MLK, Inauguration, Math Contest and A Radical Investigation!

Don't miss registering for MathNotation's First Math Contest. Registration is as simple as emailing me (dmarain "at" "gmail dot com") to request a form and the Rules. The contest is team-based (up to 6 students), is designed for both middle and high school students and should take 45 minutes or less (extra time is provided for students to enter their answers/solutions on the official answer form in Word). Look here for further info.

I would also like to thank the following blogs and/or webmasters for their graciousness in spreading the word about our first math contest:
Let's Play Math!
MathNexus
Wild About Math
Vlorbik
jd2718
Note: Take a look at jd2718 to see the latest Carnival of Mathematics. Another excellent job by Jonathan!
Homeschool Math Blog

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While we're waiting for the Inauguration on 1-20-09 (12,009 = 3 x 4003 of course), today is Dr. King's birthday, 1-19-09 and 11,909 is prime as it should be! How appropriate it is that we should be honoring today the man who paved the way for our new President...
The title of this post reminds me of an old Johnny Carson routine: Which one doesn't belong with the others! In fact, we can probably make connections among all of these if you're willing to play with words...
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In case you thought that the Math Contest would lead to a hiatus in publishing investigations and instructional strategy articles, fear not! Today we will once again examine the raison d'etre of this blog:
TEACHING BOTH PROCEDURALLY AND FOR MEANING


Part I
Consider the equation


To reinforce multiple representations (Rule of Four) we can ask students to:

Explain or show why this equation has no real solutions
(a) Graphically
(b) Numerically (TABLE)
(c) Algebraically


At this point I am including some ScreenShots from the TI-84. The bold graph is Y1:











Part II - The Extension!
Consider the equation



(a) For what value(s) of k will the above equation have one real solution? In this case, also determine an expression for that solution in terms of k. Show method clearly.

(b) For what value(s) of k will the above equation have no real solutions. Show method clearly.

(c) Demonstrate your results in (a) and (b) by choosing specific values of k for each case. Use both a graph and a TABLE to support your argument. [Use of the graphing calculator makes sense here.]

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Pedagogy
Which do you think is more helpful to students -- the graph or the TABLE? From my experience I find that both are important for comprehension and concept. They not only complement each other but each contributes something by itself. The graph not only suggests (not prove!) that the two graphs in part I do not intersect but it leads to a natural questions like: Why is the graph of y = √x + 2 above the the graph of Y1? What do the graphs suggest about the domain of each function? Explain the ERR messages!
Note: I used the word "suggest" because we want our students to understand that graphs do not prove mathematical truth.

When is it appropriate to use this approach? After you've taught the algebraic procedures of solving radical equations? Of course, part (c) of the activity asks for the algebraic explanation, but I've often used the graphical and numerical approach BEFORE teaching the procedure. I believe that it developed meaning for the traditional procedure but, in no way, did it replace the need for carefully explained instruction with a variety of examples! (The "balanced" approach!).

Further, the common reaction I've heard to this kind of instruction is that it is too time-consuming and appropriate only for the honors students. I couldn't disagree more. Developing meaning does take time and is absolutely worth it. It's all part of the "less is more" philosophy and, that, if the foundation is properly put into place, students can develop both the skills of solving radical equations and an understanding of the underlying mathematics. Enough preaching to the choir...


I hope you find this useful when building your next exploration in mathematics! Let me know...

Puff the Magic Mortgage Part II

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 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 $53.78.

(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 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) 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? 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.