Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

You are walking along a beach, and your toe hits something hard. You reach down, grab onto a handle, and pull out a lamp! It is sandy. You start to brush it off with your towel. Poof! A genie appears.

He tells you, “Thank you for freeing me from that bottle! I was getting claustrophobic. You can choose one of these purses as a reward.”

Purse A, which contains $1,000 today. If you leave it alone, it will contain $1,200 tomorrow (by magic). The next day, it will be $1,400. This pattern of an additional $200 per day will continue.
Purse B, which contains 1 penny today. Leave that penny in there because tomorrow it will (magically) turn into 2 pennies. The next day, there will be 4 pennies. The amount in the purse will continue to double each day.

1. How much money will be in Purse A after a 7-day week? Enter your answer as a number rounded to the nearest cent.

After 7 days, Purse A will contain $2,400 because it grows by $200 each day for 7 days.

2. How much money will be in Purse B after a 7-day week? Enter your answer as a number rounded to the nearest cent.

Purse B will contain $1.28. It doubles each day for 7 days, so it will have $1 \cdot 2^{7}$ or 128 cents (the 1 here represents the 1 cent for when the genie appeared).

3. How much money will be in Purse A after two weeks? Enter your answer as a number rounded to the nearest cent.

After 2 weeks, Purse A will contain an additional $1,400 for a total of $3,800.

4. How much money will be in Purse B after two weeks? Enter your answer as a number rounded to the nearest cent.

Purse B will contain $1 \cdot 2^{14}$ cents. This is 16,384 cents or $163.84.

5. The genie later adds that he will let the money in each purse grow for three weeks.

a. How much money will be in Purse A in three weeks? Enter your answer as a number rounded to the nearest cent.

After 3 weeks, Purse A will have an additional $1,400 during the third week for a total of $5,200.

b. How much money will be in Purse B in three weeks? Enter your answer as a number rounded to the nearest cent.

Purse B will have 128 times as much money (compared to the end of two weeks) for a total of $20,971.52.

6. Which purse contains more money after 30 days? Enter your answer as a number rounded to the nearest cent.

Purse B will have a lot more money. Purse A will have $1, 000 + 30 \cdot 200$ or $7,000, which is much less than the doubling purse has on day 21.

Building Character: Grit

A person in business attire climbs a mountain with a briefcase and walking stick. Trophy and bullseye icons appear in the background, symbolizing achievement and reaching goals. Orange flags and clouds complete the scene.

Grit is passion and perseverance to reach long-term goals.

Think about your current sense of grit. Are the following statements true for you?

Setbacks don’t discourage me for long.
I never stop working to improve.

Don’t worry if none of these statements are true for you. Developing this trait takes time. Your first step starts today!

Self Check

Olivia has two different job offers for babysitting next Saturday.

Job 1: She will receive $5 for accepting the job, then $10 for every hour she works.
Job 2: She will receive $1.50 for accepting the job, $3 after the first hour, $6 after the second hour, and for each additional hour her pay per hour will double.

Olivia can only work 3 hours. How much would she make at each job?

Olivia would make $\$30$ at Job 1 and $\$21$ at Job 2.
Olivia would make $\$25$ at Job 1 and $\$10.50$ at Job 2.
Olivia would make $\$35$ at Job 1 and $\$22.50$ at Job 2.
Olivia would make $\$35$ at Job 1 and $\$12$ at Job 2.

Additional Resources

Exploring Linear and Exponential Growth

The tables listed below are modeling four different functions, all showing growth. Which one does not have the same growth pattern as the others?

$x$	$y$
1	8
2	16
3	24
4	32
8	64

Table A

$x$	$y$
0	0
2	16
4	32
6	48
8	64

Table B

$x$	$y$
0	1
1	4
2	16
3	64
4	256

Table C

$x$	$y$
0	4
1	8
2	12
3	16
4	20

Table D

When determining growth, there are a few ways to check. When given a table, you could graph each, or you could check the rate of change. Linear functions have a constant rate of change, where consecutive terms have a common difference. The common difference is a value that is added to or subtracted from consecutive terms to find the next term. Exponential functions have a constant ratio or common multiplier. The constant ratio is a value that is multiplied or divided from consecutive terms to find the next term.

Table A is linear and has a slope of 8 since the change in $y$ is 8 and the change in $x$ is 1.
Table B is also linear because the change in $y$ is 16 and the change in $x$ is 2 for a slope of 8.
Table C is an example of an exponential function. There is a common ratio instead of a common difference. Notice that each $y$ -term is 4 times the previous term. Therefore, the common ratio is 4.
Table D is also linear with a common difference of 4.

Table C is the only exponential function. Therefore, it has a different growth pattern since Tables A, B, and D are linear.

Try it

Try It: Exploring Linear and Exponential Growth

The table shows the height, in centimeters, of the water in a swimming pool at different times since the pool started to be filled.

minutes	height
0	150
1	150.5
2	151
3	151.5

Does the height of the water have a common difference or a common ratio?

Common Difference

Explain how you know.

Compare your answer:

When you look at the $y$ -values in the table, you notice that they are all 0.5 more than the previous term. That means there is a common difference of 0.5.

What type of function is represented by the table?

Compare your answer:

The function is linear because the common difference means the rate of change, or slope, is constant.

Explain how you know.

Linear function

5.3.2 Exploring Different Growth Patterns

Activity

Additional Resources

Exploring Linear and Exponential Growth

Try It: Exploring Linear and Exponential Growth