MathNotations

[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 Y₁ 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 Y₁ 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ドル 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: P₁ + PI = P₂ + (P-P₁)I.
(2) Show that P₂ = P₁(1+I) by solving the equation in (1) for P₂.
(3) Explain why P₁ + PI = P₃ + (P - P₁ - P₂)I
(4) Show that P₃ = P₁(1+I)² by solving the equation in (3) for P₃ (after substituting for P₂ from (2)).

The results in questions (2) and (4) suggest the following general formula which can be verified by mathematical induction:
(**) P_x = P₁(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 P₁ and common ratio, 1+I (or Z).
(5) Explain why P = P₁ + P₂ + P₃ + ... + P_N
(6) Using (5) and the formula for the sum of the first N terms of a geometric sequence, show that P₁ = 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 Y₁ 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 P₁ 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 Y₂. 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 Y₃. 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 Y₄. 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)

Posted by Dave Marain at 1:16 PM 1 comments

Labels: advanced algebra, amortization, compound interest, exponential function, geometric sequence, investigations, mortgage, precalculus

OVERVIEW OF PART II
We will continue our investigation of mortgages. In parts (a), (b) and (c) below, you will analyze the effect of accelerating repayment by paying off the loan in one year with 2 equal payments at 6-month intervals, instead of one payment each year for 2 years, [NOTE: Some lenders do not allow this without a prepayment penalty, but we'll assume Stan the Mortgage Man is sorry for the error of his ways and wouldn't charge this.]

In (d) we will analyze data tables corresponding to the same loan of 100ドル but there will now be several payments over the course of one year (you will need to determine how many).

(a) From your knowledge of compound interest you know that if payments are made semiannually (in 6 month intervals), the interest rate is divided by 2, the number of interest periods; thus the rate would be 5% on each of these payments. Using an analysis (algebraically) similar to part of the earlier activity, show that each of these equal payments would be 53ドル.78.

(b) How much is saved in total by repaying the debt in one year by this method, compared to one payment a year for 2 years? Explain why this happens. [By the way, if your parents do make mortgage payments, ask them if they are making two payments a month and, if so, why?]

(c) What could you do to reduce the total payment even more, assuming that the debt is paid off in one year?

(d) Study the 4 tables below. The data in Y1, Y2, Y3, Y4 all relate to the loan of P = 100ドル at 10% annual rate of interest. The loan is repaid at the end of one year but is paid in several payments. We will not tell you what the meaning of each of the columns (functions) are. That's part of the challenge! Your job is to interpret the data and respond to the following questions:

(i) How many interest periods (payments) are there? How do you know? Be careful here!
(ii) Which column (function) corresponds to each monthly mortgage payment. Give reasons.
(iii) Which column corresponds to the amount of debt remaining after each payment? Give reasons.
(iv) Which column corresponds to the amount paid toward the principal (P = 100ドル) at each payment? Give reasons.
(v) Which column corresponds to the amount of interest paid at each payment? Give reasons.
(v) The total dollar amount of which column should be exactly 100ドル? Explain why.
(vi) How much is the first interest payment? How much is the first payment toward principal?
(vii) Explain the meaning of the zero value in Y4.
(viii) Which function (column) is best modeled by an exponential function of the form
f(x) = a ⋅ b^x-1? Determine the values of a and b and their relationship to the loan.

Hint: Consider a simpler example. Suppose the first few terms in a sequence or list are 3,6,12,24,... This is known as a geometric sequence because, starting with the 2nd term, the ratio of each term to the preceding term is constant: 6/3 = 12/6 = 24/12 = 2. The function that describes this sequence is 3 ⋅ 2^x-1, for x ≥ 1. Thus, every geometric sequence can be modeled by an exponential function. Use this approach to answer this part.

Posted by Dave Marain at 9:26 PM 0 comments

Labels: amortization, compound interest, exponential function, graphing calculator, investigations, mortgage