Lesson 1

This warm-up prompts students to compare four expressions with exponents. 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. To allow all students to access the activity, each expression except for \(2^3\) has one obvious reason it does not belong. Don't let students dwell on trying to explain why \(2^3\) doesn't belong. During the discussion, listen for important ideas and terminology that will be helpful in upcoming work of the unit.

Arrange students in groups of 2–4. Before introducing the warm-up, display the expression \(7^4\) for all to see. Ask students if they recognize this notation and to explain what it means. Then display the expressions in the warm-up for all to see. Ask students to indicate when they have noticed one expression that does not 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 expression does not belong and together, try to find at least one reason each expression 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 “base” or “exponent.” Also, press students on unsubstantiated claims.

This activity uses the context of a genie who gives a magic coin that doubles in number each day. This context reminds students about the need for exponential notation in thinking about problems involving repeated multiplication. For the sake of simplicity, the problem was written so that the exponent is equal to the number of days.

There may be some confusion about what time of day the coin doubles or how the exponent connects to the number of days. Clarify that the number of days is equal to the number of doublings.

Some students may misinterpret “How many times more coins does Mai have than Andre” as “How many more coins does Mai have than Andre.” Others may think they need to know exactly how many coins Mai and Andre have in order to answer this question. Suggest to students who are stuck that they first figure out how many times more coins Mai had on the 8th day than on the 5th day.

The goal is for students to understand exponential notation and use it to reason about a situation that involves repeated multiplication. Display the table for all to see. Tell students that exponents allow us to perform operations and reason about numbers that are too large to calculate by hand. Explain that the “expanded” column shows the factors being multiplied, the “exponent” column shows how to write the repeated multiplication more succinctly with exponents, and the “value” column shows the decimal value. Consider asking, “How many times larger is \(2^6\) than \(2^4\)? How does expanding into factors help you see this?”

The broken coin prompts students to think about repeated division, laying the foundation for later work on negative exponents. Understanding repeated division by 2 as being equivalent to repeated multiplication by \(\frac{1}{2}\) will later allow students to make sense of negative exponents.

Look for students who express their answers to question 2 as \(\left(\frac{1}{2}\right)^{28}\) and those who write \(\frac{1}{2^{28}}\). Ask them to share their responses later.

If students spend more than several minutes on trying to multiply \(\left(\frac{1}{2}\right)^{28}\), remind them of the more efficient exponential notation.

If students are wondering how to represent dividing repeatedly, ask if they can think of division by 2 as multiplication, perhaps by another value.

The goal is for students to understand that dividing by 2 repeatedly corresponds to multiplying by \(\frac{1}{2}\) repeatedly. Ask students to discuss their responses with their partner. Select previously identified students to share their responses to the second question, highlighting the difference between \(\left(\frac{1}{2}\right)^{28}\) and \(\frac{1}{2^{28}}\). Ask students whether they agree or disagree with either of those responses. Students should come away with the idea that repeatedly dividing by 2 is the same as repeatedly multiplying by \(\frac12\). Here are some questions to consider:

• "Why are exponents useful when thinking about the coin after many days?" (It is shorter to write than a lot of 2s.)
• "What does your partner think about the last question? Do you agree? Why or why not?"