Lesson 1

Exponent Review

1.1: Which One Doesn’t Belong: Twos (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

Which expression does not belong? Be prepared to share your reasoning.

$$2^3$$

8

$$3^2$$

$$2^2\boldcdot 2^1$$

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

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.

1.2: Return of the Genie (15 minutes)

Activity

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.

Launch

Invite a student to read the first paragraph for the class. Make sure students understand how the magic coin works before moving on to the problem statement, perhaps by drawing a picture of a coin that doubles each day. Allow 10 minutes of work time before a whole-class discussion.

For classes using the digital version, there is an applet to help visualize the growth. If projection is available, teachers using the print version can display it from this link. https://ggbm.at/xQP9xNDm

Representation: Internalize Comprehension. Represent the same information through different modalities. If students are unsure where to begin, suggest that they draw a diagram to help organize the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Mai and Andre found an old, brass bottle that contained a magical genie. They freed the genie, and it offered them each a magical $1 coin as thanks. • The magic coin turned into 2 coins on the first day. • The 2 coins turned into 4 coins on the second day. • The 4 coins turned into 8 coins, on the third day. This doubling pattern continued for 28 days. Click on the arrow to see the coins start to magically multiply. Mai was trying to calculate how many coins she would have and remembered that instead of writing $$1 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2$$ for the number of coins on the 6th day, she could write $$2^6$$. 1. The number of coins Mai had on the 28th day is very, very large. Write an expression to represent this number without computing its value. 2. Andre’s coins lost their magic on the 25th day, so Mai has a lot more coins than he does. How many times more coins does Mai have than Andre? Student Response For access, consult one of our IM Certified Partners. Launch Invite a student to read the first paragraph for the class. Make sure students understand how the magic coin works before moving on to the problem statement, perhaps by drawing a picture of a coin that doubles each day. Allow 10 minutes of work time before a whole-class discussion. For classes using the digital version, there is an applet to help visualize the growth. If projection is available, teachers using the print version can display it from this link. https://ggbm.at/xQP9xNDm Representation: Internalize Comprehension. Represent the same information through different modalities. If students are unsure where to begin, suggest that they draw a diagram to help organize the information provided. Supports accessibility for: Conceptual processing; Visual-spatial processing Student Facing Mai and Andre found an old, brass bottle that contained a magical genie. They freed the genie, and it offered them each a magical$1 coin as thanks.

• The magic coin turned into 2 coins on the first day.
• The 2 coins turned into 4 coins on the second day.
• The 4 coins turned into 8 coins on the third day.

This doubling pattern continued for 28 days.

Mai was trying to calculate how many coins she would have and remembered that instead of writing $$1 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2$$ for the number of coins on the 6th day, she could just write $$2^6$$.

1. The number of coins Mai had on the 28th day is very, very large. Write an expression to represent this number without computing its value.
2. Andre’s coins lost their magic on the 25th day, so Mai has a lot more coins than he does. How many times more coins does Mai have than Andre?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

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.

Activity Synthesis

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?”

expanded exponent value
2 $$2^1$$ 2
$$2 \boldcdot 2 \boldcdot 2 \boldcdot 2$$ $$2^4$$ 16
$$2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2$$ $$2^6$$ 64
Reading, Writing: MLR3 Critique, Correct and Clarify. Display the incorrect statement: “Mai has 6 times more coins than Andre because she had 3 more doublings, and $$3 ⋅ 2 = 6$$.” Ask students to critique the response by asking, “Do you agree with the author’s reasoning? Why or why not?” Give students 2–3 minutes of quiet think time to write feedback to the author that identifies how to improve the solution and expand on his/her work. Invite students to share written feedback with a partner before selecting 2–3 students to share with the whole class. Listen for students who refer to repeated multiplication and use the language of exponents in their feedback to the author. This helps students evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness, Optimize output (for explanation)

1.3: Broken Coin (15 minutes)

Activity

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.

Launch

Arrange students in groups of 2. Allow 10 minutes of work time, 2 minutes for partner discussion, and follow with a brief whole-class discussion. It is expected that some students will multiply 6 times for question 1. For the second question, give students time to realize that they need to use a more efficient method to describe the number.

For students using the digital activity, there is an applet to help visualize the coin halving each day.

Representation: Internalize Comprehension. Begin with a physical demonstration of the actions that occur in the situation to support connections between new situations and prior understandings. Remind students that repeated multiplication by $$\frac{1}{2}$$ is equivalent to repeated division by $$2$$.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

After a while, Jada picks up a coin that seems different than the others. She notices that the next day, only half of the coin is left!

• On the second day, only $$\frac{1}{4}$$ of the coin is left.
• On the third day, $$\frac{1}{8}$$ of the coin remains.
1. What fraction of the coin remains after 6 days?
2. What fraction of the coin remains after 28 days? Write an expression to describe this without computing its value.
3. Does the coin disappear completely? If so, after how many days?

Watch the magical coin changing for ten days with this applet.

Student Response

For access, consult one of our IM Certified Partners.

Launch

Arrange students in groups of 2. Allow 10 minutes of work time, 2 minutes for partner discussion, and follow with a brief whole-class discussion. It is expected that some students will multiply 6 times for question 1. For the second question, give students time to realize that they need to use a more efficient method to describe the number.

For students using the digital activity, there is an applet to help visualize the coin halving each day.

Representation: Internalize Comprehension. Begin with a physical demonstration of the actions that occur in the situation to support connections between new situations and prior understandings. Remind students that repeated multiplication by $$\frac{1}{2}$$ is equivalent to repeated division by $$2$$.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

After a while, Jada picks up a coin that seems different than the others. She notices that the next day, only half of the coin is left!

• On the second day, only $$\frac{1}{4}$$ of the coin is left.
• On the third day, $$\frac{1}{8}$$ of the coin remains.
1. What fraction of the coin remains after 6 days?
2. What fraction of the coin remains after 28 days? Write an expression to describe this without computing its value.
3. Does the coin disappear completely? If so, after how many days?

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Every animal has two parents. Each of its parents also has two parents.

1. Draw a family tree showing an animal, its parents, its grandparents, and its great-grandparents.
2. We say that the animal’s eight great-grandparents are “three generations back” from the animal. At which generation back would an animal have 262,144 ancestors?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

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.

Activity Synthesis

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?"
Speaking, Listening: MLR7 Compare and Connect. As students share different approaches for representing the fraction of the coin remaining after 6 days (or 28 days), ask students to identify how the approaches are alike, and how they are different. Invite students to connect the approaches by asking, “Where is the number of days represented in each approach?” Ask students to describe what worked well with their approach. These exchanges strengthen students’ mathematical language use and reasoning based on exponents.
Design Principle(s): Maximize meta-awareness, Support sense-making

Lesson Synthesis

Lesson Synthesis

The goal of the discussion is to check whether students understand that exponents indicate repeated multiplication. Consider recording and displaying student responses for all to see during the discussion.

Here are some questions to consider for discussion:

• “What does it mean when we write $$2^{42}$$?” ($$2^{42}$$ means that 2 has been repeatedly multiplied 42 times. To expand this into factors would show 42 factors that are 2.)
• “How many times larger is $$2^{45}$$ than $$2^{42}$$?” ($$2^{45}$$ is 8 times larger than $$2^{42}$$ because it has 3 more factors that are 2, so it has been multiplied by 2 an extra 3 times.)
• “What does it mean when I write $$\left(\frac{1}{2}\right)^{42}$$?” ($$\left(\frac{1}{2}\right)^{42}$$ means that $$\frac{1}{2}$$ has been repeatedly multiplied 42 times. To expand this into factors would show 42 factors that are $$\frac{1}{2}$$.)
• “Which is greater, $$\left(\frac{1}{2}\right)^{42}$$ or $$\left(\frac{1}{2}\right)^{45}$$? Why?” ($$\left(\frac{1}{2}\right)^{42}$$ is greater since multiplying by $$\frac{1}{2}$$ results in a value closer to 0 and $$\left(\frac{1}{2}\right)^{45}$$ has been multiplied by $$\frac{1}{2}$$ three extra times.)

1.4: Cool-down - Exponent Check (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Exponents make it easy to show repeated multiplication. For example, $$\displaystyle 2^6 = 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2.$$ One advantage to writing $$2^6$$ is that we can see right away that this is 2 to the sixth power. When this is written out using multiplication, $$2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2$$, we need to count the number of factors. Imagine writing out $$2^{100}$$ using multiplication!

Let’s say you start out with one grain of rice and that each day the number of grains of rice you have doubles. So on day one, you have 2 grains, on day two, you have 4 grains, and so on. When we write $$2^{25}$$, we can see from the expression that the rice has doubled 25 times. So this notation is not only convenient, but it also helps us see structure: in this case, we can see right away that it is on the 25th day that the number of grains of rice has doubled! That’s a lot of rice (more than a cubic meter)!