Eureka! The Results Are In! The Winners Are...

Ok, the secret's out. Start ordering your
Archimedes and Eureka -- Nature Abhors a Vacuum T-Shirts.

It was a difficult decision, but the results are in (although we still haven''t heard from Palm Beach County). The top 3 submissions based on the log_b²(b⁶) entries in the first ever MathNotations: Name that Mathematician Challenge are -- in no particular order --

Mathmom
Eric Jablow
tc

I came in a distant 4th (I am going to appeal this). Of course, it was only fitting that I chose the 'Arc-Man' to lead off, since one of my personal favorite posts of all time came earlier this year:
The Genius of Archimedes: Parabolas, Tangents, ....
Anyone recall I attempted my first complicated diagram, suggesting how Archimedes proved that light emanating from the focus of a parabolic surface are reflected in parallel rays (and conversely)?

By the way, one of the better explanations for kids of his discovery of the displacement principle can be found here.

Ok, here are the details:
Mathmom found the following fascinating fact here:
Archimedes invented a puzzle called the Loculus (or the Stomachion,
the Ostomachion, the Syntemachion, or Archimedes' Box). It's like a
huge, complex, tangram. In November of 2003, Bill Cutler used a computer program to enumerate all solutions. Barring rotations and reflections, there are 536
distinct solutions.

Eric actually used something called a book (for youngsters out there, here is a link to explain the meaning of this obsolete term) to find the following information about Archimedes' perspective on pure math vs. science:

ARCHIMEDES OF SYRACUSE (c. 287-212 B.C.), son of an astronomer, was
Greece's star mathematician. By avocation he desired the pursuit of
mathematics proper, and he was wholly and passionately committed to
mathematics at its "purist." But by world reputation he was an
engineer, especially in the field of military engines, even if he
protested that he derived no satisfaction from this kind of work. And
when confronted with the problem of determining whether a golden crown
was made of pure gold or was alloyed with silver, he initiated the
method of Hydrostatics for the purpose.

Scientist-professors will always be the same. Archimedes, when
heading the Weapons Research Group for the Syracuse Department of
Defense, would write letters to friends that he was yearning to return
to the Campus and do nothing but pure research for its own sake. But
he was apparently doing classified work to his last breath, literally
so. And when he found his theorem on Hydrostatics he was so excited
that he insisted on talking about it to the man in the street.

I won't actually publish tc's joke regarding one of Archimedes' engineering feats (young children might find their way to this blog), so I will leave it up to my readers to invent their own...

Finally, I promise, from now on, not to the use the name of the mathematician in labeling the image file! Duh...

Just Another Square Problem? A Means to an End...

Before announcing the thousands (or less!) winners of the Name That Mathematician Challenge, I came across a problem about dissecting a square ABCD with lines PQ and RS which are parallel to the sides of the square. (see diagram).

Naturally, I decided to make it into a deeper investigation. Students and/or readers will be asked to find the maximum value of the product of the areas of either pair of non-adjacent rectangles formed. There are many approaches here, one of which uses the famous Arithmetic Mean-Geometric Inequality. As usual you will work from the particular to the general, beginning with a specific value for the sides of ABCD.

STUDENT/READER INVESTIGATION - PART I
The given conditions about the diagram are given above.

For Part I, we will assume each side of the square has length 4.

(1) (Particular) If AP = 3 and RC = 2, determine the product of the areas of APTS and RTQC. Do the same for the other pair of non-adjacent rectangles formed. Do you believe this product is the maximum possible as we vary the positions of segments PQ and RS?

(2) (General) Show that the product of the areas of either pair of non-adjacent rectangles formed is less than or equal to 16. For example the product of the areas of APTS and RTQC is ≤ 16.

Notes:
(1) Do you think many students would guess what the configuration would be for the maximum product to occur? Is proving the conjecture much more difficult?
(2) The challenge here is to find an effective use of variables to denote the segments. There are many possibilities, some much more efficient than others.
(3) I will add additional parts to this challenge after receiving comments on Part I. How would you generalize this result further? More interestingly, there is a way to prove Part I using the AM-GM Inequality?

A New Feature: Name That Mathematician (and more...)

Update²: Only one submission thus far (and a good one!) so I will give our readers another day to email me some fascinating fact about 'A'! Ironically, Isabel, over at God Plays Dice, has a humorous post on attaching names to faces of current mathematicians. What are the odds! As with all experiments, we'll see how this challenge is responded to before making it a regular feature!

Update: Mathmom, reminded me that one can right-click on a PC and see the name of the image file. Working on a MAC, I forgot that. Well, this first picture might be a freebie, but you still need to wow me with some esoteric fact about him! Next time, I'll rename that file!! I also removed the personal info requirement (again, thanks to mathmom's astuteness!).

Starting today, we are introducing a different kind of challenge. Look at the image near the top of the sidebar. Here are the rules (read these carefully before submitting):

DO NOT NAME THE FAMOUS MATHEMATICIAN IN A COMMENT! (If you did already, I will delete it!).
Instead, send me an email at "dmarain at gmail dot com" with the following information:

(a) The name of the person
(b) One fascinating fact about her/him; it doesn't have to be what made that person famous -- I'm looking for the unusual or curious here, be it mathematical or something personal...
(c) Please cite your source for this fact (include the link) - I need to verify its authenticity

I will choose the top responses received within 24-48 hours of the time the picture first appears. Earlier submissions may receive higher ranking than later submissions. I will announce the winners in a couple of days and, of course, include the fascinating facts as well.

Note: I realize that students today have extraordinary online research skills (although one cannot enter an image in Google!), but, remember, the winners are not only determined by giving a name. Feel free to share this with students. They can enter too!

The Right Combination - A Metaphor for Teaching and Learning Mathematics?

If you were looking for a challenge here in higher math using combinations and permutations, sorry to disappoint you! I felt compelled to write this essay after watching my wife patiently attempting to teach one of my children how to open a combination lock. She doesn't think of herself as a teacher, but, she is, and, in many ways, far more skilled than I ever was.

One of the rites of passage for many middle schoolers is mastering the intricacies of the combination lock for their lockers, somewhat akin to elementary schoolers learning how to tie their shoes. Do you remember the frustration you felt the first few times you tried to solve the puzzle of these locks? Do you recall your euphoria when it magically opened? Consider all of the 'skills' involved and think of the parallels to mastering the algorithms of mathematics:

(1) Fine motor skills required to precisely turn the dial and stop at the correct number
(2) Memorizing the 3 numbers in sequence
(3) Understanding the difference between Right and Left when rotating the dial and retaining the R-L-R sequence
(4) The absolute discipline and precision required - close is not good enough
(5) The dreaded second step of the process needed in going 'past zero'
(6) The extreme feelings of frustration from failing repeatedly and the inclination to give up, yet driven to continue
(7) The elation felt in getting it the first time all by yourself, only to be followed by despair when you can't seem to duplicate the feat!
(8) The feeling of accomplishment when you can do it almost every time without anyone helping you
(9) Is there any substitute for independent practice in achieving mastery here?
(10) How important is motivation here in driving the child to continue in the face of adversity?

What about the challenges faced by the 'instructor' here? If you were the one who helped someone succeed, did you find it frustrating or did you have 'unlimited' patience? Did you have to practice it yourself first and think about breaking this 'automatic' process into simple discrete steps? Did you have to try different verbal instructions (for example, using 'down' and 'up' vs. 'left' and 'right') or different techniques of one approach failed? Did repeated demonstrations in front of the child suffice? Did the child say, "Let me do it by myself?" If you've helped several children learn to 'unlock' the combination, did you use the same approach successfully with each child? Are some youngsters simply unable to 'solve the problem' at that time and need to be given a key lock instead as an accommodation? Is making this concession detrimental to their self-esteem and eventual development or is it reasonable at that time? Will some of these youngsters be able to succeed later if given the opportunity to try again (when developmentally ready)?

Is there a metaphor here for teaching children mathematical algorithms? By the way, can you think of others skills or concepts involved in opening the lock that I overlooked? Pls share!

Now, parents, extrapolate this 'teaching' process to dozens of unique math students every day with a myriad of different algorithms over the course of a school year? Anyone can teach, right?

I realize some of you will see the flaws in this metaphor and will point out all the differences between opening the lock and solving a mathematical problem? I know the parallel is far from perfect but this is something that just struck me and I had to put my thoughts down. You know, like a journal, a diary, a blog... Your thoughts?

The Prime Rate: A Post-Thanksgiving Class Opener for Grades 6-12 that Stirs the Brain

Many math educators use warmup problems to review, challenge or set the tone as students walk in the room. Routines like this are effective in having students 'hit the ground running'. These mini-problems can be on the board, while an overhead transparency of selected answers are displayed. The instructor then has time to circulate, check homework, engage students, and get a feel for the difficulties they had with the assignment. By the way, do most of your students take these warmups seriously?

Here is a warmup that requires more active participation on the student's part. I may suggest a different one in a later post, but I'd really like readers to share some of their favorites!

Math Bee
All students stand at their seats. They are told they will be have to give the next number in sequence, according to some rule that will be explained. They will have 3 seconds to respond (can be adjusted but no more than 5). If incorrect or time runs out, they will be instructed to sit down and the next person will have to give the correct answer and so on. You may want to start them off by giving them the first number (judgment call here). I suggest you allow a maximum of 5 minutes for this activity.

Here's the problem I gave to a group of high schoolers but it is highly suitable for middle schoolers as well:

Using positive integers, think of primes ending in 1 or 3 (you may want to use the technically precise phrase 'whose units digit is 1 or 3'). For example, 21 'ends in' 1, but it is not prime. You must go in order and you are not allowed to ask the number the previous person gave! You will have 3 seconds to respond. If incorrect or time runs out, I will ask you to sit down and the next person will need to give the correct number. We will continue until there is only one star shining or time runs out and we have co-champions!

Learning Objectives:
(a) Reviews primes (How many of your students do you think would be eliminated early by starting with 1? You can always start them off with 3 if you feel this will help. Some students will begin with 11, assuming that you meant 2-digit numbers. By the way, 51 and 91, in particular, typically knock out many, if they get that far! Finally, are you thinking this question is not particularly relevant for high schoolers? Count how many questions relate to primes on the SATs!)
(b) Improves listening skills and concentration (How many of your students do you think will forget the last number given either because their minds are wandering or from trying to think ahead to their turn?)
(c) Learning how to think under pressure. (Although we know some students will 'freeze up' or be embarrassed if they are eliminated, they will not be alone! Typically, about half of the students in an above-average class will be sitting down on the first pass through the class! With a high-achieving class of very strong students, you may need to make several passes to reach a winner. If there are a couple left after 3-4 minutes, proclaim co-champions.

Let me know how this goes if you try it after the Thanksgiving break. What variations would you use to make this more effective for your students? What other kinds of problems are suitable for this 'Bee?' Were my predicted statistics way off? Did you predict that students would get past 100?
Again, please share some of your favorite warmups!

A Thanksgiving Thought...

[Reprinted from Jan 3rd, 2007]

I'd like to share a sentiment that was emailed to me by a close friend. I decided to re-publish this on the eve of Thanksgiving. You may have already seen this, but it does send a message about giving thanks for what we have. (Thank you, Elisa, for sharing this):



One day a father of a very wealthy family took his son on a trip to a farm with the firm purpose of showing his son how poor people live. They spent a couple of days and nights on the farm of what would be considered a very poor family.

On their return from their trip, the father asked his son, "How was the
trip?" "It was great, Dad." "Did you see how poor people live?" the father asked. "Oh yeah," said the son. "So, tell me, what did you learn from the trip?" asked the father.

The son answered: "I saw that we have one dog and they had four.

We have a pool that reaches to the middle of our garden and they have a creek that has no end.

We have imported lanterns in our garden and they have the stars at night.

Our patio reaches to the front yard and they have the whole horizon.

We have a small piece of land to live on and they have fields that go beyond our sight.

We have servants who serve us, but they serve others.

We buy our food, but they grow theirs.

We have walls around our property to protect us, they have friends to
protect them.

The boy's father was speechless. Then his son added, "Thanks, Dad, for showing me how poor we are."

Isn't perspective a wonderful thing? Makes you wonder what would happen if we all gave thanks for everything we have, instead of worrying about what we don't have.

Appreciate every single thing you have, especially your friends!

Life is too short and friends are too few.


HAPPY THANKSGIVING TO ALL MY READERS!

Circles, Chords, Tangents, Similar Triangles and that Ubiquitous 3-4-5 Triangle

[As always, don't forget to give proper attribution when using the following in the classroom or elsewhere as indicated in the sidebar]

The cone in the sphere problem led me to an interesting relationship in the corresponding 2-dimensional case with a surprise ending. (Only a math person would compare a math problem to a mystery novel!). The following investigation allows the student to explore a myriad of possibilities: from similar triangles to the altitude on hypotenuse theorems to Pythagorean, to chord-chord or secant-tangent power theorems, coordinate methods, draw the radius technique, etc. Sounds like this one problem might review over 50% of a geometry course? You decide for yourself! Just remember -- one person is not likely to think of every method. Open this up to student discovery and watch miracles unfold...

STUDENT ACTIVITY OR READER CHALLENGE
In the diagram above, segment AF is a diameter of the circle whose center is O, BC is a tangent segment (F is the point of tangency), BC = AF and BF = FC. Segments AB and AC intersect the circle at D and E, respectively. Lots of given there! Perhaps some unnecessary information?

(a) If AF = 40, show that DE = 32.
Notes: To encourage depth of reasoning, consider requiring teams of students to find at least two methods.

(b) Let's generalize (of course!). This time no numerical values are given. Everything else is the same. Prove, in general, that DE/BC = 4/5.

(c) So where's the 3-4-5 triangle (one similar to it, that is)? Find it and prove that it is indeed similar to a 3-4-5.

The Classic Cone Inscribed in the Sphere Problem: Developing Relationships Before Calculus

Update: View the series of videos here explaining the procedure for solving the cone in the sphere problem below as well as related questions.

Many Algebra 2 and Precalculus textbooks have begun to include those challenging 3-dimensional geometry questions involving 2 or more variables and/or constants. However, we know from the difficulty that calculus students continue to have with these, that we need to do more before students do their first optimization problems in calculus. You know the kind: Determine the radius of the __________ of maximum volume that can be inscribed in a _________ of radius R. These problems have fallen out of favor somewhat with the AP Development Committee, perhaps because they lack that real-world flavor or perhaps because they had become predictable or perhaps too hard. I would argue they have been part of the rites of passage for calc students for many generations for a reason - they blended the spatial reasoning of geometry with the need to identify variable relationships and reduce the number of conditions down to one function of one variable if possible. In other words, they help to develop mathematical sophistication. I 'cut my teeth' on these -- did you? Any calculus teachers reaching this topic yet in AP Calc?

Anyway here's an activity for you Algebra 2 or Precalculus students to prepare them for these challenges. As usual we proceed from the concrete (i.e., given numerical dimensions) to the abstract. Rather than attempt to draw the diagram, which is fairly challenging for me given the tools I have, I will describe the problem verbally. Good luck!

STUDENT ACTIVITY

(1) A right circular cone of height 32 is inscribed in a sphere of diameter 40.
Note: Students need to learn how to make a diagram of this problem situation.

(a) Determine the radius of the cone.
(b) Determine the volume of the cone. [Imagine asking students to memorize the formula!]
(c) Keep the diameter of the sphere at 40. This time, determine both the radius and volume of the inscribed cone whose height is 80/3. The numbers are messy but try to work in exact form (fractions, radicals) before rushing to the calculator to convert everything to decimals. Oh well, we all know what will happen here!
(d) Try another value for the height of the cone, keeping the diameter of the sphere at 40. See if you can produce a volume greater than in (c). Any conjectures?

(2) We could throw in an intermediate step by using a parameter R to denote the radius of the sphere, and use numerical values for different possible heights of the cone, but I'll leave that to the instructor. Instead, we'll jump to the abstract generalization:

A right circular cone of height h is inscribed in a sphere of radius R.
(a) Express the radius, r, of the cone in terms of R and h.
(b) Express the volume, V, of the cone as function of h alone (R is a constant here).
(c) Use your expression for r and your function for V to verify your results in (1).
(d) Calculus Students: You know what the question will be! Oh, alright:
Determine the dimensions and volume of the right circular cone of maximum volume that can be inscribed in a sphere of radius R. Anything strike you as interesting in this result?