More on Al's Problem

There were a few questions about the problem in my last blog... As I stated I read it in a very old math prize competition, so I am as subject to misinterpret them as anyone else...but since it is MY blog... here is how I interpret it...
The first group walks up to a pile with T cannonballs, removes 72, then takes one-ninth of the (T-72) remaining cannonballs. Subsequent groups do the same with the remaining piles and the numbers assigned.

I will leave the question hanging another day ..but for those who need more... here is a second question if you have solved the first, and await furthur challenges... I think it should be possible to change the pattern of values given to the first few people and change the number of divisions that select cannon balls to any value less than the denominator of the fractional part (9 in this case)..

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I did the simplest case, two divisions, and found it can be done if the first division takes 63 cannonballs plus 1/9 of the remainder. Then the second division will take what is left (which for the number I used for the total works out to two even shares)... This is NOT the smallest possible number of cannonballs, but that gives away too much.

How about three? Well working backwards from the solution for two, I realized that three would work if the first took 54 + 1/9 of the remainder, and the second took 63 + 1/9 of the remainder... the same pattern for larger numbers of detachments could start easily at 27, 36, 45, etc... and it seems as if it would be possible (and actually somewhat simple if I really understand) to generate a pattern for any number of divisions and any fraction we care to use in place of 1/9. Maybe if there is interest, I will address it later

A Problem for Al

I went into math education later than many, and was disappointed to find that math teachers do not, in general, love to do math problems. So I was fortunate that when I landed in Misawa, Japan to teach, they popped me right next door to an old veteran who loved problems as much as I did, and was very demanding about a clear proof and explanation. Today, after twenty years and thousands of miles between us, Al Harmon is still one of my math correspondents and an educational inspiration. He frequently sends me problems he is working on that he doesn't see a solution, or sometimes just doesn't think he has the most elegant solution. I am always flattered that he would think I could solve a problem that he couldn't,

I have another great internet advisor, Dave Renfro, who is no longer employed as a professor, but who still teaches many of us with his posts and references to old journals. Along the way he took to sharing with me the journals he thought might help me learn a little more about math, education, and their history. A few weeks ago he sent me a stack of reprints from the Philosophical Magazine from around 1825-1830. one insert included the "Examination of the scholars at Wyke-House" , a school in Middlesex, England. The scholars were tested on questions ranging from arithmetic to geometry and trigonometry, and took six days to answer the seventy-four questions. The image at top shows the prize medal presented to the winner. The first question this year asks for the student to "Numerate and "point off in periods, half-periods, etc... the numeral 123456789012345678901234567890 (raise your hand if you know what a half-period is.... and if not, look here).... and the last question asks for a proof that x+1/x = 2 cos(theta) could only have solutions when theta was a multple of 180^o....several asked for proofs of geometric or trigonometric identities.

One caught my eye as the type of problem Al might love, so I wanted to share it here, with a thanks to both Al and Dave for their continuing contribution to my education.

Problem 51, from the 1827 examination: Several detachments of artillery divided a certain number of cannon balls. The first company took 72 and 1/9 of the remainder; The second detachment 144 and 1/9 of the remainder. The third company took 216 and 1/9 of the remainder; the fourth company took 288 + 1/9 of the remainder; and so on; ---- finally it was discovered by the commanding officer commanding the brigade of guns, the the shot had been equally divided. Determine the number of detachments and the number of balls in the pile.

Ok, So the first answer (with explanation) wins a pat on the back and high praise....... I noticed an interesting pattern when solving this problem, so there is a nice generalization to any number of regiments... For instance I could write a similar problem for any given number of regiments.... . Will provide that later, if needed.

The Earth and the Turtle

A recent computer science blog reminded me of the wonderful story about what holds the earth in place. There are many versions, but the one I like best, as told by the folks at Wikipedia, comes from Stephen Hawking:

The most widely known version appears in Stephen Hawking's 1988 book A Brief History of Time, which starts:


“ A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise." The scientist gave a superior smile before replying, "What is the tortoise standing on?" "You're very clever, young man, very clever," said the old lady. "But it's turtles all the way down!"

Whether you see it as deeply metaphysical, or just poking fun at physics, you can still read it and smile.

WHY, we Flip and Multiply

A recent anonymous comment to my blog on "Division of Fractions by the Alien Method" wrote:

"This is really cool! And you are correct...I (like most other 5th graders a long time ago) memorized a method to divide fractions...I believe the mantra was "Yours is not to reason why, just invert and multiply"....or another one that students seem to use is "Keep, change, re-arrange". Although I can certainly perform the operation, and can even ask the question posed by 2/3 divided by 5/7 (If we think of it as a piece of wood with length 2/3, then I believe the question is how many 5/7's are there in the piece of wood). Unfortunately, that doesn't tell me "why" inverting the second fraction and then multiplying works....I've asked several very bright people and have never gotten an answer that sticks...can you enlighten????"

I want to make one comment about division of fractions that seems harder to visulaize than for general division, and then I hope to explain in simple terms just why "invert and multiply" works.

For every multiplication problem, there are two associated division problems; A x B = C begets C/A=B and C/B=A. Elementary teachers call these a "family of facts for C" (or did in the recent past.. educational language changes too fast for firm statments by a non-elementary teacher). So if we add units to one or both factors, appropriate units must be appended to the product. So how does this effect operations with fractions? Well if we have length, as in ANON's comment, then the division problem he states, "If we think of it as a piece of wood with length 2/3, then I believe the question is how many 5/7's are there in the piece of wood" he is dividing length by length to get a pure scaler counting how many pieces (or fractions of a piece) will fit into another. In the case he gives, the answer would be only 14/15 of a piece... becuase the 2/3 unit length is not quite enough to provide a 5/7 unit length piece...

The multiplication associated with this operation is then 14/15 of 5/7 units = 2/3 units... What about the other division in this family of facts... 2/3 units divided by 14/15 (a scaler here, not a length)will give 5/7 units length. What is this sitution describing? This seems the one most difficult for teachers and students alike. We all know what it means to divide a length into (by?) two pieces, but what sense does it make to divide it into 1/2 a piece.

We might try to make this clear to students by taking some common length (12 inches?) and see what happens if we divide it into (by) 8 pieces, then four, then two, then one, (each division is by half the previousl number)and look at the pattern of lengths. 12/8=3/2; 12/4 = 3; 12/2 = 6; 12/1= 12... I am confident most students could identify the next numbers in the sequence, 12/ (1/2) = 24, and 12/(1/4) = 48.

At this point, using whole numbers as divisors, the pattern for "invert and multiply" seems obvious, but this is far from a why for all fraction problems.

Let's look at one more case where we sneak in a related idea at the elementary level. Given a problem like 3.5 divided by .04, the student is taught to "move the decimal places enough to make the divisor (.04) a whole number. What we do is another problem (350 divided by 4) that has the same answer (87.5)as the original. Another why does that work that is not often explained.

What do the two operations have in common.... multiplication by one. In each case we have a division (fraction) operation and we simply mulitiply the fraction by a carefully chosen version of one that will make it easier to do. If we view 3.5/.04 as a fraction, then every fifth grader knows that multipliying it by one will not change its value. This is the core of what we do to find equivalent fractions... to get 3/5 = 6/10 we multiply by one, but expressed as 2/2... The decimal division problem uses the same approach... we multiply 3.5/.04 by 100/100 to get another name for the same fraction, 350/4.

Now to explain "invert and multiply" we just use the same idea... dividing fractions is simply fractions which have fractions instead of integers in the numerator and denominator. We want to multiply by one in a way that the division problem will be easier. But the easiest number to divide by is one,... so why not pick a number that changes the denominator of the fraction over a fraction to be a one... that is, multiply by its reciprocal. So for 2/3 divided by 5/7 we can write

And I hope that makes it clear.... questions and comments are gratefully received