Lesson 16: Saving for a Rainy Day

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Logan Cunningham
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Opening Exercise Mr. Scherer wanted to show his students a visual display of simple and compound interest using Skittles TM. 1. Two scenes of his video (at https://www.youtube.com/watch?

1 Opening Exercise Mr. Scherer wanted to show his students a visual display of simple and compound interest using Skittles TM. 1. Two scenes of his video (at are shown below. What questions do you have about the Skittle demonstration? COMPOUND SIMPLE COMPOUND SIMPLE Comparing Compound and Simple Compound interest is calculated on the principal amount and also on the accumulated interest of previous periods. It is often referred to as interest on interest. Simple interest is calculated on the principal, or original, amount. Unit 6: Exponential Functions & Their Applications S.175

2 In Mr. Scherer s video, 30 years go by from the first scene to the last one. 2. A. Try to write equations for the compound situation, f(t), and the simple situation, g(t), with the Skittles, where t is the time in years. B. What other information do you need to write the equation? Explain. C. How many Skittles would you expect to see at the 15-year scene for each situation? D. Fill in the table below so that the compound interest is showing exponential growth and the simple interest is showing linear growth. Year Compound Simple Year Compound Simple Unit 6: Exponential Functions & Their Applications S.176

3 3. Grayson s grandmother started a savings account for him when he was born. The table below shows the amounts in the account at different times in Grayson s life. A. Graph the data on the grid provided. Grayson s Age (in years) Money in the Savings Account (in dollars) B. How much money did Grayson s grandmother put in the account when he was born? C. Write an equation to model this data. D. If the money is left in this account and the growth rate doesn t change, how much will Grayson have when he is 21 years old? E. Is this an example of simple or compound interest? Explain your thinking. F. What is the domain for this situation? What is the range? Unit 6: Exponential Functions & Their Applications S.177

Lesson Summary adapted from NCTM s Illuminations Suppose you have $1000 and hide it under your mattress for 20 years. At the end of the 20 years, you would still have $1000.

4 Lesson Summary adapted from NCTM s Illuminations Suppose you have $1000 and hide it under your mattress for 20 years. At the end of the 20 years, you would still have $1000. However, if you had invested it in a bank at an interest rate of 10%, you would have almost $7,000 at the end of the 20 years. How is this possible? The answer is compound interest, which works in the following way. Money is first invested. Then, at regular intervals, interest is awarded to the account and becomes the investor s money. In this way, interest is earned on previously earned interest in other words, the interest is compounded. Huge difference between compound & simple Not much change between compound and simple Compound Simple Principle Unit 6: Exponential Functions & Their Applications S.178

5 Homework Problem Set 1. Complete the table below showing the banking amounts for a $2000 investment that earns 4.5% interest each year with simple and compound interest. Time (years) Principal Simple Annual Balance at the end of the year Time (years) Compound Principal Annual Balance at the end of the year Graph the data from Problem 1. Then write an equation for each set of data. Unit 6: Exponential Functions & Their Applications S.179

6 3. Let s see what happens if the bank compounds the interest four times a year or quarterly. The formula nt r A= P 1+, where A is the amount at the end of the number of years in your account, t is the number of n years the investment has been in the bank, P is the original amount invested (principle), r is the interest rate expressed as a decimal and n is the number of times per year the amount is compounded. A. What does the fraction r n represent? B. What does the exponent nt represent? C. Use this formula to compute the value of $2000 invested for 10 years with an interest rate of 4.5% with compounding four times a year. Be sure to write the equation out in the space below. D. Compare the amount in Part C to the simple interest and compound interest found in Exercise 1 at the end of 10 years. E. What is the value of the $2000 if the interest is compounded daily (365 times a year)? F. What part of the equation do you think has the biggest impact on the amount of money in the account at the end of the time period? Explain your thinking. Unit 6: Exponential Functions & Their Applications S.180

Key Terms: exponential function, exponential equation, compound interest, future value, present value, compound amount, continuous compounding.

Key Terms: exponential function, exponential equation, compound interest, future value, present value, compound amount, continuous compounding. 4.2 Exponential Functions Exponents and Properties Exponential Functions Exponential Equations Compound Interest The Number e and Continuous Compounding Exponential Models Section 4.3 Logarithmic Functions