Center for Technology and Teacher Education: Mathematics Activities

Suppose you are able to do a certain job that few people can do. An employer wants to hire you to do this job 20 times, and offers you a choice of compensate plans. Plan 1 pays you $10,000 each time you do the job, and Plan 2 offers you 1 cent the first time you do the job, 2 cents the second time, 4 cents the third time, 8 cents the fourth time, and so on. How do you want to be paid?

(4) Generate algebraic expressions for the amount paid for the nth time doing the job, and for the cumulative amount paid for doing the job n times.

(5) Use your graphing calculator to generate five lists of data – one listing the number of times the job was performed, two listing the amounts paid for each time it was performed (one for each payment plan), and two listing the cumulative amounts paid through each time the job was performed.

Hint: You can use several capabilities of your calculator to generate these lists. To create a list of times the job is done, use the sequence feature to generate a sequence in List 1 based on the expression x, with respect to variable x, starting with 1, and ending with 30, with increments of 1.

For Plan 1 you can create a list of each time’s payment by filling List 2 with a sequence with each value equal to 10,000. Using the cumulative list feature to define List 3 as the cumulative of List 2 will create a list of cumulative payments (or you can define List 3 to be equal to 10000*List 1).

For Plan 2, you can generate each time’s payment by defining List 4 to be equal to (.01) 2^{(List 1-1)}(using one of your expressions from Task 4 above), and generate cumulative payments by defining List 5 as the cumulative of List 4, or by defining List 5 as
(.01) (2^{(List 1)}-1).

(6) Examine your lists, and compare your observations with your results from Part 1.

(7) On the same screen, generate graphs for the two cumulative payment lists. Comment on how these graphs help one explore the situation, and on how they illustrate the underlying mathematics. (See Task 11 below).

(9) Write two sets of recursive equations to model the two pay plans. (Note: recursive equations can be written for both the amount paid for the nth time the job was done, and for the cumulative amount paid through the nth time the job was done.)

For the first plan, the amount a paid for the n+1^st time can be represented as a_n+1=a_n (or 10,000), and the cumulative amount A paid through the n+1^st time as A_n+1=A_n+a_n (or 10000).

For the second plan, the amount b paid for the n+1^st time can be represented by b_n+1=2b_n, and the cumulative amount B paid as B_n+1=B_n+2b_n.

If you only want to display the two cumulative amounts only, use A_n+1=A_n+10000 and B_n+1=2B_n+.01

Note 1: these formulas can be written in terms of n instead of n+1.
Note 2: Casio calculators have a summation option for recursion, hence the screenshots on the cover page do not show expressions for the cumulative amounts, yet they appear in the table and graph.

(10) Enter your equations into your graphing calculator and generate tables to help you explore this dilemma in the short run and long run. Examine the values in these tables and comment on your observations. Discuss the short and long run for this dilemma.

(11) Graph these table values, and comment on the shapes of the graphs. Compare and contrast the tabular representation and the graphical representation of this data.

(13) Discuss real world lessons that can be learned from exploring this dilemma.

(14) Compare the list method used in Part 2 with the recursive method used in Part 3.

The Consumer Price Index (CPI) is an index used to gauge inflation. Many economists think the current CPI overestimates inflation by 1.1% because of a number of biases, and many believe it should be adjusted downward to account for these biases. But, adjusting the CPI downward is controversial because it is used to determine cost of living increases for social security and other expenditures (e.g. food stamps and school lunches). The newspapers reported recently that the cost of living increase for social security recipients would be 2.1 %. Also reported was that the average monthly social security check this year is $749, and that the maximum check is $1,326.

(15) Write a general recursive equation to determine the amount of money a retiree would receive from Social Security next year, assuming a cost of living increase of i% over this year’s amount.

(16) Use this equation to generate a recursive equation for the total amount of money a retiree would receive from Social Security over the next n years, if each year there was a cost of living increase of i% each year.

The amount received in a given year, in terms of the amount received the previous year, can be expressed as: a_n+1=a_n(1+i). The total, or cumulative, amount received through year n+1, in terms of the total amount received through year n, can be expressed as A_n+1=
A_n+ a_n(1+i ). Again, these equations can be written in terms of n or n+1.

(17) Use this equation and the data given above to generate recursive tables on your calculator to determine the total amount of money a retiree receiving the average payment today would receive over the next 20 years assuming: (i) a constant 2.1% increase each year, and (ii) a CPI downward-adjusted constant 1% increase each year by using the .

(18) Determine the total amount of money a retiree receiving the maximum payment today would receive over the next 20 years assuming: (i) a constant 2.1% increase each year, and (ii) a CPI downward-adjusted constant 1% increase each year.

(19) Estimate the total amount of money such an adjustment would save Social Security over the next 20 years, given that there are approximately 45 million people receiving social security today.

(20) The 403b regulation allows teachers and other employees of non-profit organizations save for retirement on a tax-deferred basis. Money contributed to a 403b account through payroll deduction is contributed pre-tax and is not taxed until money is withdrawn at retirement or at age 591/2. Use recursion to calculate the amount of money a new teacher would accumulate after 20 years of contribution into a 403b plan if the teacher contributes $100 per month to a money market account paying 6% per year, compounded monthly.

(21) Calculate the amount of money accumulated if the teacher can contribute an additional $10 per month each year.

Write a calculator program that can determine the total amount accumulated after M months, with interest rate I, initial contribution A and increment B.

Create a spreadsheet that calculates and displays the new contribution each month, the interest received each month, and the total accumulation, for any given monthly contribution amount, interest rate, and increment.

(22) Suppose that you took out a $100,000 mortgage at an annual interest rate of 8%. The mortgage broker tells you that your monthly payment will be $733.76 if you pay off the mortgage in 30 years, or 360 months. Derive a recursive formula to determine your balance after each mortgage payment in terms of the balance after the previous payment. Use this formula to generate a table of values of your balance after each payment. Verify that a payment of $733.76 per month will pay off the loan after 360 payments. Generate a graph of the balances after each payment.

(23) Comment on the connections you see between the algebraic, numerical, and graphical representations of this situation.

(24) Use the above recursive formula to determine how many months it would take to pay off the mortgage if you made monthly payments of $600, $1000, $1500. Generate graphs for the balances after each payment, and compare the graphs and number of months to pay off the loan.

(25) Make several interpretive statements about the relationship between the term of mortgage and the payment amount.

Extension: Credit Card Debt (from Charles Lowery, Cedar Bridge Academy, Bermuda)

Credit cards usually have high interest rates. If your credit card has an interest rate of 19.8% and you have a balance of $5000 and a minimum monthly payment of $20, how long will it take you to pay off the loan? What if you paid $60 a month? What if you paid $82.50 per month? Compare the number of months required to pay off the loan with payments of $150 and $300. Discuss your results.