Lesson 4.2.1
HW: 4-29 to 4-33
Learning Target: Scholars will learn the difference between proportional relationships and other linear
relationships.
Grocery stores often advertise special prices for fruits and vegetables that are in season. You might see a
sign that says, “Special Today! Buy 2 pounds of apples for $1.29!” How would you use that information
to predict how much you need to pay if you want to buy six pounds of apples? Or just 1 pound of
apples? The way that the cost of apples grows or shrinks allows you to use a variety of different
strategies to predict and estimate prices for different amounts of apples. In this section, you will explore
different kinds of growth patterns. You will use those patterns to develop strategies for making
predictions and deciding if answers are reasonable.
As you work in this section, ask yourself these questions to help you identify different patterns:
 How are the entries in the table related?
 Can I double the values?
 What patterns can I see in a graph?
4-21. COLLEGE FUND
Five years ago, Gustavo’s grandmother put some money in a college savings account for him on his
birthday. The account pays simple interest, and now, after five years, the account is worth $500. Gustavo
predicts that if he does not deposit or withdraw any money, then the account balance will be $1000 five
years from now.
1. How do you think Gustavo made his prediction?
2. Do you agree with Gustavo’s reasoning? Explain why or why not.
4-22. Last week, Gustavo got his bank statement in the mail. He was surprised to see a graph that showed
that, although his balance was growing at a steady rate, the bank predicted that in five years his account
balance would be only $600. “What is going on?” he wondered. “Why isn’t my money growing the way
I thought it would?”
With your team, discuss how much Gustavo’s account appears to be growing every year. Why might his
account be growing in a different way than he expected? Be ready to share your ideas.
4-23. Gustavo decided to look more carefully at his balances for the last few years to see if the bank’s
prediction might be a mistake. He put together the table below.
Time since Original Deposit (in yrs)
2
3
4
5
Bank Balance (in dollars)
440 460 480 500
1. How has Gustavo’s bank balance been growing?
2. Does Gustavo’s money seem to be doubling as the number of years doubles? Explain your
reasoning.
3. Is the bank’s prediction a mistake? Explain your answer.
4-24. Once Gustavo saw the balances written in a table, he decided to take
a closer look at the graph from the bank to see if he could figure out where he made the mistake in his
prediction. Find the graph below on the Lesson 4.2.1 Resource Page or use the 4-24 Student
eTool (Desmos).
1.  There is additional information about Gustavo’s account that you can tell from the graph. For
example, what was his starting balance? How much does it grow in 5 years?
2. Gustavo had assumed his money would double after 10 years. What would the graph look like if
that were true? Using a different color, add a line to the graph that represents what Gustavo was
thinking.
3. Is it possible that Gustavo’s account could have had $0 in it in Year 0? Why or why not?
4-25. FOR THE BIRDS
When filling her bird feeder, Sonja noticed that she paid $27 for four pounds of bulk birdseed. “Next
time, I’m going to buy 8 pounds instead so I can make it through the spring. That should cost $54.”
1. Does Sonja’s assumption that doubling the amount of birdseed would double the price make
sense? Why or why not? How much would you predict that 2 pounds of birdseed would cost?
2. To check her assumption, she found a receipt for 1 pound of birdseed. She decided to make a
table, which is started below. Copy and complete her table or use the 4-25 Student
eTool (Desmos).
Pounds
0
1
2
3
4
5
6
8
Cost
$6.75
$27
3. How do the amounts in the table grow?
4. Does the table confirm Sonja’s doubling relationship? Give two examples from the table that
show how doubling the pounds will double the cost.
4-26. What makes Sonja’s birdseed situation (problem 4-25) different from Gustavo’s college fund
situation (problem 4-21)? Why does doubling work for one situation but not in the other? Consider this
as you examine the graphs below.
1. With your team:
o Describe how each graph is the same.
o Describe what makes each graph different.
2. How do the differences explain why doubling works in one situation and not in the
other? Generalize why doubling works in one situation and not in another.
3. The pattern of growth in Sonja’s example of buying birdseed is an example of a proportional
relationship. In a proportional relationship, if one quantity is multiplied by a scale factor, the
other is scaled by the same amount. Gustavo’s bank account is not proportional, because it grows
differently; when the number of years doubled, his balance did not.
Work with your team to list other characteristics of proportional relationships, based on Sonja’s
and Gustavo’s examples. Be as specific as possible.
4-29. The lemonade stand at the county
fair sells the lemonade at a price of two
cups for $3.60. Complete the table at
right to find what Paula’s family will pay
to buy lemonade for all eight members of
the family. Homework Help ✎
# of Lemonades
1
2
3
4
5
6
7
8
Price (in dollars)
3.60
4-30. Carmen is downloading music for her Pod. Each song costs $1.75. Is this relationship
proportional? Explain your reasoning. What is the cost for five songs?
4-31. Simplify each expression.
1.
2.
3.
4.
5.
6.
−3 + 7
7 + (−8)
−6 − 9
−3 + 4(−2)
4 − 2(−5)
(−7 + 3)(4 − 5 · 2)
4-32. Copy and complete each of the Diamond Problems below. The pattern used in the
Diamond Problems is shown at right.
4-33. Lucy keeps track of how long it takes her to do the newspaper’s crossword puzzle each
day. Her recent times, in minutes, were.
 8 22 19 12 18 19
1. What is the median of her data?
2. What is the mean?
10
35
12
19
16 21
Lesson 4.2.1

4-21. See below:
1. He saw that the number of years doubled, so he also doubled the amount the
account will be worth.
2. Answers vary. See the “Suggested Lesson Activity” for description of student
thinking.

4-22. His account is growing $20 each year; possible reasons: the interest rate is lower
than what he thought, or his grandmother invested some money to start with so the
account did not earn $500 in interest in 5 years.

4-23. See below:
1. It increases by $20 each year.
2. No, reasons vary but could include that the amount of money at 4 years is not
twice the amount at 2 years.
3. The pattern in the balances supports the bank; the account will reach $600 in year
10.

4-24. See below:
1. It started with $400 in the account. The account grew by $100 in the first 5
years.
2. The line should pass through (0, 0), (5, 500), and (10, 1000).
3. No. There would be no money to earn interest.

4-25. See below:
1. Yes, because it is reasonable that bulk birdseed has a constant cost per
pound. Two pounds should cost half as much, or $13.50.
2. See answers in bold in the table below.

3. See answers in the table; the price increases by $6.75 per pound.
4. Yes. You can show doubling using 1 and 2 lbs., 2 and 4 lbs., 3 and 6 lbs., or 4
and 8 lbs.
4-26. See below:
1. Sample response: The graphs are each lines. One starts at (0, 0), while the other
has a y-intercept at (0, 400).
2. Because Gustavo’s graph starts at $400 doubling does not work, but for Sonja 0
pounds of birdseed cost $0 so doubling works.
3. Proportional relationships grow by multiplying each quantity by the same
number; proportional relationships also go through (0, 0) on a graph.

4-29. See answers in bold in the table below:

4-30. Yes, it is proportional because each tune is the same cost, $8.75.

4-31. See below:
1. 4
2. –1
3. –15
4. –11
5. 14
6. 24

4-32. See answers in bold in the diamonds below:

4-33. See below:
1. 18.5 min
2. 17.6 min