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

[Note: For an exceptionally clear and definitive exposition of all things financial, the best resource I have found is MoneyChimp. There are interactive calculators to thoroughly understand the concepts in this post and much much more. More importantly, for math nerds like me, the formulas are explained and, in some cases, derived. The mathematics is accurate and the analysis is excellent. Enjoy it!]

In Algebra 2 and Precalculus (or whatever it may be entitled in your local schools), students often do compound interest problems. Typically, the author of the text and/or the instructor will derive the formula for what your original investment of P dollars will be worth in t years, if interest is compounded n times per year for t years at an annual rate given by r (a decimal for this discussion):
Compound Interest Formula: A(t) = P(1+r/n)^nt.

This is a nice practical application of exponential functions, exponential growth in particular. A similar, but more sophisticated, concept applies to annuities and amortization of a mortgage (paying off debt over time in n equal payments). In both an annuity and a mortgage, the original amount of money (whether it's the amount invested or the debt you owe) generally decreases over time. In an annuity, you receive a fixed amount at the end of each period, whereas, in a mortgage, you pay a fixed amount. In an annuity, your original investment is earning (accruing) interest (it may be possible to 'live off' the interest and not touch the principal), while you are receiving periodic equal payments that are deducted from your account. A central concept in both annuities and mortgages is that that interest is applied before receiving an annuity payment or before making a mortgage payment.

The following is the first part of an activity introducing students to the mysteries of mortgage calculations. The fact that the formulas for monthly payments or the decreasing amount of debt seem very intricate lead many to believe that this topic is too sophisticated for most secondary math students. Just give them the formulas, mention that it is related to exponential functions and let them plug it all into their graphing calculators. We know most adults, other than those in the business of lending, punch the numbers into the computer and read the results. Before calculators, bankers would look it up in those mortgage tables on some well-worn-out card. This activity may demystify a little bit of this. Students need good algebra skills, knowledge of exponential properties and functions in particular, a basic knowledge of compound interest and background in geometric sequences and series (later on). I am well-aware that sensitivity is needed here for students whose parents do not own a home, however, all students can benefit from these ideas since these principles apply to far more than a monthly mortgage payment.


STUDENT ACTIVITY
You can find many excellent web resources for mortgage calculations. You can also find the actual formulas for all of this either in your text or in other sources. Most of us would probably use the built-in applications typically found on a graphing calculator or more likely use those free mortgage calculators all over the web. In this activity you will take an active role in the process of borrowing and lending and see what lies behind those sophisticated formulas.

In actual practice, mortgages can range from less than a hundred thousand into the millions of dollars. Therefore, these loans are typically repaid over 5, 10, 15, 20, 25 or 30 years to make the monthly payments more manageable. In this activity you will be borrowing a small amount and considering an oversimplified form of repayment, leading up to more general considerations.

You will be borrowing 100ドル from a reputable lender, Stan, The Mortgage Man.
Stan is charging the going rate at the moment, which is 10% compounded annually.

(a) If you repay the loan in one year, explain why your single payment would be 110ドル.
(b) If you agree to repay the loan at the end of two years, in one single payment, explain why that single payment would be 121ドル?

Discussion: Parts (a) and (b) should remind you of the compound interest formula you've learned:
One year re-payment: 100(1+0.1)¹
Two year re-payment: 100(1+0.1)²

Surely, increasing the payment schedule to TWO payments over two years or one year, cannot be that much more difficult? Let's find out...

(c) This time you will make two equal payments over two years. Stan gives you the repayment schedule: Two equal payments of 60ドル. He explains it as follows: The loan (your debt) of 100ドル is divided into two equal payments of 50ドル each. The interest charges are 10ドル (10% of the amount you borrowed) on each payment. Explain the mistake that Stan is making (or is he trying to take advantage of an unsuspecting borrower who didn't pay attention in algebra?). We're not asking you to correct Stan's error here - just explain why his calculation is either wrong or unfair.

(d) Now that you figured out that the two equal annual payments should not be 60,ドル we will tell you what the actual payments would be according to mortgage formulas:
Each annual payment is 57ドル.62 (rounded to the nearest penny).
Show why these two payments correctly repay the loan of 100ドル and the interest that is due on each payment. Show method clearly. Use calculator as needed.

(e) Do you think you could figure out an algebraic way to determine those equal payments of 57ドル.62? You are about to...
Let A represent the equal annual payments you will make.
At the end of the first year, before you make your payment, you owe 100ドル(1.1) = 110ドル.00. Now the fun begins:
(i) Represent, in terms of A, the amount of debt (loan + interest) you will owe AFTER you make your first payment.
(ii) Represent, in terms of A, the interest you will be charged by the end of year 2 BEFORE you make your final payment.
(iii) Represent, in terms of A, your debt, AFTER making your final payment.
(iv) What should be the numerical value of your debt AFTER making your final payment? Now, write an equation and solve for A. You should come up with 57ドル.62!

(f) Are you up to the challenge of solving for the general formula for A given an original loan of $P at an annual interest rate of i (expressed as a decimal)? Of course, you are! For now, you only need to do this for TWO payments, just as we did in (e). Of course, your formula for A should be in terms of P and i.

More to follow...

Posted by Dave Marain at 9:25 AM 6 comments

Labels: advanced algebra, algebra 2, compound interest, exponential function, financial math, geometric sequence, mortgage, precalculus

Algebra teachers, like myself, are always looking for ways to help students make sense of exponents. We look through copies of the Mathematics Teacher, we go to the Math Forum and now we Google, Google ad infinitum (or some other search engine to be fair!). Here's an approach that I have found helpful. I assume the student has had some basic introduction to exponents and their properties. I call it the exponential function approach which sounds too challenging for middle schoolers but you decide if they can handle this. Students will use pattern-based thinking and graphs to make conjectures about extending powers of 2 to include zero, negative and even fractional exponents. Properties of exponents will then be used to 'justify' the conjectures. The juxtaposition of the numerical, symbolic, graphical and verbal descriptions are consistent with the Rule of Four that is now regarded as the most powerful heuristic in teaching mathematics.

Begin by making an x-y table - this is the critical piece.

Exponent (x)...........................Power (y = 2^x)
3 ..................................................2³ = 8
2 ..................................................2² = 4
1 ..................................................2¹ = 2
0 .................................................2⁰ = ??

The instructor of course is prompting the students for the powers while they are taking careful notes. At the same time the instructor is plotting these results as ordered pairs and the students do likewise. It might be helpful to let 2 or 4 boxes represent one unit on the y-axis since, at some point, the y-values will be fractional. Similarly for the x-axis (play with it first).
At this point, the instructor asks a key verbal question (you may phrase it much differently depending on the level of the group and your preference):
[While pointing to the left and right columns]
"When the exponent decreases from 3 to 2, the corresponding power of 2 is divided by ___.
Repeat this phrase a couple of more times until you reach an exponent of 0, then -1 and voila! Keep going until x = -3, plot the points and the students are seeing an exponential curve in grade 7? 8? 9?
Motivating zero and negative exponents using a function model (tables!) seems to make sense to me because it begins to create a 'function' mind-set that can be carried through all subsequent math courses. It may also help students to 'see' that the range of the function consists of positive real numbers. If you're wondering why I didn't mention turning on the graphing calculators to make the TABLE and GRAPH, I hope you can guess why. It was important for me to have students do this by hand first, then I will turn on the overhead viewscreen and we can explore with technology. Just my opinion of course but students in my classes seem to make sense of this. Of course, I don't kid myself that this approach will lead to better grades on tests of this unit! Facility with the properties of exponents only comes from considerable skill practice with paper and pencil.
For fractional exponents, I'll begin the discussion but I will have to explore further on another post or leave it to your imagination. "Ok, boys and girls, if mathematicians believed exponents could be zero or negative integers, would you be surprised if they wondered about 2^1/2? From the table and the graph, 2^1/2 should fall between ___ and ___? Do you think it will be exactly 1.5? Why or why not?
I know many of you use the exponent properties to develop this topic, but I wanted to suggest an alternative. I usually follow this discussion with arguments like: " Hmmm, I wonder what
2^1/2 times 2^1/2 would be?" etc...

Posted by Dave Marain at 5:39 AM 2 comments

Labels: advanced algebra, exponential function, exponents, functions, middle school math, patterns, Rule of Four