## Predicting Your Financial Future

- Lesson

Students often ask, “When are we ever going to use this?” Compound interest is a topic that provides an inherent answer to this question. In this activity, students use their knowledge of exponents to compute an investment’s worth using a formula and a compound interest simulator. Students also use the simulator to analyze credit card payments and debt.

The Savings Account activity sheet begins with a statement of how much certain products have increased in price over time and how hard it is to predict these increases. To get this discussion going, the day before this lesson, have students interview their parents about how much products such as a gallon of gas, a gallon of milk, a loaf of bread, etc cost when they were growing up. Alternatively, have students research the prices of these items 30 years ago. When they return to class, record their answers on the board and compute the percent increase of each item.

To calculate percent increase use the formula |

Students should notice that each item has a different percent increase. Ask students, Why were all the percent increases different? What could be the reason for this?

[There are several factors that may contribute to price increases. One possible response could be that different items are affected by inflation to different degrees.]

After this discussion, lead students to understand the need to predict the future. Inform them that in terms of invested money, accurate predictions are possible. The purpose of this lesson is to engage and excite students about financial investments, and to educate them about credit card debt.

Distribute the Savings Account Activity Sheet and have students read the introductory text and work through the first table.

Savings Account Activity Sheet

Here are the correct results for the first table:

Years Investment Has Been in the Bank | Balance at the Start of the Current Year | Interest Earned for the Year (3%) | Balance at the End of the Current Year |

1 | $100.00 | $3.00 | $103.00 |

2 | $103.00 | $3.09 | $106.09 |

3 | $106.09 | $3.18 | $109.27 |

4 | $109.27 | $3.28 | $112.55 |

5 | $112.55 | $3.38 | $115.93 |

After students complete the first table ask, "Would you want to do this calculation for each investment year from 1 to 80?" [Hopefully, students will say no.] While students may not want to do the hand calculations, tell them you just want the amount in the bank after 80 years. Pose the question, “How can we find this out?” [Possible student responses might be to use spreadsheets, computers, calculator programs, etc.] If students do not suggest it, propose looking for a pattern and developing a formula.

Then, have students complete the algebraic table and work
together to generate the formula. The completed table appears below. If
students need help with the Simplified Amount column, show them that
the entries in this column are based on the factor by grouping method.
For example: In the expression 5*x*(*x* + 1) + 7(*x* + 1), the term (*x* + 1) is a common factor of both terms, so it can be factored out. This leaves the simplified expression (*x* + 1)(5*x* + 7).

You could also complete the second row, as an example:

P(1.03) + 0.03P(1.03) | |

= (1.03)(P + 0.03P) | *** Factored (1.03) from both terms |

= P(1.03)(1.03) | *** Factored P from the binomial |

= P(1.03)^{2} | *** Rewrote the expression using exponents |

Years Investment Has Been in the Bank | Balance at the Start of the Current Year | Interest Earned for the Year (3%) | Balance at the End of the Current Year | |

Previous Balance + Interest | Simplified Amount | |||

1 | P | 0.03P | P + 0.03P | P(1.03) |

2 | P(1.03) | 0.03P(1.03) | P(1.03) + 0.03P(1.03) | P(1.03)^{2} |

3 | P(1.03)^{2} | 0.03P(1.03)^{2} | P(1.03)^{2} + 0.03P(1.03)^{2} | P(1.03)^{3} |

4 | P(1.03)^{3} | 0.03P(1.03)^{3} | P(1.03)^{3} + 0.03P(1.03)^{3} | P(1.03)^{4} |

5 | P(1.03)^{4} | 0.03P(1.03)^{4} | P(1.03)^{4} + 0.03P(1.03)^{4} | P(1.03)^{5} |

### Savings Account Activity Sheet Answers

**1a)** This formula calculates the value of the investment, the previous balance plus the interest. If *r* was used alone, the formula would only calculate the amount of interest earned.

**1b)** $6,524.08

**2a)**^{r}/_{n} is the interest rate used for each compounding period.

**2b)** Earlier we noticed that the exponent in the formula
was the number of times interest was assessed over the lifetime of the
investment. Since interest is now assessed *n* times a year and *t* is measured in years, *nt* is the number of times interest will be assessed over the life of the investment.

**3a)** $6,610.57

**3b)** It is greater because interest is being compounded more than once a year.

While this formula is helpful, it cannot handle investments that also have a monthly contribution. To perform those calculations another formula is needed.

To gain an appreciation of the convenience provided by the Compound Interest Simulator, have any students who are interested in using spreadsheets compute the amounts in the savings account after each monthly contribution of $50. Allow these students to observe the number of calculations required to find the amount in the savings account at the end of one year.

Introduce students to the Compound Interest Simulator and how quickly it works to generate these values. Have students use the simulator for Question 4 on the activity sheet.

### Savings Account Activity Sheet Answers continued

**4a)** 13 years, 11 months

**4b)** 1 year, 6 months

When comparing the two individuals who start their investments at different ages, have the students input the given information into the simulator to confirm the given predicted difference and to again demonstrate how quickly the simulator works.

As students are finding their own way to reach $1 million dollars by age 65 in Question 5, record the various strategies on the board as they are discovered. See how many different ways students can come up with. Also comment on each student’s graph on the simulator. Ask questions to help students see that each curve grows exponentially. Some of these questions could be:

- Does the graph increase at a constant rate?
[No, the graph increases at a greater rate as time increases.]

- Is the a straight line or is it curving upward?
[The graph is curving upward.]

### Savings Account Activity Sheet Answers continued

**6.** 46 years, 3 months

**7.** This will vary depending on the student’s age.

Age 15 = 23.3%, Age 16 = 24.5%, Age 17 = 25.9%, Age 18 = 27.4%

**8.** Answers will vary.

**9.** Annual *rate* is the percentage of growth an investment had over a year; the annual *yield* is the amount of money an investment made over a year.

**10.** You should start investing as early as possible.

Students will now consider credit card debt and how quickly it can get out of control. Distribute the next activity sheet:

To begin this section of the lesson, you may want to ask if any of your students have a credit card or know someone who has one. Ask those individuals, "What the Annual Percentage Rate is for the card?" The APR is the interest rate you pay for any purchases made with a credit card if you do not pay the full amount owing by the end of the monthly billing cycle. Be sure to make the observation that this APR is much higher than the interest rates banks offer on savings accounts, which students learned about in Question 8 of the Savings Account activity sheet.

### Credit Card Activity Sheet Answers

**1.** In this situation, the balance on the credit card will reach $8,500 in 7 years, 8 months.

**2.** Answers will vary.

**3.** 10 years, 10 months.

**4.** $1,791.97

**5.** $2,915

- Calculator
- Computer with Internet connection
- Savings Account Activity Sheet
- Credit Card Activity Sheet

**Assessment Options**

- Consider two investments, Plan A and Plan B:
Plan A: $20,000 investment for 5 years at 10% interest rate

Which is the better investment? In what situations would each of these plans be the best option to choose?

Plan B: $14,000 investment for 15 years at 15% interest rate with monthly contributions of $50 - Have students compare scenarios with varying compounding periods, including annually, monthly, weekly, daily, hourly, etc.

**Extensions**

- Doubling Time
A well known formula for calculating the doubling time of an investment is

^{72}/_{interest rate}. This tells us the number of years it will take for an investment to double its value if interest is compounded yearly. If interest is compounded quarterly, then divide the result by 4, and so on. Have students use the simulator and calculate the doubling time for an investment of $1,000 with no monthly contributions. Once that is done, have them change the investment amount to a different amount. The doubling time will remain the same. Tell them that the number 72 is related to this and challenge the class to explain why this doubling formula works regardless of the investment amount. - Fill out the following table, plot it, and then run a regression. This is a fun activity to do as a curve-fitting exercise.

After some trial and error, students will observe that the correct regression to choose is the power regression. The exact equation for the fitted curve isInterest Rate Years for Principal to Double when Compounded Yearly 1% 70 2% 35 3% ... 4% ... *f*(*x*) = 70.76*x*^{–0.99}, but an approximate equation is*y*=^{72}/_{x}, hence the name “The Rule of 72.” To get better results, more data points are needed. The equation above was arrived at using 4 data points. If a unit on statistics is going to be covered next, this would be a great time to introduce the concept of*sample size*. The larger the sample, the closer the results mimic the expected value. This is also known as the*law of large numbers*.

**Questions for Students**

1. For invested money, is the growth linear or exponential?

[The growth is exponential because the interest earned with each compounding is added to the original amount. This results in receiving interest on the interest on the interest... etc. Linear growth would have a constant rate of increase. Another way to state this is that invested money increases by the same percent each year, so the growth is exponential. If it were increased by the same dollar amount each year, it would be linear.]

2. Which is more important, the amount invested or the interest rate?

[When comparing investments with different percentages of growth, the interest rate is the most important. When comparing actual dollar amounts that investments will be worth in the future, it is a combination of both and depends on the length of the investment. Given enough time, the higher interest rate will always be the best choice. To see a situation where the lower interest rate yields a higher future investment value see the first Assessment question below.]

**Teacher Reflection**

- What learning styles does this lesson address?
- Did using technology enhance or hinder the mathematical idea being presented?
- What advantages are there in presenting mathematical topics in this setting?
- What were some of the ways your students illustrated that they were actively engaged in the learning process?
- What did you learn, as a teacher, by orchestrating this activity?

### Learning Objectives

Students will:

- Determine the future value of an investment using the formula
*A*=*P*(1 +^{r}/_{n})^{nt}. - Decide how much to invest to guarantee a future amount.
- Realize how damaging carrying credit card debt can become, even when making the monthly minimum payment.

### NCTM Standards and Expectations

- Generalize patterns using explicitly defined and recursively defined functions.

- Use symbolic algebra to represent and explain mathematical relationships.

- Draw reasonable conclusions about a situation being modeled.

- Approximate and interpret rates of change from graphical and numerical data.