Lesson 8

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.

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.

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).

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?

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?”

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.

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.

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.

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?

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.

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.

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 \(2^{10}\).