Lesson 4 The In-Betweeners Develop Understanding
Find the missing terms in this geometric sequence:
Write an explicit equation for the geometric sequence given in the table from problem 1.
Explore the meaning of a fraction as an exponent.
How can you determine the values of an exponential function that occur between whole number inputs?
Since the domain of a continuous exponential function includes all rational numbers, how do we interpret a fraction as an exponent?
Open Up the Math: Launch, Explore, Discuss
In previous lessons, you’ve worked with contexts for continuous exponential functions, such as the decay of medication in the bloodstream or earning interest over time in a bank account. Since time is continuous, we need to start thinking about the numbers that fill in between the values like
Let’s begin in a familiar place:
Complete the following table for a geometric sequence defined recursively as:
Write an explicit equation for the function defined by the table.
Plot these points on the graph at the end of this task and sketch the graph of
We know a geometric sequence is discrete, with a domain consisting only of the whole numbers. Travis and Tehani, two inquisitive math students, are wondering what would happen if they decided they wanted this table to be just a few points on a continuous, exponential function. Then they could create a table with more entries, maybe with a point halfway between each of the points in the table above. There are a couple of ways that they might think about filling in the half steps.
Travis makes the following claim:
“If the function doubles each time
“So for example, we can find the output at
Fill in the parts of the table below that you’ve already computed, and then decide how you might use Travis’s strategy to fill in the missing data. Also plot Travis’s data on the graph at the end of the task (problem 12).
Comment on Travis’s idea. How does it compare to the table generated in problem 1? For what kind of function would this reasoning work?
Tehani suggests they should fill in the data in the table in the following way:
“I noticed that the function increases by the same factor each time
Fill in the parts of the table below that you’ve already computed in problem 1, and then decide how you might use Tehani’s new strategy to fill in the missing data. As in the table in problem 1, each entry should be multiplied by some constant factor to get the next entry, and that factor should produce the same results as those already recorded in the table. Use this constant factor to complete the table. Also plot Tehani’s data on the graph at the end of this task (problem 12).
What if Tehani wanted to find values for the function every third of the interval instead of every half? What constant factor would she use for every third of an interval to be consistent with the function doubling as
What number did you use as a constant factor to complete the table in problem 6?
What number did you use as a constant factor to complete the table in problem 7?
Pause and Reflect
Tehani and Travis know that the values in her tables should fit the function rule written in problem 2. For example:
What does this suggest about the value of
What does this suggest about the value of
Give a detailed description of how you would estimate the output value
Use the graph to record information requested in problems 4 and 6.
Ready for More?
Consider your work with prior exponential relationships and do your best to connect a story with this expression:
In the past, I have used positive integers as exponents. For example:
Today, I found a meaning for rational exponents that are non-integer fractions, as in this example:
I can justify that
In general, our new property of exponents can be summarized as:
- cube root
- number sets (systems)
- rational exponent (fractional exponent)
- square root
- Bold terms are new in this lesson.
In this lesson, we developed meaning for using a fraction as an exponent. We began with a geometric sequence and found a way to fit data at the points halfway between whole number inputs in a way that would maintain the multiplicative behavior of the sequence. We continued to work with different fractional inputs and gave meaning to using a variety of fractions as exponents.
For each sequence, find the next two terms in the sequence and state whether the sequence is arithmetic or geometric. Then create an explicit equation for the sequence.
Rewrite each expression using rules of exponents.