Building a Linear Model
Stacks of Cups
SUGGESTED LEARNING STRATEGIES: Shared Reading,
Marking the Text, Questioning the Text, Use Manipulatives,
Create Representations, Look for a Pattern
ACTIVITY
2.4
My Notes
You are the packaging director for a paper products company. Your
company is introducing a new type of paper cups. Your design team
must design a cardboard container to use when packaging the cups
for sale. Your supervisor has given you the following requirements.
• All lateral faces of the container must be rectangular.
• T he base of the container must be a square, just large enough to
accommodate one cup.
• T he height of the container must be given as a function of the
number of cups the container will hold.
• All measurements must be in centimeters.
To help discover which features of the cup af fect the height of
the stack, your team will collect data on two types of cups found
around the of f ice.
1. Use two dif ferent types of cups to complete the tables below.
© 2010 College Board. All rights reserved.
CUP 1
Number of
Height of
Cups
Stack
CONNECT TO GEOMETRY
The carton will be a right
rectangular prism. A rectangular
prism is a closed, threedimensional figure with three
pairs of opposite parallel faces
that are congruent rectangles.
CUP 2
Number of
Height of
Cups
Stack
2. What patterns do you notice that might help you f igure out the
relationship between the height of the stack and the number of
cups in that stack?
Unit 2 • Linear Functions
99
ACTIVITY 2.4
Building a Linear Model
continued
Stacks of Cups
My Notes
SUGGESTED LEARNING STRATEGIES: Create
Representations, Look for a Pattern, Guess and Check,
Think/Pair/Share, Quickwrite
Use your data for Cup 1 to complete Items 3–6.
3. Make a graph of the data you collected.
h
Cup 1 Stack
20
18
Height (cm)
16
14
12
10
8
6
4
2
1
2
3
4
5
6
7
8
9
10
n
4. Predict, without measuring, the height of a stack of 16 cups.
Explain how you arrived at your prediction.
5. Predict, without measuring, the height of a stack of 50 cups.
Explain how you arrived at your prediction.
6. Write an equation that gives the height of a stack of cups, h, in
terms of n, the number of cups in the stack.
100
SpringBoard® Mathematics with Meaning™ Algebra 1
© 2010 College Board. All rights reserved.
Number of Cups
Building a Linear Model
ACTIVITY 2.4
continued
Stacks of Cups
SUGGESTED LEARNING STRATEGIES: Create Representations,
Think/Pair/Share, Quickwrite
My Notes
7. Use your equation from Item 6 to f ind h when n = 16 and
when n = 50. Do your answers to this question agree with your
predictions in Items 4 and 5?
8. Sketch the graph of your equation from Item 6.
h
Cup 1 Stack
20
18
Height (cm)
16
14
12
10
8
© 2010 College Board. All rights reserved.
6
4
2
1
2
3
4
5
6
7
8
9
10
n
Number of Cups
9. How are the graphs you made in Items 3 and 8 the same?
How are they different?
Unit 2 • Linear Functions
101
ACTIVITY 2.4
Building a Linear Model
continued
Stacks of Cups
My Notes
SUGGESTED LEARNING STRATEGIES: Guess and Check,
Shared Reading, Create Representations, Group
Presentation, RAFT, Self/Peer Revision
10. Remember that you are designing a container with a square
base. What dimension(s), other than the height of the stack,
do you need to design your cup container? Use Cup 1 to f ind
this/these dimension(s).
11. Find the dimensions of a container that will hold a stack of
25 cups.
• Role—team leader
• Audience—your boss
• Format—a letter
• Topic—stacks of cups
102
• the equation your team discovered to f ind the height of
the stack of Cup 1 style cups
• a description of how your team discovered the equation
and the minimum number of cups needed to f ind it
• an explanation of how the numbers in the equation relate
to the physical features of the cup
• an equation that could be used to f ind the height of the
stack of Cup 2 style cups
SpringBoard® Mathematics with Meaning™ Algebra 1
© 2010 College Board. All rights reserved.
When writing your answer to
Item 12, you can use a RAFT.
12. Your team has been asked to communicate its f indings to
your supervisor. Write a report to her that summarizes your
f indings about the cup container design. Include the following
information in your report.
Building a Linear Model
ACTIVITY 2.4
continued
Stacks of Cups
SUGGESTED LEARNING STRATEGIES: Visualization, Create
Representations, Identify a Subtask
My Notes
After reading your report, your supervisor was able to determine
the equation for the height of the stack for the specif ic cup that the
company will manufacture. T he cup will be the same basic shape
as described in your report. T he company will use the function
S(n) = 0.5n + 12.5.
13. What do S, S(n), and n represent?
MATH TERMS
The function S(n) = 0.5n + 12.5
represents a linear model. A
linear model is a representation
of data whose graph consists of
points that lie in a straight line.
© 2010 College Board. All rights reserved.
14. What do the numbers 0.5 and the 12.5 in the function S tell
you about the physical features of the cup?
15. Evaluate S(1) to f ind the height of a single cup.
The graph of a linear equation is
a straight line.
16. How tall is a stack of 35 cups? Show your work using function
notation.
Unit 2 • Linear Functions
103
ACTIVITY 2.4
Building a Linear Model
continued
Stacks of Cups
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Quickwrite, Look for a Pattern, Marking the Text
My Notes
17. If you add 2 cups to a stack, how much does the height of the
stack increase?
18. If you add 20 cups to a stack, how much does the height of the
stack increase?
20. If you were to graph the linear function S(n) = 0.5n + 12.5
and connect the points, you would see that they lie in a straight
line. T he slope of a line is a measure of the steepness of a line
and indicates a rate of change.
a. What is the slope of this line?
CONNECT TO AP
The idea of rate of change is a key
concept in calculus.
104
b. Interpret the slope of the line as a rate of change that
compares a change in height to a change in the number
of cups.
SpringBoard® Mathematics with Meaning™ Algebra 1
© 2010 College Board. All rights reserved.
19. A member of one of the teams stated: “If you double the
number of cups in a stack, then the height of the stack is also
doubled.” Is this statement correct? Explain.
Building a Linear Model
ACTIVITY 2.4
continued
Stacks of Cups
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Work Backward, Group Presentation
My Notes
21. a. The supervisor wanted to increase the height of a container by
5 cm. How many more cups would f it in the container?
b. If the supervisor wanted to increase the height of a container
by 6.4 cm, how many more cups would f it in the container?
c. How many cups f it in a container that is 36 cm tall?
© 2010 College Board. All rights reserved.
d. How many cups f it in a container that is 50 cm tall?
22. T he function S(n) = 0.5n + 12.5 describes height S in terms of
the number of cups n.
a. Solve this equation for n to describe the number of cups n in
terms of the height S.
Unit 2 • Linear Functions
105
ACTIVITY 2.4
Building a Linear Model
continued
Stacks of Cups
SUGGESTED LEARNING STRATEGIES: Quickwrite,
Debriefing
My Notes
b. How many cups f it in a carton that is 85 cm tall? Compare
your method of answering this question to your method
used in Item 21 parts (c) and (d).
c. What is the slope of the line represented by your equation in
part (a)? Interpret it as a rate of change and compare it to the
rate of change found in Item 20(b).
CHECK YOUR UNDERSTANDING
Write your answers
answers on
on notebook
notebook paper.
paper.Show
Showyour work.
5. The equation for the cost C of a cab ride of
your work.
m miles is C = 2.5m + 3.5.
Hours
Earnings
0
$75
1
$125
2
$175
3
$225
4
$275
1. If the consultant has a 36-hour contract,
how much will she earn?
a. What is the cost of a 6-mile ride?
b. What is the cost of a 7-mile ride?
c. How is the price difference between a
6-mile ride and a 7-mile ride related to
the numbers in your equation?
6. In the equation S = 0.25n + 8.5, S is the
height in inches of a stack of jumbo cups
and n is the number of cups.
2. Write an equation that shows the
consultant’s earnings E in terms of h, the
number of hours of her contract.
a. How many cups would it take to make a
stack 1 inch higher?
Use this table for Items 3 and 4.
b. How many cups would fit in a carton
that is 18 inches high?
x
y
1
1
2
5
3
9
4
13
5
17
3. Write an equation for y in terms of x.
4. Explain how the numbers in your equation
relate to the numbers in the table.
106
SpringBoard® Mathematics with Meaning™ Algebra 1
c. Interpret the slope as a rate of change.
7. MATHEMATICAL What did you learn about
R E F L E C T I O N creating a linear model?
How can you recognize and interpret a
constant rate of change?
© 2010 College Board. All rights reserved.
A consultant earns a flat fee of $75 plus $50 per
hour for a contracted job. The table shows the
consultant’s earnings.