MFM 1P – Unit 5 – Linear Relations
Term
Number
Term
1
8
2
11
3
14
4
17
5
20
x
?
64
REVIEW
DATE:
15. Describe the method for the “speedy way to graph” (i.e. without using a table of values) – Be
sure to discuss how you know where to start (to put the first point) and where to put the next
point.
16. Graph each of the following equations using the “speedy way to graph” (i.e. without making
a table of values)
1. C = 1000 + 100n
2. C = 25n + 60
3. y = 5x
4. y = 100 – 5x
MFM 1P – Unit 5 – Linear Relations
65
REVIEW
DATE:
17. Write the equation for the relationship and graph the relationship.
a)
The Recreation Centre charges a
membership fee of $25 plus $5 per class.
Equation:
c)
b) Repair Window charges a $20 service fee
plus $10/h to fix the window pane.
Equation:
Katie sells programs at the Omi Arena.
d) Megan has $400 in her bank account and
She is paid 50 cents for every program she
withdraws $25 each month.
sells.
Equation:
MFM 1P – Unit 5 – Linear Relations
Equation:
66