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# Credit Card Investigation: A Dastardly Scheme (Day 4 of 4)

Lesson 12 of 14

## Objective: SWBAT explain the meaning of the mathematical constant e and use the continuous compounding interest formula to solve real world problems.

## Big Idea: There’s a new irrational number in town! Students explore the constant e in the context of continuously compounding interest.

*51 minutes*

Today is the final day of the Credit Card Investigation. In the first two days of this investigation students learned all about interest. They learned how to calculate a percent increase or a percent decrease using only a one-step calculation. Then they applied this new knowledge to quickly compare how three credit cards’ balances grow with three different interest rates. They also explored a simplified problem where we increase a credit card payment by only $30 a month and saw how much quicker the $1000 balance we owed decreased. In the third day of the credit card investigation, students learned how interest is actually calculated. We talked about different rates of compounding and students analyzed how compounding rates can affect the amount of money we owe on a credit card. They then derived the compound interest formula by analyzing the structure of the expression used to calculate the interest for each compounding period and the patterns they observed. Today, students will extend their learning to continuously compounding interest.

In this section of the investigation, students first review how to calculate compound interest again and then continue to increase the compounding frequency in order to introduce continuously compounding interest. This then leads students to the constant *e. *Finally, students will apply their knowledge of calculating compound interest by solving some example problems.

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#### Closure: Summary time!

*10 min*

If time remains at the end of day 3, I think it would be awesome to hear what students wrote up for their responses to the last question. This is such an important concept in math that is quickly forgotten. I know the first time I taught exponential functions/logarithms I spent some time researching what the constant *e* was and tried to find a way to explain it to students that would make sense in their non-calculus world. And that was me… a math teacher… fresh out of college. I should’ve known that! But felt that I missed it somewhere along the way. My point… I am going to try to make this memorable for students. And by having they present their different views and hearing their peers talking about *e* I hope to help the concept stick a little better and to communicate to my students that this is important. We will definitely be revisiting this in the next unit… Logarithms! This activity is a great lead-in to logarithms too!

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Cat Island (Day 1 of 2): Cats can’t add but they do multiply!
- LESSON 2: Cat Island (Day 2 of 2): Cats can’t add but they do multiply!
- LESSON 3: Graphing Exponential Functions
- LESSON 4: Shifting Exponential Functions
- LESSON 5: Counting the Change: Linear, quadratic, or exponential? (Day 1 of 2)
- LESSON 6: Counting the Change: Linear, quadratic, or exponential? (Day 2 of 2)
- LESSON 7: Sorting out the Change
- LESSON 8: Stretching Exponential Functions (and your mind)
- LESSON 9: Credit Card Investigation: What is interest? (Day 1 of 4)
- LESSON 10: Credit Card Investigation: What is interest? (Day 2 of 4)
- LESSON 11: Credit Cards Investigation: How is interest really calculated? (Day 3 of 4)
- LESSON 12: Credit Card Investigation: A Dastardly Scheme (Day 4 of 4)
- LESSON 13: Exponential Functions Test Review
- LESSON 14: Exponentials Functions Unit Test