# Lesson 2 Half Interested or More Interesting Solidify Understanding

## Jump Start

Notice and Wonder

Previously, you examined the following context:

Medicine taken by a patient breaks down in the patient’s blood stream and dissipates out of the patient’s system. Suppose a dose of

Here are three representations of that context. List at least two things that you notice and one thing you are wondering about relative to these representations.

Representation #1:

Representation #2:

Representation #3:

## Learning Focus

Examine how the properties of exponents work with rational exponents.

Write equivalent exponential functions using different growth factors.

What do rational exponents and negative exponents mean in contexts?

Do the laws of exponents work with rational exponents?

How does the growth factor change if we focus on a month of exponential growth instead of a year?

## Open Up the Math: Launch, Explore, Discuss

Carlos and Clarita, the Martinez twins, have run a summer business every year for the past

“Remember how Dad said we could withdraw this money from the bank when we are

### 1.

Carlos calculates the value of the account

Year | Amount |
---|---|

### 2.

Clarita thinks Carlos is silly calculating the value of the account one year at a time, and says that he could have written a formula for the

### 3.

Carlos was surprised that Clarita’s formula gave the same account balance as his year-by-year strategy. Explain, in a way that would convince Carlos, why this is so.

“I can’t remember how much money we earned that summer,” said Carlos. “I wonder if we can figure out how much we deposited in the account five years ago, knowing the account balance now?”

### 4.

Carlos continued to use his strategy to extend his table year-by-year back

Year | Amount |
---|---|

Explanation:

### 5.

Clarita evaluated her formula for

Clarita doesn’t think leaving the money in the bank for another

Carlos remarked, “But we’ll be withdrawing our money halfway through the year. Do you think we’ll lose out on this year’s interest?”

“No, they’ll pay us a half-year portion of our interest,” replied Clarita.

“But how much will that be?” asked Carlos.

### 6.

Calculate the account balance and how much interest you think Carlos and Clarita should be paid if they withdraw their money

Clarita used this strategy: She substituted

Carlos had some questions about Clarita’s strategy:

What numerical amount do we multiply by when we use

as a factor? What happens if we multiply by

and then multiply the result by again? Shouldn’t that be a full year’s worth of interest? Is it? If multiplying by

is the same as multiplying by , what does that suggest about the value of ?

### 7.

Answer each of Carlos’s questions listed as best as you can.

Pause and Reflect

As Carlos is reflecting on this work, Clarita notices the date on the bank statement that started this whole conversation. “This bank statement is three months old!” she exclaims. “That means the bank will owe us

“So how much interest will the bank owe us then?” asked Carlos.

### 8.

Find as many ways as you can to answer Carlos’s question: How much will their account be worth in

Carlos now knows he can calculate the amount of interest earned on an account in smaller increments than one full year. He would like to determine how much money is in an account each month that earns

He starts by considering the amount in the account each month during the first year. He knows that by the end of the year the account balance should be

### 9.

Complete the table showing what amount is in the account each month during the first

Deposit | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

### 10.

What number did you multiply the account by each month to get the next month’s balance?

Carlos knows the exponential equation that gives the account balance for this account on an annual basis is

### 11.

Verify that both equations give the same results. Using the properties of exponents, explain why these two equations are equivalent.

### 12.

What is the meaning of the

Carlos shows his equation to Clarita. She suggests his equation could also be approximated by

### 13.

Answer Carlos’s question. What does the

Pause and Reflect

The properties of exponents can be used to explain why

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## Ready for More?

Use

## Takeaways

The following properties of exponents that make sense for positive integer exponents also apply and make sense for negative integer exponents and for fractional exponents:

We can interpret the negative exponential factor in

We can interpret a fractional exponential factor in

## Lesson Summary

In this lesson, we continued to explore the meaning of rational exponents, including negative integer exponents and fractional exponents. We learned that the properties of exponents can be applied to all rational exponents, not just integer exponents.

Use the rules of exponents to find three expressions that would be equivalent to the one provided.

### 1.

### 2.

Rewrite each of the expressions.