[Update as of 6-17-07: At the bottom you will now see 3 screenshots from the TI-84 showing all of the formulas used for this series of mortgage activities and the input screen for the built-in Finance Application on the TI-84 that can be used to determine the monthly mortgage payment. The first 2 screens overlap, i.e., the 2nd screen contains part of the first screen and the 4th function, Y4. You will need to refer to the index of variables below to make sense of all this. There are more details below.]

The following is the 3rd and possibly the last in this particular series of classroom activities. All three should be assigned for complete effect:

Part I: Taking the Magic Out of Mortgages

Part II: Puff the Magic Mortgage

Thought I forgot to finish this activity?

Well, with the school year over for some and ending for others, here's Mortgages Part III to think about as we look forward to making our monthly payments during the summer and plan enrichment classroom activities for the fall and spring. Part III is more ambitious and requires more sophistication on the part of the Algebra 2, Advanced Algebra or Precalculus student. As always, I am attempting to provide a completely developed enrichment lesson ready to use or modify as needed. You may want to bookmark this and return to it when teaching this unit next year.

The goals here are:

(a) Providing a more challenging application of exponential functions and their relation to geometric sequences and series

(b) Systematic development of the formulas for the equalized monthly mortgage payment as well as the portion of the monthly payment that goes toward paying off the principal, etc.

This is an activity that is particularly suited for block scheduling. If begun in a 40-45 minute period, the lesson will probably run over two periods or the last few parts can be assigned for homework. Another effective approach is to give this as a long-term individual or group project. In this case, I would recommend combining all three Mortgage activities.

STUDENT ACTIVITY

In the previous activity, you should have observed that the sequence of data values in the Y_{1} column formed a geometric sequence with common ratio 1+I, where I was the interest rate per payment period (decimal form). It's time to derive this mathematically and see how the other columns were generated and how some of those famous mortgage formulas came to be. Did you figure out that Y_{1} contained the amounts labeled P_{x} below?

The following is an index of the variables we will use . I'm using uppercase variables and X for ease of entry when instructed to enter these formulas into your graphing calculator. Note that the discussion below answers the questions from the previous activity regarding the meanings of the Y-columns in the calculator.

P = Original amount of Loan (remember, it was $100 in the previous activity)

I = Rate of interest **per payment** (expressed as a decimal)

Note: E.g., if the bank is charging 6% annual rate on your loan, I = 6/12% or 1/2% = 0.005 per month!

Z = 1 + I (to make formulas easier to write and enter into the calculator, since 1+I appears frequently when doing compound interest)

N = number of payments (e.g., N = 360 for 12 payments a year over 30 years)

X = the index used for the xth payment

P_{x} = Amount of the xth monthly payment that goes toward reducing the principal

I_{x} = Monthly interest payment

A = Level (equal) monthly payment

U_{x} = Amount of debt (Unpaid amount) remaining after Xth payment

(1) Explain the meaning of the equation: P_{1} + PI = P_{2} + (P-P_{1})I.

(2) Show that P_{2} = P_{1}(1+I) by solving the equation in (1) for P_{2}.

(3) Explain why P_{1} + PI = P_{3} + (P - P_{1} - P_{2})I

(4) Show that P_{3} = P_{1}(1+I)^{2} by solving the equation in (3) for P_{3} (after substituting for P_{2} from (2)).

The results in questions (2) and (4) suggest the following general formula which can be verified by mathematical induction:

(**) **P _{x} = P_{1}(1+I)^{X-1}**.

Recall that P

_{x}denotes the amount of the Xth payment that goes toward paying off the original loan amount P.

The next few parts require that you recall the formula for the sum of the first N terms of a geometric sequence. If you have forgotten it, research it or your instructor will review it.

(**) shows that the sequence P

_{x}is a geometric sequence with first term P

_{1}and common ratio, 1+I (or Z).

(5) Explain why P = P

_{1}+ P

_{2}+ P

_{3}+ ... + P

_{N}

(6) Using (5) and the formula for the sum of the first N terms of a geometric sequence, show that P

_{1}= PI/((1+I)

^{N}-1) = PI/(Z

^{N}-1) where Z = 1+I.

(7) Use (6) to explain why A = PI/((1+I)

^{N}-1) + PI.

(8) Simplify the result of (7) to derive:

A = PI(1+I)

^{N}/((1+I)

^{N}-1) = PIZ

^{N}/(Z

^{N}-1)

[Again, Z = 1+I]

(9) STORE the following values from the Home screen:

100 STO P

.1/12 STO I [10% annual rate divided by the number of payments during the year]

1+I STO Z

12 STO N

Note: If you haven't used the ALPHA key before, you will now! Remember: The variables listed above will store these constant values until you or some program changes them. Clearing the screen has no effect on stored variables.

(10) Enter the last formula for A (Z-form) from (8) into Y

_{1}in your graphing calculator. You may have to modify it slightly for entry purposes. The * symbol for multiplication is not necessary for most graphing calculators. Try it!

(11) Start a TABLE from X = 1 and display your TABLE. If entered correctly, the values for

X = 1 through 12 should all be the same. Why? Which column was this in Part II of the Mortgage Activity?

(12) Using ** and the formula for P

_{1}from (6) (the one in Z-form), write a formula for P

_{x}in terms of P, I, Z, X and N. Enter this into Y

_{2}. Display the TABLE starting from X = 1. Which column was this in Part II of the Mortgage Activity?

(13) Derive a formula for I

_{x}using preceding results. Again, express it in terms of P, I, Z, X and N and enter this into Y

_{3}. Which column was this in Part II of the Mortgage Activity? Explain why these values are decreasing.

(14) Derive a formula for U

_{x}using preceding results. Again, express it in terms of P, I, Z, X and N and enter this into Y

_{4}. Which column was this in Part II of the Mortgage Activity? Explain why these values are decreasing.

NEW!!

Below you will find 3 screenshots from the TI-84. The first 2 show the actual functions used to compute the 4 key quantities used for mortgage repayments. The 3rd screenshot shows the finance application screen (APPS, Finance, TVM Solver) used to input the actual data values used in this activity. Students will need to refer to the index of variables above to make sense of these functions. PMT (the monthly mortgage payment) was obtained by pressing ALPHA ENTER (SOLVE). One of the main goals of this series of activities was to show students how they could obtain the formulas that are hidden behind this 'cool' application. Ask your students to explain why PMT is displayed as a negative amount!

Y1 = The payment toward principal function, i.e., the portion of the xth monthly payment that is applied to the loan principal (increasing function)

Y2 = The monthly interest payment (decreasing function)

Y3 = The fixed monthly mortgage payment (constant function, thus the variable x does not appear)

Y4 = The debt function, i.e., the amount still owed on the principal after the xth payment (decreasing function)

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