Lesson 6

The purpose of this warm-up is for students to reason about powers of positive and negative numbers. While some students may find the numeric value of each expression to make a comparison, it is not always necessary. Encourage students to reason about the meaning of the integers, bases, and exponents rather than finding the numeric value of each expression. 

Display one problem at a time. Give students up to 1 minute of quiet think time per problem and tell them to give a signal when they have an answer and a strategy. Follow with a whole-class discussion.

There may be some confusion about how negative numbers interact with exponents. Tell students that exponents work the same way for both positive and negative bases. In other words, \((\text- 3)^2 = \text- 3 \boldcdot \text- 3\).

Students may also struggle to interpret the notation difference between two expressions such as \((\text- 4)^2\) and \(\text- 4^2\). Tell students that the interpretation of the first expression is \((\text- 4)^2 = \text- 4 \boldcdot \text- 4\) and the interpretation of the second expression is \(\text- 4^2 = \text- 1 \boldcdot 4 \boldcdot 4\).

Poll students on their response for each problem. Record and display their responses for all to see. If all students agree, ask 1 or 2 students to share their reasoning. If there is disagreement, ask students to share their reasoning until an agreement is reached. After the final problem, ask students if the same strategies would have worked if all of the bases were 10 to highlight the idea that powers with a base of 10 behave the same as those with bases other than 10.

To involve more students in the conversation, consider asking:

• “Do you agree or disagree? Why?”

• “Who can restate ___’s reasoning in a different way?”

• “Does anyone want to add on to _____’s reasoning?”

Extend the definition of non-positive exponents to other bases. The work of this activity is parallel to the work with the powers of 10 table in the previous lesson. Notice students who transfer their understanding of powers of 10, especially non-positive exponents, to exponential expressions with bases other than 10.

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. Encourage students to look for similarities to work they have already done with base 10. Give students 10 minutes of work time, followed by a whole class discussion.

The important idea for this activity is that the same logic that we used to analyze powers of 10 applies to other bases as well. Students will work mostly with bases that are whole numbers and positive fractions.

Here are some sample questions for whole-class discussion:

• “How does working with powers of 2 compare to working with powers of 10?” (The properties apply in the same way, but computing the value of the exponents is more difficult.)
• “What happens when you use exponent rules when one of the exponents is 0?” (The rules apply in the same way. \(2^0 = 1\) in the same way that \(10^0 = 1\).)
• “How does \(10^{\text- 3}\) compare to \(2^{\text- 3}\)?” (Both values are positive and less than 1. Since \(10^{\text- 3}\) has greater numbers in the denominator, it is less than \(2^{\text- 3}\).)

Introduce new (or update current) visual displays to illustrate exponent rules for positive, rational bases. Display them for all to see throughout the unit. An example would be to update the rule \(10^n \boldcdot 10^m = 10^{n+m}\) with \(x^n \boldcdot x^m = x^{n+m}\) with a guiding example that uses a base other than 10.

This activity gives students a chance to practice using exponent rules to analyze expressions and identify equivalent ones. Five lists of expressions are given. Students choose two lists to analyze. As students work, notice the different strategies used to analyze the expressions in each list. Ask students using contrasting strategies to share later.

Arrange students in groups of 2. Tell students to take turns selecting an expression, determining whether the expression matches the original, and explaining their reasoning. Give students 10 minutes to work followed by a whole-class discussion.

Some students may struggle with comparing \(\text-5^9\) to \(5^{\text-9}\). Ask these students to explain the difference between \(10^3\) and \(10^{\text-3}\), and relate that to the difference between \(5^9\) and \(5^{\text-9}\). If needed, explain that \(\text-5^9\) is a large negative number, whereas \(5^{\text-9}\) is a very small positive number.

The key idea is that the exponent rules are valid for nonzero bases other than 10. Select students to share which expressions did not match the original in the lists they chose to study. Consider focusing on questions 3 and 4 to assess how students have generalized in terms of variables and whether students have internalized the 0 exponent. Select previously identified students to share the different strategies used to analyze the same list.

Here are some questions to consider for discussion:

• “What mistakes might lead to the expressions that are not equivalent to the original expression?”
• “How do you know that expression is not equivalent to the original?”
• “Did anyone think about that expression in a different way?”