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# Investments, Loans, and Mortgages - Day 2 of 2

Lesson 8 of 18

## Objective: SWBAT solve problems using geometric sequences.

*50 minutes*

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Yesterday we worked on finding the value of an account at the end of a certain amount of time with periodic deposits. For homework, students were to work on questions #3-5 on the worksheet. I start class by **going over these questions together**. Here are some notes on each question that I want to address if they don’t come up during our discussion.

**#3** – Students tend to do a good job with this one since it was very similar to #1. However, some will be **unsure if the very last payment should be compounded or not **(i.e., many had exponents 1 to 48 instead of 0 to 47). Of those who correctly use the exponents 0 to 47, some will plug 47 in for *n* in the partial sum of the geometric series formula. I like to let other students remind them that *n* refers to the number of terms, not the greatest exponent.

**#4** – A lot of my **students bring up inflation** in their answer for Problem 4. I like to clarify that in this problem we are not really accounting for inflation. However, I also commend them for recognizing that it would affect the real value of the money. I make sure that they udnerstand that it will not affect how many dollars you actually have. I gave the example of putting $100 in an envelope and it sitting in a drawer for 50 years versus putting $100 in the bank for 50 years. The $100 in the envelope is still $100 after 50 years even though the inflation is happening. However, the $100 in the bank is a new sum of money because it earned interest – and the interest is what we are focusing on in this problem.

**#5** – The **most common misconception** for this problem is that students will use #1 and #2 as guides to find the amount that would have to be deposited every quarter (about $492) in order to have $10,000 at the end of five years. Instead, I want my students to find the lump sum that should be invested at the beginning of the five year period that untouched in the bank would be $10,000 at the end.

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#### Extend

*35 min*

Question #6 is certainly a challenging problem (even for a math teacher). It is important to review the conceptual underpinnings of the task before attempting to teach it. (I have included the teacher notes in order to provide some background on my approach to this problem. Like #2, I want to **set it up as a finite geometric series** before we set up the equation to find the monthly mortgage problem.) In the video below I outline how I use question #5 as foundation for setting up this geometric series.

We will discuss Problem 6 as a class. After our sequence is set up, we can use the formula for the *n*th partial sum of a geometric series to set up an equation to solve for *M*.

**Teacher note**: The formula will work out a little easier if M/(1.0075^360) is the first term since the common ratio will be 1.0075 instead of (1/1.0075). In my teacher notes I did it the more difficult way since that is how it came up in my class, but reversing the order of the terms might make it easier. Finally, we solve the equation for *M* in order to find the monthly payment to pay off the mortgage.

A great way to **end this lesson** is to calculate how much money you will actually pay for the $100,000 house. The monthly payment is about $805 and you make 360 payments, so it is almost $290,000. This always begins an interesting discussion with students!

Here is the homework assignment I give to summarize all of these topics over the last two days.

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: The Skyscraper Problem
- LESSON 2: The Fractal Tree
- LESSON 3: Describing Sequences and Series
- LESSON 4: Arithmetic Sequences
- LESSON 5: Geometric Sequences
- LESSON 6: The Fractal Tree Revisited
- LESSON 7: Investments, Loans, and Mortgages - Day 1 of 2
- LESSON 8: Investments, Loans, and Mortgages - Day 2 of 2
- LESSON 9: Mathematical Induction
- LESSON 10: Formative Assessment Review: Sequences and Series
- LESSON 11: Formative Assessment: Sequences and Series
- LESSON 12: The Limit of a Sequence
- LESSON 13: Area Under a Curve - Day 1 of 2
- LESSON 14: Area Under a Curve - Day 2 of 2
- LESSON 15: Binomial Expansion
- LESSON 16: Unit Review: Sequences and Series
- LESSON 17: Unit Review Game: Pictionary
- LESSON 18: Unit Assessment: Sequences and Series