SWBAT solve problems using geometric sequences.

How can geometric sequences help us solve problems involving financial mathematics?

30 minutes

I love when I can teach something useful and relevant to my students. This is an example of one of those lessons; most students in my class will take out a loan or buy a house at some point in their life. Today we get to investigate the math behind the calculations for these areas of finance.

**I begin this lesson by giving an anecdote** about the magical day after you graduate from college and get a letter from your student loan lender saying that you owe $234 per month for the next 81 years of your life. Today and tomorrow we will investigate where that number comes from and how it relates to the work have been doing in this unit. Not all of my students are familiar with what a mortgage is, so I will be sure to explain that to them as we kick things off.

As they start working my students are sitting in their table groups. I will give them about 20 minutes to work on problems #1 - #2 from this worksheet. (This is a challenging task and it is important to do the math before giving it to your students in order to interact responsively. I have attached a teacher version of the notes worksheet so you can see how I approach everything with my students.)

Here are some things I will** keep an eye out for** as students work through these problems:

- The interest rates are confusing to many students - 8% annual interest, when compounded semiannually, is 4% per half of a year. Many students were thinking it was 8% per time period.
- The problem becomes much more accessible when answers are not evaluated completely. For example, the amount of money on January 1, 2015 is $636.48, but it is more useful in the form 300(1.04) + 300(1.04)^2.
- Some students will think additively instead of multiplicatively. Yes, you can write $312 as $300 + $300(0.04), but it is much easier to write it as $300(1.04).
- Students will be able to write out the series correctly, but not knowing how to find the sum. Specifically there were having difficulty identifying it as geometric or arithmetic.

15 minutes

After the group work portion, it is time to** share our thinking** with the entire class. In the video below, I discuss my teaching moves for going over question #2. The teacher notes also include many of the important points that I want to get across as we discuss.

After we get our final solution, I **assign questions #3 - #5 from the same worksheet for homework** and let students know that we will continue work on this tomorrow. This has been a challenging day for them, but I remind them that we are just finding the *n*th partial sum of a geometric series - something they did the days before with no problem. The context is the difficult part but we want to focus on the structure of the problem (**MP7**) to help us out.