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.
- How do you think Gustavo made his prediction
- 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 |
- How has Gustavo’s bank balance been growing?
- Does Gustavo’s money seem to be doubling as the number of years doubles? Explain your reasoning.
- 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).
- 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?
- 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.
- 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.”
- 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?
- 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 |
|
|
|
- How do the amounts in the table grow?
- 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.
With your team:
a. Describe how each graph is the same.
b. Describe what makes each graph different.
c. 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.
d. 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.
4-27. IS IT PROPORTIONAL?
When you are making a prediction, it is important to be able to recognize whether a relationship is proportional or not.
Your Task: Work with your team to read each new situation below. Decide whether you think the relationship described is proportional or non-proportional and justify your reasoning. Be prepared to share your decisions and justifications with the class.
- Carlos wants to buy some new video games. Each game he buys costs him $36. Is the relationship between the number of games Carlos buys and the total price proportional?
- A single ticket to a concert costs $56, while buying five tickets costs $250. Is the relationship between the number of tickets bought and the total price proportional?
- Vu is four years older than his sister. Is the relationship between Vu and his sister’s age proportional?
- Janna runs at a steady pace of 7 minutes per mile. Is the relationship between the number of miles she ran and the distance she covered proportional?
- Carl just bought a music player and plans to load 50 songs each week. Is the relationship between the number of weeks after Carl bought the music player and the number of songs on his player proportional?
- Anna has a new video game. It takes her five hours of playing the game to master level one. After so much time, Anna understands the game better and it only takes her three hours of playing the game to master level two. Is the number of hours played and the game level proportional?
Assignment: 4.2.1 Homework
In Lesson 4.2.1, you learned that you could identify proportional relationships by looking for a constant multiplier. In fact, you have already seen a relationship with a constant multiplier in this course. Today you will revisit the earlier situation that contains a proportional relationship.
Kaci Cheese Task (4-35)
Assignment: 4.2.2 Homework
Robert decided to graph the situation, as shown below. The distance Robert can travel using one gallon of gas is called the unit rate. Use Robert’s graph to predict how far he can drive using one gallon of gas. That is, find his unit rate.
pound of cheese
pound of bananas
pound of deli roast beef
pound of sliced turkey
In the previous lessons, you studied different ways to represent proportional relationships. You organized information into tables and graphs. You also wrote equations modeling the proportional relationships. Proportional relationship equations are of the form y = kx, where k is the constant of proportionality. Today you will find connections between different representations of the same proportional relationship, explore each representation more deeply, and learn shorter ways to go from one representation to another. As you work today, keep these questions in mind:
