5.5 Exponential Change

Last updated over 1 year ago

26 Nsɛmmisa

Hyɛ no nsow a efi ɔkyerɛwfo no hɔ:

Akwankyerɛ

Intro/Warm-Up:

Learn It

Practice It

ACTIVITY: MATH: Exponential Decay in Purchasing Power

APPLICATION: Analyze Investments for Exponential Change

5.5 Exponential Change

Last updated over 1 year ago

26 Nsɛmmisa

Hyɛ no nsow a efi ɔkyerɛwfo no hɔ:

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

Akwankyerɛ

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

Intro/Warm-Up:

REVIEW: Calculating Percent Change

Let’s review percent increase and decrease. Try to perform each of the following calculations in one step.

Sierra walks into a store and spends $12. If there is a 7% sales tax, how can Sierra calculate her total bill in one step?

Ang finds a great sale on a new pair of headphones. The original cost was $89 but the item is 25% off. How can Ang calculate his total bill in one step?

Learn It

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

Write an equation that represents the following situation

Starting value = 125
Rate = 6% growth per year
Years = 10 years

Write an equation that represents the following situation

Starting value = 200
Rate = 2.5% decay per year
Years = 5 years

Practice It

Use what you know about writing exponential functions with percent growth or decay to complete the following problems.

Farid makes a $12,000 investment that yields an average yearly return of 4%. If he makes no additional investments, how much will his investment be worth in 15 years? Screen shot desmos

You purchase a car for $36,000. You did some research and found that this particular model of car depreciates in value by around 15% per year. What is the value of your car after 6 years?

ACTIVITY: MATH: Exponential Decay in Purchasing Power

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When Sarah turned 18, she took $500 from her savings and opened a ROTH IRA. Sarah chose to invest her money in a low fee index fund. Based on historical data, Sarah predicts she can expect an average 7% return on her money each year. If Sarah’s prediction is correct, how much will her $500 investment be worth when she is 68 years old? (Assume she makes no other contributions to the account after the first time)

Equation:

Answer:

Screen shot Desmos for full credit

Before Sarah does open her account, she sees her grandmother for the holidays. Grandma wants to contribute to Sarah’s ROTH and even matches her contribution, giving her an additional $500 to invest. Now that Sarah has an initial investment of $1000, how much would Sarah have at age 68 based on her predicted rate of return from historical data?

Equation:

Answer:

Screen shot Desmos for full credit

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 where the balance over time can be modeled by the equation

, where x represents the number of years since the investment started and y represents your total balance after x years.

a. What is the starting balance of your investment?%

b. What is the rate of growth of your investment? %

is an equation that can be used to represent the purchasing power of $100 after x years of inflation. What is the rate of inflation used to make this calculation? %

APPLICATION: Analyze Investments for Exponential Change

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.

Write an equation that represents your investment where x represents the number of years that has passed since you started and y represents your account balance after x years.

Review

To calculate return on investment, use the formula:

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Complete the table below by using your equation from question #15 to calculate the Money column and the percent change formula for the % Growth column. The original value for every row will be $250 when calculating percent growth.

Years Money % growth

0 250 0

How much did your % Growth from Initial Investment change for each 10 year period?

Time Interval % Growth

0 to 10

10 to 20

20 to 30

30 to 40

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)

5.5 Exponential Change

5.5 Exponential Change

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?

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?

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.

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?

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?

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?

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?

5.5 Exponential Change

5.5 Exponential Change

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?

Before we do an analysis, which option do you think will have the greatest return aftera. 10 years?b. 20 years?c. 30 years?

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.

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?

Before we do an analysis, which option do you think will have the greatest return aftera. 10 years?b. 20 years?c. 30 years?

Which investment option made the most money after 10 years?A. Option 1B. Option 2C. 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?

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 1B. Option 2C. 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?

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?

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?

Which investment option made the most money after 10 years?
A. Option 1
B. Option 2
C. Option 3

Which investment option made the most money after 10 years?
A. Option 1
B. Option 2
C. Option 3