Updates Week of 1-14-08

Lots of loose ends...

(0) CARNIVAL OF MATH XXIV AT ARS MATHEMATICA
Nice links including a couple from MathNotations.

(1) The math contest problem from the other day now has one possible solution posted - you need to scroll down quite a ways. There are some interesting comments to look at also, but my solution is not in the comments section. Check out tc's and Joshua's approaches.

(2) A SPECIAL THANK YOU TO SOL at wildabout math.
I am very grateful to Sol for highlighting MathNotations.
Sol has an excellent site that reveals his passion for mathematics and sharing it with students and adults alike. This is a must for anyone who loves math!

(3) A SPECIAL THANK YOU TO DENISE at Let's play math!
Denise selected MathNotations as one of her favorite blogs in her Math Bloggers Hall of Fame post from a few weeks ago. Denise knows she will always be one of my favorites as well!

(4) MATH CASTS
I'm awaiting the Mimio technology I mentioned a few weeks ago. When I receive it, I plan on learning how to use it to produce some rudimentary videocasts or 'math casts.' I've seen a few high quality ones online and I feel like this is a direction that makes sense for me. Surely this makes more sense than for me to stand in front of a digital camcorder and pretend to be teaching a lesson! This has recently been discussed over at Sol's excellent site, wildaboutmath. He has already developed some nice math videos I recommend you watch. This is a promising direction for the future that is currently in its infancy. Another excellent example of high quality math videos can be found at isallaboutmath. I've contacted the author of these outstanding videos that present math lessons in Flash, all with a touch of history.

(5) If you have a youngster who is struggling to learn the times tables or wants to get ahead, I strongly recommend you test drive Timez Attack, which I recently reviewed. I don't get anything from this other than the satisfaction of knowing I might be helping some child or frustrated parent out there. I believe that this is an exceptional product, particularly for the child with learning disabilities. Parents, math supervisors, educators, and directors of special services and other child study team members may want to check this out.

Leonardo Pisano -- The Greatest Mathematician of his time? No 'Fibbin'!

I didn't announce this week's Mystery Math Contest to see how that would play out. To be fair to those 200 or so subscribers to MathNotations who get a feed redirected through Feedburner, I guess I should mention it from now on, since the feed only shows new posts, not changes in the sidebar.

Anyway, we have two winners this week who correctly identified the one and only Fibonacci. BTW, this extraordinary gentleman had more names than George Foreman's sons! TC below mentioned one of his other names...

After visiting several websites, I found that he had devised some interesting word problems that seem to have survived to this day in one form or another. So now we know the rest of the story!

The following is from his most famous work, Liber abaci:

(1) A spider climbs so many feet up a wall each day and slips back a fixed number each night, how many days does it take him to climb the wall.
(2) A hound whose speed increases arithmetically chases a hare whose speed also increases arithmetically, how far do they travel before the hound catches the hare.
(3) Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given.


Here are our winners...

Mathmom!
Here's the info she supplied (the math symbols went a bit awry when I copied this from her email):

The mystery mathematician is Fibonacci. One interesting fact is that his real name was Leonardo Pisano.

Liber quadratorum, written in 1225, is Fibonacci's most impressive piece of work, although not the work for which he is most famous. The book's name means the "Book of Squares" and it is a number theory book which, among other things, examines methods to find Pythogorean triples. Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n² + (2n+1) = (n+1)². Fibonacci also proves many interesting number theory results such as: there is no x, y such that x² + y² and x² - y² are both squares, and x⁴ - y⁴ = z² has no non-trivial integral solutions.

He also defined the concept of a congruum, a number of the form ab(a + b)(a - b), if a + b is even, and 4 times this if a + b is odd. Fibonacci proved that a congruum must be divisible by 24 and he also showed that for x, c such that x² + c and x² - c are both squares, then c is a congruum. He also proved that a square cannot be a congruum. The Liber quadratorum alone ranks Fibonacci as the major contributor to number theory between Diophantus and the seventeenth century French mathematician Pierre de Fermat.

And our 2nd winner is
TC
who offered:

That's Leonardo Bigollo of Pisa.

This description of a famous sequence popularized by another famous Leonardo's code is interesting:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

It is almost a rite of passage for most math students to discover the closed form expression to this (featuring the Golden ratio, of course).

Timez Attack - A Review and a View from an 11-year old

After doing my usual random search for web sites that might help my son learn his multiplication facts, I came across Timez Attack from Big Brainz. The company allows the user to download a free version which I did and I was immediately impressed. My 11-year old has struggled to retain his facts and learning software rarely engages him for more than a few minutes. However, after discovering how to navigate the game, he stayed with it for almost a half hour. Since attentional issues make it hard for him to sustain focus, I considered this promising. In addition to observing how he played the game, I tried it myself for awhile.

Having seen dozens of computer learning software products both online and on CD-ROM, I've reached the point where I can rapidly identify quality and Timez Attack is quality in every sense. I decided to email Ben Harrison, the president of the company, and share my thoughts about his product, the only item his company sells at this point. I explained that I would like to write an objective review of Timez Attack for MathNotations after I had an opportunity to try it myself and to observe a child using it. I asked him if he would be willing to give me access to the full version and he agreed. I haven't had time to thoroughly play around with this new version, but my son has been on it for about a week and he likes it. To derive the maximum benefit of this software, I feel I need to establish consistency for him to be on it for at least 15-30 minutes a day for about a month but that has not yet happened. I already see better retention but he is still only on a beginner level.

I strongly urge you to go to the Big Brainz website. There you will see an audio and video presentation of the software. From this home page you can download the free version or purchase the full version for $39.99.

PROS:

(1) Full video game format replete with creatures, a benevolent monster and exceptional graphics that maintain the child's interest. This is the only piece of educational software that I have seen that combines the professional graphics you expect from a video game with solid educational principles. Ben and his crew clearly put a great deal of thought and planning into this.

(2) The free version is outstanding. It is rare to see something of this quality given away. My son was engaged by the free version, although the full version has additional features that will sustain interest over a longer period of time. I may be incredibly naive, but I see this as more than a marketing ploy. To me it says something about the commitment of this company to actually help children learn. The software has already been well-reviewed, but it's possible you haven't heard about it yet. I strongly urge you to download the free version and just try it for yourself or allow your child to play around with it.

(3) The child has to demonstrate mastery of facts before moving on. Thus if working on the 2's, a mistake requires that the child respond correctly before the next fact is presented. Moreover, if, for example, the child corrected 3x2, the program might return again with the same problem or move onto 9x2, then return to 3x2 many times until the child demonstrates they can do the whole set without error and within a time limit. Remember how you used those flash cards - you made two piles and you'd keep going back to the troublesome pile until you got them all. Well, this simulates that and does it better! Throughout, the child gets hints but they must pay careful attention to see it. Further, each time the game is played, the program smartly retains those facts the child struggled with and reinforces frequently. Again, we're talking mastery here!

(4) The program saves the child's information, picking up exactly where the child left off from the previous play.

(5) The child is required to work through a consistent routine of capturing two cute little creatures, revealing the answer, then hurling an object at a wall revealing a picture of the multiplication fact (the child sees two rows of four objects when encountering 4x2). Only when both actions are taken does the friendly ogre allow the child to enter the answer. If incorrect, the ogre gently pushes the player and the child needs to try again (with hints). There is no violence here and the software is perfectly suitable for younger children who may be ready to jump ahead in their learning.

(6) I'm particularly impressed by an ancillary effect of this game. The child must concentrate in order to follow the correct sequence or to to see the answer when it's presented for an instant. I've already seen improvement in my son's ability to focus and attend as a result of this.

(7) Ben and his team seem committed to developing the best product possible, choosing to perfect this game rather than develop other games at this time. He has responded immediately to all of my questions in the several emails I sent. In fact, I shared my son's suggestions about the possibility of other creatures or aliens! He told me that some of this is already in the latest version of the full game and they are continuing to revise the game to make it even more engaging.

My instinct is that this software could be of benefit to all learners but particularly those children who have attentional or other learning issues. I asked Ben about the age range for which the software is intended and he replied:
"Timez Attack is fairly age-independent. It works extremely well for anyone who needs to learn multiplication. The value of the tool increases in cases where there the student is a teen, adult, or has ADHD, Autism, or other learning challenges. The value increases not so much because it works better but because it still works just as effectively where all other solutions have completely failed. It’s really fun when you see that happening."

It is important for me to say that this does not replace classroom learning -- it enhances it. The software could be used at home or in the classroom. Because it is not web-based, a school or classroom would have to purchase a site license, then have it networked. Several purchasing options are offered. However, the company allows the school to download the free version and network it. To quote Ben:
"As for schools, the free version is meant for them as well. They can install it on all their computers—and preferably on their server so that they can access all the built-in reporting/tracking tools."

You may also want to read the review on the website Super Smart Games.
In particular, you can read the interview they did with Ben Harrison - well worth reading. Also you can see videos of the game in action but nothing replaces downloading the game and playing it!

I can only add that I would love to see this same quality and commitment to excellence brought to many other educational software products. It's unfortunate but products such as Timez Attack are generally not seen as cost effective by most software publishers. I'm looking forward to Big Brainz developing other learning tools as well as continuing to perfect this one.

An Exceptional Math Contest Problem

Update: The solution I emailed to the director of the Math League is at the bottom after you scroll down a ways. Note how this ties into the difference of squares problem I posted the other day!

A very good friend of mine who runs one of the most successful math league contests in the US has given me permission to reprint the question below with the following attribution:
“Copyright Mathematics Leagues Inc 2007. May not be reproduced without permission of the copyright holder.”

This question was the last question on a recent contest. It certainly made me think! I felt the need to be rigorous about justifying one of the steps and that was the reason for the
M² - N² = 12 post from the other day. There are at least 3 methods that have been found thus far, according to my friend. Teachers often send in their own solutions and some even get included with the official solutions. My feeling is that questions like these are vital for our students intending to pursue higher mathematics, not just for those who happen to participate in contests. I also believe that variations on great questions like this will have an eternal life...

THE QUESTION

There's exactly one real number a for which ax² + (a+3)x + (a-3) = 0 has two positive integer solutions for x. What are these values of x?

Have fun with this!
For more information about the Math League, visit here.

Scroll down to see the solution I sent to the directors of the Math League...











First of all, I'll state the answer I obtained before giving the details:

A real number a which produces two positive integer solutions is a = -3/7 which does lead to the solutions x = 4 and x = 2. Proving that this is the only real value of a is far more challenging and I suspect that there's a much easier solution than the one I found. Anyway, here's my approach:

In order to simplify the original equation, divide both sides by a to make the leading coefficient equal to 1:

x^2 + (1+(3/a))x + (1 - (3/a)) = 0
To simplify further, let b = 3/a:
x^2 + (1+b)x + (1-b) = 0
From a well-known rule about the roots of a quadratic equation, the sum of the roots in this case equals -1-b and the product of the roots is 1-b. Since the roots must be positive integers, it follows that b is an integer less than -1. Remember for later:
b must be negative!

From the quadratic formula, we obtain the roots to be:
x = ((-1-b) +- SQRT(b^2+6b-3))/2.
In order for the roots to be integers, the expression b^2+6b-3 must be a perfect square.
I will rewrite this expression by completing the square:
(b+3)^2 - 12 and this must equal a perfect square , call it N^2.
Let b+3 = M, then we must have M^2 - 12 = N^2 or
M^2 - N^2 = 12. It's easy to guess a difference of perfect squares equal to 12, namely 16-4, but I will now prove this is the only solution:

Factoring, we have (M+N)(M-N) = 12 where M,N are integers.
The only pairs of factors to consider are: 12,1; 6,2; 4,3 as well as their negatives.
In order for M,N to be integers, the factors of 12 must BOTH be EVEN so the only possibilities are 6 and 2 or -6 and -2..
If M+N = 6, M-N = 2, then, by adding, M=4 and N=2. From b+3 = M, we would have b = 1, but this doesn't work since b must be negative (see above!)
Therefore M+N = -6 and M-N = -2. From this we have M=-4 and N=-2.
Then b = M-3 = -4 - 3 = -7.
b =-7 leads to the quadratic equation: x^2 + (1+(-7))x = (1-(-7)) = 0 or
x^2 - 6x + 8 = 0. Factoring we obtain the roots to be x = 4 and x = 2.

Since b = 3/a, it follows that -7 = 3/a or a = -3/7.

A Visualization for Factoring the Difference of Two Squares

M² - N² = (M+N)(M-N)

Given that my difference of squares problem from yesterday may have been overly ambitious for middle schoolers (and I will have more to say about that in the comments section from that post), I thought it was worth reviewing a diagram that many of you have probably seen before. There are many similar geometric representations of standard factoring and distributive formulas in algebra, but this one has always been one of my favorites. It would be more effective if I had been able to shade rectangles R and S using different colors, but I did the best I could on short notice.
It's often a good exercise for algebra students to invent similar diagrams for other formulas, although the use of manipulatives such as algebra tiles can be even more effective.

M^2 - N^2 = 12...Prove there is only one solution in positive integers and much more

Update:
(1) See the visualization for the difference of squares posted on 1-9-08.
(2) Read the comments in this post for considerable clarification and instructor guidelines and suggestions. Mathmom's and Eric's comments are particularly insightful.

This post can be developed into an activity for prealgebra through first-year algebra students (or even 2nd year algebra). The last part is more challenging.
The focus here is on developing a method/strategy that can be used to solve similar Diophantine equations. The other objective is to introduce the ideas and methods of proof. This problem may later be used to solve a recent math contest problem for which I obtained permission to discuss on this blog. I am fully aware that many students will 'solve' these equations by Guess-Test methods, but they need to go further.

STUDENT/READER PROBLEMS/ACTIVITIES

(a) Prove there is only one solution in positive integers for the equation:

M² - N² = 12

Note: If we omit the word positive, what would the solution(s) be?

(b) Determine all positive integer solutions:

M² - N² = 15

(c) Determine all positive integer solutions:

M² - N² = 36

(d) Let's investigate for what positive integer values of P, M² - N² = P has NO solutions in positive integers.

(i) Determine at least 5 positive integer values of P for which the above equation has no positive integer solutions.

(ii) (More challenging) Describe all values of P for which the above equation has no solutions. Justify your result.

Note: All students should have success with (i), although some may struggle to find 5 values. Part(ii) should challenge the student who has finished the other parts in rapid order and sits there complacently!

Additional Comment: If P is itself a perfect square, our equation is obviously related to the most famous equation in geometry. Thus, if P = 9 or P = 16, for example, students should recognize something! For this reason you may want to have students consider these values when doing this investigation. More to come...

Mystery Math Idol Week of 1-1-08 Revealed and a Proof!

Believe it or not, there were actually two winners this week. This probably translates to a total of two individuals in the math blogosphere who actually enjoy this feature! I will continue this feature until we run out of mathematical icons, however, by indirect proof, one can prove that the list is infinite:

Assume there are only finitely many mathematical geniuses to choose from:
M1, M2,M3,...Mk.
Since such mathematicians 'multiply' like the rest of our species, there will always be at least one other offspring from this list who is different from anyone on the list. This contradiction proves that the Name That Mathematician Feature will be around for awhile!

From looking at the picture in the sidebar, you now know that our mystery man is none other than (drum roll please...)

Ferdinand Eisenstein

Winner #1 (and still overall champion): LYNX

Here is her contribution:

Ferdinand Eisenstein was the only one of 6 children to survive. The rest died of meningitis. During school (aged 11-13) he solved 100 proofs during the time most students were expected to solve 11 or 12. He died of tuberculosis at the age of 29. (It seems many mathematicians have their lives cut short. What might have been discovered sooner if they had lived?) Gauss would later say the three top mathematicians were Archimedes, Newton, and Eisenstein. (Quite a compliment if I say so myself.) [http://fermatslasttheorem.blogspot.com/2005/07/ferdinand-gotthold-max-eisenstein.html]

In his autobiography, he writes, "As a boy of six I could understand the proof of a mathematical theorem more readily than that meat had to be cut with one's knife, not one's fork. "


Co-Winner: VLORBIK

that's eisenstein, of course

(the only one so far i knew

right away just by looking ...).

fun fact? well, according to

http://en.wikipedia.org/wiki/Ferdinand_Eisenstein

gauss rated him with newton & archimedes.

'Feed Me, See More' MathNotations in '08!

Puns are bad way to kick off the New Year, but, given the RSS problems I've been having with my posts feed I had to do something to turn it around! Well, my first New Year's resolution was to fix the problem and I believe I have. If you had not been receiving my feed for some time, it should be ok now. Of course, many of you have re-subscribed using the icons in the sidebar.

Some updates on 1-1-08:
(1) Please share your thoughts about your favorite math teacher. Perhaps the comments section of this post from the other day is not an appropriate forum but that's all I had available at this point. Someone probably has already created such a website!
(2) The Bingo post is getting some attention even during holiday time. I will be expanding on that but I await your thoughts.
(3) Did you notice the new Mystery Math Idol to kick off '08? This should prove difficult as he is not a household name for most, but his genius is still being recognized by algebraists and number theorists (hint, hint, wink, wink).
(4) The State of the Blog address -- I'd be interested in your thoughts about what you enjoy best about MathNotations. What changes would you like to see? What to keep? I've gone off in dozens of new directions but the meat and potatoes of this blog is still discussions of problems, concepts and instructional strategies.
(5) I'll be introducing Mimio technology shortly, perhaps some Mathematica Demonstrations, and reviewing an exceptional piece of software for helping students master their Times Tables. More videos too! Coming soon!

Have a healthy and happy!