# A Conceptual Introduction to Exponential Functions: Collaboration, Questioning, and Financial Advice!

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## Objective

Students will be able to further their experience with exponential functions through analyzing a real world document and providing âmathematical financial adviceâ to a client. Oral communication and collaboration are highly emphasized in this real-world lesson!

#### Big Idea

This lesson will provide students with the opportunity to investigate a conceptual understanding of exponential functions. Students will begin with a mathematical model and then extend their knowledge to providing "mathematical" financial advice based off

## Entry Activity

12 minutes

Set up:

1)   Create 25-30 sticky notes with rational and irrational values ranging from -4 to

-Include decimals, fractions, irrational numbers, and integers – any real number for that matter!

-Make 2 copies total, on different colors of sticky note paper

2)    Have coordinate grid on the board ready to go as students enter

3)    1” sticky notes work the best because they do not take up as much space on the coordinate grid.

Each student should get 2-3 sticky notes as they enter (exactly as in previous days).  It is important that all students’ receive sticky notes of both colors, so that they can plot on each of the two functions.

On the board, I project the graph grid and instruct the students to graph y=2^x with the __________ colored domain and y=(1/2)^x with the __________ colored domain.  Just as in previous lessons of this unit, check both graphs for outliers and address what possible mistakes were made in their creation.  The students will then be able to compare the graph of a familiar function (y=2^x) with that of a totally new one (y=(1/2)^x).

(See the attached picture of the end result - it is really cool!)

The following discussion (starting in thinking groups and then transitioning to the whole group) revolves around introducing the concept of exponential growth vs exponential decay.  It also illustrates to the students how different types of exponents affect the outputs in each case.

I have found this entry activity to be very powerful as the students continue to grapple with the complexity of exponentials.  Showing them two graphs at one time fosters a great deal of critical thinking – especially when answering the following questions:

3 Questions/Instructions:

1)  Collaborate to compare/contrast and describe each graph in detail.  List any key features that you find.

2)  How has your knowledge of exponential functions been changed by this activity?

3)  Think of 1 real life situation that could be modeled by each curve.  Describe.

The students will be able to answer these questions in 4-5 minutes.  Whole class debrief of the questions usually results in the realization that the functions cross paths at (0,1)… hence why anything to the zero power is 1.  I usually have several students who cry out “Oh!  I get it now!  I see WHY anything to the zero power is one.”  Other notable talking points will inevitably come out, and I am ALWAYS surprised by the great answers students have for #3.  It is really cool to see them start to understand exponential functions after several days of work!  Several honors level students may even attempt to formulate an equation to model the events.  I encourage you to try your own questions for this activity and share them in the comments section of this lesson!

20 minutes

(Since it is Friday when I am teaching this lesson, I decided to close the week in a way that is "non-traditional" and fit with our day's theme of striving for conceptual understanding.)  Considering there is only 30-35 minutes left of class, I decided to divide the class into pairs as quickly as possible.  For me, this was as easy as dividing each thinking group in half.  I quickly passed out the activities "entry document" and explained the following task at hand:

For the last part of class, you will serve as a financial adviser with a conceptual understanding of exponential functions, growth, and the data provided to you on the spreadsheet.  Your job is to explain the best strategy of investing to a recent college graduate looking to start saving for retirement.  Do not overwhelm your client with mathematical jargon - but strive to break down the facts and explain to them the "concept" of what is most beneficial for their future, and why.  You have 20 minutes to work before giving a 3-4 minute presentation to a group of your peers.  Good luck!

*The Dave Ramsey excerpt comes from a well respected financial advisor who advocates for the understanding of the mathematics behind long term investing.

## Student Presentations

15 minutes

Students share results in an oral presentation with their peers.  I typically group the class into groups of six (somewhat by ability level) so that there are 3 presentations in each.  If time, debrief the key points with the class in a culminating discussion… However, as you will see, the most beneficial conversations occur at the small group level.

Note:  You can also follow this activity up with a journal reflection – It is a great way to assess individual understanding in the context of a group investigation.  If you try it, please let me know how it goes!