# Lesson 5

Negative Exponents with Powers of 10

## 5.1: Number Talk: What's That Exponent? (10 minutes)

### Warm-up

The purpose of this Number Talk is to elicit strategies and understandings students have for dividing powers. These understandings help students develop fluency and will be helpful later in this lesson when students investigate negative exponents. While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem. It is expected that students won't know how to approach the final question. Encourage them to make their best guess based on patterns they notice.

### Launch

Display one problem at a time. Give students 1 minute of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Allow students to share their answers with a partner and note any discrepancies.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

### Student Facing

Solve each equation mentally.

$$\frac{100}{1} = 10^x$$

$$\frac{100}{x} = 10^1$$

$$\frac{x}{100} = 10^0$$

$$\frac{100}{1,\!000} = 10^{x}$$

### Activity Synthesis

Ask students to share what they noticed about the first three problems. Record the equations with the solutions written in place:

$$\frac{100}{1} = 10^2$$

$$\frac{100}{10} = 10^1$$

$$\frac{100}{100} = 10^0$$

Ask students to describe any patterns they see, and how they would continue the patterns for the last problem:

$$\frac{100}{1,\!000} = 10^{x}$$

Tell them that in this lesson, we will explore negative exponents.

Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

## 5.2: Negative Exponent Table (10 minutes)

### Activity

Students extend their understanding of exponents to include negative exponents and explain patterns in the placements of the decimal point when a decimal is multiplied by 10 or $$\frac{1}{10}$$. Students use repeated reasoning to recognize that negative powers of 10 represent repeated multiplication by $$\frac{1}{10}$$ and generalize to the rule $$10^{\text- n} = \frac{1}{10^n}$$ (MP8).

A table is used to show different representations of decimals, fractions, and exponents. The table is horizontal to mimic the structure of decimals. As students work, notice the strategies they use to go between the different representations of the given powers of 10. Ask students who use contrasting strategies to share later.

### Launch

Tell students to complete the table one row at a time to see the patterns most clearly. Ask a student to read the first question aloud. Select a student to explain the idea of a “multiplier” in this context. Give students 5–7 minutes to work. Follow with a whole-class discussion.

### Student Facing

Complete the table to explore what negative exponents mean.

1. As you move toward the left, each number is being multiplied by 10. What is the multiplier as you move right?
2. How does a multiplier of 10 affect the placement of the decimal in the product?  How does the other multiplier affect the placement of the decimal in the product?

3. Use the patterns you found in the table to write $$10^{\text -7}$$ as a fraction.

4. Use the patterns you found in the table to write $$10^{\text -5}$$ as a decimal.

5. Write $$\frac{1}{100,000,000}$$ using a single exponent.

6. Use the patterns in the table to write $$10^{\text -n}$$ as a fraction.

### Anticipated Misconceptions

Some students may think that, for example, $$\frac{1}{1,000,000} = 10^{\text-7}$$ because the number 1,000,000 has 7 digits. Ask these students if it is true that $$\frac{1}{10} = 10^{\text-2}$$.

### Activity Synthesis

One important idea is that multiplying by 10 increases the exponent, thus multiplying by $$\frac{1}{10}$$ decreases the exponent. So negative exponents can be thought of as repeated multiplication by $$\frac{1}{10}$$, whereas positive exponents can be thought of as repeated multiplication by $$10$$. Another key point is the effect that multiplying by 10 or $$\frac{1}{10}$$ has on the placement of the decimal.

Ask students to share how they converted between fractions, decimals, and exponents. Record their reasoning for all to see. Here are some possible questions to consider for whole-class discussion:

• “Do you agree or disagree? Why?”
• “Did anyone think of this a different way?”
• “In your own words, what does $$10^{\text- 7}$$ mean? How is it different from $$10^7$$?”

Introduce the visual display for $$10^{\text- n} = \frac{1}{10^n}$$ and display it for all to see throughout the unit. For an example that illustrates the rule, consider displaying $$10^{\text- 3} = \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} = \frac{1}{10^3}$$.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections. For example, annotate the patterns students notice to help synthesize how negative exponents can be thought of as repeated multiplication by $$\frac{1}{10}$$, whereas positive exponents can be thought of as repeated multiplication by $$10$$.
Supports accessibility for: Visual-spatial processing
Conversing, Representing, Writing: MLR2 Collect and Display. During the whole-class discussion, capture the vocabulary and phrases students use to describe their strategies for converting between fractions, decimals, and exponents. Listen for students who make connections between repeated multiplication and the placement of the decimal point. Record this language onto a visual display that can be referenced in future discussions. This routine provides feedback to students that supports sense-making and increases meta-awareness of mathematical language.
Design Principle(s): Support sense-making; Maximize meta-awareness

## 5.3: Follow the Exponent Rules (20 minutes)

### Activity

This activity requires students to make sense of negative powers of 10 as repeated multiplication by $$\frac{1}{10}$$ in order to distinguish between equivalent exponential expressions. If students have time, instruct them to write the other expressions in each table as a power of 10 with a single exponent as well. Look for students who have productive debate regarding the interpretation of, for example, $$(10^2)^{\text-3}$$ versus $$(10^{\text-2})^3$$ and ask them to share their reasoning later (MP3).

### Launch

Arrange students in groups of 2. Give students 15 minutes of partner work time followed by a whole-class discussion. Ask students to explain their reasoning to their partner as they work. If there is disagreement, tell students to work to reach an agreement.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention

### Student Facing

1. Match each exponential expression with an equivalent multiplication expression:

$$\left(10^2\right)^3$$

$$\left(10^2\right)^{\text -3}$$

$$\left(10^{\text -2}\right)^3$$

$$\left(10^{\text -2}\right)^{\text-3}$$

 $$\frac{1}{(10 \boldcdot 10)} \boldcdot \frac{1}{(10 \boldcdot 10)} \boldcdot \frac{1}{(10 \boldcdot 10)}$$ $$\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)$$ $$\frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} }\boldcdot \frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} } \boldcdot \frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} }$$ $$(10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)$$
2. Write $$(10^2)^{\text-3}$$ as a power of 10 with a single exponent. Be prepared to explain your reasoning.
1. Match each exponential expression with an equivalent multiplication expression:

$$\frac{10^2}{10^5}$$

$$\frac{10^2}{10^{\text -5}}$$

$$\frac{10^{\text -2}}{10^5}$$

$$\frac{10^{\text -2}}{10^{\text -5}}$$

 $$\frac{ \frac{1}{10} \boldcdot \frac{1}{10} }{ \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10}\boldcdot \frac{1}{10} }$$ $$\frac{10 \boldcdot 10}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10}$$ $$\frac{ \frac{1}{10} \boldcdot \frac{1}{10} }{ 10 \boldcdot 10\boldcdot 10\boldcdot 10\boldcdot 10 }$$ $$\frac{ 10 \boldcdot 10 }{ \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10}\boldcdot \frac{1}{10}}$$
2. Write $$\frac{10^{\text -2}}{10^{\text -5}}$$ as a power of 10 with a single exponent. Be prepared to explain your reasoning.
1. Match each exponential expression with an equivalent multiplication expression:

$$10^4 \boldcdot 10^3$$

$$10^4 \boldcdot 10^{\text -3}$$

$$10^{\text -4} \boldcdot 10^3$$

$$10^{\text -4} \boldcdot 10^{\text -3}$$

 $$(10 \boldcdot 10 \boldcdot 10 \boldcdot 10) \boldcdot ( \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10})$$ $$\left(\frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left( \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right)$$ $$\left(\frac{1}{10}\boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left(10 \boldcdot 10 \boldcdot 10\right)$$ $$(10 \boldcdot 10 \boldcdot 10 \boldcdot 10) \boldcdot (10 \boldcdot 10 \boldcdot 10)$$
2. Write $$10^{\text-4} \boldcdot 10^3$$ as a power of 10 with a single exponent. Be prepared to explain your reasoning.

### Student Facing

#### Are you ready for more?

Priya, Jada, Han, and Diego stand in a circle and take turns playing a game.

Priya says, SAFE. Jada, standing to Priya's left, says, OUT and leaves the circle. Han is next: he says, SAFE. Then Diego says, OUT and leaves the circle. At this point, only Priya and Han are left. They continue to alternate. Priya says, SAFE. Han says, OUT and leaves the circle. Priya is the only person left, so she is the winner.

Priya says, “I knew I’d be the only one left, since I went first.”

1. Record this game on paper a few times with different numbers of players. Does the person who starts always win?
2. Try to find as many numbers as you can where the person who starts always wins. What patterns do you notice?

### Anticipated Misconceptions

Some students may struggle to distinguish between $$\left(10^{\text-2}\right)^3$$ and $$\left(10^2\right)^{\text-3}$$. Breaking down each expression into parts that emphasize repeated multiplication will help to illustrate the difference. For the first expression, $$10^{\text-2} = \frac{1}{10} \boldcdot \frac{1}{10}$$. The outer exponent of 3 means that $$\frac{1}{10} \boldcdot \frac{1}{10}$$ is multiplied repeatedly 3 times. Similarly for the second expression, $$10^2 = 10 \boldcdot 10$$ and the outer exponent of $$\text- 3$$ means that the reciprocal of $$10 \boldcdot 10$$ is multiplied repeatedly 3 times. In the end, both expressions are equal to $$10^{\text-6}$$.

### Activity Synthesis

Select students who had disagreements during the activity to share what they disagreed about and how they came to an agreement. Consider asking:

• “How is $$(10^{\text-2})^3$$ different from $$(10^2)^{\text-3}$$? How are they the same?”
• “Do you agree or disagree? Why?”
• “Could you restate __'s reasoning in a different way?”

It is important for students to understand that the exponent rules work even with negative exponents. To see why, the whole class discussion must make a clear connection between the exponent rules and the process of multiplying repeated factors that are 10 and $$\frac{1}{10}$$. Contrast the expanded version of $$\left(10^{\text-2}\right)^3$$ and $$\left(10^2\right)^{\text-3}$$. For $$(10^{\text-2})^3$$, there are 3 factors that are $$10^{\text-2}$$, where $$10^{\text-2}$$ is two factors that are $$\frac{1}{10}$$, so $$\displaystyle \left(10^{\text-2}\right)^3 = \left( \frac{1}{10} \boldcdot \frac{1}{10} \right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.$$ For $$(10^2)^{\text-3}$$, there are 3 factors that are $$\frac{1}{10^2}$$, so $$\displaystyle (10^2)^{\text-3} = \left( \frac{1}{10 \boldcdot 10} \right)\left(\frac{1}{10 \boldcdot 10}\right)\left(\frac{1}{10 \boldcdot 10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.$$Both $$\left(10^{\text-2}\right)^3$$ and $$\left(10^2\right)^{\text-3}$$ are equal to $$10^{\text-6}$$ in the same way that $$\text-2 \boldcdot 3 = 2 \boldcdot \text-3 = \text-6$$.

Listening, Conversing: MLR8 Discussion Supports. As students share the disagreements they had with their partner, press for details in students’ statements by requesting that students elaborate on an idea or give an example of their process of reconciliation. For example, ask students, “How did you explain your reasoning to your partner in a way that convinced them?” In this discussion, it may be useful to revoice student ideas to model mathematical language use by restating a statement in order to clarify, apply appropriate language, or involve more students.
Design Principle(s): Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

The purpose of the discussion is to take a step back in order to see that negative exponents are not something new and different, but rather a natural part of the decimal place value system that we have been exploring for years.

Remind students that in elementary school, we learned about our place value system and saw that it was possible to write very large numbers in a very small space because of positional notation. We learned that the value of a number is the sum of the numbers of each base 10 unit (ones, tens, hundreds, and so forth). We sometimes wrote things like $$456=4\boldcdot 100+5\boldcdot 10+6\boldcdot 1$$. We can express this with exponents as $$456=4\boldcdot 10^2+5\boldcdot 10^1+6\boldcdot 10^0$$. We now extend the discussion of place value by considering ways to write very small numbers in a manner consistent with what we did with large numbers. Ask students:

• “How would you write 2,796 as a sum with powers of 10?” ($$2\boldcdot 10^3+7\boldcdot 10^2+9\boldcdot 10^1+6\boldcdot 10^0$$)
• “What about small numbers? How do you write the place value units of 0.1, 0.01, and 0.001 with powers of 10?” ($$10^{\text-1}, 10^{\text-2}, 10^{\text-3}$$)
• “Then how would you write 0.2796 as a sum with powers of 10?” ($$2\boldcdot 10^{\text-1}+7\boldcdot 10^{\text-2}+9\boldcdot 10^{\text-3}+6\boldcdot 10^{\text-4}$$)
• “Think about the meaning of exponents. How is $$10^3$$ related to $$10^{\text-3}$$?” (Exponents tell us to repeatedly multiply by a base. Whether the base is 10 or $$\frac{1}{10}$$, the structure of repeated multiplication is the same. $$10^3$$ is multiplication by 10 repeated 3 times and $$10^{\text-3}$$ is multiplication by $$\frac{1}{10}$$ repeated 3 times.)
• “Who would need to work with very large numbers? Who would need to work with very small numbers?” (Astronomers might need to work with very large numbers. Biologists, physicists, engineers and others might need to work with very small numbers.)

## 5.4: Cool-down - Negative Exponent True or False (5 minutes)

### Cool-Down

When we multiply a positive power of 10 by $$\frac{1}{10}$$, the exponent decreases by 1: $$\displaystyle 10^8 \boldcdot \frac{1}{10} = 10^7$$This is true for any positive power of 10. We can reason in a similar way that multiplying by 2 factors that are $$\frac{1}{10}$$ decreases the exponent by 2: $$\displaystyle \left(\frac{1}{10}\right)^2 \boldcdot 10^8 = 10^6$$
That means we can extend the rules to use negative exponents if we make $$10^{\text-2} = \left(\frac{1}{10}\right)^2$$. Just as $$10^2$$ is two factors that are 10, we have that $$10^{\text-2}$$ is two factors that are $$\frac{1}{10}$$. More generally, the exponent rules we have developed are true for any integers $$n$$ and $$m$$ if we make $$\displaystyle 10^{\text-n} = \left(\frac{1}{10}\right)^n = \frac{1}{10^n}$$
Here is an example of extending the rule $$\frac{10^n}{10^m} = 10^{n-m}$$ to use negative exponents: $$\displaystyle \frac{10^3}{10^5} = 10^{3-5} = 10^{\text-2}$$ To see why, notice that $$\displaystyle \frac{10^3}{10^5} = \frac{10^3}{10^3 \boldcdot 10^2} = \frac{10^3}{10^3} \boldcdot \frac{1}{10^2} = \frac{1}{10^2}$$which is equal to $$10^{\text-2}$$.
Here is an example of extending the rule $$\left(10^m\right)^n = 10^{m \boldcdot n}$$ to use negative exponents: $$\displaystyle \left(10^{\text-2}\right)^{3} = 10^{(\text-2)(3)}=10^{\text-6}$$To see why, notice that $$10^{\text-2} = \frac{1}{10} \boldcdot \frac{1}{10}$$. This means that $$\displaystyle \left(10^{\text-2}\right)^{3} =\left( \frac{1}{10} \boldcdot \frac{1}{10}\right)^3 = \left(\frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left( \frac{1}{10} \boldcdot \frac{1}{10}\right)\boldcdot \left(\frac{1}{10}\boldcdot \frac{1}{10}\right) = \frac{1}{10^6} = 10^{\text-6}$$