Lesson 1 The In-Betweeners Develop Understanding

Jump Start

1.

Find the missing terms in this geometric sequence:

$x$	$1$	$2$	$3$	$4$
$y$	$5$			$320$

2.

Write an explicit equation for the geometric sequence given in the table from problem 1.

3.

I know $2^{3} \cdot 2^{5} =$ , because .

4.

In general, $2^{m} \cdot 2^{n} =$

Learning Focus

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?

Technology guidance for today’s lesson:

Convert Rational Exponents to Radicals: Casio ClassPad Casio fx-9750GIII

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 $2^{2}$ and $2^{3}$ in an exponential function. These numbers are actually quite interesting, so we’re going to do some exploring in this task to see what we can find out about these “in-betweeners.”

Let’s begin in a familiar place:

1.

Complete the following table for a geometric sequence defined recursively as: $f (0) = 4$ and $f (n) = 2 \cdot f (n - 1)$

$x$	$0$	$1$	$2$	$3$	$4$
$f (x)$	$4$

2.

Write an explicit equation for the function defined by the table.

3.

Plot these points on the graph at the end of this task and sketch the graph of $f (x)$ .

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 $x$ goes up by $1$ , then half that growth occurs between $0$ and $\frac{1}{2}$ and the other half occurs between $\frac{1}{2}$ and $1$ .

“So for example, we can find the output at $x = \frac{1}{2}$ by finding the average of the outputs at $x = 0$ and $x = 1$ .”

4.

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).

$x$	$0$	$\frac{1}{2}$	$1$	$\frac{3}{2}$	$2$	$\frac{5}{2}$	$3$	$\frac{7}{2}$	$4$
$f (x)$	$4$

5.

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 $x$ goes up $1$ , and I think this property should hold over each half- interval as well.”

6.

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).

Tehani’s table:

$x$	$0$	$\frac{1}{2}$	$1$	$\frac{3}{2}$	$2$	$\frac{5}{2}$	$3$	$\frac{7}{2}$	$4$
$f (x)$	$4$

7.

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 $x$ increases by $1$ ? Use this multiplier to complete the following table.

$x$	$0$	$\frac{1}{3}$	$\frac{2}{3}$	$1$	$\frac{4}{3}$	$\frac{5}{3}$	$2$	$\frac{7}{3}$	$\frac{8}{3}$	$3$
$f (x)$

8.

What number did you use as a constant factor to complete the table in problem 6?

9.

What number did you use as a constant factor to complete the table in problem 7?

Pause and Reflect

10.

Tehani and Travis know that the values in her tables should fit the function rule written in problem 2. For example:

a.

$f (\frac{1}{2}) = 4 \cdot 2^{\frac{1}{2}} = \underset{―}{}$ and $f (\frac{3}{2}) = 4 \cdot 2^{\frac{3}{2}} = \underset{―}{}$

b.

$f (\frac{1}{3}) = 4 \cdot 2^{\frac{1}{3}} = \underset{―}{}$ and $f (\frac{2}{3}) = 4 \cdot 2^{\frac{2}{3}} = \underset{―}{}$

c.

What does this suggest about the value of $2^{\frac{1}{2}}$ ?

d.

What does this suggest about the value of $2^{\frac{1}{3}}$ ?

11.

Give a detailed description of how you would estimate the output value $f (x)$ , for $x = \frac{5}{3}$ .

12.

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: $50 {(1.03)}^{\frac{1}{2}}$ .

Takeaways

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 $2^{\frac{1}{2}} = \sqrt{2}$ as follows:

In general, our new property of exponents can be summarized as:

Vocabulary

Lesson Summary

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.

Retrieval

1.

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.

a.

$4, 12, 36, \dots$

b.

$4, 12, 20, \dots$

c.

$160, 40, 10, \dots$

d.

$20, 40, 60, \dots$

Rewrite each expression using rules of exponents.

2.

$7^{2} \cdot 7^{3}$

3.

$\frac{13^{5}}{13^{2}}$

4.

$234^{0}$

5.

$3^{- 7}$