Lesson 13

This warm-up prompts students to compare expressions. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about characteristics of the expressions in comparison to one another. 

Arrange students in groups of 2–4. Display the questions for all to see. Ask students to indicate when they have noticed one expression that doesn’t belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their reasoning why a particular question doesn’t belong and together find at least one reason each question doesn't belong.

Ask each group to share one reason why a particular expression does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as “exponents.” Also, press students on unsubstantiated claims.

The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure (MP7) to compare exponential expressions. To this end, encourage students to rewrite expressions in a different form rather than evaluate them to a single number.

For students who are accustomed to viewing the equal sign as a directive that means “perform an operation,” tasks like these are essential to shifting their understanding of the meaning of the equal sign to one that supports work in algebra, namely, “The expressions on either side have the same value.”

Before students start working, it may be helpful to demonstrate how someone would figure out whether or not an equation is true without evaluating each expression. For example:

Is \(4^2=2^3\) true? Well, let's see. We can rewrite each side like this:

\(\displaystyle 4 \boldcdot 4 = 2 \boldcdot 2 \boldcdot 2\)

Then we can replace one of those \(2 \boldcdot 2\)'s with a 4, like this:

\(\displaystyle 4 \boldcdot 4 = 4 \boldcdot 2\)

Now we can tell this equation is not true.

These problems can also be worked by directly evaluating expressions, which is fine, as it serves as practice evaluating exponential expressions.

Invite students who evaluated the expressions and students who used structure or the meaning of exponents and operations to present their work. Compare and connect the strategies by noting where the use of structure might prove more efficient, where evaluating might be simpler, where thinking about the meaning of exponents and operations can make the true or false determination simpler. 

Some guiding questions to highlight the meaning of exponents:

• “Can we switch the order with exponents like we can with addition and multiplication—specifically, are \(a^b\) and \(b^a\) equivalent? How do you know?” (No, you can try different values of \(a\) and \(b\) or use the meaning of exponents to see that \(a\) multiplied \(b\) times is not always the same as \(b\) multiplied \(a\) times.)
• “What change can we make to the equation \(3+3+3+3+3=3^5\) to make it true?” (Change addition to multiplication on the left, or change the exponent to multiplication on the right.)
• “Your friend claims the equation \(\frac12 \boldcdot  \frac12 \boldcdot  \frac12 \boldcdot  \frac12 = 4 \boldcdot \frac12\) is true. What do you think they are misunderstanding? How can you convince them it is false?” (Possibly by saying that the left side shows multiplying \(\frac12\) 4 times, but the right side shows multiplying 4 by \(\frac12\), which means adding 4 copies of \(\frac12\), not multiplying them. You can show similar examples with other numbers, but the best way to convince them is to talk about what exponents and multiplication mean.)
• “Can we show that \(8^2 = 4^3\) is true without evaluating both sides? What understanding about the meaning of exponents and operations can help us?” (\(8^2\) means \(8\boldcdot 8\) or \(4\boldcdot 2\boldcdot 4\boldcdot 2\), which also equals \(4\boldcdot 4\boldcdot 2\boldcdot 2\) or \(4\boldcdot 4\boldcdot 4\). Another way to write \(4\boldcdot 4\boldcdot 4\) is \(4^3\). We are using the understanding that we can multiply in any order, or the commutative and associative properties of multiplication.)

In this activity, students search for numerical expressions that are equivalent. Students construct arguments and critique the reasoning of others (MP3) as they explain to their partner why they think two expressions are equivalent and respond to their partner’s arguments about equivalence.

Arrange students in groups of 2. Partners work for 10 minutes, alternating every question which partner is explaining why a match is a match and listening to the explanation. If their partner disagrees, the partner explains why they don’t think the match is equivalent. 

Encourage students to determine matches by looking for structure in the expressions and applying the meaning of exponents. It is not always necessary to evaluate the expressions in order to find equivalent expressions.

Invite students to describe their strategies for finding matches. As students respond, record the equivalent expressions using an equal sign. 

Consider asking students:

• To share any expressions they had to work on to agree with their partner
• How they could find a match without evaluating every expression
• To describe some ways to recognize equivalence in expressions