##
* *Reflection: Rigor
Investments, Loans, and Mortgages - Day 1 of 2 - Section 2: Share

Today, even my brightest students had difficulty finding a solution to this problem. There are many different things going on at once, but almost everyone was on the right track. To me, that is the hallmark of a really good problem - it was challenging to all but everyone was making meaning.

*Rigor: When everyone breaks a sweat...*

# Investments, Loans, and Mortgages - Day 1 of 2

Lesson 7 of 18

## Objective: SWBAT solve problems using geometric sequences.

*45 minutes*

#### Launch and Explore

*30 min*

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.

*expand content*

#### Share

*15 min*

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.

*expand content*

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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