Unit 5, Lesson 8
Unknown Exponents
Algebra 2
8.1
Warm-up
Standards Alignment
Building On
HSA-REI.A.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
HSA-REI.B.4.b
Solve quadratic equations by inspection (e.g., for ), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as for real numbers and .
Instructional Routines
NoneMaterials
NoneActivity Narrative
This activity reminds students that they have learned to solve equations involving various operations, and that the process of solving an equation involves maintaining the equality of each statement and thinking about what value of keeps the equation true. This work prepares students to think about how to solve equations when the unknown value is an exponent.
Launch
NoneActivity
NoneSolve each equation. Be prepared to explain your reasoning.
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 backward, or performing an inverse operation).
Tell students that in this lesson we will look at some equations where the value we want to find is an exponent.
8.2
Activity
Instructional Routines
MLR7: Compare and ConnectMaterials
NoneActivity Narrative
This activity motivates a need to find exponents to solve problems in a discrete geometric context. Students are given a visual pattern in which the number of objects increases exponentially, then asked to find the step in the pattern when a certain number of objects would appear.
Monitor for students who use these strategies:
- Keep multiplying by 4 and count the number of times it takes to reach 262,144.
- Use a calculator to evaluate for different values of until it equals 262,144.
- Reason about known values and skip ahead in the pattern more than 1 step at a time.
- Create a table of values or a spreadsheet (with the step number being the input and being the output) and extend the table until the output is 262,144.
- Write an equation for , graph it, and use the graph to see where it reaches a value of 262,144.
Making graphing or spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
8.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
To Gather
Scientific calculatorsActivity Narrative
In this activity, students 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 for different values of . 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 who use these strategies, and invite them 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.
- is equivalent to . If is 1, then the expression on the left has a value of 3, so must be less than 1.
Students should consider what available tools would be most helpful for estimating the solutions (MP5).
8.4
Activity
Optional
Instructional Routines
NoneMaterials
To Gather
Scientific calculatorsActivity Narrative
In earlier activities, students try to find unknown exponents in contextual problems. This activity allows students to reason about exponential equations such as and 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 backward and use their understanding of integer exponents. For example, to find when equals 256, they need to think about what power of 2 has a value of 256.
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 bacteria population reaches 1,000 thousand?” (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 has a value of 200 (or ), how do we find or estimate ?” (We could:
- Multiply by 5 repeatedly until the number reaches 200 or close to it, and count the number of times we multiplied.
- Divide 200 by 5 repeatedly until it reaches 1, and count the number of times we divided.
- Use a calculator to evaluate , trying different values of until the result is (or approximates) 200.
- Graph and estimate where the graph has a value of 200.)
- “How is the process of solving for an exponent like or unlike solving for a variable in an equation like or ?” (Like in solving other equations, sometimes we could see what value of makes the equation true without doing too much. For example, we can tell that if , then is 2. But other times, there’s no quick way to “undo” the exponentiation or to isolate the .)
Tell students that in upcoming lessons they will learn another way to find the value of an unknown exponent in exponential equations.
Student Lesson Summary
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 is the number of decades since the population was 1 thousand, then , or just , represents the population, in thousands, after decades. To answer the question, we need to find the exponent in . We can reason that since , 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: , or . We know that , so the exponent 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 in or in . In upcoming lessons, we’ll learn more ways to find unknown exponents.