Lesson 5

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.

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. 

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.

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.

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.

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 \(\frac{1}{10} = 10^{\text-2}\).

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 \(10^{\text- 3} = \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} = \frac{1}{10^3}\).

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).

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.

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 \(10^{\text-6}\).

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 \(\text-2 \boldcdot 3 = 2 \boldcdot \text-3 = \text-6\).