Visual Calculus?

Just came across a really interesting follow up to my recent Calculus without Limits blog..

To tempt your interest, here is a problem:

A bicycle rider is riding in a perfect circle and his wet tires leave two concentric circles on the pavement. (if that is difficult to visualize, see here) What is the area between the outer circle and the inner circle.(the area of the annulus). You are allowed to ask for one measurement (other than the answer) that will allow you to solve the problem. What is that measure.

I came across this little gem following up on some information from a blog at Arjen Dijksman’s Physics Intuitions.
Following up on that and searching around led to a truly interesting MAA article by Tom Apostol about a really interesting approach to a “visual calculus” by Mamikon Mnatsakanian. A really great read for calculus teachers, and students.

Who'ddd

Glancing through an old MAA Mathematics Magazine in a list of prominent contributors to the early science of analytic geometry was the name the Dutch politician, Johann Hudde.....

Johann WHO, dude? How do they list a guy with Pascal and Fermat and Descartes and I never heard the name?

It appears that the rise of Leibniz/Newtons calculus based on limits essentially obliterated the record (for most of us) of a brief period when the early calculus flourished with a method that was based on analytic geometry and no Limits...that’s right, calculus without limits.

I searched for Hoode and was hooked when I found a quote that described a comment about him by Leibniz

Leibniz in particular was impressed with Hudde’s work, and when Johann Bernoulli proposed the brachistochrone problem, Leibniz lamented:
If Huygens lived and was healthy, the man would rest, except to solve your problem. Now there is no one to expect a quick solution from, except for the Marquis de l’Hˆopital, your brother [Jacob Bernoulli], and Newton, and to this list we might add Hudde, the Mayor of Amsterdam, except that some time ago he put aside these pursuits .

When I found the whole article, it turned out to be an interesting historical, mathematical journal entry, "The Lost Calculus (1637-1670), Tangency and Optimization without Limits", by Jeff Suzuki in the MAA Mathematics Magazine, Dec, 2005.


In A History of Mathematics, by Cajori, we find that Hudde was the first to use three variables in analytic geometry.

The article by Suzuki would seem to be a wonderful historical read for calculus teachers who, like me, never heard of Hudde’s rule.

Time, and Trig, and Conics, Oh MY!

A while back I wrote about the equation of time, recalling in part,

It came from a lesson in trig on simple harmonic motion. We were talking about things that demonstrated sinusoidal behavior, and one bright young man suggested that the height of the sun at noon would be an example. I sort of agreed with a comment about "not exactly at noon.. but" and then the little guy was confused.. "You know, I said, like today ."(it was Feb 12) "I think the sun was about 12 minutes late or so."
Slow looks at each other, then back to me... the three letter word look,,,,,"Huh?"
"You know, that's what the analemma is for, telling if the sun is early or late.".....

Same look, compounded by the wild eye..."HUH?"

See the whole blog here

So anyway, today I opened the blog over at "Square Circle Z" by Zac,(who seems to prefer to be called, "the mysterious Zac") and it showed a really nice use of the addition of two trig functions to explain the two components of the Equation of time. A nice exercise for people who tend to patienly explain to students who ask "What's that good for?"...(my personal first instinct is to throw things, but many administrations frown on that)

Can Statistics Really be 120 Years Ahead of Science

Scientists have beeen able to use analysis of the 54 Genes that are known to be associated with height to predict the height of a subject. Unfortunatly the new and scientific approach is far less accurate than the statistical model created by Francis Galton 120 years earlier.

The study appears in the European Journal of Human Genetics (2009) 17, 1070–1075;Predicting human height by Victorian and genomic methods Here is the abstract,

"In the Victorian era, Sir Francis Galton showed that ‘when dealing with the transmission of stature from parents to children, the average height of the two parents, … is all we need care to know about them’ (1886). One hundred and twenty-two years after Galton's work was published, 54 loci showing strong statistical evidence for association to human height were described, providing us with potential genomic means of human height prediction. In a population-based study of 5748 people, we find that a 54-loci genomic profile explained 4–6% of the sex- and age-adjusted height variance, and had limited ability to discriminate tall/short people, as characterized by the area under the receiver-operating characteristic curve (AUC). In a family-based study of 550 people, with both parents having height measurements, we find that the Galtonian mid-parental prediction method explained 40% of the sex- and age-adjusted height variance, and showed high discriminative accuracy. We have also explored how much variance a genomic profile should explain to reach certain AUC values. For highly heritable traits such as height, we conclude that in applications in which parental phenotypic information is available (eg, medicine), the Victorian Galton's method will long stay unsurpassed, in terms of both discriminative accuracy and costs. For less heritable traits, and in situations in which parental information is not available (eg, forensics), genomic methods may provide an alternative, given that the variants determining an essential proportion of the trait's variation can be identified. "

From the details of the work, "In a population-based study of 5748 people, we find that a 54-loci genomic profile explained 4–6% of the sex- and age-adjusted height variance, and had limited ability to discriminate tall/short people, as characterized by the area under the receiver-operating characteristic curve (AUC). In a family-based study of 550 people, with both parents having height measurements, we find that the Galtonian mid-parental prediction method explained 40% of the sex- and age-adjusted height variance".

Galton's approach was not "just" the average of the parents heights, but involved the deviation of the midparent (the average of the parents when the female was scaled up by 1.08) from the average midparent. The offspring, Galton determined, would be only 2/3 as far away from the mean as their mid-parent, on average. It is this "regression toward mediocrity" that begat our present term for regression. In his words, "We can define the law of regression very briefly. It is that the height-deviate of the offspring is, on the average, two thirds of the height-deviate of its mid-parentage." [from :Regression towards mediocrity; (1885), p. 252]

Math Becomes Fashionable

At about age three my grandaughter, Zoe, became very fashion conscious. She would change clothes three to four times a day, draping a scarf/tablecloth/towel around her shoulders to go with some of the oddest collections of tops and skirts that could be imagined. She would study every commercial with "Sarahaca Jessica Parker" as she named her, and told my beautiful wife Jeannie, "Nan, she's SOOO fashionable."
Well, Zoe is a little older now, and even more fashion conscious, so I'm sure she will be thrilled to know that her old grandfather is finally fashionable. I know because it is in the Wall Street Journal.

Math becomes fashionable, focus shifts
Mathematician Stanley J. Osher, whose firms Cognitech Inc., Luminescent Inc., and Level Set Systems Inc. are all solving real world problems, says it’s an “incredible time for mathematicians".

See the story here.

The Mathematical Journals, They Are A Changing

Today I was reading some notes from a Mathematics Teacher magazine that is almost exactly one-hundred years old, June 1910 (thanks to Dave Renfro for the copies). In a regular monthly section, New Books, I was surprised at the materials suggested for the mathematics teacher.
Several were totally to be expected. "The Principles of Education" by W. C. Rudiger seemed totally appropriate, as did a summary of David E Smith's "Rara Arithmetica", which has become a classic for Math Education historians. "Attention and Interest", by Felix Arnold was a report on the psychology of education with the novel (at least in modern terms) idea that, "the reader is left free to draw his own conclusions and theories from the data given."
But alongside this were a couple of "outliers" in present-day terms. The review of "The Ethics of Jesus" by Henry C. King ended with the statement, "It is interesting reading and an invaluable book for teachers." Also in the list of books reviewed, "Manual of Gardening" by L. H. Bailey, and "How to Keep Hens for Profit" by C. S. Valentine. Just what the starving math teacher might need to keep the wolf from the door.

The beauty of data visualization

"David McCandless turns complex data sets (like worldwide military spending, media buzz, Facebook status updates) into beautiful, simple diagrams that tease out unseen patterns and connections. Good design, he suggests, is the best way to navigate information glut -- and it may just change the way we see the world."

A must see for AP stats students.... but be careful..the mind sees patterns...we WANT it to see patterns, and it will... even if????

Hexa-flexa-fun

Ok, I'm the teacher... I know how it works...have for years.... and yet..... Totally amazed again each time...

This blog is for Zoe and Xander, my grandchildren, in the hope that they can learn to enjoy the beauty of mathematics too.



If this gets you excited, here is the way to make your own..


Here is the link for the patterns mentioned in the construction, or color your own.

Here is a Wikipedia link for those of you who want to learn a little about history of the relatively new idea of flexigons in general.

The Collected (so far) Almost Pythagoras

Several people have asked me lately about the collected (a strong word for my loose ideas) "Almost Pythagorean" ideas. I finally found some time to set down and write about a few that come to mind. Some of the material has appeared in this blog already. You can find the draft copy here. I'm sure there are probably as many lost in the dusty cluttered corners of my memory. If you have others, please drop me a note or comment here.

Another "Almost Pythagorean" Relationship

A while back Arjen Dijksman at "Physics Intuitions" posted a blog reflecting on my post about Almost Pythagorean Triangles . Then he went me one better with a new relation that is pretty incredible. In essence, he shows that for ANY triangle, ABC with opposite sides a, b, and c, there is a relation a²+b²=c²+2 t², where t length of the tangent from point C to the circle with diameter AB.

He has some nice clear graphics to show the development.. check it out..and with a change of sign the equation generalizes to triangles where c lies inside the circle with diameter AB.

This is, of course, too close to the law of cosines to be ignored.. and it is easy to see that t² = a*b*cos(C)... in fact as soon as we set t²=AC*DC, and recognize that angle BDC is a right angle, we have Cos(C)= DC/BC... thus AC*DC = AC*BC*(DC/BC)= ab Cos(C).

Bottoms Up Factoring, A Question of History

A few years ago(Dec 29, 2006) a lady named Nancy Kitt sent a question to the Teacher2Teacher Service at the Math Forum and asked about a factoring method called the Bottoms Up method:

One of my former students showed me the following method to factor
trinomials.
I want to know HOW and WHY this method works.
3x^2 + 14x + 8 Multiply AC, that is 3 x 8 = 24
Now look at B = 14. We are looking for two numbers
multiplied together to give 24 and added to give 14. The numbers will be
+12 and +2.

(x + 12)(x + 2)--- put the two factors 12 and 2 inside the parentheses,
but put x as the first term in both parentheses.

Now, since A was 3, divide the two factors 12 and 2 by 3

(x + 12/3) (x + 2/3)

12 will divide by 3 giving 4.

2 does not divide by 3. Therefore, multiply the x by 3, giving the final
factorization of (x + 4) (3x + 2).

(she followed this with a second example)...

This is the COOLEST method I've ever seen. However, I have NO CLUE HOW
or WHY it works!!!!!!

I want to use this method this semester, and I'd like to have an idea why
it works?????

I responded (helpfully, I hope) with a post to explain the substitution method

Well, the secret is that 8 = 24/3...

If you consider that the solutions of x^2 + bx +c = 0 are the same as the
solutions of 2x^2 + 2bx + 2c etc... then you are a step closer to
understanding the solution....

If we take 3x^2 + 14x + 8 = 0 and let x=u/3 (or u=3x) and substitute we get


(3(u/3)^2 + 14 (u/3) + 24/3) = 0 and now if we simplify the first term
we get

u^2/3 + 14 u/3 + 24/3 = 0

now if we multiply all terms by 3 we get

u^2 + 14u + 24... and solve to get the two solutions you had, u=12 and u=2,
but remember that we wanted x, not u, and x=u/3 thus the final solution...
(And then I added two other methods that are not well known or understood)

Then I posted a second note in case she might want some historical information...

Just a little addendum on the history of this method (I was writing up an
article on factoring and thought of your question). The substitution of Z=ax to
make a solution pliable dates back to the ancient Babylonian clay tablets
according to Boyer's History of Mathematics. They used it in order to make
a trinomial (ax)² + b(ax) =ac so that they could solve using their method
of completing the square. The idea of factoring had to wait a LONG time
until Thomas Harriot came up with it around 1600-1621 (he died in 1621 but
his method was not published until 1631, ten years after his death)..

By the way, I can not find any reference to "bottoms up" name for this...
can you help ME?

So several years later, I still wonder... does anyone have a clue how/why this term was applied, or any other detail about the history?

"A Three Year Old is NOT Half a Six Year Old

"Very many people go through their whole lives having very litte sense of what their talents may be, or if they have any to speak of."

The Ted Talks introduction describes the talk this way:

"In this poignant, funny follow-up to his fabled 2006 talk, Sir Ken Robinson makes the case for a radical shift from standardized schools to personalized learning -- creating conditions where kids' natural talents can flourish."

As an early believer in revolution I really loved the statement, "Reform is no use anymore because that is simply improving a broken model. What we need is not...evolution, but a revolution in education."

Listen, then find out what it is about education that you take for granted.

Cloths of Heaven

William Butler Yeats



Had I the heavens' embroidered cloths,

Enwrought with golden and silver light,

The blue and the dim and the dark cloths

Of night and light and the half-light,

I would spread the cloths under your feet:

But I, being poor, have only my dreams;

I have spread my dreams under your feet;

Tread softly because you tread on my dreams.

My version of his "fireman" story I heard while teaching in a small rural community in northern Michigan in the mid 80's.  A teacher from another district was trying to make the point that we needed to stress "thinking" to all the students, not just the best ones.  He reminded us that in this area, we all traveled a lot on precarious roads that iced over frequently and led to dangerous driving conditions.  He said to imagine for a moment, that the worst happened. You lost control for a minute and the car slid off the road and you have been seriously injured.  The local volunteer ambulance arrives to try to rush you to an emergency room some thirty to fifty miles away, and as you look up into the eyes of the volunteer EMT who holds your life in your hands, the one thing you should expect for sure is that he was NOT one of your AP or Honors students. They are all off to the big city working toward a distinguished career.  This guy was much more likely the kid who cracked jokes in the back row of the General Math class, and asked when we were ever going to use this stuff.  But at that moment, as he is the only one between you and that light at the end of the tunnel, you are going to hope you taught him to think.

Not All Equal

Robert Talbert at "Casting Out Nines" had a link to a recent study that suggests US students have more misconceptions about the equal sign than students from other countries.

“About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students,” note Robert M. Capraro and Mary Capraro of the Department of Teaching, Learning, and Culture at Texas A&M."

My first thought was that maybe this was too much fuss about nothing. They are talking about young people who confuse the equal sign as an operator, sort of ...2 + 5 = ...

But it seems that the mis-conception is restrictive on future math success.

"They have been trying to evaluate the success of math education through students’ interpretation of the equal sign. They have published several articles on this topic, with the most recent one published in the February 2010 issue of the journal Psychological Reports.

Students who exhibit the correct understanding of the equal sign show the greatest achievement in mathematics and persist in fields that require mathematics proficiency like engineering, according to their research."

Here is a Video about the report.



ADDENDUM: I found a link to some material from authors about a comparison between US and Chinese texts here

Pi, A little more than 3.14

This one is for my student, Joe Sullivan, who memorized Pi farther than any student I ever taught... See you in a few weeks, kiddo.

Japanese and US computer scientists claim to have calculated Pi to five trillion decimal places... If you had all the others memorized, you can go back to work now....
See more at Shecky's Math Frolic.

Schooling is NOT a Natural Environment for Learning

The title of this blog is a quote from Sue Van Hattum, aka "Math Mama Writes", who runs a "math salon" for young people and their parents in her home in California. Her teaching approach and ideas are part of a documentary in progress called "Class Dismissed" which is soon to be released. On their web site I found this quote, which I like a lot;

"Class Dismissed will challenge its viewers to take a fresh look at what it means to be educated, the difference between education and schooling, while offering up a radical new way of thinking about the process of education."

There is a revolution happening out there in mathematics education, without grades. Hat tip to Sue, a revolutionary heroine.

Valedictorian Speaks Out Against Schooling in Graduation Speech

This entire post was cut from SwiftKick... thanks to John D. Cook at The Endeavor for the link.

"Last month, Erica Goldson graduated as valedictorian of Coxsackie-Athens High School. Instead of using her graduation speech to celebrate the triumph of her victory, the school, and the teachers that made it happen, she channeled her inner Ivan Illich and de-constructed the logic of a valedictorian and the whole educational system.

Erica originally posted her full speech on Sign of the Times, and without need for editing or cutting, here's the speech in its entirety:

Here I stand

There is a story of a young, but earnest Zen student who approached his teacher, and asked the Master, "If I work very hard and diligently, how long will it take for me to find Zen? The Master thought about this, then replied, "Ten years . ." The student then said, "But what if I work very, very hard and really apply myself to learn fast -- How long then?" Replied the Master, "Well, twenty years." "But, if I really, really work at it, how long then?" asked the student. "Thirty years," replied the Master. "But, I do not understand," said the disappointed student. "At each time that I say I will work harder, you say it will take me longer. Why do you say that?" Replied the Master, "When you have one eye on the goal, you only have one eye on the path."

This is the dilemma I've faced within the American education system. We are so focused on a goal, whether it be passing a test, or graduating as first in the class. However, in this way, we do not really learn. We do whatever it takes to achieve our original objective.

Some of you may be thinking, "Well, if you pass a test, or become valedictorian, didn't you learn something? Well, yes, you learned something, but not all that you could have. Perhaps, you only learned how to memorize names, places, and dates to later on forget in order to clear your mind for the next test. School is not all that it can be. Right now, it is a place for most people to determine that their goal is to get out as soon as possible.

I am now accomplishing that goal. I am graduating. I should look at this as a positive experience, especially being at the top of my class. However, in retrospect, I cannot say that I am any more intelligent than my peers. I can attest that I am only the best at doing what I am told and working the system. Yet, here I stand, and I am supposed to be proud that I have completed this period of indoctrination. I will leave in the fall to go on to the next phase expected of me, in order to receive a paper document that certifies that I am capable of work. But I contest that I am a human being, a thinker, an adventurer - not a worker. A worker is someone who is trapped within repetition - a slave of the system set up before him. But now, I have successfully shown that I was the best slave. I did what I was told to the extreme. While others sat in class and doodled to later become great artists, I sat in class to take notes and become a great test-taker. While others would come to class without their homework done because they were reading about an interest of theirs, I never missed an assignment. While others were creating music and writing lyrics, I decided to do extra credit, even though I never needed it. So, I wonder, why did I even want this position? Sure, I earned it, but what will come of it? When I leave educational institutionalism, will I be successful or forever lost? I have no clue about what I want to do with my life; I have no interests because I saw every subject of study as work, and I excelled at every subject just for the purpose of excelling, not learning. And quite frankly, now I'm scared.

John Taylor Gatto, a retired school teacher and activist critical of compulsory schooling, asserts, "We could encourage the best qualities of youthfulness - curiosity, adventure, resilience, the capacity for surprising insight simply by being more flexible about time, texts, and tests, by introducing kids into truly competent adults, and by giving each student what autonomy he or she needs in order to take a risk every now and then. But we don't do that." Between these cinderblock walls, we are all expected to be the same. We are trained to ace every standardized test, and those who deviate and see light through a different lens are worthless to the scheme of public education, and therefore viewed with contempt.

H. L. Mencken wrote in The American Mercury for April 1924 that the aim of public education is not "to fill the young of the species with knowledge and awaken their intelligence. ... Nothing could be further from the truth. The aim ... is simply to reduce as many individuals as possible to the same safe level, to breed and train a standardized citizenry, to put down dissent and originality. That is its aim in the United States."

To illustrate this idea, doesn't it perturb you to learn about the idea of "critical thinking." Is there really such a thing as "uncritically thinking?" To think is to process information in order to form an opinion. But if we are not critical when processing this information, are we really thinking? Or are we mindlessly accepting other opinions as truth?

This was happening to me, and if it wasn't for the rare occurrence of an avant-garde tenth grade English teacher, Donna Bryan, who allowed me to open my mind and ask questions before accepting textbook doctrine, I would have been doomed. I am now enlightened, but my mind still feels disabled. I must retrain myself and constantly remember how insane this ostensibly sane place really is.

And now here I am in a world guided by fear, a world suppressing the uniqueness that lies inside each of us, a world where we can either acquiesce to the inhuman nonsense of corporatism and materialism or insist on change. We are not enlivened by an educational system that clandestinely sets us up for jobs that could be automated, for work that need not be done, for enslavement without fervency for meaningful achievement. We have no choices in life when money is our motivational force. Our motivational force ought to be passion, but this is lost from the moment we step into a system that trains us, rather than inspires us.

We are more than robotic bookshelves, conditioned to blurt out facts we were taught in school. We are all very special, every human on this planet is so special, so aren't we all deserving of something better, of using our minds for innovation, rather than memorization, for creativity, rather than futile activity, for rumination rather than stagnation? We are not here to get a degree, to then get a job, so we can consume industry-approved placation after placation. There is more, and more still.

The saddest part is that the majority of students don't have the opportunity to reflect as I did. The majority of students are put through the same brainwashing techniques in order to create a complacent labor force working in the interests of large corporations and secretive government, and worst of all, they are completely unaware of it. I will never be able to turn back these 18 years. I can't run away to another country with an education system meant to enlighten rather than condition. This part of my life is over, and I want to make sure that no other child will have his or her potential suppressed by powers meant to exploit and control. We are human beings. We are thinkers, dreamers, explorers, artists, writers, engineers. We are anything we want to be - but only if we have an educational system that supports us rather than holds us down. A tree can grow, but only if its roots are given a healthy foundation.

For those of you out there that must continue to sit in desks and yield to the authoritarian ideologies of instructors, do not be disheartened. You still have the opportunity to stand up, ask questions, be critical, and create your own perspective. Demand a setting that will provide you with intellectual capabilities that allow you to expand your mind instead of directing it. Demand that you be interested in class. Demand that the excuse, "You have to learn this for the test" is not good enough for you. Education is an excellent tool, if used properly, but focus more on learning rather than getting good grades.

For those of you that work within the system that I am condemning, I do not mean to insult; I intend to motivate. You have the power to change the incompetencies of this system. I know that you did not become a teacher or administrator to see your students bored. You cannot accept the authority of the governing bodies that tell you what to teach, how to teach it, and that you will be punished if you do not comply. Our potential is at stake.

For those of you that are now leaving this establishment, I say, do not forget what went on in these classrooms. Do not abandon those that come after you. We are the new future and we are not going to let tradition stand. We will break down the walls of corruption to let a garden of knowledge grow throughout America. Once educated properly, we will have the power to do anything, and best of all, we will only use that power for good, for we will be cultivated and wise. We will not accept anything at face value. We will ask questions, and we will demand truth.

So, here I stand. I am not standing here as valedictorian by myself. I was molded by my environment, by all of my peers who are sitting here watching me. I couldn't have accomplished this without all of you. It was all of you who truly made me the person I am today. It was all of you who were my competition, yet my backbone. In that way, we are all valedictorians. "

I am now supposed to say farewell to this institution, those who maintain it, and those who stand with me and behind me, but I hope this farewell is more of a "see you later" when we are all working together to rear a pedagogic movement. But first, let's go get those pieces of paper that tell us that we're smart enough to do so!

Mountain Males Most at Risk For Death by Lightning

Lightning kills more than twice as many per-capita in Wyoming than any other state. In fact, at a micromort of 2.02, or 2.02 deaths per million, Wyoming is more dangerous than the next three highest ranking states, Utah (.7), Colorado(.65) and Florida (.56). Florida leads the nation in total deaths with 126 in the period from 1990 to 2003, but trials the others on a per-capita basis. Utah has had only two deaths in the same period, the last in 2007, but its small population places it second in per-capita mortality.



What most intrigued me about my recent introduction to lightning mortality was its severe gender bias. Guys are much more likely to feel that tingle, or at least to die from it. In 2009, 28 of the 34 deaths in the US were male, or 82%. In 2008 it was 79% males (22/28)and in 2007 it was 89% (40/45)in a bad year for lightning deaths.

When you ask people why guys are biting the electrical bullet so much more often than females, the usual response falls into two categories, occupation and recreation. Most people perceive men to be more likely to be outside by these two causes. The data I give doesn't tell me what the occupations were, but few of the deaths were fishermen or golfers. This year, one postal worker was killed, but it was a female, and a crabber in Texas (male) was also killed; but then so was a prisoner walking across a prison yard... ("an abrupt change of sentence"). In fact, the most common activity was ...wait for it...."standing under a tree". The most unusual death this year so far, at least to me, was also the oldest; a seventy year old male climbed to a hilltop in a thunderstorm "to get better cell phone reception." "Hello, this is the afterlife calling."

A Rational Triangle Idea

Last week Sol at "Wild About Math" posted a link to the Wolfram Demo site related to a novel relationship that is not well known to students/teachers. For every rational number,q, on the open interval from 0 to 1, there is a Pythagorean Triple.

Actually the demo may oversell the idea a little, or perhaps I don't completely understand it, but I don't think it enumerates all the Pythagorean triples, but it is still a nice idea to introduce to students because many of the ideas can be proved with nothing beyond a good grasp of Alg II (for non-US folks, that's about a 14-16 year old students grasp of Algebra).

I had written about the relation back at the end of 2009. If you haven't read that one, and are not familiar with the idea, you might want to start there.

Anyway, Sol's post got me thinking again about how I would approach this idea with students. The first big idea is that if you pick a point on the y-axis between y=0 and y=1, with a rational y-coefficient, then the line through that point and the point (-1,0) will cross the unit circle in the first quadrant in a point (p,r) whose x and y coordinates are both rational. Since the point is on the unit circle, that means that p²+r² = 1. But since both p and r are rational, they can be written with a common denominator, Wolfram uses (a/c, b/c) and so (a/c)²+(b/c)²=1. If we multiply both sides of the expression by c², we get a²+b²= c² and so {a,b,c} is a Pythagorean triple.

So what might we ask our young charges to do with this? Well, first, do we believe that ANY rational on the positive y axis REALLY will produce a rational intersection on the unit circle? The proof requires no more than the solution of a quadratic. If the students are not up to challenging an open proof, have them start by confirming that some given rational y-intercepts will produce a rational intersection on the unit circle, and then go on to find the Pythagorean triple associated with that rational. For example, Wolfram's demo seems to associate the rational 1/2 with a (3,4,5) triangle... can they show why? Can you find the triple associated with 3/7?

I mentioned above that I didn't think this could produce ALL the Pythagorean triples... and we might want to ask our students if they think it could. One of my concerns about the Wolfram presentation is that the point (0, 1/3) is associated with the triple (8,6,10). Why not (4,3,5)??? how did they arrive at the non-primitive triple they chose, and not, for instance, (12,9,15) or ??? pick your favorite. I knew that they had used (3,4,5) in the first example (0, 1/2)...

In fact, would (9,12,15) show up later if I plowed through enough rational fractions? I suspected it wouldn't. First, the symmetry of the situation seemed to play against it. In simplest form, the x coordinate for any version of a (3,4,5) triple must be either 3/5 or 4/5; or at least so it seemed to me.

I think I would challenge my students to work backwards from a given triple and see if they could find the related rational y-intercept. Since we know (see my earlier blog) that the x-coordinate was (1-t²)/(1+t²), we could take any primitive triple, say (5,12,13) and set either the x value = 5/13 or 12/13 and get the two (and only two) values that would generate appropriate intersections on the unit circle. As we do this, we discover that the rational always turns out to be
where H is the hypotenuse of the triple, and L is the leg we chose to use for the x-coordinate ratio. For (5,12,13) we get the two rational values 2/3 (this is number 3 on the Wolfram demo) and 1/5 (number 7, but labeled (24, 10, 26) ) ... Bright students should be challenged to prove that any Primitive Pythagorean triple, will also be such that the ratio (H-L)/(H+L) give will produce a square of a rational number.

So maybe the Wolfram demo is just good enough (and just bad enough) to motivate our students, after all, they love exposing our flaws.

Nice Martin Gardner Quote

I was reading through a new post at "Math Frolic" by "Shecky" when I noticed a nice quote he had from Ron Graham, the gymnastic- Juggling-Mathematician, about Martin Gardner:

-- In partial remembrance of Martin Gardner (1914-2010) who, in the words of mathematician Ronald Graham, “...turned thousands of children into mathematicians, and thousands of mathematicians into children.” :-)

How nice, and how true.