Lesson 17

The purpose of this activity is to reason through some examples, and the meaning of exponents, that an expression in the form \((a^b)^c\) is equivalent to \(a^{bc}\). Monitor for students who reason about the last question by replacing 8 with \(2^3\) or vice versa. When students notice that they can use properties to rewrite expressions, they are noticing and making use of structure (MP7).

Encourage students to reason about the expressions without using a calculator or evaluating them.

Ensure that students found correct matches for the first question, and invite a few students to articulate how they knew they had found a match without evaluating the expressions. Use at least one example from this activity to make sure students are comfortable with the rule: \((a^b)^c\) is equivalent to \(a^{bc}\). For example, demonstrate that \((3^5)^2=3^{10}\) because they both equal \(3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3\). Then, for the last question, make explicit a chain of reasoning such as: \(2^{15}\),  \(2^{3 \boldcdot 5}\),  \((2^3)^5\), then \(8^5\) pointing out where this property comes into play. (These steps could also be listed in reverse, first replacing 8 with \(2^3\).)

The purpose of this activity is to familiarize students with the terms annually, semi-annually, quarterly, and monthly, and use the meaning of these terms to solve some problems. In each problem, students perform a few numerical computations and then generalize by writing an expression, which is an example of expressing regularity in repeated reasoning (MP8).

Provide access to calculators. First, ask students to complete as much of the table as they can, using what they understand about the meaning of the given words. Ensure that students complete the table correctly before proceeding with the rest of the activity.

Invite selected students to share their responses. Emphasize that an expression like \(\dfrac{g}{12}\) is simply saying that no matter what the annual fee, represented by \(g\), we can represent the monthly bill with \(\dfrac{g}{12}\). Because no matter the annual fee, we know that we should divide it by 12 to calculate the monthly bill. Possible questions for discussion:

• “What do the numbers in each row of the table have to do with each other?” (The product of the pair of numbers in each row is 12.)
• “How did you decide which operation to use?” (I drew a tape diagram representing 360 and realized I needed to split it into 12 equal groups, so I divided by 12.)

In this partner activity, students take turns matching a description or expression to a representation. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Provide access to calculators. Arrange students in groups of 2. Tell students that for each item in column A, one partner finds a matching representation in column B and explains why they think it matches. 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 item in column A, the students swap roles. If necessary, demonstrate this protocol before students start working.

Much discussion takes place between partners. Invite students to share how they did mathematical work.

• “What were some ways you handled the terms monthly, quarterly, and semi-annually?”
• “Describe any difficulties you experienced and how you resolved them.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”