# Graphing Exponential Decay Functions

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## Objective

SWBAT graph an exponential decay function.

#### Big Idea

This lesson helps to build understanding about percent decrease over time and how it can be modeled using an exponential function.

## Warm Up

10 minutes

For class today, I bring a ball that will bounce well on the classroom floor.  I ask students to watch very carefully as I drop the ball.  I like to stand on a desk or a chair (just to really get student's attention) and then drop the ball.

I give students a minute to write down as many observations as they can about what they notice about the ball as it bounces each time.   Then I ask one or two students share out what they wrote.  Because this is so open ended, expect a variety of responses:

• height of bounces
• number of bounces
• color of the ball
• etc.

Next I show the second slide of exponential_decay_warmup to students and introduce the Golf Ball context. I ask the class to record the data about the height of the bounces individually. Then I ask one student to share their data. I work to engage the class in a discussion about how this data connects to what they saw in the opening demonstration (MP2, MP3).

During the ensuing discussion,  I guide students towards the understanding that the bounces get smaller and smaller every time.  Theoretically, this would continue forever.  In real life, the ball eventually just stops bouncing.  I explain to students that they have been studying exponential growth up until now.  This bouncing ball scenario models something called exponential decay.  Finally, I have students turn and talk to share what they think exponential decay might mean.

## Launch

8 minutes

I have students read the exponential_decay_launch question to themselves and write down their response. Before doing a think-pair-share, I use a non-verbal cue to assess which members of the class think the car is worth nothing after 5 years.  Allow the pairs to discuss the reasoning behind their choice.

After the partners talk have them work together to describe this situation mathematically. Expect some students to struggle with this question because there is no dollar value for the car.  Encourage students to make up a value that will be easy to work with to see how the calculations work (if not all students do this, I emphasize the students that do.  The ability to "test out" numbers to help visualize a situation is a big idea in mathematics (MP2))

After a few minutes. have several groups share out their findings.  Try to call on groups that had modeled the situation correctly (determining the 20% loss), so that other students do not become confused.  As the students present, help the class to recognize how this example is connected to the bouncing ball. Make sure that students understand that the car retains some value after five years because you are taking 20% of smaller and smaller numbers each  year. It is important for students to understand the difference between exponential and linear depreciation (MP2).

*NOTE: I like to use the above context because, although students at this age don't own cars, they seem to have a "real-world sense" of how depreciation works.  They have heard adults talk about how the value of a car drops so quickly after you first buy it.  This lesson then serves to help them understand what happens to the value after that initial drop.

## Direct

12 minutes

In this section of the lesson I will be leading the class using exponential_decay_direct. Here are my bullet point instructional notes.

Slide 1

• Help students to see that if 20% of the value is gone then 80% of the value is left.  In doing their original calculations many students probably found 20% then subtracted and then repeated that process.  Show students how multiplying by 80% (0.8) each time accomplishes the same thing.
• Students can then connect this idea to the formula they had learned for exponential growth.  Ask students to turn and talk and identify the difference in the formula.  Students will notice that the formula now has a minus sign because 20% is being taken off of the price.  The (1 - 0.2) leaves us with (0.8) which connects back to the first bullet on this slide.

Slide 2

• Let students evaluate the function for each value of x over the 5 years if the original value is \$10,000 (some students may have already used this value).   All students can now use their 6 values to make a sketch of the exponential decay graph.
• Ask students what they notice about the appearance of this graph in comparison to the exponential growth graph?

## Closure

10 minutes

This Closing Activity will assess how each student grasped the content of the day's lesson.  The first question gives you some information about the students ability to build a function that applies to a given situation.  The second question will show you if the students can evaluate the function appropriately.  The third question will help you determine if students are continuing to look at the math in context.  In other words, students can "mathematize" a situation but they have to be able connect the mathematics back to the context (MP2).