Lesson 7

Negative Exponents

These materials, when encountered before Algebra 1, Unit 5, Lesson 7 support success in that lesson.

7.1: Math Talk: Powers of Ten (10 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for dividing powers of ten, and rewriting powers of ten using an exponent. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to divide expressions using exponents that have the same base. In this activity, observations about subtracting exponents in the first three equations suggests a method for solving the last equation. Students may be puzzled by the last question. The purpose of including this equation is to conclude that the solution must be -1 because the left side can be rewritten $$\frac{10^2}{10^3}$$. The rest of this lesson offers more opportunities to make sense of negative exponents, so it’s not necessary for all students to feel completely comfortable with this idea, yet.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Solve each equation mentally:

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

$$\frac{1000}{x}=10^1$$

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

$$\frac{100}{1000}=10^x$$

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?” Draw particular attention to methods involving expressing 100 and 1,000 as powers of ten. For example, if we rewrite $$\frac{1,000}{x}=10^1$$ as $$\frac{10^3}{x}=10^1$$, we can remember our properties of exponents that tells us the denominator must be equal to 102. The same property of exponents can be used to deduce that the solution to the last equation is -1.

7.2: Maintain the Pattern (10 minutes)

Activity

The purpose of this activity is to extend the definition of exponents to include negative exponents. The meaning of zero exponents is also revisited. When students use tables to explore expressions with negative exponents, they are noticing and making use of structure (MP7).

Launch

Arrange students in groups of 2. Tell students to work on their own and then share their reasoning with their partner after each has had a chance to complete the problem. Follow with a whole-class discussion.

Student Facing

Complete the table.

exponential form number form calculations
$$2^5$$
16
$$\frac{2^4}{2}=2^{4-1}=2^3$$ $$2^3$$
$$\frac{2^3}{2}=2^{3-1}=2^2$$ $$2^2$$ 4
2 $$4 \boldcdot \frac12=2$$
1 $$2 \boldcdot \frac12=1$$
$$2^{\text-1}$$ $$\frac{1}{2}$$
$$\frac{1}{4}$$ $$\frac12 \boldcdot \frac12 = \frac14$$
$$2^{\text-3}$$
$$2^{\text-4}$$
$$\frac{1}{32}$$

Activity Synthesis

Display the completed table, and resolve any questions students have. Questions for discussion:

• “How would you make sense of $$2^{\text-6}$$?” (I know $$2^{\text-5}=\frac{1}{32}$$, and I know that $$2^{\text-6}$$ would be the result of multiplying by another $$\frac12$$. So, $$2^{\text-6} = \frac{1}{64}$$.
• “How could we figure out the value of $$7^{\text-2}$$?” (Possible response: We could rewrite it as $$\frac{7^0}{7^2}$$, and then $$\frac{1}{49}$$.

7.3: Matching Equal Expressions (20 minutes)

Activity

In this partner activity, students take turns using exponent rules to analyze expressions and identify equivalent ones. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Launch

Arrange students in groups of 2. Four lists of expressions are given. Depending on the time available, you might ask students to work through all four lists, assign only a few lists to work through, or allow students to select which lists they would like to work through.

Tell students that for each original expression, one partner finds an equivalent expression in the list and explains why they think it is equivalent. The partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next expression in column A, The students swap roles and the partner finds another equivalent expression in the list and explains why they think it is equivalent. If necessary, demonstrate this protocol before students start working.

Give students 10 minutes to work followed by a whole-class discussion. As students work, notice the different strategies used to analyze the expressions in each list. Ask students using contrasting strategies to share during whole-class discussion.

Student Facing

Take turns with your partner to match the original expression with an equal or equivalent expression in the list.

• For each match that you find, explain to your partner how you know it’s a match.
• For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Which expressions equal $$8^0$$?

• 1
• 0
• $$8^3 \boldcdot 8^{-3}$$
• $$\frac{8^2}{8^2}$$
• $$11^0$$

Which expressions equal $$5^{\text-2}$$?

• $$\text-5^2$$
• $$\frac{5^{0}}{5^2}$$
• $$\text-2^5$$
•  $$\frac{1}{5^2}$$
• $$5^{\text-1} \boldcdot 5^{\text-1}$$

Which expressions equal $$3^{10}$$?

• $$3^5\boldcdot3^2$$
•  $$\left(3^5\right)^2$$
•  $$3^7 \boldcdot 3^3$$
•   $$3^{13} \boldcdot 3^{\text-3}$$
•   $$\frac{3^{10}}{3^{0}}$$

Which expressions are equivalent to $$x^{\text-4}$$?

• $$\frac{x^9}{x^5}$$
•  $$\frac{x^5}{x^9}$$
•  $$\frac{x^{3}}{x^{-1}}$$
•   $$x\boldcdot x^{\text-5}$$
•   $$\frac{1}{x^4}$$

Activity Synthesis

Much discussion takes place between partners. Invite students to share how they decided which expressions were equal to the given expressions.

• “What were some ways you handled . . . ?”
• “Describe any difficulties you experienced and how you resolved them.”

Once all groups have completed the matching, discuss the following:

• “Which matches were tricky? Explain why.”