Totally Clueless Challenge #2 - By All Means!

It's been awhile but something this good is always worth waiting for!
TC has sent me some fascinating challenge problems for our readers. If you are now sick of watching amateur videos on the Arithmetic and Geometric Mean Inequality, it's time to raise the bar. The following involves a well-known generalization of these means but the results are worth your efforts, particularly parts (c) and (d) below.

If a and b are positive, we can define their generalized mean to be:

GNM = ((a^k + b^k)/2)^(1/k)

This would look far prettier in LaTeX but I'm hoping it's readable. In words, we're looking at:
The kth root of the arithmetic mean of the kth powers of a and b.

(a) What is another name for the result when k = 1? (we're starting off easy here!)
(b) What is another name for the result when k = -1? (slightly harder algebraically)
(c) Ok, now for the real challenge for you Calculus lovers:
What is the limit of GNM as k-->0? The result is totally cool!
(d) TC's Super Bonus: Show that the limit of GNM as k-->∞ is the maximum of a and b.

Note: These have been slightly edited from tc's original problems, but they are essentially the same. Solutions may be posted in a couple of days although the notations will be hard to render. I might just have to do another video or wait for that special technology I mentioned earlier! We're hoping some of you will tackle the harder ones and comment!

The Kite Problem Revisited -A View From the 'Exterior'

A post from 7-17-07 generated some wonderful comments and solutions. I decided to bring it back using a slightly different diagram. This time I'm asking for a solution path using exterior angles, a tool students often overlook. The most efficient approach may still be the one tc used, involving the sum of the interior angles of a quadrilateral, but the challenge here is to find another way...





Heres' the problem:

Assume Q, S and T are collinear. Determine the value of a+b+c.

Note: Again, there are many wonderful approaches here. Try to use the suggested one...

Mystery Math Idol #2 Revealed - 'There's Something About Mary'...

Our winners:
tc
mathmom

Mary Everest Boole was a remarkable woman. Her uncle, Colonel Sir George Everest, to whom she was very close, was the Surveyor General of India. He was largely responsible for completion of the trigonometric survey of India along the meridan arc from the south of India extending north to Nepal. The completion of the Indian survey allowed the subsequent survey of Mt. Everest (at the time un-named) and calculation of its summit height. It was later renamed in honor of George Everest.

Through her uncle, Mary met George Boole, an already famous mathematician. Mary enjoyed her time with Boole both socially and intellectually and they soon married. Even though Mary was 17 years younger than George, they were still very close companions and had a very successful marriage. During the next nine years, Mary and George had five daughters. Yet, this happiness would not last for long. Tragically, George caught pneumonia and died leaving Mary alone with her youngest child only six months old.

Mary Everest Boole was a miraculous woman who, widowed for fifty years, raised her five daughters and made countless contributions towards the mathematical education of many girls and boys. Mary considered herself a mathematical psychologist. Her goal was to try "...to understand how people, and especially children, learned mathematics and science, using the reasoning parts of their minds, their physical bodies, and their unconscious processes."

Mary was recognized in England as being an outstanding teacher. One of Mary's pupils was to write later, "I thought we were being amused not taught. But after I left I found you [Mary] had given us a power. We can think for ourselves, and find out what we want to know." Many of Mary Boole's contributions can be seen in the modern classroom today.

tc found this fascinating anecdote:
Niece of a famous George AND wife of a famous George! (Note the
Boolean operator)! A maven of mathematical pedagogy! - A particularly
apt choice for this blog!

I found the following quote particularly interesting:
"I have had homicidal impulse at the touch of other stimuli. When I
was quite young, I used to speculate on the problem why I did not try
to kill someone who worried me. It was not love of my parents that
hindered me; in those moods I was incapable of fear. It was not regard
for God; I considered that God made me as I was and could not
reasonably be angry with anything I did. It was -- I always came back
to the same conclusion -- it was that I thought that if I killed
anyone the police or the hangman or someone would stop my working for
algebra. Besides I felt that all stormy passions in themselves
interfered between me and algebra. Hate and revengefulness,as well as
love and fear, vanished, like burned paper, when they threatened to
interfere between me and algebra (34)."

- from The Forging of Passion into Power (London: C.W. Daniel Company,
1910), found online at http://www.troubling.info/boole10.html

mathmom found the perfect web site to search (sorry, I'm not sharing that at this time!) and commented:
The only woman I could think of before searching was Ada Lovelace and I didn't think the photo was her, though I'm pretty bad with faces. If I'd found a different photo of Boole, I might not have been sure it was the same woman.

I'll leave Mary's picture up there for a couple of weeks. I will probably do this contest biweekly from now on due to 'overwhelming' response!

Does doubling an integer double the number of factors? A Deeper Investigation for Middle School

NOTE: PLS READ THE COMMENTS FOR A DETAILED DISCUSSION OF THIS PROBLEM, WHICH SHOULD PROVE QUITE CHALLENGING FOR MOST MIDDLER SCHOOLERS.

The previous activity I posted regarding integers that have exactly four factors might lead to some interesting discussion regarding a general description of such numbers. All of these kinds of problems could be handled by simply giving students the general rule for determining the number of factors of any positive integer. I have given this well-known number theoretic formulation in earlier posts, so I won't review that at this time. However, there is a greater benefit to be derived from having students investigate these relationships. The following activity should be adapted to meet the needs of your students.

Some educators react to these kinds of deeper investigations with reactions like:
(a) I have a curriculum to cover. I don't have time for this.
(b) Unless this kind of question appears on state testing, it's simply not practical for me to do this.
(c) My students are just not ready for this kind of thinking.
(d) Dave, you're out of the classroom now, so you're forgetting the realities of most classrooms. Some students don't know their basic facts and you want me to do higher-order thinking! Gee, Dave, are you forgetting we have classified children mainstreamed in our classes? Get real!
(e) Dave, stop suggesting HOW we should teach and just give us the problem. You're trying to impose your style on others - it doesn't work - we each bring our own style to a lesson.

[Comment: I have strong reactions to some of the above, but then I'd be arguing with myself! I'll respond to some of these in the comments section or devote an entire post to these critical issues if my readers decide to respond to this.]

I'm certainly not suggesting that these kinds of explorations should BE the curriculum. There must be a balance between these problem-centered approaches and skills development. I am suggesting there needs to be some time devoted to deeper cognitive processes to foster mathematical development. The following investigation is far from one inch deep! I may continue it later but I'm hoping some will suggest extensions, make comments or report back how it played out in real classrooms (also how it was adapted/revised).


INVESTIGATION/READER CHALLENGE

Part I
1. The number 6 has 4 factors: 1,2,3,6 (or in paired form: 1,6;2,3).
Suggested Questions:
If we double the number 6, what do you think will happen to the number of factors? Will it increase or stay the same? Will the number of factors also double?
Mathematicians, like scientists, make conjectures or educated guesses, but not wild guesses! We need some evidence or data on which to base our conjectures. With your partner, fill in (and possibly extend) the following table, then formulate your conjecture using correct mathematical language:
Positive Integer.................Number of Factors
6...........................................4
12.........................................____
24........................................____
48.......................................____

Do you think we have enough data to make a conjecture or should we continue the table? Record your observations and then state your conjecture or 'rule'.

Teacher Tip: Depending on the maturity of the group and their experience with these kinds of formulations, you may want to start them off with a prompt:
If we double a positive integer, the number of factors _______________.

Many students will be convinced they have found a mathematical rule that will always work. That's one of the objectives of this investigation: To help them understand that
(a) Pattern recognition does not a rule prove!
(b) The conjecture is based on starting from the number 6. There is no basis for assuming that their 'rule' will be valid if we start from a different positive integer!

Part II
This time have students start from a different integer: 18
Make a table similar to the one above, again doubling the integer in the left column.
Again, record your observations and then state your conjecture or 'rule'.

Suggested Questions:
Do you think there is a more general rule that covers all cases or are there simply different rules for different integers? If you were going to investigate this further, what other kinds of starting positive integers would you try?

Teacher Tip: Asking many questions stimulates student thinking and leads to more questions and deeper thought processes on their part. A free interchange for a couple of minutes is invaluable here to have students come to see that mathematical research requires persistence and an attitude of inquiry. As Ms. Fribble from the Magic School Bus would say: ASK QUESTIONS! (or something like this!).

Part III - Start from an odd integer this time: 15
Part IV: To be continued...

Middle School or SAT Math Activity - The Four Factors Problem

There are countless problems involving the factors of a positive integer we're seeing in middle school classrooms and on standardized tests these days. They are often used as challenges or warm-ups and questions similar to the one(s) below have appeared frequently on this blog. Students become more proficient with this type of question by doing many variations repeatedly over time. As they mature, they will come to appreciate a more general approach to finding the number of factors of any positive integer. Number theory is now included in most states' standards so there needs to be some time devoted to this topic on a regular basis.

STUDENT PROBLEM/READER CHALLENGE

This problem/activity is often best implemented in small groups. Each member of the group should make their own list and then compare, however, they might want to divide the labor by having some students do the numbers up to 50 and others do the rest.
Suggested Time for Activity: 15-20 minutes (the problem can be explored further for homework or a challenge, then revisited the following day for 5 minutes).

The number 12 has 6 positive integer factors: 1,12;2,6;3,4.
(a) List all positive integers up to and including 100 that have exactly four factors.
(b) Higher-order: These numbers fall into 2 categories. Describe these categories.

Alternate Problem (shorter time needed): What is the largest 2-digit positive integer that has exactly 4 factors?

Mystery Math Idol Week of 12-3-07

As much as I revere Archimedes, life is a series of hellos and goodbyes (Billy Joel has a way with words). Again, if you think you know who the famous woman in the sidebar is, email me at "dmarain at g-mail dot com".

Please do not put your answer in the comments section of this post - we do not want to give it away until the contest is over! I will give up to 48 hours depending on how many emails I receive.

Don't forget to include some curious, fascinating or esoteric fact relating to her mathematical career or some compelling personal anecdote. From what I've read of her, the possibilities are many! Please cite your sources (link) for verification.

If I don't get a response within 12 hours, I will post a hint either in the sidebar or in this post.

Hopefully, this time, one cannot right-click, ctrl-click or view source to find her name!
I know those of you with advanced search skills will find her fairly quickly, but, remember, the name is not enough!

Happy Hunting!

The Dissection of the Square Problem - Part III - A Video Surprise

Ok, we've been discussing the square dissection problem for a few days now. The original problem was here and Solution Method I was here.

Now don't start laughing at these crude, low-tech videos! The 1st video is 2-3 minutes long. It provides a quick overview of the squares problem and a demonstration of why the product of the areas is the same for either pair of non-adjacent rectangles. It ends abruptly with part of the derivation I gave in the previous post.

The remaining presentation is split into 3 segments in order to control the file size (Blogger has a problem with larger files). In these segments, I derive a formula for the maximum product using the Arithmetic-Geometric Mean Inequality (AM-GM) and apply it to the square dissection problem. The first of these clips reviews this important and useful inequality, presenting a standard algebraic derivation. The 2nd of these 3 clips presents a geometric derivation and the final clip shows how we can derive a formula, m⁴/16, for the maximum possible product of the areas in the general case of a square of side length m.

As I say on the video, there's no SmartBoard technology, no Mimio, no whiteboard with vivid color dry erase markers! Just an old chalkboard my kids use in the basement, some inexpensive chalk and sheets of paper towel to erase. I wore my favorite 'Pop-Pop' sweatshirt - sorry, no formal wear.

Let me know if you find this helpful (once you stop chuckling!). At least you'll be able to match a name to a face and it isn't Archimedes! If I get few if any comments, I'll know that my 15 minutes of fame have expired.

Solutions and Discussion re Squares problem

Since I only received one comment from the squares problem, I guess readers were either bored by this question or are awaiting my solution(s)!

Solution I: Divide side AB into segments of lengths 2+x, 2-x. Here, x can be any real between -2 and 2. Similarly, divide BC into segments of lengths 2-y and 2+y. One can demonstrate that the order here is irrelevant. Then the product of the areas of either pair of non-adjacent rectangles can be expressed (after rearrangement of factors) as
(2+x)(2-x)(2+y)(2-y) = (4-x²)(4-y²).
From the restrictions on x and y, it follows that x² is greater than or equal to zero and less than 4. Similarly for y.
Therefore, (4-x²)(4-y²) ≤ 4⋅4 or 16.
This also demonstrates that the maximum product occurs when x=0 and y=0! QED

Solution II (using the Arithmetic-Geometric Mean Inequality): We will prove a general result for squares of side m. This will be forthcoming and there may be a visual surprise! Stay tuned!