# Exponential Growth and Interest Day 1 of 2

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## Objective

Students will be able to build the interest formula and use it to model situations involving interest.

#### Big Idea

Let's build an exponential model around something all kids are interested in... money.

## Warm Up and Homework Review

5 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Exponential Growth and Interest which asks students to solve an exponential problem in context.

I also use this time to correct and record the previous day's Homework.

## Introduction to Interest

10 minutes

This is the first of a two-day lesson.

We begin our lesson by recalling the basic exponential model.  We will be building the formula for finding compounded interest off of this basic model.  I remind the students that if b>1 then we have a growth model, and if it is between 0 and 1 then it is a decay model.

Next, it is important to establish that our students understand interest.  That may seem obvious but if someone doesn’t know then this whole lesson is going to be difficult for them.  I give a real life example like car payments.  If a car costs \$15000 and you get a loan for it, are you really only going to pay \$15000?

Using the scenario of investing \$100 at 8% interest per year,  the students complete the first task of finding the amount in the account after 4 years (Math Practice 8).  Some may want to add \$8 each year.  This is an excellent opportunity to hold an opportunity to have a class discussion.  A couple of strategies to discuss this error in thinking would be to work the problem that way myself and have the students debate on whether I am correct, or hand pick a student with a correct answer to present their thinking.  Both methods will give the students an opportunity to critique some faulty mathematical thinking as well as allow those students who made the mistake to correct their own thinking with being singled out (Math Practice 3).

The second question, writing a function model, provides a challenge (Math Practice1).  Students will want to write f(x)=100(0.08)x.  The obvious issue with this is that it will give you the interest and not the total amount in the account.  Scaffolding can be provided here by way of leading questions.  The goal is to get the students to identify that you need to use 1.08.  A possible method is to try a couple of samples in the model and see if it creates the table.  Since it doesn’t, a discussion can ensue regarding how to make it work.

The final question is just an application of the function.  I like to remind students that functions are awesome because they allow us to quickly find information that would take a long time without the function.

## Building the Compound Interest Formula

15 minutes

Once students have a decent understanding of our basic yearly interest model, we are going to start talking about interest rates that are compounded more than once per year (Math Practice 1).  This is an important place to discuss vocabulary such as biannually, quarterly, etc.  The next task is for the students to determine what it means to take an interest rate and compound it say quarterly.

Once students understand the concept of compounding interest, they will investigate how it changes the amount of interest received by changing how the interest is compounded (Math Practice 4).  I have the students make a prediction before they begin.  Once they have investigated the same interest on the same principal using different methods of compounding, we discuss their conclusions as a class.  Please note that we have yet to put this all into one formula

Now that they have looked at compounding interest, it is time to write a general formula.  Here is the general formula to use as a basis: A=P(1+r/n)nt  where P = principal, r = interest rate, n= # time compounded per year, t= years.  The students’ version does not need to end up with this exact formula as long as what they come up with is equivalent in content.

Scaffolding will be extremely important here.  Classes with more advanced students will require little beyond well worded questioning to push them in the right direction.  These classes would be more likely to create their own version of the formula.  You may want to give the students some time to work as pairs and then have them present as a class.  Another option would be hold a class discussion up front.  This will lessen each student's personal participation but can be a tolerable option if a decent number of students would be afraid to try by themselves anyway.   Classes with reluctant learners require a step by step guided approach.  I would start with the general formula and relate what they have investigated to each part.  I know there are a ton of choices here but I truly believe that each teacher and each class is different and there is no one size fits all approach that will work here.

Please note that this section will run into the second day of this lesson.

## Exit Ticket

5 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

The goal of this Exit Ticket is to ensure that students understand how to use the interest formula that we built in class.