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REVIEW: Calculating Percent Change
Let’s review percent increase and decrease. Try to perform each of the following calculations in one step.
OBJECTIVES & STANDARDS
Math Objectives
Write percent change exponential functions given the necessary values
Model a real world situation using exponential functions
Find the rate of growth and decay given the equation
Common Core Math Standards
Personal Finance Objectives
Analyze return on an investment given starting investment, rate of return and time periods
Explain what inflation is and how it affects purchasing power and investments
Explain how depreciation affects the value of a car
National Standards for Personal Financial Education
Investing
2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
4a: Describe the impact of inflation on prices over time
DISTRIBUTION & PLANNING
Distribute to students
OBJECTIVES & STANDARDS
Math Objectives
Write percent change exponential functions given the necessary values
Model a real world situation using exponential functions
Find the rate of growth and decay given the equation
Common Core Math Standards
Personal Finance Objectives
Analyze return on an investment given starting investment, rate of return and time periods
Explain what inflation is and how it affects purchasing power and investments
Explain how depreciation affects the value of a car
National Standards for Personal Financial Education
Investing
2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
4a: Describe the impact of inflation on prices over time
DISTRIBUTION & PLANNING
Distribute to students
Exponential Growth and Decay using Percents
Remember that the b value in the general form
represents the growth or decay of the initial value. If the b value is 1, then no growth or decay happens.
If you are told that the initial value changes by a percent, then we can represent the b value as growth or decay where:
Growth is 1 + r
Decay is 1 - r
r is the decimal form of the percent
Use what you know about writing exponential functions with percent growth or decay to complete the following problems.
Now that we know how to use 1 + r and 1 - r to write percent growth and decay equations, we can use this to determine the rate of change of a given equation.
You make an investment of $250 and know that it has an average annual growth rate of 7%. Let’s do an analysis to see how much money you’ll have over a period of time and how much that return has grown from your initial investment, assuming your growth rate remains unchanged.
After seeing the growth of this investment over 4 ten year periods, what conclusion can you draw about how time affects an investment with an exponential rate of growth?
Level 2 *DESMOS**
There are several factors to consider when deciding what investment is best for you. Let’s analyze three investments with different starting values, growth rates, and investment periods.
You can make one of the following investments:
Option 1: $50 initial investment with an average annual growth rate of 9% starting at age 30
Option 2: $50 initial investment with an average annual growth rate of 8% starting at age 20
Option 3: $100 initial investment with an average annual growth rate of 7% starting at age 25
Before we do an analysis, which option do you think will have the greatest return after
a. 10 years?
b. 20 years?
c. 30 years?
Level 3 -BONUS
Every investment that we’ve seen so far involved making a one time deposit and then letting your investment grow. What if we wanted to keep contributing to the investment account every month like many people do for retirement? In this scenario, the initial value will be changing every year so it’s easier to use a spreadsheet.
You are going to invest $1200 at the start of every year into an account that has an annual rate of return of 3%.
Make a copy of this spreadsheet
Complete column C for each year. The initial value will be the total amount in the account from the previous year plus any additional deposits that you’ve made. Think carefully about what the exponent should be for this calculation.
Complete column D by subtracting the total account investment (the money that you deposited) from the total in the account to find how much interest you’ve earned on your investment.
Share and Paste a copy of your spread sheet (and screen shot)
Why are you not able to use an exponent of 10 in this situation?
How would you change this situation if you wanted to use one exponential equation with an exponent of 10?
Refer back to questions 7 and 8. Plot the value of Sara’s ROTH at age 68 when she invested $500 and $1,000 to start. Pick one additional initial investment amount and add that to your graph then draw a curve to represent the exponential growth.
Your friend is considering how much he should invest in his new IRA. He is debating if he should invest the full $800 he was planning to or just $400 and use the extra money to buy a couple of pairs of sneakers he had been eyeing. What advice would you give? Use math to justify your answer.
Which investment option made the most money after 10 years?
A. Option 1
B. Option 2
C. Option 3
Which option yielded the most total money by the time you were 60 years old?
After seeing the growth of these investments over time, what is one piece of advice you would give to a friend who is making a long-term investment like this?
