# Different Time Intervals and Exponential Functions

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## Objective

SWBAT write exponential functions to model growth and decay using different units of time. Students will be able to change either the multiplier or the coefficient in the exponent to change the units of time in an exponential function.

#### Big Idea

One quantity doubles every hour and one doubles everyday--how you can write functions to describe their behavior using the same units?

## Warm-Up

30 minutes

The first two problems are basic review of the key skills that students should understand thoroughly by this point. I make this clear to them by stating it explicitly and I reinforce this by circulating to make sure that each student works on these two problems.

The third problem is important for previewing today’s lesson. The idea is to highlight the fact that both situations have the same rate of increase, but that it happens over a different interval of time. The big question is how does this affect the function, if we want to use the same units for both functions. Students might write the exact same function for both situations, which reveals the key misconception. Ask them, “How does the number of years affect the equation? Do these two different tuitions increase at the same rate?”

The question about multipliers is important. Students will likely say that the two functions have the same multiplier.

30 minutes

## Sharing and Closing

10 minutes

The purpose of this closing is, as always to make sure that students have made sense of the key idea of this lesson, which is that the units of time matter when setting up the function and that there are two different ways to deal with this. The three possible functions are presented in the exit ticket to highlight these ideas.

Even if students did not master these new ideas during the lesson, they can figure some stuff out during the closing, if they understand how to check some data points. I start this lesson closing by asking students to tell me how they can check to see whether the functions fit this situation. We start by creating a quick class data table based on the initial function, with some simple input-output pairs, like  and  if the units of time are hours. Students can then plug these pairs into the given functions to verify that they work.

The third function was created by finding the 24th root of 2 to get the new multiplier. Students might or might not fully understand this today, but at least they will realize that we can change the multiplier. If students are ready for this, you can discuss it in more detail or if a few students figure it out you can have them explain it to other students.