##
* *Reflection: Connection to Prior Knowledge
Graphing & Modeling with Exponents - Section 3: An Exponential Model

My students had WAY more trouble writing an explicit formula for this problem than they should have. I found that while many could complete the data table fairly quickly, they couldn't move past a recursive formula for h(n). You can see what I'm talking about in this student's work. It is typical of 90% of the class.

After a few relatively simple suggestions failed to help them, I decided it was time for a deeper and more direct intervention. I decided to use an example of linear data to simplify the problem and connect to their prior knowledge. They all quickly recognized the constant difference from term to term, we formulated a recursive formula, and then explained how the explicit formula is developed out of it. (See the whiteboard here.)

Next, we moved back to our exponential data. This time we have a constant *ratio*, rather than a constant *difference*. By carefully and continually referring back to the linear data, I was able to help them see that in this case we have repeated *multiplication* from an initial value rather than repeated *addition*. At this point most were able to see that our formula would be exponential. (See the whiteboard here.)

Once they had the formula, the rest was pretty easy. Check out this student's work for a good example of exactly what I was hoping for from everyone.

*Connection to Prior Knowledge: Making the function explicit*

# Graphing & Modeling with Exponents

Lesson 5 of 14

## Objective: SWBAT graph simple exponential functions with integer bases. SWBAT use an exponential model to analyze a real-world situation.

## Big Idea: How high will the basketball bounce and will it ever stop? An exponential model sheds light on the question!

*48 minutes*

#### Sprints!

*8 min*

Continue practice with Sprints 7 & 8. Previously, students have had a full 2 minutes to complete one sprint, but by now I typically find that this is too much time. So, I'll reduce the time by about 30 seconds to keep everyone on their toes. Check out this quick video for the rationale behind the timing of the sprints.

So as not to discourage them, we might discuss how to compare their score on a 90-second sprint to their previous score on a 120-second one. We'll also discuss, again, some strategies for increasing efficiency. Hopefully, they're still seeing improvement!

*expand content*

Hand out the worksheet called Exponential Equations - Graphically. Today, students will work individually or in small groups to make graphs of these functions *without* technology. The point is to recall the difference between exponential growth & decay, as well as the effect of changing the base of the function.

Students will begin by working individually, but once everyone has one or two functions graphed correctly, I'll let them begin collaborating on the rest.

I've provided unscaled axes, and I'm sure students will have some trouble coming up with an appropriate scale. It's helpful to remind them that they do *not* have to use the same unit on both axes, but they *do* have to use each unit consistently on each axis. Since exponential functions grow so quickly, I recommend a larger unit on the x-axis than on the y-axis. You can see my solutions document for an example (it's too bad the pretty colors didn't show up in the scan!).

These graphs should be completed quickly, since this material belongs to the Algebra 1 curriculum. Once they are, I begin a discussion in which I will call on a student to share his or her solution with the class via the document camera. After taking a minute or two to enjoy the symmetry of the figure, we'll talk a bit about the difference between exponential growth & decay. We'll also talk about the effect that changing the base has on the graph. The discussion will not take more than five minutes; again, this should be somewhat familiar to most students.

*expand content*

#### An Exponential Model

*20 min*

Now, students should begin work on the reverse of the worksheet. This real-world problem requires them to create an exponential function to model an example of exponential decay. (**MP 4**) Students will use the given information to complete a data table, create an exponential equation to model the data, and draw the graph of the equation. This kind of problem *should* be familiar from Algebra 1, but it's important to re-activate that knowledge!

Students are also asked one question that would be simple with logarithms, but is not so simple without them. I expect students to solve this *without* logarithms because this question is intended to create a *motivation* or *need* for logarithms. (**MP 1**)

After about 15 minutes, students should be ready to share/compare their data and the equations they've created. We can discuss them briefly before class ends. Homework will be to make an accurate graph and answer the remaining questions.

#### Resources

*expand content*

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: Opening Day Exponential and Logarithmic Functions Unit
- LESSON 2: Rational Exponents
- LESSON 3: Simplifying Exponential Expressions, Day 1
- LESSON 4: Simplifying Exponential Expressions, Day 2
- LESSON 5: Graphing & Modeling with Exponents
- LESSON 6: Comparing Growth Models, Day 1
- LESSON 7: Comparing Growth Models, Day 2
- LESSON 8: Comparing Growth Models, Day 3
- LESSON 9: Logarithms - Napier's Wonderful Invention!
- LESSON 10: Getting to Know Logarithms
- LESSON 11: Properties of Logarithms Day 1 of 3
- LESSON 12: Properties of Logarithms, Day 2 of 3
- LESSON 13: Properties of Logarithms, Day 3 of 3
- LESSON 14: The Number e