Delayed Gratification
I was supposed to teach my algebra class “the distance formula” (Note to any and all: how can I display mathematical formulas & symbols in this space?)
Anyway, I wasted most of the period having them hunt for Pythagorean Triples (three numbers such that the sum of the squares of the first two equals the square of the third,
ie a2 + b2 = c2. An example is 8,15, and 17, since 82 + 152 = 172 (64 + 225 = 289).
Then I ran a contest, almost a spelling bee, seeing who could come up with the most triples for a,b,c <= 100. When I get kids cheering and screaming about numbers, I think I am okay
Finally I got to finding the distance between two points in the coordinate plane by using the Pythagorean Theorem. There was a little fuss when I wondered out loud whether we would get a formula, but I gave in, and briefly developed the distance between two points in terms of their x- and y- coordinates. But I insisted that they use the Pythagorean Theorem in their homework, and not until the end of next week will I start using the distance formula in class.
I fight this battle every year. I want kids to appreciate the distance formula as a convenient way to express the Pythagorean Theorem in the plane, not as yet another formula in their toolboxes. There are kids who only want to memorize formulas, but for those who go along with me, I am convinced that they will be able to use the distance formula more freely in the future since they will really understand it.
(as for me, I can use the distance formula. but when I calculate distance mentally, I “see” the right triangle drawn)
JD2718 
I believe you hit the nail on the head when it comes to Math. You can have them memorize all the formulas you want but if they can’t apply it to the real world then it is a waste of time building on the subject. At least in science I can apply the math to real world applications.
For example, very few of my sophmores/juniors understand the difference between a dependent & independent variable until I explain to them that you cannot control time (independent) but you can control the temperature of a room (dependent).
In physics, many schools teach 9th graders conceptial physics, applied math and this seems to help the students pickup their Math skill, or so I’m told.
In my school it appears the real problem is the math teachers as it seems to be a revolving door. In the last three years my school has replaced almost the entire Math staff, mostly due to poor results. It doesn’t help that many of them were recruited from overseas (and as nyc educator said intergalatically) and have poor english skills or are unprepared for the American/urban student. Of course the poor math abilities of the incoming students is and has always been a problem.
Here’s a puzzle, easy for you, I’m sure, but maybe appropriate for students.
Below are a few Pythagorean triplets in which b + 1 = c.
3,4,5
5,12,13
7,24,25
9,40,41
So what’s the formula for a in these cases? And, of course, what’s the (straightforward algebraic) proof?
http://sixthform.info/steve/wordpress/index.php?p=13
Thank you so much. Now I will have to figure out a bunch of little stuff, but it’s got to be worth it, to end up with mathy type.
For a math guy, I really don’t know much at all about computers.
John,
those are a lot of fun. Kids get excited by this stuff (a, 60, 61), and afterwords are pretty much “into” participating while I demo the algebra behind it. Then they can make up their own. It’s been a couple of years since I ran this. I’ll try it tomorrow. Thanks.