# Lesson 8

Unknown Exponents

## 8.1: A Bunch of $x$’s (5 minutes)

### Warm-up

This activity reminds students that they have learned to solve equations involving various operations, and that the process of solving involves maintaining the equality of each statement and thinking about what value of \(x\) keeps the equation true. This work prepares students to think about how to solve equations when the unknown value is an exponent.

### Student Facing

Solve each equation. Be prepared to explain your reasoning.

- \(\frac {x}{3}=12\)
- \(3x^2=12\)
- \(x^3=12\)
- \(\sqrt[3]{x}=12\)
- \(\sqrt{3x}=12\)
- \(\frac {3}{x}=12\)

### Student Response

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### Activity Synthesis

Invite students to share their responses and reasoning. Point out the different ways in which students found the value of an unknown variable in equations (such as relying on a familiar math fact, working backwards, or performing an inverse operation).

Tell students that in this lesson we will look at some other equations where the value we want to find is an exponent.

## 8.2: A Tessellated Trapezoid (15 minutes)

### Activity

This activity motivates a need for finding exponents to solve problems in a discrete geometric context. Students are given a pattern where the number of smallest trapezoids changes exponentially—growing by a factor of 4. They are asked to find the number of trapezoids after a certain number of growth steps, and to find the step number that would produce a certain number of trapezoids, essentially looking for the unknown exponent \(n\) in \(4^n\) when the expression has a certain value.

When students answer the question about the smallest trapezoids and try to find the step number that produces a certain number of trapezoids, monitor for students using any of the following strategies to share during the whole-class discussion:

- Keep multiplying by 4 and count the number of times it takes to reach 262,144.
- Use a calculator to evaluate \(4^n\) for different values of \(n\) until it equals 262,144.
- Reason that \(4^4\) is 256 and \(256 \boldcdot 256\) (or \(4^4 \boldcdot 4^4\)) is around 63 thousands, which is the rough number of trapezoids at Step 8. Because 262,144 is about 4 times 63 thousands, the step number would be 9 at that point.
- Create a table of values or a spreadsheet (with the step number being the input and \(4^n\) being the output) and extend the table until the output is 262,144.
- Write an equation for \(y=4^x\), graph it, and use the graph to see where it reaches a \(y\) value of 262,144.

Making graphing or spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

*Action and Expression: Internalize Executive Functions.*Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Invite 1–2 students to share who have organized their work in sequential ways that clearly show the relationship between the step number, the exponent, and the related output number (and possibly a picture of the resulting shape). Record their thinking on a display and keep the work visible as students continue to work. If students struggle to articulate how one would reach an equation for 262,144, invite them to reflect on the process they used for smaller numbers as shown in the display.

*Supports accessibility for: Organization; Attention*

### Student Facing

Here is a pattern showing a trapezoid being successively decomposed into four similar trapezoids at each step.

- If \(n\) is the step number, how many of the smallest trapezoids are there when \(n\) is 4? What about when \(n\) is 10?
- At a certain step, there are 262,144 smallest trapezoids.
- Write an equation to represent the relationship between \(n\) and the number of trapezoids in that step.
- Explain to a partner how you might find the value of that step number.

### Student Response

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### Anticipated Misconceptions

Students may try to count the trapezoids in Step 2 and either miscount or not see the structure to predict what happens at the next step. Ask them what happens to each trapezoid in Step 1 going to Step 2. How can this be used to figure out how many trapezoids there are in Step 2? What about at the next step?

### Activity Synthesis

Focus the discussion on the second question. Select previously identified students to present their responses and strategies, in the order listed in the Activity Narrative. If no students brought up that finding the step number that produces 262,144 trapezoids is essentially finding the exponent \(n\) so that \(4^n\) is 262,144, bring this up.

Draw students’ attention to the merits and drawbacks of the different strategies for finding an unknown number. Connect the discussion to the solving work they did in the warm-up. Discuss questions such as:

- “Which strategies seem effective and efficient? Which seem to take the most steps or be more prone to error?”
- “Which strategies might be helpful if the number of trapezoids is pretty small, say 64? What if the number of trapezoids is much, much larger than 262,144?”
- “You solved different equations in the warm-up. Can any of those strategies be helpful here?”

## 8.3: Successive Splitting (15 minutes)

### Activity

Previously, students reasoned about an unknown exponent in a discrete growth context where the exponents were whole numbers. In this activity, they find exponents in a continuous growth context, where the unknown exponents may not be integers.

Students may choose similar strategies for finding the value of an exponent as in the previous activity, but they will find that these strategies do not produce exact answers making it necessary to make estimates and then check them, or to use alternative strategies.

In the first question, students are likely to use a calculator to evaluate the expression \(100 \boldcdot 3^h\) for different values of \(h\). In the second question, to estimate the time when the colony would reach a certain population, students may estimate using a calculator (perhaps repeatedly, to get increasingly precise answers) or estimate using a graph. Monitor for students using these strategies to invite to share during the whole-class discussion.

Also monitor for students who can reason that the number of hours in the last question is less than 1 by reasoning concretely and abstractly (MP2):

- The population triples in 1 hour, so it must take less than 1 hour to double.
- \(100 \boldcdot 3^h = 200\) is equivalent to \(3^h = 2\). If \(h\) is 1, then the expression on the left has a value of 3, so \(h\) must be less than 1.

### Launch

Students may be familiar with the fact that some bacteria reproduce by binary fission (each parent cell splitting into two cells). Tell students that some species of bacteria reproduce by multiple fission, which means each cell splitting into 3 or 4 cells.

Arrange students in groups of 2. Ask students to read the activity statement and complete the first question. After a few minutes of quiet think time, ask them to share their answers and reasoning with their partner before selecting 2–3 students to share how they determined the population for the three different times.

*Action and Expression: Internalize Executive Functions.*Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Present the first set of questions and allow students to work before presenting the last set of questions dealing with approximation. Check in with students after the first 2–3 minutes of work time in approximating. Invite 1–2 students to share their work for approximating the number of hours to produce 1,000 bacteria before asking students to approximate for 200 bacteria. Record their thinking on a display for all to see and keep the work visible as students continue to work.

*Supports accessibility for: Organization; Attention*

### Student Facing

In a lab, a colony of 100 bacteria is placed on a petri dish. The population triples every hour.

- How would you estimate or find the population of bacteria in:
- 4 hours?
- 90 minutes?
- \(\frac12\) hour?

- How would you estimate or find the number of hours it would take the population to grow to:
- 1,000 bacteria?
- double the initial population?

### Student Response

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### Student Facing

#### Are you ready for more?

A $1,000 investment increases in value by 5% each year. About how many years does it take for the value of the investment to double? Explain how you know.

### Student Response

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### Anticipated Misconceptions

If a student is not sure how to find the values for the population which are not whole numbers, encourage them to either use graphing technology or use the calculator to estimate values of the population at different times.

### Activity Synthesis

Invite students to share their strategies for answering the second question. If no student mentions graphing as a strategy, bring it up and demonstrate, if needed.

Clarify the distinctions in how we approach the two sets of questions. Make sure students see that:

- In the first question, we can find the population of the bacteria easily since we know the exponent and the base. For example, the population of bacteria, \(p\), after 5 hours is \(p = 100 \boldcdot 3^5\). We can simply evaluate the right side of this equation.
- In the second question, what we want to know is the value of the exponent. For example, to answer “In how many hours will the colony double?” means finding \(h\) in the equation: \(200 = 100 \boldcdot 3^h\). This equation can be rewritten as \(2 = 3^h\). We have seen multiple ways to find or approximate the solution, and some ways are more involved than others.

Heading into future lessons, a guiding question will be is there a better way to find the value of \(h\) than by approximating?

*Speaking, Representing: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: “I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “I would graph the function” can be restated as a question such as, “How would graphing the exponential function help you find the number of hours it would take the population to grow to 1,000 bacteria?” This will help students listen and respond to each other as they share strategies.

*Design Principle(s): Support sense-making*

## 8.4: Missing Values (10 minutes)

### Optional activity

This activity is optional. In earlier activities, students try to find unknown exponents in contextual problems. This activity allows students to reason about exponential equations such as \(2^5 = {?}\) and \(2^{?}=1024\) in a straightforward, decontextualized task. Consider using this activity if it is needed to further motivate the upcoming work on logarithms.

When completing a blank cell in the bottom row of a table, students may choose to write an expression with an exponent. Encourage students to evaluate the expression and write a numerical expression without an exponent. To complete a blank cell in the top row, students need to work backwards and use their understanding of integer exponents. For example, to find \(x\) when \(2^x\) equals 256, they need to think about what power of 2 has a value of 256.

### Launch

*Representation: Internalize Comprehension.*To support working memory, provide students with scrap paper or mini whiteboards to explore the relationships that may help them complete the table. Demonstrate writing a number as the product of two exponents with like bases, and as repeated multiplication. To further support this process, provide sentence frames for students to use to guide their exploration process. For example, “I noticed _____ so I . . .” and “If _____ then _____ because . . . .”

*Supports accessibility for: Memory; Organization*

### Student Facing

Complete the tables.

\(x\) | -1 | 0 | \(\frac12\) | 1 | 5 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

\(2^x\) | \(\frac{1}{32}\) | \(\frac14\) | \(\frac12\) | 4 | 16 | 256 | 1,024 |

\(x\) | \(\frac13\) | \(\frac12\) | |||||||
---|---|---|---|---|---|---|---|---|---|

\(5^x\) | \(\frac{1}{25}\) | \(\frac15\) | 1 | 5 | 125 | 625 | 3,125 |

Be prepared to explain how you found the missing values.

### Student Response

### Anticipated Misconceptions

Students may use a calculator and put down approximate decimal answers for some of the roots. Ask them if there is an expression that gives the exact value. The sequence of buttons they used on the calculator should help write down these expressions.

### Activity Synthesis

Invite students to share how they reasoned about the missing values in the top and bottom rows. Allow the class to hear as many strategies as time permits.

If any students used properties of exponents to reason about unknown exponents, highlight these strategies. For example, to reason about 1,024, students might say, “I know that 4 times 256 is 1,024. Because 4 is \(2^2\) and 256 is \(16^2\), which is \(\left(2^4\right)^2\) or \(2^8\), 1,024 is \(2^2 \boldcdot 2^8\), which is \(2^{10}\).

## Lesson Synthesis

### Lesson Synthesis

Invite students to reflect on their process for finding an unknown input in an exponential function. Ask students questions like:

- “How is finding the step number that would produce a certain number of trapezoids similar to finding the hours when a population reaches 1,000?” (They both involve figuring out the value of an exponent.)
- “What are some ways to determine the exponent if we know the base and the value of the exponential expression? For example, if we know that \(5^x\) has a value of 200 (or \(5^x = 200\)), how do we find or estimate \(x\)?” (We could:
- Multiply by 5 repeatedly until the number reaches 200 or close to it and count the number of times we multiply.
- Divide 200 by 5 repeatedly until it reaches 1 and count the number of times we divide.
- Use a calculator to evaluate \(5^x\), trying different values of \(x\) until the result is or approximates 200.
- Graph \(y=5^x\) and estimate where the graph has a \(y\) value of 200.)

- “How is the process of solving for an exponent like or unlike solving for a variable in an equation like \(5x = 200\) or \(x^5 = 200\)?” (Like in solving other equations, sometimes we could see what value of \(x\) makes the equation true without doing too much. For example, we can tell that if \(5^x = 25\), then \(x\) is 2. But other times, there’s no quick way to “undo” the exponentiation or to isolate the \(x\).)

Tell students that in upcoming lessons they will learn another way to find the value of an unknown exponent in exponential equations.

## 8.5: Cool-down - Video Viewers (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

Sometimes we know the value of an exponential expression but we don’t know the exponent that produces that value.

For example, suppose the population of a town was 1 thousand. Since then, the population has doubled every decade and is currently at 32 thousand. How many decades has it been since the population was 1 thousand?

If we say that \(d\) is the number of decades since the population was 1 thousand, then \(1 \boldcdot 2^d\), or just \(2^d\), represents the population, in thousands, after \(d\) decades. To answer the question, we need to find the exponent in \(2^d = 32\). We can reason that since \(2^5 = 32\), it has been 5 decades since the population was 1 thousand people.

When did the town have 250 people? Assuming that the doubling started before the population was measured to be 1 thousand, we can write: \(2^d = 0.25\) or \(2^d = \frac14\). We know that \(2^{\text-2}=\frac14\), so the exponent \(d\) has a value of -2. The population was 250 two decades before it was 1,000.

But it may not always be so straightforward to calculate. For example, it is harder to tell the value of \(d\) in \(2^d = 805\) or in \(2^d = 4.5\). In upcoming lessons, we’ll learn more ways to find unknown exponents.