Some Algebraic Reasoning (Math Contest/SAT-Type) to Whet Your Appetite

While we're waiting for Carnival of Mathematics #11 and Part II of Recursive Sequences, here are a few algebra problems to store away for your students...
Using math contest problems (AMC, MathCounts, etc.) in the classroom can be very helpful in raising the level of expectation for many students. One thing experienced teachers know is that students will only learn how think at higher levels when we provide these kinds of challenges. Math educators are always seeking more examples of this nature. Look no further than released math contests, samples of which can be found online. Use these problems as models which can be revised to match the skill and conceptual level of your group. This is not easy but it's worth the effort. Assuming that more basic level math students would have little or no chance at these (or will become too frustrated) can be a self-fulfilling prophecy. Those of you who have been following this blog since its inception 6 months ago, know that I challenged just such a group of 9th graders this year. Deciding what prerequisite skills were needed and how to develop some of the challenges incrementally required considerable thought and planning but their engagement in the activities and a feeling of accomplishment on their part made it worthwhile.

Questions 1 and 2 below are appropriate for 1st or 2nd year algebra students. The objective of these conceptually-based problems is to develop algebra sense rather than provide mechanical practice with algorithms.

1. For how many values of x is
(x-3)(x-4)(x-5)(x-6) = (3-x)(4-x)(5-x)(6-x)?
(A) 0 (B) 2 (C) 3 (D) 4 (E) more than 4

2. For how many values of x is
(x-3)(x-4)(x-5)(x-6)(x-7) = (3-x)(4-x)(5-x)(6-x)(7-x)?
(A) 0 (B) 3 (C) 4 (D) 5 (E) more than 5

3. What is the greatest integer value of N, less than one million, for which
√(1+√N) is a positive integer?

Note: Is estimation worthwhile here as a starting point? How would some students use a calculator to 'solve' this? Watch them! Do algebra textbooks provide methods and practice for solving positive integer problems? Are there special methods one needs for these?

An In(de)scribable Rectangle in a Rectangle - A Geometry Challenge

Now that the summer months have arrived, I thought it was time for a geometry challenge problem to chew on. Although this is a departure from the lesson plans I have been writing, it's still an enrichment experience. I'm often asked by students and parents how one becomes better at solving 'hard' math problems. My response is: "Keep trying hard problems!" One can only improve at problem-solving by challenging one's mind. Also, learn from others - we all learn from good models. There are no shortcuts here. Some frustration is healthy and if you want more cliches, let me know!

OVERVIEW
This question should definitely challenge your geometry students. It was brought to my attention by a teacher via a student who was given this by his honors geometry teacher. I'd provide proper attribution if I knew the original source. However, it is possible to go beyond this question and generalize. There are endless problems one could generate from inscribing Figure A in Figure B. Rectangles in rectangles, other than special cases (square in a square) are not often seen by students.

In addition, strong algebra skill and a graphing calculator would be useful. Use of Geometer's Sketchpad (or traditional drawing tools) would also make sense here as a fairly accurate construction of the diagram (better than my crude attempt) would be highly instructive and students enjoy 'solving' the problem this way. Of course they need to understand that such a solution is not mathematically valid!

THE PROBLEM
In the diagram below, ABCD is a rectangle with AB = 8 and BC = 6. Rectangle PQRS is inscribed in ABCD, i.e., the vertices of PQRS lie on the sides of ABCD. If PQ = 8, what is the length of QR?

Notes:
(a) Figure not drawn to scale! Drawing this was not fun!
(b) Someone out there will argue that side PQ could coincide with side AB by my definition of inscribed. After all, the diagram is not drawn accurately! I should have added that the rectangles share only those vertices in common!
(c) Students often begin by assuming that PQ is parallel to the diagonal AC. Careless use of similar triangles could lead to an answer of 2. The only problem is that the actual answer is 2.2085 rounded! Does PQ have to be parallel to AC? In fact, is it even possible here? The instructor might begin with assuming parallelism and asking students to see where that leads and if the conclusion makes sense.
(d) What might a mathematician do to extend this numerical problem? Would they consider the issue of a unique solution here, i.e., is the given length of PQ enough to produce only one such inscribed rectangle? What is the range of possible values for PQ (assuming that PQ represents the longer dimension)? Could PQ be 10 or more? Explain. Could PQ be 6 or less? How could we generalize this result further?
(e) As always the disclaimer: My results need independent verification - I depend on my astute readers to check them and correct any careless errors. You are always my best editors!

Take any number, Add Three, Divide the Result by -1. Now Repeat this! Recursive Sequences and Functions Part I: Grades 7-12

Here is the link to the Carnival of Math Edition X.

The following is the first in a series of investigations in recursive sequences and functions for middle school and secondary students. This apparently advanced topic is accessible to prealgebra students at an introductory level. The first few parts of the investigation below are appropriate for the younger students. The remaining parts require more algebraic facility and reasoning. The problem in the title of this post doesn't begin until more than halfway down the page (after some background is developed). Do not skip the background below since it's referred to frequently in the activity. My personal experience is that this topic is highly engaging to students. Considering the connection between recursion and fractals, this topic is certainly part of most standards-based curricula. The terminology of recursion (recursively-defined sequences, recursive description, recursive function, recurrence relations, etc.) is quite confusing at first. Many confuse these ideas with iteration, a general term for describing repetitive algorithms.
Finally, from a pedagogical point of view, please note how the Rule of Four is implemented in the activity below: We start with a verbal description of the rule of formation of a sequence (in natural language), followed by a concrete numerical representation of the terms, followed by symbolic representation. One could also depict the terms graphically on a number line or in the coordinate plane if the function model is used for the sequence.
----------------------------------------------------------
I should probably save this for the new school year but it's hard for me to suppress ideas when they begin to crystallize. I've been thinking for some time about how we can introduce recursive functions in prealgebra through advanced algebra and beyond. I enjoy taking sophisticated ideas and reducing them to basic principles, then developing lessons that explore the topic in some depth. Moreover, this particular topic reveals the interconnectedness of mathematics in a particularly elegant and beautiful way.

Background (Needed for the Investigation Below!)
Consider the sequence 1,2,4,8,...
Elementary students can generally guess the most likely value for the next term, 16. They also are expected to identify the 'rule' of forming the 'next' term, namely doubling or multiplying by 2. This is an important stage in their development of algebraic reasoning - abstraction or generalization. In addition, they should begin to recognize that the terms of the sequence can be described generally as powers of 2, even though a formal introduction to exponents normally begins in 7th grade.
Middle school students should progress to the function table format of a sequence:
n.....a_n
1.....1
2....2
3....4
4....8
5....16
...

Elementary and middle school students should be able to verbalize in natural language that 'you double the terms'. As educators, we need to lead them to a more formal relationship by a line of Socratic questioning like: "Double what? To get what?" Students should then be able to express the idea that each term is twice the previous term. We can ask, "Which term doesn't follow that rule?"
To symbolically describe this sequence, we can write:
a₁ = 1
a_n+1 = 2 ⋅ a_n, n = 1,2,3,...
This is known as a recursive description of the sequence. Try it - replace n by 1,2, and 3 and see if it produces the terms above.
[Note: Later on, in more advanced algebra, students should be able to express this as a recursive function: f(1) = 1; f(n+1) = 2f(n), n = 1,2,3,...]

The closed or general form requires a knowledge of exponents but is accessible to 7th graders
a_n = 2^n-1, n = 1,2,3,... Try it!
(If you're questioning my sanity (you wouldn't be the first!) about introducing such sophisticated mathematics to general 7th graders, well, I do have a legitimate basis for this curricular decision - more later...).

Powers lend themselves naturally to a recursive description and this is why I begin with the above example. Recursive thinking develops when we ask questions like:
If we know what 2⁵ is, how would we obtain 2⁶?
To deepen this understanding further:
If we know what 2⁹⁸ is, how would we obtain 2¹⁰⁰?
Does the exponent key on a calculator help students see these relationships? Not really! The calculator is useful to demonstrate powers and exponents but not for this discussion. Later on, the graphing calculator can be used to enter recursively-defined functions (after they've learned the ideas!).

If you're very familiar with recursively defined sequences and functions, you've probably left this page already! However, the idea of an operation or function being defined in terms of itself is a beautiful and very important notion in mathematics. This type of thinking was necessary for Mandelbrot to develop the notion of fractals, which defines a process in which each stage is defined in terms of the preceding stage or stages - that is recursive thinking!

Ok, by now you're wondering what happened to the title of this blog!
THE PROBLEM:
TAKE ANY NUMBER, ADD THREE, DIVIDE (OR MULTIPLY) THE RESULT BY -1. NOW REPEAT THIS SEQUENCE OF OPERATIONS ON THE RESULT YOU OBTAINED.

Student Activity:
1. Start with the number 6 and follow the instructions above. Repeat this 2 more times. List the first 4 terms of the sequence obtained. Write a brief description of what you observe about this sequence.
2. This time start with a different integer. Again, list the first 4 terms of the sequence obtained and your observations.
3. By now you've concluded that the sequence obtained will alternate in the form a,b,a,b,...
Which one of the original operations (add 3, multiply result by -1, etc.) do you believe is causing the sequence to repeat like this?

The remaining parts require algebra background.

4. If the first term is x, verify algebraically that the sequence will alternate.
5. You've now determined that the sequence appears to be repeating but not constant like a,a,a,a,... For what value of x, the first term, will the sequence be constant, i.e., all terms will have the same value?
6. Write a recursive definition (refer to how we did this for powers) for our sequence whose first term is x. We'll start you off:
a₁ = ____
a_n = __________, n = _______
7. BACKGROUND
The algebraic formulation of the recursive description in #6 was fairly straightforward, since it is just a symbolic representation of the verbal "Take any number, add three, then divide the result by -1." As useful as this may be, we often a need a general formula for the nth term as a function of n, rather than in terms of preceding terms of the sequence. That last sentence was fairly complex, so here's an illustration:
Let's assume the first term is 6. Then the nth term can be described as:
a_n = 6 if n = 1,3,5,7,... (i.e., n is odd)
a_n = -9 if n = 2,4,6,... (i.e., n is even).
This would allow us to find any particular term, say the 100th term, without knowing the values of preceding terms. Such a description is known as a general description or the closed form of the sequence. Such a formula is often very hard to determine, whereas the recursive form is easier to formulate. In our problem, the general formula for the nth term had to be given in two cases or piecewise as mathematicians term it.
It is possible to give a single formula for all of the terms of our sequence as a function of n:
a_n = -7.5(-1)ⁿ - 1.5, n = 1,2,3,...
Verify this formula for our sequence above: 6,-9,6,-9,6,-9,...
8. Change the original problem to:
Take any number, add 2 to it and multiply the result by -1. Repeat.
(a) Starting with an original value of 6 (as the first term), list the first 5 terms of the sequence.
(b) Write a recursive description for this sequence.
(c) Write a piecewise formula for the nth term as in the background example above.
(d) Write a single formula (closed form) for the nth term in terms of n for this sequence.

9. (More Challenging)
A sequence is defined verbally by:
Take any number, add k to it and multiply the result by -1.
(a) If the first term is x, write a recursive description for this sequence.
(b) Write a piecewise formula for the nth term.
(c) (Super Challenge) Write a single formula (closed form) for the nth term in terms of n for this sequence.

A Statement of Personal Philosophy re Math Wars, etc...

Although I have long espoused national math standards (as is well-documented on this blog), I have steadfastly refused to engage in the vitriolic rhetoric of the Math Wars. Despite the fact that I live 7 minutes from Ridgewood, NJ, a community that has recently found itself in the eye of the Math War storm, despite the fact that I was directly involved in developing the Mathematics Content Standards for New Jersey (working with Joe Rosenstein), despite a myriad of reasons for me to comment on this latest conflagration, I choose not to. This blog is all about MATHEMATICS CONTENT and PEDAGOGY. Mathematics is essentially pure and I will not taint it with political issues. If you want to read controversial statements about what we should teach and how to teach it, there are dozens of blogs and forums you can visit. You will not find it here. If you want meaningful discussions of real mathematics that transcend the moment in which we live, keep coming back. I hope to not disappoint you. I might not get the volume of readers by this decision, but my conscience will be clear...

Stay tuned for an in-depth treatment of recursive functions for grades 7-14. I hope you will enjoy Part I.

Carnival of Math: Tenth Edition!

Welcome to the Tenth Carnival of Mathematics

After reading other Carnivals of Math and hosting this one, I'm wondering if anyone feels there might be a need to have two Carnivals of Mathematics:
A Carnival of Mathematics for middle and secondary grades and
A Carnival of Advanced Math for undergraduate and graduate level as well as for the research mathematicians out there.
Another approach might be to have both a Carnival of Math Education (a category into which my blog might naturally fall) and A Carnival of Mathematics.
I may be way off here and outvoted by the vast majority of readers who may simply prefer to pick and choose the posts that interest them, but there does seem to be a clear demarcation between these categories (in the non-algebraic sense of course!). The Carnival may need to reach a critical mass before this would be practical but I'd be interested in your comments here.

With the above in mind, I will begin with math blogs that focus on middle and secondary grades...

jd2718 has a wonderful variation on the Four Fours Puzzle. He has an uncanny knack for taking a good problem and adding enough complexity to it to boggle the mind! This problem is still open-ended and waiting for more ideas...

Alane over at Math Notes demonstrates divisibility tests for 7 and 11 and provides easily understood explanations for why these rules work. Her other post introduces students to perhaps their first mathematical proof, the classic "irrationality of √2" by contradiction. To assess their understanding of indirect proof one might modify the problem to "Show that √3 is irrational."

In Patty Paper Trisection, Denise, at Let's Play Math, challenges her readers to prove that Math Trek's origami trisection referred to in Carnival #9 really works. Denise has an engaging writing style that invites her readers to challenge themselves. She sees math problems as puzzles, a view shared by many who have a passion for our subject. This post is designed for students and teachers in grades 7-12 as well as homeschoolers.

Murray Bourne over at squarecirclez offers us a practical application of semi-log graph paper in plotting the dramatic increase in the ranking of You Tube in just a year and a half. The vertical scale is equally spaced, marked in powers of 10 -- logs base 10 to the rescue! Students will eat this up! He also is promoting a fascinating change in standard math notation (thanks, Murray, for promoting the name of my blog!)

Mark D from the Universe of Discourse shares a recurrence form for binomial coefficients that is far more efficient than the traditional factorial definition. He suggests that this ancient relation (published nearly 1300 years ago) has not gotten the recognition it deserves. Since the last student project in my BC Calculus class focused on efficient formulas for approximating pi (Ramanujan's formula in particular), your post fit right into the discussion.

Vlorbik on Math Ed has a fascinating post on Textbooks and Notations.
He contends, and I concur, that current texts over-stress natural language (as in spelling out the meanings of symbols in English) for set-theoretic formulas, conditional probability in particular. My philosophy has always been to introduce concepts and formulas in colloquial language to which students can relate, then move on to the formal symbolism of mathematics as early as possible. Students need to appreciate the efficiency of symbolic notation and how it provides a universal language for mathematical discourse, not subject to interpretation! Once again, great justification for the name of my blog! I knew there was a method to my madness (aka, dumb luck!).

On the technical research side of mathematics we have a couple for you to digest...

The Unapologetic Mathematician writes about the importance of category theory for undergrad math majors. Categories have become significant in contemporary mathematics. For background, read the Wikipedia article on category theory.

Michi at Michi's Blog presents a technical piece in the area of homological algebra (If only I could recall anything Professor Dyer was trying to teach me in algebraic topology 40 years ago!). The posting deals with combinatorics and coding of a very important tool for his current research - looking at extended algebraic structures in group cohomology.

Michi also recommended Terry Tao's blog. I particularly enjoyed his Advice on Mathematical Careers.

And now for something completely different...

A monthly feature, Who's Counting?, on ABC News.com is authored by the internationally recognized mathematician and author John Allen Paulos. His specialties are statistics and logic but he is also well-known for the popularizations of mathematics he has written (Innumeracy, etc.). He is a Professor of Math at Temple University and he knows how to make math interesting and meaningful. Read through some of his articles from the past 2 years. There's considerable food for thought in these articles and enough material there for projects for Statistics/AP Stat classes for every month of the school year! Not to mention that it makes for fascinating reading. He is a gifted writer who weaves a beautiful web.

I want to personally thank all of our contributors who were considerate about replying by June 13th! Further, those who responded to my gmail account provided some fascinating insights about their passion for mathematics. I felt right at home...

There are so many outstanding math bloggers out there. I can never do justice to all of them. This Carnival is just the tip of the iceberg. One that I've recently discovered is Mathematics Weblog. The author has concisely summarized all of the Carnivals to date and his discussion of math humor is worth the read (he reviews books like Comic Sections and Mathematics Made Difficult). If the books are as humorous as their titles, they're worth looking at!

I haven't yet mentioned any of my recent postings on this Carnival as I wanted to celebrate others' blogs, not mine. However, I'm working on a way to introduce and develop recursive functions and linear recurrence relations for grades 7-12. It will be entitled --
"Take any number, Add Three, Divide the Result by -1. Now Repeat this!" I hope you will look for it and share your comments as we approach our summer break (for me, a more permanent break!). Also, all of those 'beautiful' mortgage formulas I've been alluding to in the series of posts on applications of exponential functions are now displayed as screenshots from the TI-84. Those posts have received many visits and I'm not sure if it's more for the math or more for mortgage advice (believe me, you don't want advice from me on that!).

Update: Submitted late but I decided to add it on 6-15:
An interesting brain teaser for the frontal lobes on SharpBrains. To solve it you need to analyze balance-scale relationships among 3 quantities (spades, clubs and diamonds). Some might try this mentally using logic, others may want to set up algebraic equations. Have fun!

Would you believe, another couple of late additions that I discovered in my web travels--
Best of the Web - Math Blogs
(Of course my blog didn't make the cut!!)

Not Even Wrong - A Random Collection of Stuff (a nice summary of some technical math blogs from a Columbia math professor I believe)
There's no end to this so I had better stop...


Stay tuned for our next Carnival on June 29th over at Grey Matters.

Carnival of Math Edition X Coming on June 15th

Reading Carnival of Math #9 over at jd2718 has been a uniquely lexicographical experience! I am not that creative! For an excellent synopsis of all previous Carnivals of Math, visit the Mathematics Weblog.

Invitations are now open for you to submit 2 or 3 of your favorite posts from the past few weeks either by:
(1) Emailing me at dmarain at gmail dotcom (spelled out to discourage those mean little spam robots)
(2) Filling out the blog carnival form

Those of you who have had the opportunity to visit this blog know that I am an educator (soon to be retired) who is deeply committed to developing conceptual understanding of mathematics in our K-14 students (I've been in classrooms at all these levels in my career) and challenging our students to delve beneath the surface of this beautiful subject. While I invite technical math bloggers to contribute, I particularly welcome submissions from like-minded educators who are trying to shine a light on the vast darkness of education. If only Escalante had a blog...

If emailing pls include the exact phrase Carnival of Math Submission in the subject line.
Also, to do justice to each of your posts I would greatly appreciate some personal background and a description of each submission including
(a) a brief overview of the mathematical content, particularly if highly technical or advanced
(b) your general target audience (e.g., "This posting was designed for ...")
(c) Your name (optional of course, but it adds a personal touch when I refer to you as Evelyn over at...)
(d) From where did your passion for math originate? (Mine came from my dad and a book by Ivan Niven that my mom bought for me as a teenager). Also, what part of mathematics do you enjoy teaching or pursuing the most. (For me, it's number theory and developing problem-solving strategies and enrichment activities in math).

PLS TRY TO SUBMIT YOUR LINKS BY WED JUNE 13TH. As those in education know, at this time of the school year, there is a time crunch. I'll do my best to include all of your submissions but I will definitely include all who meet the above deadline. Also, feel free to provide links to other of your favorite math blogs (in the email). Many excellent math bloggers may not even be aware of the Carnivals. This will also help me if there is a low number of submissions at this time of the year.

Mortgages - Third installment

[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 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)