SWBAT use sequences and series to model financial scenarios.

Show me the money! Allow students to engage in a productive struggle with the geometric series formula by using it to model various financial situations like car loans, interest bearing accounts, and periodic savings.

30 minutes

The homework from the previous day was a set of review problems. I check these problems with the homework rubric and then ask students which problems they would like to see worked out at the board. Because there is a unit test the following day, I take time to review any requested problem.

The Warm Up Periodic Savings is an introductory periodic savings problem with significant scaffolding. Students work in their table groups to calculate the final balance on a savings account "the long way" by considering each deposit as a separate investment. The goal is for students to see the balance as a geometric series and consider how to get the same result in a more efficient way. This approach is intended to enhance students' "ability to *decontextualize*—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents **[MP2]**."

It may be the case that students recognize that the repeated calculations could be completed with a spreadsheet. If so, I encourage students to use a spreadsheet on their graphing calculator and give positive reinforcement about choosing an appropriate tool **[MP5]**.

When students have had a chance to work through the problem, I use a quick poll to students asking for their final balance. Students are generally able to get the correct balance because the warm-up has a great deal of scaffolding. I then ask if anyone found a shortcut method. If not, I let them continue to do it the "long way," as I discuss in the **Multiple Strategies & Shortcut Strategies** reflection.

50 minutes

In this two-day activity, students work in ability-based groups of three to solve financial problems involving series and communicate their process using precise mathematical language [MP3, MP6]. The "deliverables" for this two-day project are as follows:

- Each individual will submit 3 of the 5 problems, with a clear explanation of how they went about solving each problem.
- Each group will create their own unique problem that can be solved with geometric series. This problem will be presented and solved in a poster or video from each group explaining in detail how they went about solving one of the problems.

I kick off the activity by placing students in their groups, explaining the activity and then handing out a copy of Financial Project Student Instructions and the Problem Solving Demo Rubric to each student. We discuss the rubric and I answer any questions that come up.

Each group receives a set of five Interest Cards. These cards are printed on color-coded card stock, with the most straightforward question in green and the most challenging question in red. Each card has a financial problem (determine the monthly payment for a loan, determine the balance in a savings account, calculate the future value of an annuity) that can be solved with a geometric sequence or series model **[MP4]**. Individual students must select three of the five problems to solve and submit. I ask students to clarify the following in their submission:

- What is going on in the problem. This should include a discussion of the number of individual deposits or withdrawals, the appropriate interest rate, the number of compounding periods and how often the interest is compounded.
- The solution strategy they chose and how they knew it was a good choice
- The answer they came up with and how they knew it was reasonable.

I expect that many students will solve the problems the "long way," treating each month's quantity as a sequence and then adding up all the subtotals in the end rather than using a series formula. This way of solving interest problems is much more concrete than using the geometric series formula and I do not discourage it **[MP1]**. I eventually want them to see that the structure of many of the problems lend themselves to applying a geometric series formula **[MP7]**. On day two of the activity I will try to convince them of the efficiency of using this formula - at least for the problems that involve a large number of compounding periods.

When students have submitted their individual problems to me, they work with their group to develop a problem of their own. By the end of this period they should have a draft of this problem and the beginnings of a solution strategy.