Lesson 8

The purpose of this warm-up is to encourage students to relate expressions of the form \(a^n \boldcdot b^n\) to \((a \boldcdot b)^n\) by exploring the structure of the factors (MP7). Students should notice that the factors in the expanded form of \(5^3 \boldcdot 2^3\) can be rearranged and multiplied to show the factors in the expanded form of \(10^3\). Evaluating and expanding expressions with exponents helps prepare students for the next activity in which they more generally explore products of bases with the same exponent.

Give students 2 minutes of quiet work time followed by a whole-class discussion. 

Consider asking some of the following questions to focus the conversation on the common exponents:

• “What connections do you see between the two expressions?” (The product of the bases in the first expression is equal to the base in the second expression: \(2 \boldcdot 5 = 10\). The exponents are the same in both expressions.)
• “Is there a way to tell just by looking at the expressions that they would be equal? How?” (Since there are 3 factors that are 5 and 3 factors that are 2, group the 2s and 5s together to get 3 factors that are 10.)

Highlight student explanations that clearly show the connection between \(2^3 \boldcdot 5^3\) and \(10^3\) by inspecting their factors.

Students use repeated reasoning to discover the rule \((a \boldcdot b)^n = a^n \boldcdot b^n\) (MP8).

Arrange students in groups of 2. Encourage students to share their reasoning with their partner as they work to complete the table. Give students 10–12 minutes of work time followed by a whole-class discussion.

Some students may write \(2x^4\) instead of \((2x)^4\). Similarly, students may write \(a \boldcdot  b^n\) instead of \((a \boldcdot b)^n\). Ask these students to explain the difference between \(3 \boldcdot 4^2\) and \((3 \boldcdot 4)^2\).

Ask students to share their reasoning about whether \(2^3 \boldcdot 3^4\) can be written with a single exponent. The key takeaway for the discussion should be that the exponents need to be the same to combine the bases into a single base with that exponent.

Introduce a visual display for the rule \((a \boldcdot b)^n = a^n \boldcdot b^n\). Display it for all to see throughout the unit.

At this point, students have worked with many different patterns involving exponents. This activity gives students an opportunity to deepen their thinking by generating different equivalent expressions using the rules of exponents. The process of generating different expressions requires students to understand the numerous ways numbers can be broken into factors and how to combine those factors and express the result using exponents.

Arrange students in groups of 2–3. Provide students with tools for creating a visual display. There will be several rounds in which students will have to generate multiple expressions equivalent to a specific number. For the first round, students have to generate expressions that are equal to 3,600. Show students how to set up their display with an example, such as the one shown here. As a whole class, come up with an expression equivalent to 3,600 using each of the three rules. If time allows, discuss using prime factorization as a strategy. For an example using the first rule: \(3,\!600 = (600 \boldcdot 6) = (2^3 \boldcdot 3^1 \boldcdot 5^2)(2^1 \boldcdot 3^1) = 2^{3+1} \boldcdot 3^{1+1} \boldcdot 5^2 = 2^4 \boldcdot 3^2 \boldcdot 5^2\). Some simpler examples include:

Display these examples for all to see while students are working to generate their own expressions equivalent to 3,600. Set a timer for 1 minute (or other amount, depending on time available) and let students work.

When time is up, pair each group with another group for scoring. It would be beneficial to choose 2 groups to use as examples and demonstrate this process for the whole class:

• A group gets 1 point for every unique expression they found that is equivalent to 3,600. (If the two groups found the same expression, neither group gets a point for it.)
• 2 points for every unique expression that uses negative exponents.
• Students can challenge the other group’s expressions if they think they don’t really equal 3,600, or if the group didn’t use any of the three rules.

In the second round, shift the students’ attention to the number \(\frac{1}{200}\) and have them create another visual display (perhaps on the back of their first visual display). Again, the group gets 1 point for each unique expression equivalent to \(\frac{1}{200}\) except if it uses negative exponents, in which case it gets 2 points.

Play as many rounds of this game as time allows. In subsequent rounds, groups pair up with a different opponent. Consider using the following numbers in different rounds if time permits: 810,000; \(\frac{1}{64}\); 3,375. Leave a few minutes for a brief whole-class discussion.

In a whole-class discussion, ask “Explain in your own words: What did you learn about exponents from this activity?”

The main point is that numbers can be broken down into their factors in many ways and the exponent rules can be used to express the same value in many ways.