My two children get what the payoff to reading well is immediately and completely. My two children get what the payoff to writing well is as well: they understand that it is fun now and it will be important later on if they want to have options to be able to write quickly, clearly, and coherently.
But math. Math textbooks are remarkably dry. How can I persuade them that math can be fun, that they will be able to learn and calculate interesting things if math is their friend, and that their options later on will be much, much greater if only they apply themselves to math now?
So far, I only have twentythree problems that I regard as interesting and amusing enough to hand them in an attempt to propagandize for math. But I want more: I want to have one hundred...
Posted by DeLong at December 15, 2003 04:01 PM  TrackBackBrad DeLong's Collaborative Website: OneHundredInterestingMathCalculations:
 [1. World War II Bomber Pilot Survival Odds]
 [2. How Many Extraterrestrial Civilizations Are There?]
 [3. Gravity and "Weighing the Earth"]
 [4. Economic Growth Since 1500]
 [5. Exponential Growth and Human Populations]
 [6. How Much Blood Is There in the World?]
 [7. Julius Caesar's Last Breath]
 [8. The Birthday Fact]
 [9. The FalsePositive Problem]
 [10. The GrassIsGreener Paradox]
 [11. The AllKnowing Alien Paradox]
 [12. Repeating Decimals]
 [13. Introduction to Compound Growth]
 [14. Elementary Ballistics: The Kinematics of Falling Bodies]
 [15. Elementary Ballistics: What Goes Up Must Come Down]
 [16. How Rich Is Fitzwilliam Darcy?]
 [17. The Clock Hands Problem]
 [18. Sunscreen, or the Freak Mutant NearAlbino Problem]
 [19. The Distributive Law, or the GetOutoftheWay Problem]
 [20. The Federal Reserve Problem]
 [21. The Ancestor Problem]
 [22. Strategy Secrets of ENRON]
 [23. The Muddy Parent Problem]
In the comments to your first problem, someone recommends this book: http://www.amazon.com/exec/obidos/tg/detail//0521568234/ No good?
Posted by: ogged on December 15, 2003 04:19 PMWell, from calculus I remember something about a 50gallon barrel which is being filled through one pipe at a certain rate while it is simultaneously being emptied through a smaller pipe at a different rate. I suggest you avoid that one.
Posted by: Zizka on December 15, 2003 04:20 PMDo you need help?
About Enron  I tried 4 times to add and the comment does not appear.
Let's say that Enron bet a dollar, and after a day it got the outcome. If it failed for 15 days it will have to bet ~$36 500 for a day in order to get a 50% chance of profit of $1 and ~50% chance of getting the dollar back. But for this day it will miss $1 from the last bet if it has the opinion to invest the money at 1%APR. Hence even if the 15th bet is a garanteed to win, the company will have missed bigger income.
Hence, Enron had to skip this strategy even if the chance is little above 50%.
Similar logic can be applied to roulette  even 10min betwinning delay will lead you to a point of missing bigger interest than the total profit of the bet.
There is another problem : "The Concord fallacy" (I think this is the name)  after you lost money that you accounted as "invested", that is "not lost in fact", it is hard to leave the game and to recognize them as a real loss.
Posted by: GB on December 15, 2003 04:27 PMWell, I can't help, but let's just say that the appreciative audience for this endeavor extends beyond your nuclear family . . . .
Posted by: Williamsburger on December 15, 2003 04:42 PMMy daughter (7) has enjoyed some things out of "Family Math." You might look at "Family Math: The Middle School Years" (ISBN 091251129X) published by the EQUALS group at Lawrence Hall of Science, though it's based on activities that families can do together rather than written problems.
http://www.lhs.berkeley.edu/equals/equals.html
I don't have specific problems with numbers and answers and stuff. But I have some suggestions for topics:
Maybe some of these topics were there, but I didn't see them, or recognize them from the titles.
1. Simpson's Paradox and related paradoxes of aggregation (Each of UC Berkeley's academic depts are meeting affirmative action goals, but whole university the aggregate of the departments, does not).
2. Gambler's ruin and related other gambling strangenesses and paradoxical problems.
3. Monty Hall problem!
4. Geometric tilings and symmetry
5. Simple noneuclidean geometry problems simplests possible: parallel lines in hyperbolic, euclidean and circular worlds
6. math of perspective drawing (projective geometry but DON'T say that word!!)
7. geometry of impossible figures
8. math of paper folding origami
9. prisoners dilemma (that might have been there, though)
All these have some elements that every kid will understand and be naturally attracted to: money, wining/losing/power and/or visual interest.
The visial aspect works even with graduate students. After hearing a lecture by an economist explaining the implications of Simpsons' paradox for mulitple equilibria, we grad students were puzzled. The economist's presentation was so opaque and so covered with the scab of obscure symbols (as Hobbes would say) and interminable formulas that we were going to forget the lecture forever as soon as we walked out the door. So we asked the mathematician in the audience afterward.
MATH: Oh, well, it was given by an economist, of course you couldn't understand it. I'm not sure understanding is the point when economsits speak.
Then the mathematican held out his hand
MATH: Imagine there is a string dangling from my fingers. Can you represent that as a continuous function in threespace?
US: Uhhuh.
Then the mathematician slammed his hand down onto the table.
MATH: Now think about that string, will it now be a smooth continuous function in the twospace of the table top?
US: No.
MATH: OK, that was the lecture.
So, we still didn't *quite* understand, but at least we remembered and I did some followup reading in kind of a half*ss*d way.
I had to be a home schooler for about three weeks last spring. If you check with some of the home school supply stores you'll find a quite a few books. The folks who run the stores are also pretty good resources. They should also help you tailor books to your kids' levels of achievement.
Also they have project books that can make math interesting by practical examples.
The fact that you have 23 problems is very interesting numerically. Perhaps problem 23a could be "My Firsst Look at the Back of a Dollar Bill."
Posted by: pops on December 15, 2003 05:07 PM
Do they have a copy of the annotated Alice? There's a lot of math and logical problems in there, but once you read the footnote they're in you want to figure them out.
Posted by: julia on December 15, 2003 05:18 PMA book I remember very fondly from high school days is George Gamow's "One Two Three . . . Infinity : Facts and Speculations of Science".
Many people feel that Gamow could have gotten two Nobels in physics but for his death, and that does not count his collaboration with Crick/Watson on the protein coding problem ... but he writes in simple prose and explains the complex elegantly with great charm and wit. The book could supply multiple cases to your collection.
AG
Posted by: Arun Garg on December 15, 2003 05:42 PMMy son says that the books 'The Number Devil' & 'Math Curse' are THE ONES to get kids to have fun with math. He read them multiple times, cause they were that fun!
My kids just loved Raymond Smullyan's books of mathematical puzzles.
Posted by: Tim Lambert on December 15, 2003 05:49 PMThe majority of the old Mathematical Games by Martin Gardner provides an ideal basis for developing a love of mathematics painlessly.
While most of the math problems that you have posted are facinating, I've always regarded word problems to be an anathema to an appreciation of mathematics. Having gone from D's in algebra to a PhD in physics, I found that an appreciation of math developed on a need to know basis. If you need math to do something you want to do, math becomes a wonderful thing. From this standpoint I have two suggestions:
Do they program yet? If so, the judicious introduction of computer graphics programming or computer game programming in conjunction with the "Graphics Gems" series of books and "Physics for Game Developers" should resolve the problem rapidly. The beauty of the math needed for programming is that an immediate feedback is provided. Particularly if an elegant language such as Python (which is free as well) is used.
Another wonderful math toy which is ideal for appreciating math is MathCad(the academic version is reasonably cheap), a "magic sheet of foolscap" which provides numerical and symbolic math in mathematical notation. It also supports unit tracing, an invaluable tool which is easily learned via MathCad.
Several people above recommend good books.
Another is _Playing with Infinity_ by Rosza
Peter, which has nice illustrations of a whole
lot of things. (Why is 10/9 = 1.11111....?
Suppose you have chocolates that come with a
coupon, and ten coupons get you a chocolate,
how is this the same as ten chocolates for the
price of nine?)
I remember around that age trying to figure out
how much extra distance I went if I did a lap around a track in one of the outer lanes, and realizing that it didn't matter how big a track
it was (2*pi*width of lanes per lap).
Dave MB
Some other people brought up the books on infinity  in that vein, even though it's not a calculation so much as a proof, you could demonstrate the existence of multiple infinities. I found that pretty fascinating as a kid.
Another infinity puzzle could be a resolution to Zeno's paradoxes, or something with infinite summations  how far does a fly travel if it flies back and forth between two people who are walking towards each other, etc.
Posted by: Annonymous on December 15, 2003 06:53 PMThere's an interesting book coming out soon (my roommate is reviewing it now) called "Experimental Mathematics," which explores how computers can help mathematicians guess and prove new theorems. It's chock full of interesting observations.
Brad, I must question somewhat your strategy for making math interesting. Your examples imply that interesting math problems are those that have some relevance to the real world. Some math, like music, is great simply because of its beauty. I'm not sure how much of this type of math exists at this level, but you shouldn't exclude the possibility.
I think the usual trick for summing a geometric series would make a nice little entry.
Posted by: Matt on December 15, 2003 06:59 PMI second Matt's motion that "the usual trick for summing a geometric series would make a nice little entry". After all, it's simple, and more importantly, by showing an infinite sum can converge to a finite number you put that highly confused Zeno in his rightful place.
Posted by: Bill on December 15, 2003 07:11 PMI think a lot of these problems are interesting, but I think its also possible to miss the interest in math for itself. We homeschooled (until my son started 9th grade) this year, and did a lot of playing with numbers, more than organized math at the start. Also, remember that math isn't just arithmatic and problems with numbers.
A really good computer game that is about mathimatical thinking is The Logical Journey of the Zoombinis.
This progam seems to be defunct, but there is an archive of interesting problems.
http://mathforum.org/mathmagic/
Books: Insides, Outsides, Loops and Lines by Herbert Kohl. This is topology for kids.
Calculus by and For Young People by Don Cohen. He's on the web at http://www.shout.net/~mathman/
This isn't for formal calculus, but playing with concepts of limits, etc.
Mathematics: A Human Endeavor by Harold Jacobs. This is a different sort of text book.
We used to play games in the car. The easiest one was guessing, with the person thinking of the number saying higher or lower. We started with 1100, and as time went on we increased the range. Then there is guessing a fraction. That can get pretty hard for the person thinking of the number to figure out whether the guess is higher or lower, unless the denominator is limited. I've missed exits trying to figure it out. You can find variations, trying to stump someone on whether the fraction you're thinking of is equivalent to the one they said or not.
Another book: Beyond Facts and Flashcards: Exploring Math with Your Kids by Jan Mokros
I think part of my point is that its not just a matter of interesting problems that are outside of everyday life, but it can be playing with math as you go about your life. That is what this last book is about.
Posted by: Cindy on December 15, 2003 08:05 PMCourant and Robbins, "What is Mathematics"
Actually written entirely by Robbins from lectures given by Courant.
Posted by: A. Zarkov on December 15, 2003 08:30 PMYou're hampered in your "math is fun" efforts because math isn't fun. And I say this as someone with five semesters of calc/postcalc math. Best to stick to convincing them that a) it's not at all bad, and b) it's amazingly useful. Two true facts that don't require forced jollity.
Posted by: Mike Kozlowski on December 15, 2003 08:39 PMBrad,
Have you read Donald Knuth's "Surreal Numbers: How two exstudents turned on to pure mathematics and found total happiness" ? A brief description is available on Knuth's web page.
Knuth wrote it to try to demonstrate how real math research was nothing like the dry stuff you find in textbooks.
PS Any comments on Everett Ehrlich's piece in the Sunday WaPo applying Coase theory to the Dean campaign ?
CNN recently had the headline "Largest prime number discovered". Why, oh why, can't we have a better press corps? What they meant was "Largest _known_ prime number discovered" since there are infinitely many primes.
I read a proof for this when I was 13 or 14 and was amazed by the way it was proved. Assume that there are fintely many primes 2, 3, 5, 7... Now multiply them all together and add 1. Call the result n. Now n cannot be divisible by 2, 3, 5, 7, etc since it will always leave a remainder of 1. This means that n must be divisible by some prime not in our original list (and may be prime itself), so our initial assumption that we'd listed all the primes is wrong and there are infinitely many primes.
The proof is given in Hoffman's "The Man who Loved Only Numbers". Your children might enjoy that book too.
I think the fact that the discovery of the 40th Mersenne prime made CNN is proof of the fascination many feel for these integers. Also, that hundreds (thousands?) of people devoted clock cycles to the Great Internet Mersenne Prime Search (GIMPS) suggests that a bunch of us think primes are pretty cool.
Posted by: Ernest Hammingweight on December 15, 2003 09:03 PMIn the infinite series vein (or not), you could do the one where there's the bug flying back and forth between 2 approaching trains, with constant speed & the ability to turn around instantaneously  how far does it fly before the trains meet?
Someone mentioned the exam paradox in the comments to your Newcombe's paradox problem  I second that.
Have they been introduced to the wonders of induction? There are plenty of inductiontype word problems involving things like hats and pirates and killing people. Also, there's the whole genre of people who always lie and people who always tell the truth. There's a book by Raymond Smullyan with a bunch of puzzles like that that I liked as a kid  Alice in Puzzleland.
I don't know if you're ready to get into derivatives and Taylor series and so on, but the proof of e^(i*pi) + 1 = 0 is pretty cool. Another nifty proof is showing that there are infinitely many primes by assuming the opposite, multiplying them all together, and adding 1 (on preview I see this has already been mentioned). And then there's Cantor's diagonalization. You might want to save some of these for when they're actually taking the math classes, though.
Posted by: Dan on December 15, 2003 09:09 PMCantor's "diagonal" proof of the difference between the "cardinality" (I think is the term) of the infinite sets of all real numbers and all integers (or all rational numbers) is nicer than it has any right to be (and connected to #12 on your list), although some people will tell you it isn't really math, because proof by induction isn't really proof. Keep a list of these people, and remember to ignore them in all matters.
It's an utterly useless result, but the proof of this weird thing is very common sense, and it's one of the few results that one could consider "cool", given certain disclaimers about social ostracism and being stuffed into one's locker and so on. "Did you know there are more real numbers between 1 and 1.0000001 than there are integers between 1 and infinity?" Neat. There's a geometrical argument that does the same thing in picture form, but I can't remember how it goes.
I also like Richard Gott's argument that we need to colonize other planets right now, because out extinction is at hand, based on extrapolation from a single data point (the current age of the human species), and the Copernican principle. Another argument that sounds like sleightofhand, but actually isn't (too much). He predicted the date of the fall of the Berlin Wall completely accurately, although  here's the trick!  not very precisely. His full argument is actually grimmer than the article I linked to, as, for the same reason you always find yourself in the longest line at the supermarket, the most likely time for a given member of a species to be alive is at the time of peak population, we are, if we don't colonize space, within spitting distance of a major population decline, if you put some rather generous error bars around "spitting distance".
This isn't math, but Freeman Dyson's idea of searching for extraterrestrial life by "pitlamping" is a pretty good example of how supersmart people just think differently than the rest of us. He might still be on the lecture circuit with this one, so be sure to see him if he comes around, or if he turns it into a book (there's really no writeup of it on the internet which does justice).
Also, quantum computing is way cool, although you have to stop thinking about it the moment it starts making sense, or it will all fall apart. And, again, not really math as such. But far out.
Posted by: Andrew Northrup on December 15, 2003 09:39 PM... And html tags don't work. For some reason, I saved the whole thing before MT ate it, and so you can see it with links here:
http://www.thepoorman.net/archives/002237.html#002237
Also, none of these things are actually "calculations", so I'm not really answering the question, I guess. In the spirit of plowing on regardless, I'll recommend "Pi In The Sky" by John Barrow, a very nice history of numbers and mathematics.
Posted by: Andrew Northrup on December 15, 2003 10:09 PMI hope these math problems are for amusing yourself only and that you are not running experiments with them on your kids. I am not an education specialist but I think kids should be exposed to only tried and tested material and I am afraid some of your example math problems are not that. The sliding parent and dawg problem was not very sound. I looked at pilot survival problem and that didn't look right either. So I think you shouldn't expose the kiddos to material without consent of their teachers. (I'm sure you don't but I'd rather risk sounding like, uhm, well I can't think of a word, than not having said this.)
I may have more comments later about my personal view that kids need to be told that math is more than manipulating numbers, it is more about logic. Well at least as much.
Andrew Northrup writes:
>
> I also like Richard Gott's argument that we need to
> colonize other planets right now, because out extinction
> is at hand, based on extrapolation from a single data
> point (the current age of the human species), and the >
> Copernican principle.
Well, unfortunately that argument is a steaming heap, despite what you might have read from his original paper in _Nature_. As an antidote, do a quick search at Google on (Sowers Gott Doomsday Demise) and read the PDF that pops up as #1 (a paper by Sowers).
(To be too brief to be fair, Gott suggests that his own birth is most likely to have occurred in the inner N% of all births of our species, so that a 95% confidence interval for the endpoint of the species is between (nowgenesis)/0.975 and (nowgenesisis)/.025. The naive version of this is truly wrong.)
Now, there may be reasons not to consider the Sowers argument as decisive against all possible versions of the Doomsday Argument, but this gets to some really fine points, and argues against its inclusion in DeLong's Top 100 (as it were).
Posted by: Jonathan King on December 15, 2003 10:47 PMMy father used to sketch opamp circuits on napkins and envelopes. That was how he did math. He was always trying to draw me in to the work he was doing. That, and competing with my older sister, made me look forward to algebra.
How about the polling problem. As in "How can one get meaningful results off a sample size of a couple of hundred?" A good intro to some basic statistics. The math can be pretty hard if you want to solve it completely but most of the details can be glossed over pretty easily.
A relevant problem going into the election year.
Posted by: Gryphon on December 15, 2003 11:01 PMHere are the comments I promised, quickly, about math being more about logic than numbers, numbers being important as they are:
Well, let me put it this way: For example, American economy (and therefore the world economy, and therefore the general state of affairs around the world) would have been in much better shape today had Mr. Bush at al been taught about "necessary and sufficient conditions" (in fact I have a hunch that Bush team is made up of math haters, especially Mr. Bush himself, along with Mr. Rumsfeld, as he did things with army logistics process that completely defied any math or logic).
And perhaps the easiest way to tell about necessary and sufficient conditions is to give examples:

Let us say that there are two different events or states of the world A and B. There are four possible relationships between A and B:
1 A is neither necessary nor sufficient for B, i.e., for B to exist, or to be true: It means A and B are not related. They can exist / hold true independently from each other. Simple enough.
2 A is necessary but not sufficient for B: It means if B is True then A is also true but A being true does not mean B is also true. Example: Wind is necessary but not sufficient for a sailship to sail safely on wind power alone. Or, for example, a ship's leaving Marseille is necessary but not sufficient for that ship to arrive safely in Rome. This too is also simple enough.
3 A is sufficient but not necessary for B: It means if A is true then B is true also but B can also exist without A. For example, a sailor climbing down a cargo ship's lower chambers and not coming back and another sailor going down and not coming back and then a third sailor with a gas mask going down and finding the previous two sailors dead due to gas poisoning is sufficient but not necessary to conclude that the ship's lower chambers contained poisonous gas. (This has actually happened, several years ago, aboard a cargo ship with Rumanian flag  the captain obviously didn't know about mathematics of necessary and sufficient conditions.)
4 A is necessary AND sufficient for B (A is true 'if, and only if, (iff)' B is true.): It means if A is true then B is also true AND if B is true then A is also true. It means A and B are both true or they are both false  neither of them can exist or be true without the other. Example: The necessary and sufficient condition for an object to float in sea water is that its total weight is less that the weight of the volume of sea water that it displaces.
And then there are practical problems, for example, problems of perception / appearance. An object that appears to be floating in the sea may in fact be sitting on underwater rocks. Life is not easy. Neither is making observations.
What are the necessary and sufficient conditions for having solved a problem?

Text above comes from my web page on Problem Based Learning, a certain approach to education and learning, which originated in medical education ( and I seem to think it would be excellent for education and learning in strategy).
In fact doctors and soldiers seem to understand necessary and sufficient conditions much better than people in other professions. I guess they learn it the hard way.
What was the professional background of Howard Dean again?
Johnathan King  I'm having trouble loading the Sowers file, but if I remember correctly (a big "if"), it seemed like he was mostly missing the point of Lott's argument. Here's the experimental proof: knowing nothing about Broadway, he accurately, not precisely, predicted the runs of various plays based simply on how long they had been running so far. Of course, it's only common sense to expect that a show that just opened won't have a run of a thousand years, but Gott's point is: why is it common sense? And can we quantify our confidence about common sense? And  if you are confident in assuming that there's nothing about the lifecycle of these things that makes your assumption of a roughly Guassian probability function incorrect, which, give the assoutofuandme proviso, turns out to be a reasonable assumption for such diverse and unpredictable things as walls partitioning East and West Berlin and Andrew Lloyd Weber plays  you can quantify it.
I don't think Gott's argument for colonizing space is very persuasive, but I don't think there's anything logically wrong with it, as far as it goes. And while it's certainly not the most important argument in the history of mathematics, it's definitely one of the coolest.
[Disclaimer  I'm not a mathematician.]
Posted by: Andrew Northrup on December 15, 2003 11:37 PMWow! I'm checking out the Surreal Number thing by Knuth ASAP. That sounds like fun, and I am too cheap right now to spring for the Conway book.
I hope you get a lot of suggestions on this thread.
Have your kids made a mobius strip yet? OK, well have they cut one in half (lengthwise). Well, OK, have they cut one into thirds? I saw an internet page on mobius strip experiments, but I can't find it now. Will inform upon discovery, since the responses above have inspired me to do some fooling around myself with this stuff over Christmas.
My experience teaching kids math is that you either have to give them some interesting motivation money, power or visual interest. They also love the sheer fun neato aspect. Anything they can show off to their friends will be a sure hit.
Hence: mobius strips (if they haven't already done them)
The birthday problem is always a sure thing (what are the odds X number of people in room have the same probability). Especially if they have more than the magic number (what is it, 24, 32 ???) in their class at school.
A few finite Ramsey theory problems are also nice (among X people are Y mutual friends, etc.) Some of Smullyans problems books are these. His problems can be V hard for kids though, so start with the simplest and stope when frustration or disinterest sets in.
Music and algebra and baby number theory go very well together. Teaching triplets, time signatures does wonders for developin an interest and ability in fractions.
Speaking as a fellow economist, who is familiar with the professional tendancies and tastes that come with it your problems might be on the dry side. As another poster said, this stuff needs some yuks, paradoxes and visual fun in the mix.
Posted by: jml on December 15, 2003 11:56 PMTo clarify, I'd say that Gott's argument is a good introduction to the magical world of statistics and probability, and that objecting that it doesn't really prove we need to move to Mars right now is sort of like objecting that there isn't really a train leaving Chicago at 45 mph. Given his assumptions, it's a fun thing to think about.
Posted by: Andrew Northrup on December 16, 2003 12:11 AMI may be wrong but many people seem to think of colonizing space as a matter of transporting the entire population of earth to escape a doomsday. That is not possible. If a doomsday does occur, almost entire earth population will be gone.
Even in colonizing America and Australia, overwhelming majority of world population remains outside those continents.
But saving the human species is very different from saving the entire human population. In fact human species would probably survive doomsday even right here on earth. But, just in case:
A flower sending out its spores is doing something like colonizing space. When you look at it from the point of the flower and its life span and the modest attributes of the spores, it looks a hopeless endeavor.
But by God it works!
Posted by: Bulent Sayin on December 16, 2003 12:12 AMhttp://www.cuttheknot.org/do_you_know/moebius.shtml
Posted by: jml on December 16, 2003 12:26 AMStatistics can be interesting/useful/mindboggling. There's the Monty Hall thing. (Of course it doesn't take into account the psychology of the host.) There was also a really counterintuitive statistics problem I remember: A king has 2 children. One child is a male. What is the probability that the other child is female? It's not 1/2, it's 2/3. Trying the "highlight here for the answer":There are 4 total combinations of sexes of the children: MF, MM, FM, or FF. (You could think of one being the elder sibling and the other the younger if you want to make a chart.) We know one child is male, so we can rule out FF. That means there's a total of 3 possibilities. 1 out of 3 possibilities is that the other child is also male (MM). 2 out of 3 possibilities that the other child is female (MF,FM), thus a 2/3 chance. Now that I think about it, I'm not sure that problem would encourage a love of statistics, it's kind of hard to accept.
Posted by: scott h. on December 16, 2003 12:29 AMStatistics can be interesting/useful/mindboggling. There's the Monty Hall thing. (Of course it doesn't take into account the psychology of the host.) There was also a really counterintuitive statistics problem I remember: A king has 2 children. One child is a male. What is the probability that the other child is female? It's not 1/2, it's 2/3.
The reason?: There are 4 total combinations of sexes of the children: MF, MM, FM, or FF. (You could think of one being the elder sibling and the other the younger if you want to make a chart.) We know one child is male, so we can rule out FF. That means there's a total of 3 possibilities. 1 out of 3 possibilities is that the other child is also male (MM). 2 out of 3 possibilities that the other child is female (MF,FM), thus a 2/3 chance. Now that I think about it, I'm not sure that problem would encourage a love of statistics, it's kind of hard to accept.
Posted by: scott h. on December 16, 2003 12:33 AMI like the Towers of Hanoi problem. Or the variant with a chess board, a penny, and the king's promise to double the penny for every square.
Ask your kids how much money will they have at the end of the board. They will be shocked by the answer  and it is a good example of the growth of the geometric series.
Posted by: Timothy Klein on December 16, 2003 12:57 AM"You're hampered in your "math is fun" efforts because math isn't fun. And I say this as someone with five semesters of calc/postcalc math."
You're completely wrong. Math IS fun, and I say this as someone with graduate level training in the subject.
Brad, I'd have to second the suggestion made earlier on here that too much emphasis on applications can be an interest killer, as it tends to foster the idea that mathematics is only useful as a tool, rather than as an end in itself. Have you thought about introducing your kids to, say, a simple proof of the infinitude of the primes? How about giving them an application of the pigeonhole principle? Or showing them something about continued fractions? What about a gentle introduction to the ideas of group theory, via a discussion of the symmetries of geometric shapes? Or selfsimilarity and sensitive dependence on inital conditions, using Field and Golubitsky's "Symmetry in Chaos" for the pretty pictures, and the logistic equation X(k+1) = k*X(k)*(1X(k)) for a simple example?
There's plenty of mathematics that doesn't involve calculus or tedious calculation at all, and I for one have tried strenuously to avoid those branches of mathematics that do. The topics I've suggested all have the merit of elucidating significant mathematical principles that can be grasped without a great deal of technical sophistication beforehand.
Posted by: Abiola Lapite on December 16, 2003 01:38 AMWonderful books: "Anno's Hats" and several others all by Mitsumasa Anno.
Useful math: there are several easily understandble proofs of Pythagoras, including one invented by Einstein as a youngster, another by President Ulysses S. Grant, I believe.
I think it's great fun tht Pythagoras works for right "triangles" of any number of dimensions. Thus the length of the line from one corner of a box to the diagonaly opposite one will be the root of the sum of the squares of the three sides of the box.
What an interesting set, I may have to borrow it some time if you don't mind : )
Posted by: Colin Gregory Palmer on December 16, 2003 02:39 AMWhat an interesting set, I may have to borrow it some time if you don't mind : )
Posted by: Colin Gregory Palmer on December 16, 2003 02:52 AMOne thing that you find scientists and engineers doing all the time is estimating the magnitudes of various physical quantities with 'back of the envelope' calculations and then going on to figure out how the quantities will vary under different circumstances. It's hard to overstate how important this is. The problem is, you need some facility with units of measurements, you need to know a lot of numerical facts, and you need to know a lot of the underlying physics. It's hard to make the first two interesting and the last two take experience.
A typical (nontrivial) example: Given that a cubic meter of air (at room temperature and on the surface of the earth) has a mass of about 1.8 kilograms and a cubic foot of water weighs about 60 pounds, what does this tell you about the inertial air resistance that a car meets at 60 mph compared with the inertial water resistance that a submarine meets at the same speed? Does viscosity matter (you have to calculate the Reynolds number)? What if you go to 50,000 feet above sea level? And so forth.
It's clearly an Rrated (if not Xrated) calculation, but I wonder if there's a version that would get the point across.
Posted by: Matt on December 16, 2003 04:48 AMSecond the recommendation of Martin Gardner  I grew up with his books and was interested in math  I can hypothesize that these two things are related. If your kids like Lewis Carroll, Gardner's "Annotated Alice" has some fun math in it IIRC, also his novel (or short story? It's been a while...) "And he Built a Crooked House" is very entertaining and mathy.
Posted by: Jeremy Osner on December 16, 2003 04:48 AMGambling > teach them poker, blackjack, craps. Show them how to calculate odds in order to win. If you have gambling problems in your family, you might want to skip this one.
Risk> in general. Decision making. The 90's bubble
Finance > reading a balance sheet; investing
Statistics > Start with everyday examples, like the ones cited in the news. Or you could try to find this book: http://www.amazon.com/exec/obidos/tg/detail//0393970833/ at the Berkeley library.
Physics >throwing something and estimating where it will land; designing paper airplanes
Logic >applying logic to stupid arguments. Spotting fallacies. Republican talking points are a rich source. You can go use this as a springboard to set theory.
Feynman's books inspired me, briefly, to learn more about math.
Cryptography
Internet problems > show how math relates to network problems. Maybe using math to calculate how peer2peer applications work/scale.
Econ >duh
I don't know what your specific goal is. Is it to get them to like math as a means ("math is useful") or as an end ("math is cool")? I think you'd have more luck in showing them that math matters as a means. Getting them to like math as a end is a lot harder.
No offense, but your 100 problems seems heavy handed. (Disregard the previous statement your kids actually seem into them) Your best bet is to start with other subjects that depend on math, like econ or stats. When you cover more ground in said subject, show them how/why they need math to get to the next level. If you're going for the "math really matters" argument, Google up some stats on careers and how math ties into them. But I'm not sure it really matters even then. For most nonengineering professions, you aren't required to know anything beyond simple calculus. I'm not sure you can really get someone to like something if they're not really into it. I know a lot of people who are really into certain subjects, but their kids aren't.
I never liked math when I was a student. Mostly because it dealt with abstract problems, like Zizka's example. It wasn't until I was interested in other subjects, like the ones above, that I started reviewing my old textbooks.
The human body is 88% (or something like that) water. When I worked with the U.S. Geological SUrvey, we once calculated that at rush hour, more water passes over Chain Bridge (crosses the Potomac near Washington, DC) than under it.
Obviously adaptable to other locations.
Posted by: Timothy Wyant on December 16, 2003 05:10 AMIn the Journal of Irreproducible Results (still exist?), some contributors once tackled the problem of whether it makes sense to run to get out of the rain. They modeled the human body as a cube, and calculated the amount of water hitting the front and top surface as a function of rainfall rate and speed afoot. Complication  what if you are running downhill?
One of my college math professors (Wade Ellis, Oberlin) once wanted to build a house with panels in the siding that would allow him to change the color of the house when the whim struck. (Who says Ohio isn't a happenin' place?) His contractor suggested that they install thermopane windows  which have two panes of glass that are separated to reduce thermal conductivity  and fill the interpane gap with colored fluid. Dr. Ellis pointed out that water pressure depends on depth, so that despite the narrowness of the column, the fluid would blow the glass out in the contractor's scheme. The contractor was impressed. He claimed that this was the first time in his experience that a college professor knew something useful.
Posted by: Timothy Wyant on December 16, 2003 05:23 AMWow! I second that web site recommended by jml:
http://www.cuttheknot.org/content.shtml
I copy below a paragraph from their content page, hope they won't mind:
"Raymond Smullyan, a Mathematician, Philosopher and author of several outstanding books of logical puzzles, tells, in one of his books, a revealing story. A friend invited him for dinner. He told Smullyan that his teenage son was crazy about Smullyan's books and could not wait to meet him. The friend warned Smullyan not to mention that he is a Mathematician and that Logic is a part of Mathematics because the young fellow hated Mathematics."
And their "Manifesto" is very telling as well.
Posted by: Bulent Sayin on December 16, 2003 05:56 AMPerhaps too similar to 1). An approximation of your odds in hitting a hole in one in golf.
Posted by: theCoach on December 16, 2003 06:05 AMI think Matt's comment about the game of estimating magnitudes is good.
Fermi used to play it at Los Alamos during the Manhattan project, and my fvourite of his questions is "How many railroad locomotives are there n the United States?" The available fact base was that there was one railway track to be crossed between Los Alamos and Albequerque, and you might or might not see a locomotive while you were making the trip.
John Kenneth Galbraith's version was adding up columns of numbers: with very little practice you can add up a column with less than 10% error just by looking at it.
Oh, another great one that's math but not a calculation (Stolen from Stephen Pinker, but I think it's a fairly classic logical experiment).
You have a deck of cards with numbers on one side, and letters on the other. Given the hypothesis that "All cards with a 3 on one side have an L on the other side", which of the following cards do you have to check:
A card with a 3
A card with a 5
A card with an E
A card with an L
And then repeat the question replacing the cards with people at a bar, and the hypothesis with "All drinkers need to be 21 or older":
A drinker
A nondrinker
A 17year old
A 30year old
Following is a puzzle that relies on a straightforward application of the pigeonhole principle:
"Consider a chess board with two of the diagonally opposite corners removed. Is it possible to cover the board with pieces of domino whose size is exactly two board squares?"
Posted by: Abiola Lapite on December 16, 2003 07:32 AM I have good memories from Martin Gardner's book(s) "Mathematical puzzles" (I think). I remember one trick: write 3 digit number (123 for example). Write it twice (=> 123,123). Now divide the big number to 7; 11 and 13. You will get the original number (123). Why this happens?
My additions:
Can you devise similar trick with 2 digit numbers, or with numbers written 3 times?
GB,
What you have just run across is the class of Sheharazade Numbers (Bucky Fuller's term). These are the products of three consecutive primes  and are named because the three you have supplied multiply out to 1,001, the number of nights in Sheharazade's tales.
The 1,001 is of course the reason Gardner's trick works: abc * 1001 = abc,abc.
I second that Smullyan and Knuth´s "Surreal Numbers" are excellent.
Abbott´s "Flatland" is a classic to make kids think about dimesnions. In a similar area are the wonderful "The Shape of Space" by Jeffrey weeks and "Geometry, Relativity and the Fourth Dimension" by Rudy Rucker.
A childrenfriendly application of pretty good math is combinatorial game theory, there are several good books on that. For example, you can find an easy proof that Hex has no draw under
http://www.cs.ualberta.ca/~javhar/hex/hexyproof.html You can use that to tell your kids about Nash´s proof that the game can´t end in a draw, which brings me to another point: Teaching kids elementary gametheory is useful, mathematical and interesting. Dixit and Nalebuff wrote a book that even smart kids can comprehend. "Games of Live" by Karl Sigmund, a evolutionary GT focused book, shows how useful math is in biology.
Interesting mathematical material can be found in "Mathematical Snapshots" by Hugo Steinhaus, a cheap Dover book. If they are highschool level, the should read "Introduction to Mathematical Thinking" by Friedrich Waismann at all cost.
Knot Theory is a nice application of algebraic topology that one can present in a "concrete" fashion. Sossinsky wrote a nice book about it (review: http://www.maa.org/reviews/knotstwist.html ).
The most important application of number theory is cryptology, a topic that should appeal to kids for it´s connection to mystery, adventure and puzzling. There is enough accessible literature on that.
Two nice books on the application of mathematics to understanding politics: "Mathematics and Politics" by Alan Taylor, " Chaotic Elections!" by Donald Saari (review: http://www.maa.org/reviews/saari.html )
There is a nice series on theoretical computer science for kids: "Computer Science Logo Style" by Brian Harvey.
Posted by: Michael Greinecker on December 16, 2003 08:51 AMHomework is boring. Just tell them to use http://www.homeworkster.com instead.
Posted by: anon on December 16, 2003 08:53 AMOne of my children announced the other day that
he hates numbers (of what value are weighted average problems) (why should he learn about lines
and point slope form  a discussion of the
budget constraint didn't help much here) so he's
decided that math and science are boring. He's
in 8 honors. However, he's decided that he doesn't like words either. English and Latin are
boring too. He likes History, Study Hall and Gym.
I told hinm that HIstorians worry about words and
also use statistical techniques to try to resolve
outstanding questions. Sigh. We continued on with our walk. I hope it's just a phase.
I'll second Arun Garg on One, Two, Three, Infinity, by George Gamow, which I read in the late 50’s, when I was in grade school. One of the images in the book, describing the fourdimensions of spacetime, stuck with me for all those many years. I bought another copy as mine had vanished, and it holds up very well.
Posted by: Masaccio on December 16, 2003 09:00 AMThe sequence of Scheherezade numbers is:
6, 30, 103, 385, 1001, ... do these numbers have any interesting properties in common other than being the product of 3 consecutive primes?
I also like calculations of volumes. Assuming everyone in the US drinks 8oz of orange juice each morning, is the volume of OJ drunk each morning greater than or less than the volume of Evans Hall (the math building at UC Berkeley); how many orange trees are needed to produce that much orange juice; how many tractortrailer smeis are needed to supply the grocery stores in LA each day; etc.
Posted by: David Margolies on December 16, 2003 09:02 AMIf didn't develop Alzheimer's all of a sudden again, my recollection is that Abbot's "Flatland" contained social  political implicaitons that were not too democratic for my taste.
Posted by: Bulent Sayin on December 16, 2003 09:08 AMOk, here's my favorite math problem. It's similar to one posted by stephen above but the story has a bit more flair.
"Imagine the earth is perfectly round, and we are able to take a copper band and fit it around the equator so that it is perfectly snug up against the earth. Now, take that copper band and add one foot, so that the band is now loose and there is a gap between the earth and the band. Is that gap big enough to slide your hand under?
Answer: Yes. We know c = 2*pi*r, and dc = 2*pi*dr. So if dc = 12 inches, then dr = approximately 2 inches, so you could stick your hand in it.
The cool thing is that this works for the earth, for a basketball, and for the solar system. Expand the copper band by 12 inches, and it increases the radius by approximately 2 inches."
Posted by: John McConnell on December 16, 2003 09:27 AMThe earth rotates on its axis in 23 hours and 56 minutes. A day is 24 hours. What's up?
Posted by: Daniel Klenbort on December 16, 2003 09:29 AMAllow me to be the second or third person to STRONGLY recommend "Mathematics: A Human Endeavor":
http://www.amazon.com/exec/obidos/tg/detail//071672426X/
I enjoyed math as a child mostly in spite of all of the teaching texts on the subject. "A Human Endeavor" was the shining exception to that rule.
Posted by: Doctor Memory on December 16, 2003 09:35 AMAllow me to be the second or third person to STRONGLY recommend "Mathematics: A Human Endeavor":
http://www.amazon.com/exec/obidos/tg/detail//071672426X/
I enjoyed math as a child mostly in spite of all of the teaching texts on the subject. "A Human Endeavor" was the shining exception to that rule.
Posted by: Doctor Memory on December 16, 2003 09:37 AMI think somebody already mentioned moebius strips, but if you really want to blow their minds, try to get them to visualize 4 dimensions by sewing the strip together along its one side, producing a Klein bottle.
Posted by: Jefe Le Gran on December 16, 2003 09:46 AMYou can get older editions of "Mathematics: A Human Endeavor" a lot cheaper from http://www.fes.follett.com/products/catalogsearch.htm The first edition, which I have, is only $1.10 + shipping.
Posted by: Cindy on December 16, 2003 10:02 AMAnnonymous writes:
>
> Oh, another great one that's math but not a calculation
> (Stolen from Stephen Pinker, but I think it's a fairly classic
> logical experiment).
This is *supposed* to be known as the Wason Card Selection Task. I trust that Pinker had a pointer to one of the original papers by Wason on this task. My favorite one is:
Wason, P. C. (1968). Reasoning about a rule. Quarterly Journal of Experimental Psychology, 20, 273281.
But that's not (alas) a paper to hand to a 10 yearold, and the problem, while neat, isn't really a mathematical calculation.
To add the real thread here, a book that might help Brad DeLong think of other cool calculations to do would by Mosteller's "Fifty Challenging Problems in Probability with Solutions". A really beautiful book, I think. Some of the problems are beyond what the blogitor is looking for, but many are close to "off the shelf". Alas, my copy of the book is off the shelf, or I'd give some examples.
Posted by: Jonathan King on December 16, 2003 10:18 AMMr DeLong,
You might want to have a look at this Russian book:
"Mathematics can be fun" or its earlier incarnation "Mathematics for everyone", by Y.I Perelman. The book used to be published by Mir publishers, and is probably out of print now, but it is likely that a good university library will have it. In general, there are a lot of books published in the erstwhile Soviet Union on Math and physics which are really good.
Hope this helps.
Pardon if this is a repeat, but the Very Smart Keith Devlin, in his "Language of Mathematics" offers a wonderful example to illustrate "what math is all about"  or more precisely, mathematical induction. Here's the proposition:
The sum of a series of odd numbers is equal to the square of the number of items in the series. We can see by observation that this is true in simple cases: e.g., 1+3+5 = 9 = 3^2 . Can it be shown thqt it must be true for all numbers? The answer is yes, and rather simply: the proof takes just a couple of steps and requires nothing more sophisticated than the capacity to factor a quadratic. I inflicted this on a 13yearold last year. As ever with kids, I had no clue as to whether she was receiving it or not  but then later, I saw it on a white board at her school, in her handwriting. Details are at Devlin 4750.
Oh, and while I am at it  she thought the ancestor problem was cool. We also discussed "the population implosion"  think of your ancestors as a giant inverted pyramid, with its apex on the top of your head.
Posted by: Buce on December 16, 2003 11:00 AMIt's been my experience that "math" concentrates too much on esoteric algebraic bullshit and not enough with the stepbystep math of physics  eg. calculating how many square meters of solar panels would be required to meet the US's electrical power needs (answer: an array roughly the size of Rhode Island).
IOW awareness of physical units ($/person, W/m2) and how to move among them is more important than abstract stuff.
Posted by: Troy on December 16, 2003 11:01 AM
The rule of 72: if a quantity is compounding at n% annually, it takes roughly 72 / n years to double. Useful for thinking about investments and populations.
I suspect the most obvious usefulness of math is in dealing with money. How much interest will you pay over the lifetime of a mortgage? How much can you save by making a prepayment?
Posted by: Russil Wvong on December 16, 2003 11:04 AMIt's been my experience that "math" concentrates too much on esoteric algebraic bullshit and not enough with the stepbystep math of physics  eg. calculating how many square meters of solar panels would be required to meet the US's electrical power needs (answer: an array roughly the size of Rhode Island).
IOW awareness of physical units ($/person, W/m2) and how to move among them is more important than abstract stuff.
Posted by: Troy on December 16, 2003 11:06 AM
Before glasnostperestroika, I heard that Russians were a bookreading nation. I hope that didn't change but I am not keeping my hopes too high  there is expansion of TV, I guess, and book prices probably skyrocketed after change of regime, and unemployment, and ...
Every body loves you Brad!
You type in "REQUEST FOR HELP" and you get sixty eight thousand responses. I'm jealous!
(I'm usually not nice like this  I'm probably thinking in the back of my mind something like "time to move on" ... I wonder what people are thinking about things like "shareholder vigilance" and "socially responsible investing"... but how can I not come back once in a while to the blog of the economist WSJ writers mention top of the list?)
Posted by: Bulent Sayin on December 16, 2003 11:10 AMRe rule of 72  my dad was a credit manager in the precalculator age. He taught me this one along with a bunch of of other finance/math tricks based on similar simplifications  30 day months, 360day years, etc. (bur I confess I have forgotten most of them). He did point out that 69 probably worked better than 72but was computationally less convenient. He did not point out (and I'm not sure he knew) the logarithmic basis for all this. He did not point out, but he certainly equipped me to understand, how much better off we might have been if our math system had started with base12 rather than base10.
Posted by: Buce on December 16, 2003 11:14 AMSomeone mentioned the Klein bottle. Check out "Experiments in Topology" by Stephen Barr Dover book cheap. A book of topology projects, I remembered that I had bought some time ago, and tried some the experiments, and dug out last night. Shows how to make mobius strips from three dimensional models of Klein bottles. Mobius disks. Not for the clumsy. But if you can handle paper, scissors and scotch tape, just the thing.
If the kids are too old for the neato, showing cool stuff to their friends, or visual interest factor, then find math tools to help them with a subject that they are interested in. One of my flute students has fallen in love with the blues. So suddenly the math behind the blues scale, syncopated triplets and fifthlets over a regular marchtime time signature has become *very* interesting to her. An amazing burst of interest in some one who has shown no interest in math at all (probably because her brother and sister are better at it than she is). So you have
music algebra and baby number theory
sports statistics
drawing projective geometry
computers logic, boolean algebra
history statistics
geography topology and geometry
nature diffy Q, graph theory (for taxonomy and phylogeny)
This a great discussion. Reading it has probably cut down greatly on my shopping time for nieces and nephews. I know just the person for "The Math Devil"
Posted by: jml on December 16, 2003 12:06 PMI was always a fan of the volume problems. Usually taking the form of things like how many ping pong balls will fit in a 747. It can also be changed to how many golf balls will fit, which is weight dependent and not volume.
Posted by: Matthew Peters on December 16, 2003 12:16 PMNot sure how mathy this will seem to the kids, but this puzzle is a good introduction to symmetry. You are playing a game in which players take turns putting pennies on a round table. When placing a penny, it must not overlap any of the pennies already there, and the penny must not fall off the table. Last player with a valid move wins. Should you play first or second?
Posted by: Matt on December 16, 2003 12:29 PM
Two interesting math problems for kids are as follows.
(1) How tall must a mirror be for you to see your entire figure in it? (Ans: half your height)
(2) A rope at ground level goes all the way around the earth at the equator. How much longer would the rope have to be if you were to raise it to one foot above ground level at all points along the equator? (Ans: 6.283 (or 2*pi) ft)
Andrew Northrup writes:
>
> I don't think Gott's argument for colonizing space is very
> persuasive, but I don't think there's anything logically
> wrong with it, as far as it goes. And while it's certainly not
> the most important argument in the history of
> mathematics, it's definitely one of the coolest.
But it's bogus. I promised myself I wouldn't do this, but here goes. Imagine you have two urns with balls in them. One of them has 10 balls, and one of them has 1 million balls in it. Further suppose that each ball is stamped with its serial number, so the balls in the "FEW" urn are stamped with the numbers 1, 2, 3...10, while the balls in the "MANY" urn are stamped with the numbers 1, 2, 3....1,000,000. Suppose you are handed a ball and asked to tell which urn it came from; you believe (or are told) that it's just as likely that the picker selected the ball from either urn. So p(FEW)=p(MANY)=0.5. You look at the ball, and note that it has serial number "7". Which urn is this ball more likely to have come from? The answer here is easy to see: the urn with few balls. The probability that you'd draw the ball marked "7" from the FEW urn is 1/10, and the odds you'd draw the ball from the other urn is 1/10^6. This is not controversial.
Now let's suppose that the balls in the two urns are not numbered, but blank. Helpfully, though the person who takes them out stamps a consecutive serial number on the ball. That person gives you the last ball he drew, and asks you which urn he was picking from. You look at the ball, and see the number "7" on it. So which is it? Did it come from the urn with 10 balls, or the urn with 10^6 balls? Obviously, you can't tell, since the one thing you know for certain is that both urns are large enough to give you a ball marked "7", and, given the sampling procedure, both would do so with probability 1 on some draw.
Now lets toss the balls away for a minute and look at the Gott argument. Gott says that to predict how long the human race will last, we should not think we are particularly special, and that we should assume that at some point our species will die out, since that's true for virtually all of the species that have ever lived. If you or I or Gott aren't special, we are probably living at a point that is in the middle 95% of the history of the species, and suggests that we should construct a confidence interval on our extinction that runs from the length we expect if we have 95% run our course to the point where we have 5% run our course. It should be clear that this corresponds fairly to an argument where every human who will ever live has a serial number, and our serial number was randomly drawn from the urn, and so is likely to be somewhere in the middle, so that the highest serial number is likely to be in his confidence interval.
But this argument is silly. If we are serial number 70 billion in the species, then the preceding person was 70 billion 1 and the next born will have serial number 70 billion + 1. In other words, our serial number is NOT a random draw from the urn. All we know is that we were the 70 billionth ball, and so what we know about the number of balls in the urn we were drawn from is that there are at least 70 billion balls in it.
I could go on and on, but I won't. :)
Posted by: Jonathan King on December 16, 2003 01:49 PMMathew Peters's "golf balls in a 747 is a weight problem" reminds me of one of the best old chestnuts: How much does a city weigh?
It weighs nothing, because the weight excavated from the basements is necessarily the same as the weight of the added structures. If it were more, they would sink into the ground, if it were less they would pop up and fall over.
There is a small amount of room for variance from this because of buildings spiked down into bedrock (which are not at all common) and for light wooden structures which can sit up on the surface of the soil.
This latter only works for a few tens of years though. Over the long haul Archimedes rules.
Posted by: David LloydJones on December 16, 2003 01:50 PMI think the best textbooks for math are by:
1) Harold Jacobs. His textbooks are unbelievably good, especially his geometry book(s). He has three books out:
a) Elementary Algebra. (1979) one edition
b) Geometry. Three editions, 1974, 1987, 2003
c) Mathematics: A Human Endeavor. Also three editions, latest 1994
2. Paul Foerster. His books are also very good, though not as "delightful" as Jacobs
a) Elementary Algebra
b) (Advanced) Algebra and Trigonometry
c) Precalculus
d) Calculus
e) Calculus Explorations.
magic penny:
this illustrates how compound growth works. good way for kids to conceptualize long term investing. if you take a penny on april 1 and double it every daya 100% return (so day 2 you have 2 cents, day three 4 cents etc.) you end up with a little under $11 million.
magic penny:
this illustrates how compound growth works. good way for kids to conceptualize long term investing. if you take a penny on april 1 and double it every daya 100% return (so day 2 you have 2 cents, day three 4 cents etc.) you end up with a little under $11 million.
puzzles and curious problems, by Henry Earnest Dudney, edited, Martin Gardner, and more puzzles etc, by HED, ed Gardner. There is the coconut problem,which is most easily solved, by using an imaginary coconut.
Posted by: big al on December 16, 2003 03:38 PMGeorge Polya is great  my favorite of his is "How to Solve It".
Posted by: cafl on December 16, 2003 04:37 PMI forgot to mention phi,the golden ratio, see if the can figure out the math of that. A related problem is to start with two clumns. Put a number at the start of each, best to start small,to get the next numbers in the cols, Col A=previous A + prev B Col B = new A + prev B continue checking new ratio B/A .What is the relation to phi.
1:1
2:3
5:8 etc
Johnathan King  "If we are serial number 70 billion in the species, then the preceding person was 70 billion 1 and the next born will have serial number 70 billion + 1. In other words, our serial number is NOT a random draw from the urn. All we know is that we were the 70 billionth ball, and so what we know about the number of balls in the urn we were drawn from is that there are at least 70 billion balls in it."
But if we're serial number 70 billion, WHY are we 70 billion? It comes after 70 billion1; 1955 came after 1954, and it didn't stop Gott from predicting the fall of the Berlin Wall. Either there's something special about our choice of 70 billion  in which case the Copernican principle doesn't apply  or there isn't, in which case I think we should call it "random". Suppose there either exists, or will at some point exist, a number N which is the total number of balls that will ever exist. Whether we pull these balls out of urns or are assigned them in a series, I think Gott would still say that, in addition to our 100% certainty that N>70 billion, we have a 95% certainty that ~72 billion < N < 2.8 trillion. It's possible that we aren't in there, but then we have done an improbable thing, or else we've biased our choice or assignation such that the argument isn't applicable. This could be done as a thought experiment as well  hand out balls to a series of people, and determine that 95% should have N within their 95% calculated confidence interval, if that works out correctly. It shouldn't matter whether they get them out of urns or in a series. (Of course, there's no reason to think that being born is anything like drawing balls from any number of urns or being assigned it in any sequence, or anything like that  I can't prove or disprove that I would be me if I was born 20 billion people ago, or even say that such an idea makes any sense. But, as far as it goes, it seems consistant.)
On the other hand, you're probably right. If Gott's theory has not been successfully disproven as of 1997, careful application of his own theory reveals that, with 95% certainty, it will be disproven in from 2 months to 240 years from now, and so is almost certainly false. Specious, but comforting. And now my head hurts.
Posted by: Andrew Northrup on December 16, 2003 05:47 PMMath, as opposed to arithmetic, can be lots of fun. I do agree, unfortunately, that most math books are terribly dry.
But the other big problem is that most primary math educators don't (from my childhood experience) really get what makes math fun for mathematicians. They are just teaching methods without any motivation that could appeal to children or adults for that matter. Actually, it is rare to find a mathematics educator below the university level that "gets" it.
I believe that there are two elements that make math enjoyable (a) the pure contemplation of abstract relationships and (b) the excitement of being able to deduce truth about the world through symbols. Arithmetic, and later algebra, is more pertinent to the latter. As a lazy student, my earliest application of "Algebra I" was to figure out the minimum grade I needed to maintain a particular average. Simple, but very exciting! Such power at my fingertips. The former is best conceptualized through puzzles (e.g. Rubik's cube) and is often a popular pastime for those who manage to enjoy it mostly because they never realized they were doing math.
But school math (particularly at the grade school level) was taught to me more as a speed/endurance contest on the handexecution of some specific algorithms (long division for instance). I still cannot imagine what could have made it appealing.
I don't know if I would recommend Polya's How To Solve It in this context. It's also quite dry. But it does have a unique emphasis on heuristic: the art of how mathematician's minds work rather than on how the proof of their theorems can be discerned. I wonder if anyone has had success introducing this book to kids.
The Great Generals Problem: I have no idea if this is apocryphal, but see here for the story about Deming, Fermi, and General Groves (search down):
http://www.endsoftheearth.com/EOEJokes.htm
Fermi problems in general are awfully useful to be able to do: How many gas stations are there in the United States? How many piano tuners in Chicago? How many grains of sand on 'a beach'? etc. People vary widely in whether they think these are *fun* though, but I got a kick out of learning to do arithmetic where 3*3=10 and any other number should be immediately rounded to 1, 3, or 10.
Link to philosophy and physics. Ask them if they know where numbers come from and why we can describe realities with numbers that cannot be experimentally confirmed. Introduce them to Godel’s Incompleteness Theorem and what this meant for Bertrand Russell who was on the verge of tying it all together in a tidy finite little bundle of epistemology. Well, maybe when they’re a little older, but truly the magic of numbers is well hidden in traditional education.
After you or your children have trimmed their respective toe or fingernails ask them to calculate the total weight and/or daily growth of toe and/or fingernails of the total population of the world. Calculating the daily total weight of these two common human traits (let alone our animal friends) begins to draw into focus issues of population and conservation. Extrapolated political implications could emerge but that is probably a
streach. Keep up the good work.
After you or your children have trimmed their respective toe or fingernails ask them to calculate the total weight and/or daily growth of toe and/or fingernails of the total population of the world. Calculating the daily total weight of these two common human traits (let alone our animal friends) begins to draw into focus issues of population and conservation. Extrapolated political implications could emerge but that is probably a
streach. Keep up the good work.
After you or your children have trimmed their respective toe or fingernails ask them to calculate the total weight and/or daily growth of toe and/or fingernails of the total population of the world.This obviously requires measurement and other data. Calculating the daily total weight of these two common human traits (let alone our animal friends) begins to draw into focus issues of population and conservation. Extrapolated political implications could emerge but that is probably a
streach. Keep up the good work.
After you or your children have trimmed their respective toe or fingernails ask them to calculate the total weight and/or daily growth of toe and/or fingernails of the total population of the world.This obviously requires measurement and other data. Calculating the daily total weight of these two common human traits (let alone our animal friends) begins to draw into focus issues of population and conservation. Extrapolated political implications could emerge but that is probably a
streach. Keep up the good work.
If you figure out how to get your (or any) kids interested in math, can you please publish the answer on your blog? Thanks.
Posted by: Scot Johnson on December 16, 2003 08:42 PMPaul: you're quite right about most teachers not knowing math from arithmatic. I had lots of teachers who enjoyed teaching figgerin, but it wasn't until I got to college that I had a _math_ teacher. What a difference that made.
Posted by: Matt on December 16, 2003 09:08 PM(Pardon, I hope this is not a repeat) Estimating problema few years ago, a student of mine was examining the Economist's MacDonald Hamburger Indexthe celebrated measure of purchasing power (dis)parity. In dollar terms, the price of a Big Mac in Shanghai was jawdroppingly cheap. The student pointed out that there must be a quantity at which it would make sense to rent a 747 and fly em in to Seattle. Adding new meaning, as he had to say, to the term "Chinese takeout," oh heh heh.
Posted by: Buce on December 16, 2003 09:34 PMMaybe Lord of the Rings could inspire some math problems:
1. Assuming that each would start from Baraddur and travel alone, and that no two would visit the same city, calculate the routes of the Dark Riders that would take them to every town and major fortification (except for Minas Morgul and Isengard) in MiddleEarth.
2. Assuming that enough data can be extrapolated from the story, calculate the force of the explosion of Orodruin (in megatons).
3. Assuming Gandalf weighs 180 lbs, what is the average flight speed of a laden Gwaihir? (Sorry, I couldn't resist.)
Any other ideas?
Posted by: Alan K. Henderson on December 16, 2003 11:37 PM''If didn't develop Alzheimer's all of a sudden again, my recollection is that Abbot's "Flatland" contained social  political implicaitons that were not too democratic for my taste.''
I´m too young for Alzheimer, but I may be mistaken too when I remember it as a critic of the closedminded culture of Victorian England.
Posted by: Michael Greinecker on December 16, 2003 11:54 PMYou may very well be right, Michael Greinecker. I read it very very quickly about 20 years ago in part out of curiosity in part to please the friend who recommended it, who spoke sort of favorably of some of the arrangements of social hierarchy therein, very rigid like a cast system, no social mobility, and so I may have missed the fact that it was a critic  assuming I recall the book correctly, I think I'll check internet to find out. Unless I come back screaming "oh my god I have Alzheimers!", it is the same book we are talking about.

Oh wow a hundred thousand responses to "REQUEST FOR HELP"! Is this all cheers to you Brad, or some of it is cheers to me because I said I was considering to move on?
Hey folks make it a hundred and fifty thousand responses and you won't hear from me again as of that moment  well maybe except under pseudonym!
Posted by: Bulent Sayin on December 17, 2003 01:12 AMI started drawing up a short list of suggestions, and it turned into a long list, so I put it on my own blog. So go read it there:
http://www.steelypips.org/principles/2003_12_14_principlearchive.php#107162833445843012
Posted by: Chad Orzel on December 17, 2003 05:00 AMYour kids are right. Math between arithmetic and calculus is boring. I suggest you (quickly) teach them quadratics and other polynomials and then go straight to the function concept and then to the differential calculus of polynomials. With this you can teach them about motion with constant acceleration as well as some economics.This will give them a taste of the usefulness of math. You can backfill the rest of algebra and trig later. I tried this with my twelveyear old nephew and it worked.
Posted by: J Rossi on December 17, 2003 09:34 AMMatt "Paul: you're quite right about most teachers not knowing math from arithmatic."
It's far worse than that. I believe that few really get the fundamental distinction between a mathematical entity and its representatione.g., the difference between a number and its representation as a string of decimal digits. This despite all the sets/counting/romannumeral/otherbases lessons that ostensibly hammer home this point but are usually taught without any motivation whatsoever.
As for the difference between a mathematical operation (e.g. addition) and an algorithm to compute it ("5+8 makes 3, carry the 1...") I'm not certain the subject is adequately dealt with until graduate school (joking a little).
Posted by: Paul Callahan on December 17, 2003 10:54 AMOne aspect of mathematics as a toool of truth is its use in the construction of illustrative graphics.
The excellent Edward Tufte has an index of his Graph of the Day series at http://www.edwardtufte.com/tufte/newet. The whole page is full of good stuff. Graphs of the Day are about halfway (0.5 plus or minus 0.15) down the page.
Brad,
Great post, even greater comments, and a wonderful gift to your children!
Turn your kids onto "Flatland: a romance in many dimensions." Edwin (?) Abbot (unsure of spelling, google it and enjoy!)
Mathematics is the language of relationships. It is used when words and pictures are not sufficient.
Arithmetic (which nearly killed my interest at an early age) is to mathematics as grammar is to literature. Good for you for introducing vectors and other more interesting and engaging elements of the field. I strongly agree with other commenters that some elementary calculus should be introduced ASAP (do you kids know that your car's speedometer is a derivaitve/differential machine? that cubist painting represents an artists rendering of the special theory of relativity?)
I'll keep my eye out for interesting problems.
Good luck,
Dave
Posted by: David Rankin on December 17, 2003 12:56 PMHere's one for the "Star Trek" fan in everyone:
The two principal ways in which the Earth's surface
is shielded from cosmic rays and the solar wind is
by the Earth's magnetic field and its atmosphere.
(a). Atmospheric pressure is usually given as
14.7 psi, with 0.4536 kilos to the pound and
0.0254 meters to the inch. What is the mass
of atmosphere above 1 square meter of the
Earth?
answer: 14.7 psi * 0.4536 kilo/lb *
(1/0.0254 in/m)^2 = 10335 kilo/m^2.
(b). The "saucer section" (or "primary hull") of
the USS Enterprise from ST:TOS is 417' in
diameter. What is the area of the top or
bottom surface of the primary hull
(neglecting all that fancy contouring)?
answer: area = pi r^2 = pi (D/2)^2. In
feet the area is 136,572 sq ft, or 12,688
sq m.
(c). Assume that the Enterprise requires
radiation shielding at least as good as
what we get from the earth's atmosphere.
What mass of metal is needed on the top
surface of the primary hull?
Answer: 12,688 sq m * 10,335 kilo/m^2 =
131 million kilos or 131,000 metric tons.
Of course, the same weight of shielding is
needed on the bottom surface as well.
(d). Assume that lead shielding is used, with a
density of 11.34 gm/cm^3. What is the
required thickness of the lead shielding?
Answer: 11.34 gm/cm^3 * (100 cm/m)^3 *
(0.001 kilo/gm) = 11,340 kilo/m^3. To get
to 10335 kilo/m^2, you would need
(10335 kilo/m^2)/(11340 kilo/m^3) = 0.91 m,
or about 3 feet of lead.
All that reads a bit like Philip Blait's
which is a very good web site about astronomy. I understand he published books as well.
Astronomy, in fact, apart from P. Blait's work, could be an effective vehicle for teaching math to kiddos, the ones who like astronomy, that is. Just a thought.
Inspired by these comments, I just bought Martin Gardner's "The Colossal Book of Mathematics" at the local bookstore. It covers a lot of the topics mentioned here. My own kids are still a little young for most of it, but I'll enjoy it myself, dammit!
Someone above brought up the topic of the growth rate of fingernails. I have read that the San Andreas fault moves (averaged over the long term) at basically the same rate that fingernails grow. Given this, and that LA is on the west side of the fault, and SF on the east, how long until they are side by side?
Paul, Re: "I believe that few really get the fundamental distinction between a mathematical entity and its representatione.g., the difference between a number and its representation as a string of decimal digits."
Funny you mention this. My aforementioned first math teacher was one Don Tucker at the University of Utah. Early in his calculus course, he taught us how to count, introducing the idea that a natural number is an equivalence class of sets. Who would have thought that learning calculus would involve relearning to count? But Tucker knew he had a lot of cleanup to do.
Tucker was universally acknowledged great teacher and a real character. In his office, he once pointed to university wide teaching award, a plaque mounted with a purple arrow, and said "The university president has given me the purple shaft."
Darrell Huff of "How To Lie With Statistics" fame has written a dynamite tome for examples of an interesting nature called "The Complete How to Figure It." Pick the nature of the calculation and there's an example to lead your imagination.
Posted by: pt martin on December 17, 2003 08:31 PMSome good math Etc. in no particular order:
1. Suppose it were to rain for 40 days and 40 nights at a constant rate, ignoring that the earth is a oblate spheroid not a sphere, and ignoring land masses, at what rate would it rain (stopping at exactly 15ft above mt everest on the last second of the last day)?
How Much water is that?
2. Why is this false:
a + b = c
(a + b) * (a  c) = c * (a  c)
a^2 + ab  ac  cb = ca  c^2
a^2 + ab  ac = ca + cb  c^2
a * (a + b  c) = c * (a + b  c)
a = c
3. There are 3 black hats and 2 white hats in a container. 3 Blindfolded men come and pick a hat randomly and put it on. Taking off the blindfolds man A can see man B and man C, man B can see man C, and man C can see noone. Asked man A do you know which color hat you have, replies he, nay. Asked man B do you know which color hat you have, replies he, nyet. Asked man C do you know which color hat you have, replies he, Aye. Which color hat does Man C wear and how does he know that?
4. 1/9 = .11111111...
.11111111... * 9 = .999999999... = 1
5. Hexaflexagons. Hexahexaflexagons.
6. John Conways Life.
7. Phi the golden mean = 1.61803398875
1 / Phi = 0.61803398875
Phi^2 = 2.61803398875
phi = sqroot(1 + .25) + sqroot(.25)
1 / phi = sqroot(1 + .25)  sqroot(.25)
'In a random sequence of 1's and 0's, which fourbit strings appear soonest on average, and which take the longest? Why?'
Of course 0001, 1000, and their complements show up fastest, but I've found even people very good at math who didn't believe it.
Posted by: Thompson on December 18, 2003 01:43 PMIn animation, I use lots of math. Imagine you are making a 3 minute animation. At 30 frames a second, the rate of video, you have to determine how many frames each specific scene or action will need to be to fit the timeline. If I am doing something in time to music I have to calculate where to make the changes with the beats by converting the beats per minute into frames per beat. The entirety of any animation is structured using math. And 3D software such as Maya is lots and lots of complicated math. CG characters such as Shrek and Nemo are made out of...geometry.
I think Maya has a Personal Learning Edition available on their website for free download.
You can also rent movies with 3D animation and look at the special features which often show how computers are used to make these wonders. And of course beneath all the interfaces, it's all math.
So, animation can be used as a way to interest kids in math.
Posted by: Elli Mayhem on December 18, 2003 10:43 PMAlmost everything I could brainstorm has already been mentioned so I'm going to try to think laterally: what would I, as a mathematiciantobe, have wanted to have been exposed to in my youth?
Elementary computability theory is surprisingly easy to teach nowadays: in particular, you can get to the halting problem without much difficulty. After that, it's a little hard to prove anything without recourse to the uber Black Magic Theorem: the smn theorem (also known as the Parameter Theorem to the cognoscenti, or the S&M theorem to the bored). If you can somehow justify that, though, you can get to the Recursion Theorem and some truly bizarre results.
[Relativizations are easy to explain conceptually, but you can't really *do* anything with them at this level.]
Smullyan has a great exposition of the Godel Incompleteness Theorem in one of his books (I think it's the ending of What Is The Name Of This Book?), if you're interested in other logical techniques. He and Martin Gardner are on my list of essentials in this domain.
I second the motion above to do some elementary group theory. Lagrange's theorem is about the limit of what you can expect at this level, but there are some nifty proofs (don't remember what, sorry) that can be done to show this.
If you can do group theory, you can also do basic stuff on the fundamental group: How do you show a circle is not a line? Again, details have to be foregone but the basic concepts are pretty straightforward. Once you have that, you can move on to algebraic topology: Brouwer's Theorem, the Hairy Ball theorem, the fact that at any given moment in time there are two places with the same temperature and pressure, &c
There's also a nifty video out there somewhere of the inversion of the sphere, but I don't know where offhand; anyone else know?
The Cantor set (and other selfsimilar structures) is an excellent thing to get into. You can't really describe it as "a perfect set of measure zero", but saying it's a set of noninteger dimension is just damn cool. You can also throw in the Koch curve  what is its area versus what is its length is an *excellent* use of convergent/divergent geometric series  the Sierpinski gasket &c
[I wouldn't advise trying to *prove* fractional dimensionality, though; I'm not convinced that the essential mathematical concepts can (or should) be mastered at this level.]
[You can also mention the CantorLebesgue function if you want to thoroughly weird them out ;)]
You can certainly give them Cantor's proof of the uncountability of the reals; it gets really trippy when you show the countability of the rationals, too. But there are similar results from the early days of set theory that are also accessible: settheoretic notions of cardinality, the BernsteinSchroder theorem (which is completely elementary, although tricksy), the Russell paradox, variants of BuraliForti (what is the size of the universe?) and so forth.
[Note to Andrew Northrup: ...although some people will tell you it isn't really math, because proof by induction isn't really proof. It's true that there are such people, but that's irrelevant: this is a proof by contradiction, not induction, and the people who dispute such proofs can make some claim to be legitimate mathematicians. Not that I agree with them, but at least Intuitionism/Constructivism is more or less consistent.]
Once you have the uncountability of the reals, you can start looking at transcendental numbers. I think you can even squeak through Liouville's Theorem (the transcendental one, not the complex or differential one) with an application of the pigeonhole principle, although I confess it's been a while since I've examined the proof. If you get that, or at least a variant of that, you can start doing continued fraction approximants and an explicit construction of a transcendental number.
[To really make that cool, though, you'll have to explain algebraic numbers first, which could be a bit tricky. Certainly basic field theory isn't that hard  start out with "clock arithmetic" and go from there  but to motivate algebraic extensions you'll need to introduce polynomials as objects of interest in their own right.]
Finally, if you're *really* into messing with their heads  and what good father isn't?  you might be able to start looking at limitations of axiomatization. Again, the proofs of much of these will be beyond the grasp of your kids... but nonstandard models of arithmetic are a wild and woolly place, and quite fun. [Again, "infinite integers" just rates high on the coolometer.] In particular, you can use this to motivate the completeness axiom of the reals (how the heck can we stop this from happening?), or just junk our usual notion of number and look at nonstandard models of analysis, which dovetails nicely back into the surreal numbers.
[See both Knuth's Surreal Numbers, Conway's On Numbers and Games, and, if you're more technically inclined, Goldblatt's Lectures on the Hyperreals for details.]
Oh, and hey... how about that Axiom of Choice and Continuum Hypothesis? Godel's L, anyone? ;)
Posted by: Anarch on December 19, 2003 01:05 AM{Let's see if this works this time}
Almost everything I could brainstorm has already been mentioned so I'm going to try to think laterally: what would I, as a mathematiciantobe, have wanted to have been exposed to in my youth?
Elementary computability theory is surprisingly easy to teach nowadays: in particular, you can get to the halting problem without much difficulty. After that, it's a little hard to prove anything without recourse to the uber Black Magic Theorem: the smn theorem (also known as the Parameter Theorem to the cognoscenti, or the S&M theorem to the bored). If you can somehow justify that, though, you can get to the Recursion Theorem and some truly bizarre results.
[Relativizations are easy to explain conceptually, but you can't really *do* anything with them at this level.]
Smullyan has a great exposition of the Godel Incompleteness Theorem in one of his books (I think it's the ending of What Is The Name Of This Book?), if you're interested in other logical techniques. He and Martin Gardner are on my list of essentials in this domain.
I second the motion above to do some elementary group theory. Lagrange's theorem is about the limit of what you can expect at this level, but there are some nifty proofs (don't remember what, sorry) that can be done to show this.
If you can do group theory, you can also do basic stuff on the fundamental group: How do you show a circle is not a line? Again, details have to be foregone but the basic concepts are pretty straightforward. Once you have that, you can move on to algebraic topology: Brouwer's Theorem, the Hairy Ball theorem, the fact that at any given moment in time there are two places with the same temperature and pressure, &c
There's also a nifty video out there somewhere of the inversion of the sphere, but I don't know where offhand; anyone else know?
The Cantor set (and other selfsimilar structures) is an excellent thing to get into. You can't really describe it as "a perfect set of measure zero", but saying it's a set of noninteger dimension is just damn cool. You can also throw in the Koch curve  what is its area versus what is its length is an *excellent* use of convergent/divergent geometric series  the Sierpinski gasket &c
[I wouldn't advise trying to *prove* fractional dimensionality, though; I'm not convinced that the essential mathematical concepts can (or should) be mastered at this level.]
[You can also mention the CantorLebesgue function if you want to thoroughly weird them out ;)]
You can certainly give them Cantor's proof of the uncountability of the reals; it gets really trippy when you show the countability of the rationals, too. But there are similar results from the early days of set theory that are also accessible: settheoretic notions of cardinality, the BernsteinSchroder theorem (which is completely elementary, although tricksy), the Russell paradox, variants of BuraliForti (what is the size of the universe?) and so forth.
[Note to Andrew Northrup: ...although some people will tell you it isn't really math, because proof by induction isn't really proof. It's true that there are such people, but that's irrelevant: this is a proof by contradiction, not induction, and the people who dispute such proofs can make some claim to be legitimate mathematicians. Not that I agree with them, but at least Intuitionism/Constructivism is more or less consistent.]
Once you have the uncountability of the reals, you can start looking at transcendental numbers. I think you can even squeak through Liouville's Theorem (the transcendental one, not the complex or differential one) with an application of the pigeonhole principle, although I confess it's been a while since I've examined the proof. If you get that, or at least a variant of that, you can start doing continued fraction approximants and an explicit construction of a transcendental number.
[To really make that cool, though, you'll have to explain algebraic numbers first, which could be a bit tricky. Certainly basic field theory isn't that hard  start out with "clock arithmetic" and go from there  but to motivate algebraic extensions you'll need to introduce polynomials as objects of interest in their own right.]
Finally, if you're *really* into messing with their heads  and what good father isn't?  you might be able to start looking at limitations of axiomatization. Again, the proofs of much of these will be beyond the grasp of your kids... but nonstandard models of arithmetic are a wild and woolly place, and quite fun. [Again, "infinite integers" just rates high on the coolometer.] In particular, you can use this to motivate the completeness axiom of the reals (how the heck can we stop this from happening?), or just junk our usual notion of number and look at nonstandard models of analysis, which dovetails nicely back into the surreal numbers.
[See both Knuth's Surreal Numbers, Conway's On Numbers and Games, and, if you're more technically inclined, Goldblatt's Lectures on the Hyperreals for details.]
Oh, and hey... how about that Axiom of Choice and Continuum Hypothesis? Godel's L, anyone? ;)
Posted by: Anarch on December 19, 2003 01:08 AMgrrrrr... stupid doubleposting. I swear, it wasn't there when I checked ten minutes ago!
Forgot: there's also a nifty geometric proof that the Minkowski sum of the Cantor set with itself is the whole interval [0, 2].
And here's a link on sphere eversion (not inversion, as I mistakenly said above):
Sphere eversion
There's a video called "Outside In" that shows explicitly how the sphere eversion works, but I haven't seen it yet, and the one link I found to it is busted.
Posted by: Anarch on December 19, 2003 02:45 AMGee, how about "100 Great Problems of Elementary Mathematics":
http://store.yahoo.com/doverpublications/0486613488.html
Posted by: mdflood on December 19, 2003 08:44 AMGee, how about "100 Great Problems of Elementary Mathematics":
http://store.yahoo.com/doverpublications/0486613488.html
Posted by: mdflood on December 19, 2003 08:49 AMGee, how about "100 Great Problems of Elementary Mathematics":
http://store.yahoo.com/doverpublications/0486613488.html
Posted by: mdflood on December 19, 2003 10:48 AMGee, how about "100 Great Problems of Elementary Mathematics":
http://store.yahoo.com/doverpublications/0486613488.html
Posted by: mdflood on December 19, 2003 10:51 AMAnd, of course, there is always the yet unproven Riemann Hypothesis, which, in the opinion of one mathematician, represents the solution to a matrix that corresponds to a quantum system whose classical dynamics are chaotic, unlike our own, whose classical dynamics are deterministic.
I'll add my hopefully distinct 2 cents, but I'd like to know how your "children get what the payoff to writing well ...". You summarize the utility precisely as I now see it. I didn't get it growing up and now suffer. I did enjoy math and now benefit. I want my children to benefit from both. So if you can, please reply with or post some guidance.
My two cents.
I'd emphasize funtions, plots, sets and charts since they are fun and visual. (There are now lots of applets and such, e.g. Mathcad & Xcell, that can be used for demonstrations.) The usefulness can be shown in solving money problems and exposing "sucker bets". Eventually tie as much together as possible, e.g. "e" is related to trigonometry and subsequently complex numbers, Fourier/Laplace Transforms, linear predictive filters ...
Schools do a horrible job of teaching this stuff in a coherent fashion  I've long thought this deserves a good book or two.
Posted by: apav on December 23, 2003 12:49 AMThis belongs on the page with item #12, but I wasn't able to post it as a comment there:
Once you have solved the challenge posted by Matt (a different Matt) on 2/5/03, you can fairly easily derive the rule for how many digits repeat in a repeating decimal. Here is the rule; it should be fairly easy to see why it works.
Let x/y be the (reduced) fraction to be represented as a decimal. If y has no prime factors other than 2 and 5 (including the trivial case y = 1), there is no repetition. If y is a prime number other than 2 and 5, the number of repeating digits is equal to the smallest n for which y divides 10^n  1. If y is composite, and has prime factors other than 2 and 5, let P be the set of such prime factors, and let N be the set of repetition intervals n determined by applying the rule of the preceding sentence to each p in P (substituting p for y); the number of repeating digits is then equal to max(N).
Posted by: Matt on December 30, 2003 10:34 AMThis belongs on the page with item #12, but I wasn't able to post it as a comment there:
Once you have solved the challenge posted by Matt (a different Matt) on 2/5/03, you can fairly easily derive the rule for how many digits repeat in a repeating decimal. Here is the rule; it should be fairly easy to see why it works.
Let x/y be the (reduced) fraction to be represented as a decimal. If y has no prime factors other than 2 and 5 (including the trivial case y = 1), there is no repetition. If y is a prime number other than 2 and 5, the number of repeating digits is equal to the smallest n for which y divides 10^n  1. If y is composite, and has prime factors other than 2 and 5, let P be the set of such prime factors, and let N be the set of repetition intervals n determined by applying the rule of the preceding sentence to each p in P (substituting p for y); the number of repeating digits is then equal to max(N).
Posted by: Matt on December 30, 2003 10:37 AM