Friday 30 May 2008

If you needed a reason ...

If you needed a reason to feel good today, this could be it.

Ok, it’s just getting to senior exams, and getting anything out of a senior right now is like squeezing Ketchup out of a Heinz bottle, you're not sure what you get out is worth the effort you put in. You are starting to think the next generation just isn’t going to make it…. Then you hear about someone like Neil Sauter of Blissfield, Michigan. ...Ok, I never would have known.. I keep my head well buried in my math books and never see the news, but my wife calls from the staes and says, “Hey, you’ll never believe the kid I just met” And tells me the story.

Neil is 25, and when he graduated from Grand Valley State University in 2006, he was looking for adventure. The Peace Corps sounded good, being the kind of kid who wanted to make a difference in the world AND have some adventure; but they turned him down. (all the kids go “ahhhhhh”). Neil has cerebral palsy, and it has effected his legs, so he has trouble walking… pay attention folks, he has trouble walking, tightness in the thighs, ankles turning in… walking is HARD…pay attention.. it matters in the story.

So Neil decides that he wants to pick his own challenge, something where no one else can tell him “No!”. He decides, in his words, “I would take a stand for people with disabilities right here in Michigan." How??? Well, for starters he decides to walk across Michigan to promote Cerebral Palsy…. No… you still don’t get it…

First, when Neil says across michigan, he doesn’t mean, like, go from Detroit over to Muskegon or something… no ticky tacky cake walk across the bottom of the state for this boy… he is going to walk all the way up the lower Peninsula to the Big Mac Bridge, then across the Upper Penisula to the Wisconsin Border…. NOOOOO wait.. you still don’t get it.. The kid with CP in his legs is going to walk the LONG way across Michigan, 830 miles, ____ ON STILTS____

Ok, so now enter my wife and her group of friends. You see, while I take care of all the hard stuff, like teaching kids math, Jeannie has the easy job of bringing peace into the world and feeding the hungry… Every week she and her friends get together and talk about how to make the world a better place… what THEY can do..Thinking globally, acting locally; trying to get people to realize how powerful they can be, and one of them decides that Neil is a guy with a story to tell and they invite him to attend. Neil gets an evening of entertainment with some beautiful ladies(mostly,except for Joey and Cleve), and I get to hear a story that pulls me out of my bout of self-pity.

Neil is off to Norwood today, another 24 miles to notch off, but the last bit along Eagle Highway is beautiful, and then next week he is off to the Bridge. Along the way he is raising money for a fund he created that buys the tools and technology to allow CP sufferers to live a fuller life. He even has a grant from the government to match what he raises when he reaches a specific leve. He could use a hand.. so if you are camped out somewhere across the Upper Peninsula of Michigan and you were just thinking,now what could we do for excitement this week, then think… “Hey, we could do a fundraiser for CP and pass it on to Neil when he comes by”..Or maybe just buy him a burger, the boy looks a little thin in the picture. Maybe you would want to walk a few miles with him for company.. not much between all those pine trees up there, and I understand the boy is an amusing and interesting story teller, and a really talented juggler.

And if you don’t live along the way in the UP, and not many of you do, you can make a donation on his web site, or mail a check to:
Fresh Earth Peace Project
Box 402
Elk Rapids, Michigan

and we will get it to him. Either way it is totally tax deductable..and maybe matched by the government.

Ahh, Kids today… don’t they make you proud? Good luck Neil.. Left, right, left, right..

Wednesday 28 May 2008

Taking things to a New (and some old) Length

While reading "Zero to Lazy Eight" by Alex and Nicholas Humez and Joseph Maguire, I came across the unusual measure of length called a smoot. I looked it up on Wikipedia and found this:

"The smoot is a nonstandard unit of length created as part of a Massachusetts Institute of Technology (MIT) fraternity prank. It is named after Oliver R. Smoot,(class of 1962), an MIT fraternity pledge to Lambda Chi Alpha, who in October 1958 was used by his fraternity brothers to measure the length of the Harvard Bridge between Boston and Cambridge, Massachusetts.
One smoot is equal to his height (five feet and seven inches ~1.70 m), and the bridge's length was measured to be 364.4 smoots (620.1 m) plus or minus one ear, with the "plus or minus" intended to express uncertainty of measurement. Over the years the "or minus" portion has gone astray in many citations, including the
Oliver Smoot ’62 is used to measure the bridge in 1958.

commemorative plaque at the site itself. Smoot repeatedly lay down on the bridge, let his companions mark his new position in chalk or paint, and then got up again. Eventually, he tired from all this exercise and was thereafter carried by the fraternity brothers to each new position. Everyone walking across the bridge today sees painted markings indicating how many smoots there are from where the sidewalk begins on the Boston river bank. The marks are repainted each year by the incoming associate member class (similar to pledge class) of Lambda Chi Alpha.
(Image from *

Markings typically appear every 10 smoots, but additional marks appear at other numbers in between. For example, the 70-smoot mark is omitted in favor of a mark for 69 (boys will be boys!) The 182.2-smoot mark is accompanied by the words "Halfway to Hell" and an arrow pointing towards MIT. Each class also paints a special mark for their graduating year.
The markings have become well-accepted by the public, to the point that during the bridge renovations that occurred in the 1980s, the Cambridge Police department requested that the markings be maintained, since they had become useful for identifying the location of accidents on the bridge.The renovations went one better, by scoring the concrete surface of the sidewalk on the bridge at 5 feet and 7 inch intervals, instead of the conventional six feet.

Google Calculator also incorporates smoots, which it reckons at exactly 67 inches (1.7018 meters). Google also uses the smoot as an optional unit of measurement in their Google Earth software.

Oliver Smoot later became Chairman of the American National Standards Institute (ANSI) and President of the International Organization for Standardization (ISO)."  

He returned to MIT on October 4, 2008 for a 50th anniversary celebration,including the installation of a plaque on the bridge. Smoot was also presented with an official unit of measurement: a smoot stick.

Speaking of stange measurements and their origins... two ancient measurements that really have a common origin are the cubit, and the ell. Both, in some form, measure the distance from the elbow to the finger tips. At one time almost every country had an Ell, and almost none of them were the same. During the monarchy of Edward I, he made a law that every town have an ellwand, a rod of length one ell, that was used as an official measure. The cubit is the same measure but the word is from the Latin cubitum (elbow), and is related to the Latin word that means to lie down. By whichever name it is a very old unit of measure and frequently known by a word relating to "forearm" or the elbow, such as, Greek pekhys and Hebrew ammah.

Monday 26 May 2008

The NEW Friendly Numbers

Friendly Numbers

Up until the spring of 2008, if you asked me what "friendly" numbers were, I would refer you to my listing for amicable or aliquot numbers. Two numbers were amicable (friendly) if the sum of the factors of each equaled the other. 220 and 284 are the oldest pair known, and they date back to Pythagoras. The numbers were inscribed on "magic charms" in the middle ages which were sold to insure the fidelity of ones lover. Other stories suggest that the gift of 220 Goats from Jacob to Esau in the Biblical story was an expression of love made significant by the use of one of the pair. The proper divisors of 220, 1, 2, 4, 5, 10, 11, 20, 22, 44,55, and 110, add up to 284 and the same is true the other way around. Until very recently, at least to the best of my knowledge, this was what people meant when they referred to "friendly" numbers. This is how Simon Singh describes them in his book Fermat's Enigma; Western mathematics knew only the one pair until 1636 when Fermat discovered a second pair; 17,296 and 18,416 . This was also how Hoffman described them in his book about Paul Erdos, The Man Who Loved Only Numbers, and how Alfed Posamentier had used the term in Math Charmers: Tantalizing Tidbits of the Mind. These were world class mathematician/authors. I assumed I was in line with the current usage.

Then recently, I came across a reference on Mathworld that described them as the ratio of the sum of all the divisors (including the number itself) divided by the number. For example, 8 can be divided by 1, 2, 4, and itself, 8. The sum of its divisors is 15, so its ratio, is 15/8; this is sometimes called the "abundance" of the number. Do not confuse this with the much older term "abundant" for a number for which the sum of the proper divisors (factors, or divisors not including the number itself) is greater than the number itself.. For example the proper divisors of 8 are 1, 2, and 4, which total 7, so 8 is NOT abundant, but deficient.

The classification of numbers as being deficient (the sum of the proper divisors is LESS than the number), abundant (the sum is greater than the number) and perfect (the sum is equal to the number, as in 6 and 28) goes back at least to Nicomachus (about 100 CE) who separated the even numbers into abundant (it was the mistaken belief for a long time that all odd numbers were deficient) or perfect.

When the ratio of the sum of the numbers divided by the number equals two, the number is perfect, so all perfect numbers are friendly under this new usage. Other numbers that are mutually friendly besides 6, 28 and the rest of the perfect numbers include 30 and 140, with a ratio of 12/5, as well as 80 and 200, whose ratio should be 2.325, if I calculated right. There are many numbers, such as all the primes, that are known to be solitary, that is, they have no friends. All the primes have a ratios of (p+1)/p, so their ratio would get smaller towards a limit of 1 as the size of the prime grew larger. There are other numbers, some relatively small like 10 and 14, that we do not know if they are friendly or solitary. I wondered as I computed these if there is a number or a pair with the greatest ratio? (write if you know, please)

I am not sure, and am presently searching to find the first use of the newer use of friendly numbers, but it seems to exist at least since the 1970's (surprised me!) from a citation on Sloans integer sequence site for ...Anderson, C. W. and Hickerson, D. Problem 6020. "Friendly Integers." Amer. Math. Monthly 84, 65-66, 1977. Ok, if you know about this stuff, drop me a line and set me straight.

Sunday 25 May 2008

Over Protected and Under Educated?

I’ve been thinking a lot about how kids grow up today, and how it impacts on their education, which should not be surprising since teaching is what I do. I’m sure there has never been a generation that didn’t ask that same old question, “What’s the matter with kids today?” Several articles and a couple of TED broadcasts seemed to resonate with what I’ve been thinking lately.

Robert Siegler, a psychology professor at Carnegie Mellon University in Pittsburgh, has done some research that suggests that children’s early concept of a number line is more logarithmic than linear, and that’s bad for math achievement. As they grow older, they usually develop a linear conceptualization of the number line but it happens in bits, and how long it takes to happens seems to be closely related to how well they do later in mathematics. The ones who seem to develop later spend more time with video entertainment while the ones who develop earlier, and therefore have a better chance of being successful in math, are the ones who play card games and board games. There are exceptions to every rule, but most mathematicians tend to have a game aspect to their mathematical learning.

Sir Ken Robinson, is an internationally recognized leader in the development of creativity and innovation. In one of his TED talks, he suggests that creativity is as important in literacy, and we should treat it so. I think he is right, but I think the problem, or at least a part of the problem, is the lifestyle of young children today. Don’t take my word for it, watch the TED video below by Gever Tulley creator of the "Tinkering Schools".
Tulley thinks that our children are over protected, and suggest five dangerous things we should give our children (he admits he has none of his own). I’m going to give my two cents worth about some of these, so if you want to, just skip down and run the video first…

Tulley suggests that you let your children play with fire. He makes some suggestions about what they might pickup, but relapses back to a faith based mantra… You don’t know what they are going to learn, but they are going to learn.
His second dangerous recommendation for children is a pocket knife … OF course they will cut themselves... and while you are wondering if you were a terrible parent to give them something so dangerous, they are proudly peeling back the bandages to show their cut to their friends on top of the garage as they take turns jumping off the roof and onto the limb of the tree a few feet away...(you won't find out until they fall, and incredibly, many of us never do).

I still remember the wisdom of my father-in-law telling me the story of his experience as a child at a carnival near Perry, Michigan… after paying a dime at the tent that said “Life Saving Advice Inside”... they entered a string of other curious visitors. The line passed by an old guy with a pocket knife and a stick that was quickly turning into wood he repeated... "Whittle from you, never cut you”| and through the next tent flap which led... outside... I’m not sure how many times he told me that story in the forty years I’ve known him, but it was more than once for each of those years. Ten cents when you are a child for a story you could still be telling at in your 90's... a bargain...

The third dangerous thing he recommends is a spear. I’m not sure I see the benefit of a spear, but our brains do seem to be wired for throwing things, and when you let one part of the brain go soft, other parts seem to follow. So throw a spear or a baseball, girls and boys NEED to play baseball, even if they don’t compete in organized sports, a game of catch can build both body and mind....Throwing and catching are incredible analytic teaching methods visual acuity, 3 d analysis, and attention and concentration skills.

Number four was to give them your broken appliances to take apart. The land fill can wait a couple of days. Seeing what things look like and trying to figure out what they do is BIG.. and the curiosity is BIGGER..I suspect it develops a belief that they can KNOW difficult things.. and an awareness that it may not come all at once. I worry that kids today don’t seem to persevere at difficult tasks. I suspect that a kid who takes apart a toaster or spin dryer (with parental supervision even) will look at all the “black boxes” he encounters differently. They will always be wondering what might be “in there” and how it could work…. Did I mention that curiosity is GOOD!

And finally, about ten or twelve, let them drive the car. Find a good safe place out in a vacant lot, sit them in your lap if you have to, and let them control the steering wheel and any other parts they can reach while you roar across the lot at 15 mph. Being in charge of this huge piece of moving steel can give them a sense of control… Kids need to feel they have some control over their world… If you really believe that children are the future, you better help them feel they can be in control.

Here are a couple of extras that I think are important for life and math skills… Teach them to play games... NOT video games,, they'll do that on their own.. Teach them to play card games, board games... Monopoly, Parcheesi, Snakes and Ladders are GREAT childhood games, even for kids UNDER 8 years old.. Teach them to juggle, to ride a unicycle, to do card tricks, to play a musical instrument, to speak a foreign language. All those things take discipline and focus, both of which are essential for success in education and life. When my youngest son was learning to play soccer in elementary school, we would go out in the front yard in the evening and take turns seeing who could juggle the ball with our feet for more touches. It only took a month or so and he had gone past the best I would ever do. When a parent watches their kid go past them in anything, it’s a great feeling.. that’s the natural order of the universe, each generation gets a little better and smarter.. I worry if we are doing that lately.

Wednesday 21 May 2008

I will Derive

OK, one for all my weary calculus students, trying to make it to the end of the year after the AP exam... OK, you think the singing is bad, wait 'till you see the dancing.

Tuesday 20 May 2008

Education, Crime, and CCTV

Imagine three towns, all within a few miles of each other, all with approximately the same population and the same distribution of wealth. In a certain year, there were 39 burglaries in the town center of A, only 25 in town B, and 36 in town C. The town council of town A, concerned about their growing reputation as an unsafe place to live, try an experiment with closed circuit TV on the streets of downtown. The two other cities choose not to act.

In the following year, crime drops in town A to only 28 crimes, a reduction of 28% over the previous year. Not only has crime gone down since the cameras were introduced, but in the two neighboring towns crime has increased. Town B has seen a 20% increase to 30 burglaries, and town C is now the crime capital of the tri-cities area. Their burglary total has climbed to 42 for the year, an increase of almost 17%.
Now imagine the town meetings in towns B and C as shopkeepers clamor for the use of CCTV in their downtown areas, and what will be the result? Surely the proof of their effectiveness is clear to everyone. Crime went down in the city that used them, even in the face of a rising crime wave everywhere else in the area.

But then, there is that one nagging doubt; what if the changes from year to year are just the random fluctuations of chance. What if the probability of burglary in all the cities is totally unchanged? And the truth is.... the numbers were randomly created. I had computer software pick 100 numbers randomly with equal probability of being 1, 2, or 3 to represent the burglaries in the three towns. Then, picking the one that was highest (they would be the most likely to adapt a change) I simply repeated the randomization a second time and got the numbers for a second year.

In the statistician's lingo, this is called regression to the mean. The same idea keeps test prep book producers and tutors in the big money. When you go to take courses like the SAT or ACT, there is a certain amount of chance involved, the questions on each form vary slightly, and the topic, or just the wording may favor one student and handicap another. Then most students do a certain amount of guessing on the exams. The probability of a raw guess being right is only 1/5, but if you can eliminate one crazy disclaimer, then you stand to profit. In the end, some kids do better than they expected, and some kids do much worse.

If they all took the test again, the kids who did very much better than they thought, would probably drop back down toward whatever their true ability level was, and the kids who did poorly on the first try would probably go up... but, the kids who did well are probably NOT going to take the test again. Only the ones who did very low compared to their expectations will pay to test again. They will probably buy a review book and may even go so far as to sign up for a tutoring program. Then what happens. BOOM... their score on the second test goes up... some percentage will score MORE than they had reason to expect due to a little luck being on their side now... and off they go to extol the virtues of ACME Study Course...

It isn't exactly fraud, and some may actually teach kids some stuff, but it sort of reminds me of the guy who sends free stock advice to 10,000 people. Half are told that stock A will go UP, the others that it will go down. After a few weeks he is right with half those people, so he sends the "SEE, I told you so." letter to the 5000 he got right. This time it is stock B, and 2500 people are told it will go up, and the other 2500 are told it will drop... and sure enough, he is right on half those. NOW he has them ready, and he offers to send them his weekly tip sheet for ONLY $xx.. for the one year subscription. Well, you think, he has been right twice in a row... better jump on board.

Just a footnote on the CCTV and Crime connection; heard a news interview with a member of one of the British Police force, (don't recall which one) and he asked an interesting question. If you install CCTV and the crime rate increases, does that mean they don't work.... or that they DO? He suggested that many small crimes go unreported because people don't feel the police will be able to do anything, rocks through windows or vandelism in general is an example. But suppose with the presence of CCTV, they think, "Maybe the idiot was caught on camera." The crime gets reported, is still unlikely to be solved, and it looks like both the crime rate and the conviction rate went down. To answer questions about CCTV, he suggests you have to look at crime across the spectrum and learn what kinds of crime CCTV will deter or help bring a conviction, and what it won't. Then he said a most amazing thing... NO ONE knows how many cctv cameras are in Britain... There is a number (in the millions) that floats around that came from a study on three streets in a suburb of London... count the cameras, divide by the population in that area, multiply by the population of Britain... sounds close enough... Ok, there is a lesson in sampling bias in that sentence, find it.

Friday 16 May 2008

The B-2 Theorem

In my B2 Pre-calc class today, we re-discovered a theorem about Pascal's triangle that I had not known. It began, appropriately enough, with a question Jacob C. asked about dealing cards from a standard deck; "How many 13 card hands can be dealt that contain exactly two suits. " As we were working through the problem, I began by attacking the somewhat easier problem, "how many hands can be dealt with only hearts and diamonds, but at least one of each." We began writing out the possibilities of 12 hearts, 1 diamond plus 11 hearts two diamonds ..etc To make life easy, for this short while let’s let (n,r) mean “n choose r” , the combinations of n things taken r at a time. So we needed to find (13,1)(13,12) for the first part, 12 hearts and 1 diamond. Then we needed to add on (13,2)(13,11) + (13,3)(13,10)…. And all the way down to (13,1)(13,12). One of the clever ones quickly realized that each of these pairs were just the same number due to the symmetry of Pascal's triangle, and so we were really looking for (13,1)2 + (13,2)2... etc. While some of the kids were adding these on their calculators, I wrote out several lines of the arithmetic triangle and began to write the sums of squares on the right....

As I wrote the totals of each row, 1, 2, 6, 20, 70.. it struck me that they were all the center number of an even numbered row, (2n,n). I remembered them from working with Catalan’s Numbers (another cool pattern that shows up in Pascal’s triangle). About the time the first students were coming up with an answer, I asked them to check (26, 13) and compare it to the answer they got for the actual squares of the thirteenth row…

Close, but not right, was the reply.... huh??? … , oh yeah, we had avoided the case of (13,0) and (13,13) because we wanted to ignore the case where all were hears or all were diamonds, so the answer to our mini-problem was (26,13) - 2; and the only thing needed to solve the original problem was to multiply by 6, to account for all the ways we could pick two suits to be in the hand out of the four possible suits.

When I showed them the result, and we checked a couple of more cases to be more sure, I admitted that I had never seen this theorem. One kid suggests it should be a test question… I countered with, “and extra credit for the person who comes up with the best name for it. Several played to my ego, “Ballew’s theorem, of course!” but then they thought they might deserve partial credit, and hence the name, B-2 theorem, at the top.

Unfortunately, we were not the first to stumble across this little gem. I haven’t had time to chase it down fully, but it may actually date back to the Chinese around the 12th century. So fame and fortune will have to wait, but when you walk in the footsteps of greatness, you’re taking pretty big steps; so congratulations class, I’m proud of you, and it will always be the B-2 theorem when I teach it. Dennis was going to send me a class picture we took on his phone, so if it turns out, I will add that later,

While I was searching for the history of the sum of the binomial coefficients, I came across another place where the triangle is related to squares. One of those theorems we teach when we get to sequence and series in high school is the sum of the integers, 1 + 2 + 3 + … + n, and the sum of the squares of the integers, 12 + 22 + 32 + ….. + n2. Usually we present the formula for this last without proof since it occurs before they are introduced to inductive proofs. As I was researching I came across this neat little relation to the arithmetic triangles. To find the sum of the squares of the first ten integers, just go down to 10 at (10,1) and turn right and follow the diagonal down two numbers to (12,3) and add this to the number on the diagonal above it (11,3), the sum of 220 + 165 = 385 which is the same as 12 + 22 + … + 102

In general you can find

And I think they can accept that as evidence, at least until we get to inductive proofs.

Wednesday 14 May 2008

l'Hospital's Rule

Back to MathWords

In 1694, Johann Bernoulli wrote a letter to Guillaume-Francois-Antoine de l'Hospital that included the theorem now known as l'Hospital's Rule (The alterante spelling "l'Hôpital" is often used in France) . There is little doubt for most math historians that a) Bernoulli first discovered it, and b) that L'Hospital first published it. l'Hospital is sometimes discredited because he published someone else's theorem, and paid for the privilege. My calulus teacher in college would rant about l'Hospital being an "inept" mathematician and "buying his fame", and gave me the impression that he had published it as if it were his own creation. The truth is, in the 1696 differential calculus book in which he published the theorem, L'Hospital thanks the Bernoulli brothers for their assistance and their discoveries. And in addition, he was far from inept as a mathematician. The MacTutor History of Math site comments that, "L'Hôpital was a very competent mathematician and solved thebrachystochrone problem."

L'Hospital never called the rule by his own name, and in fact, it appears that noone else did for several hundred years. Jeff Miller's web page on the first use of mathematical terms gives the first citation for the use as "de l'Hospital's theorem on indeterminate forms is found in approximately 1904 in the E. R. Hedrick translation of volume I of A Course in Mathematical Analysis by Edouard Goursat. The translation carries the date 1904, although a footnote references a work dated 1905 "

The rule is a method for finding the limiting behavior of a rational function whose numerator and denominator tend to zero at a point.(The rule is somewhat expanded today from its original form and can be used if both functions diverge to infinity also) In a traditional Calculus course, a student might use the rule to find, for example, the limit of the function as the value of x approaches 2. This meets the conditions since both the numerator and denominator approach zero as the value of x approaches two. The rule states that in such cases, one can take the derivative of the numerator and denominator independently and then find the limit of this ratio. Since the derivative of the numerator is 2x, and the derivative of the denominator is 1, the ratio of the derivatives is the experession 2x/1 or just 2x; and the limit as x approaches two for this function is just 2(2) = 4. In the area NEAR x=2, the value of the original ratio is very near 4.

The Stolz-Cesaro Theorem

Until recently I had never heard of this discrete analogue of l'Hospital's rule, and I thank the folks at Topological Musings blog for the lesson. The adjusted rule can apply to sequences (l'Hospital's rule is for continuous functions) under certain circumstances, and allow us to calculate the limit of the ratio of two divergent (they both go to infinity) seqences. If we think of the function above as a sequence in which the numerator ( x2-4) diverges to infinity as x grows larger and larger, and likewise the denominator (x-2) also grows without bound as x goes toward infinity, then the Stolz-Cesaro theorem says that .

So for out example, we need to find ((n+1)2 -4) - (n2-4) for the numerator, and ((n+1)-2)-(n-2) for the denominator. The numerator simplifies to (2n+1) and the denominator to (2n+2).. since these both still meet the conditions of the theorem, we can apply it once more to get 4/2 = 2... Using l'Hosptials rule for the same function as x-> infinity, we get (2x/x) and applying it once more we get two by l'Hospital as well.... (nice if they both get the same limit)..

The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro.

Tuesday 13 May 2008

Average, Percent,

Average, Percent,


Other Misunderstood Math Terms

“Misunderstood? Surely, you jest!” you reply…. But hear me out.. Okay, I agree that almost every seventh grader knows how to compute averages, but the question is, do they understand them… Consider the following:
A school is asked to report the average student/teacher ratio. Now we could do that two ways; the school could count how many students there are, then count how many teachers there are, and in the end, divide the number of students by the number of teachers to get the average… and that is how it is normally done. But you could also have each student count the number of students in each class they go to (assuming one teacher per class) and then average the class sizes reported by all the students. That takes a little more work, but it should give us the average number of students per teacher also… except, in most cases the two numbers will be different.

Let’s illustrate with a simple (no,, I mean REALLY simple) example. Suppose there is a school where there are only two teachers and two classes. The total enrollment is 6 kids in one class and 4 in the other. From the school point of view, there are 10 students and two teachers, so the average is 5 students per teacher. But when we survey the students, the class sizes reported by the ten students is {6,6,6,6,6,6,4,4,4,4} for an average of 5.2 students per teacher. Go ahead, make up your own numbers, but the only way to get it to agree is if EVERY class has the same number of students….and now the big question…. Which is the CORRECT average??? (“pssst… say ‘both’ ”)… But if they are not the same number, then they can’t both be the average of the same thing… Now you start thinking…. Tick… tick… tick… So what is is that each is the average of?

Ok, let’s go to something even easier; fifth grade percentages. Here is the problem:
Three weeks ago 87% of the students were in favor of the new football coach. Then the team lost and now only 67% of the people support the football coach. Ok, So which is correct Newspaper headline:
…..Support for Coach drops 20%
……Support for Coach drops 23%
…….Support for Coach drops 30%

All of them are a valid percent.. but the question with percentages, like the one with averages, and like most of the questions in applied math, resolve down to the object of a preposition…..percent of WHAT?
See, Math really is Hard.

Sunday 11 May 2008

Do We Really Need Scientists to Tell Us This?

Scientific American has finally told us what they must think we were too stupid to figure out on our own. The Platypus, with its wide flat body, and its duck bill, webbed feet, and beaver like tail, is a strange animal... and guess what, its genetic code is strange too. Thank you, Scientific American.

"The platypus (Ornithorhynchus anatinus) is an odd-looking creature ... A new study indicates that the distinctive mammal's genetic code is an eclectic brew of bird, reptile and mammal."

Great, A screwy looking animal has screwy genes... "Who would a thunk it".

Ok, then I'm clicking through the BBC (gotta love the Brits) and found this headline for the collection:

Great tits cope well with warming
By Richard Black
Environment correspondent, BBC News website

Honest. They even have a picture

Saturday 10 May 2008

Trust Me, I’m on NPR

“A few weeks ago, devoted listeners of National Public Radio were treated to an episode of the award-winning radio series The Infinite Mind called "Prozac Nation: Revisited." The segment featured four prestigious medical experts discussing the controversial link between antidepressants and suicide. In their considered opinions, all four said that worries about the drugs have been overblown. “ Ok, that’s good right, I mean, to most of us National Public Radio is the most trustworthy source out there. After all these are our kind of people, right? Maybe the Republicans think they are a little too liberal, but that’s just because they are out there presenting an honest evaluation of the issues, right? I mean that is right, isn’t it??? Ummm but then… I read this article on which follows the quote above with the notice that, “All four of the experts on the show, including Goodwin, have financial ties to the makers of antidepressants. Also unmentioned were the ‘unrestricted grants’ that The Infinite Mind has received from drug makers, including Eli Lilly, the manufacturer of the antidepressant Prozac.”
Now just because they take big bucks from the drug companies didn’t mean they lied, does it? I mean if they were up front about their connections, they are still credible experts who have meaningful opinions. But then I read, “The Infinite Mind's Web site states, "Our independence is perhaps our greatest asset." Perhaps, indeed. Neither Goodwin nor the show's producers responded to our repeated requests for interviews and queries about their funding.”
Ok, so maybe it’s just a one-off as the British say, something that slipped by NPR on this one show…. Well, not so says the article…It is all over the media. “Gary Schwitzer, a professor of journalism at the University of Minnesota, is the publisher of, a Web site that reviews health care news for balance, accuracy, and completeness. Schwitzer and his team of reviewers have looked at 544 stories from top outlets over the two-year period from April 2006 to April 2008. Journalists had to meet several criteria in order to receive a satisfactory score, among them: They had to quote an independent expert—someone not involved in the relevant research—and they had to make some attempt to report potential conflicts of interest. Half the stories failed to meet these two requirements, Schwitzer says.”
Yikes! I mentioned the Cowboy Comic Will Rogers a few posts ago in regard to a statistical effect which bears his name, and now I remember one of his famous lines, “All I know is what I read in the papers.” Well, Will, don’t believe half of it.

Friday 9 May 2008

The Rules of Three

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."
The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to vidi, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “
The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)20. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three. If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of gumballs is less than 1/3.

Thursday 8 May 2008

A Good Day to Die

Richard Edward Arnold died today, May 8, 2008, just seven days before his 90th birthday. In my youth, I knew him as the Tennessee Plowboy, or just Eddy, and at five I walked around the house with a broom "guitar" and sang his Grand Ol' Opry hits. I'm sure at one time I knew the words to both these songs by heart, and even now they come back to mind quickly.

We both grew a lot between then and now, and I stopped singing around the age of seven (none too soon for my family and friends, I am sure). Along the way I never completely got away from country music, and ended up married to a gal who writes cowboy poems. One I like is aptly titled, "A Good Day to Die.", so here are a couple of verses.

The coffee get's thick near the bottom of the pot

.....That's how I like it...

When the fire's on the wane, the coals are most hot,

....That's how I like it...

When I take off my boots and wiggle my toes,

Then I stretch out on my bedroll in well-earned repose,

And pat my dog on his head and feel his wet nose

....That's how I like it....

When the prairie flowers' blooms are a treat to my eye,

And the cloud-herds stir up white puffs in the sky

Well, then life is so perfect, it'd be a good day to die,

Yes, Lord, you know that's how I like it.....

Good night Eddy, and thanks for all the tunes....

Friday 2 May 2008

Geometry and Understanding

During a conversation with a fellow teacher, he made the statement that he really didn't like geometry, and went on to suggest it might not be a useful course for students. I was somewhat taken aback, and came out with what is probably more an article of faith than a tested fact... "Mathematically, if you really understand it, you can demonstrate it with geometry". Don't be misled; I love the power that comes from algebraic abstraction, but I find that algebraic "proofs" just lack a little something for convincing most students. Something visual seems to have more impact.

The teacher responded by asking (challenging?) for an example, and my first thought was the Pythagorean theorem since it is so ubiquitous in all areas of mathematics. You remember, a2 + b2 = c2... that one. Now there may be more existing proofs of this theorem than any other in existence. So I drew a right triangle, made three copies and formed them into a square as shown in the figure. Then, with the power of modern educational technology, I moved two of the triangles to get the second figure. Q.E.D. as we say; thus it is shown. Any student who knows the geometric idea of the area of a square can see that the white area in both pictures is the same since the same amount has been removed from it (four congruent triangles). In the first the white area is a square whose sides are formed by the hypotenuses (hypotenii?), c, of the four triangles. In the second it is divided into two smaller squares, one with sides of length a, one with sides of length b. The area then can be expressed as a^2 + b^2 or as c^2, and it is the same area. Somehow I envision a good sixth grader understanding this (and the next sixth grader I see is in for a sit down/talk to so I can find out).

Afterwards it struck me that this was not the best of examples. The Pythagorean theorem is essentially a geometric idea, its about triangles after all. What about an idea from math that has nothing obvious to connect it to geometry . Partitions seemed to fit the bill, and I thought of a simple example that I was sure elementary kids could play with and confirm, and I hope be convinced by a simple geometric argument.
Ok, so for the uninitiated, a partition of a number is just writing it out as a sum of smaller numbers. In order to count them we usually write them from largest addend to smallest. For example 6 = 3 + 2 + 1, so that is one partition of six. There are obviously others, 2+2+2, or 3 + 1 + 1 + 1 would be two more. A common, and not trivial, question is what is the total number of partitions of six? Most elementary students can take an organized attack and come up with the answer. If we include 6 itself, there are eleven of them. 6; 5+1; 4+2; 4+1+1; 3+3; 3+2+1; 3+1+1+1; 2+2+2; 2+ 2+ 1+ 1; 2+1+1+1+1; and of course 1+1+1+1+1+1.
But an interesting thing pops out if we look at some special limitations. There are exactly 3 ways to partition six into three terms; 4+1+1; 3+2+1; 2+2+2. There are also exactly three partitions that use three as the largest number; 3+3; 3+2+1; 3+1+1+1 . Coincidence? Not a chance. The same thing would happen if you partitioned 25 into four terms (or five or six or pick your favorite number). There are exactly the same number of partitions of a number into n differen terms as there are partitions with n as the biggest value. If you have never seen this proof, try to make it clear to yourself why it is true before you read on.
OK, it seems true, but how do we relate that to geometry, and a visual proof? To the rescue comes a 19th Century British Mathematician, Norman M Ferrers, who was a fellow at Gonville and Caius (pronounced Keys) College just down the road here at Cambridge. He seems to have been the first to make a simple discovery about partitions using dots. Here is the Ferrer diagram for two of the unique partitions of six into three groups.

Reading the number of dots in each row we see that 6 = 4 + 1+1; but if we read down the columns we see 3+1+1+1. In the second diagram we see rows of 2+2+2; but columns of 3+3. Can you see that this would apply to any diagram? If there are three rows that add up to some number, then the columns will give us the same total with a three as the largest number. The same thing would be as clear with any other number of rows.

See, Geometry makes it easy to understand, and to explain. Geometry is good. Now go do your homework!

That "Divisible by 11" Rule

Over some tweets about division mental tricks lately I got a question about why the somewhat well known test for eleven works.

If you are not one who grew up BC(before calculators)  working with mental arithmetic, some of these may not be known to you, so here is the one about which they were asking. 
If you take a number, say 31492435 and want to know if it is divisible by eleven, the old-school common trick was to take every other digit and find the sum, then do the same with t he remaining digits.  If these two totals differ by a multiple of eleven, then it is divisible by 11. ( I like a more effective one if you suspect that tests divisibility by 7, 11, and 13 all at once, and  I think is a simpler approach that I will explain after answering the primary question.)

So to begin, I'd like to explain why the more common test of "Casting Out Nines" works.

The secret behind casting out nines is hidden in the following set of equalities:

10 = 1*9+1

100 = 11*9 +1

1000= 111*9+1

10000= 1111*9+1

So if we have a number like 1234, which is 1000+200 + 30 + 4, we can rewrite it as 1(111*9 + 1) + 2(11*9+1) + 3(1*9+1) + 4. Distributing the numbers powers of ten gives me 111*9 + 1+ 22*9+2 + 3*9+3 + 4 collecting all the terms that are multiples of nine gives us 136*9 + 10. Since nine will divide evenly into the 136*9, the remainder is the remainder when ten is divided by nine, and of course we know that 10 = 1*9+1, so the remainder is one.

For elevens the pattern is slightly different. Notice that :

10 = 1*11 –1

100 = 9*11 +1

1000= 91*11-1

10000= 909*11+1

100,000= 9091*11-1

….. and each multiple of ten alternates being one more than or one less than a multiple of 11. So if we expand 1234 as before into 1000+200 + 30 + 4, we can rewrite it as 1(91*11 –1) + 2(9*11+1)+ 3(1*11-1) + 4 . Expanding as before we get some powers of eleven, and the remaining numbers are –1 + 2 – 3 + 4 = 2; so the 1234 divided by 11 leaves a remainder of 2.

It is very similar to the method of casting out nines, except that we alternate adding and subtracting instead of adding all the terms. Most people find it easier to start at the right since the constant term is always added, but you must be able to deal with small negative numbers. For instance if we use 4152, we will do 2 – 5 + 1 – 4 which yields –6. We know there can not be a negative remainder, so what to do. Simply remove this amount from the base, 11, and the answer is revealed. The remainder is 5.

T. Just be aware that if you get a negative number, it means the remainder is the elevens compliment (11-r) of that number. If you just use the sums of the  alternate terms, you don't know the remainder for certain, but it still tests divisibility.

My favorite method that I mentinoed above, uses the idea that 7*11*13 = 1001, so for big numbers, it can be quicker to subtract out multiples of 1001, or 10010, or 100100, etc.

For 4153 you can quickly see by inspection that it is not divisible by 2, 3, or 5. If you reduce it by subtracting 4*1001 = 4004 to get 149. Since 7 goes into that with remainder of two, the original number has a remainder of two on division by 7 also. And since 1+9 - 4 is not zero, or a multiple of eleven then it is not divisible by 11, and a quick mental inspection says that 149 is not divisble by 13 either, so we've eliminated a lot of numbers quickly in testing for primes.