Algebra Text?

Hmm, Well in a few days will you post it. I can only get there by guessing.

So just picking Pythagorean triples (a,b,c) 3,4,5; A=1 and B=7 works (as Jasen said.) Then for 5,12,13; A=7 and B=17 works. But that?s the end of the line for simple triples that I know.

Maybe I'm missing the obvious, but I see no closed-form solution as soon as you say "integer"... you have to guess. ...Jim Thompson

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Dang... never mind I see it!

George H.

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Well look at my two solutions, and your math brain should be able to see the connection... and then the general solution is 'obvious'. (Of course it's obvious only in hind sight.)

George H.

This is getting pointless. If you want teachers to teach, let them teach. Before you criticize their methods, why not learn them. Or you can come here and whine.

HA. That just shows you poor you are at logic which is part of math. So you are wrong.

Look, you are being silly. You know nothing about teaching. You are not willing to learn about teaching. So why are you actively criticizing teachers who do know? Oh right, you are a *math* wiz, so you must know more than everyone else including the subjects you never studied.

But you still know nothing about teaching grade school. Do you really thing they are the same?

There's your problem. You know nothing about teaching school and you think those who do know are just union drones. Get out and learn something about the subject at hand.

Yes, very Larkinesque.

Yes, you are quite the logician and debating expert too. How about you stick to the subjects you know something about or at least have an open mind about learning something new?

A wonderful one, indeed!

Agreed.

Not by any means an unusual "construct," yet well hidden.

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Me too. Gaaaack! ...Jim Thompson

Frankly, my opinion is that one who aspires to teach math, should learn /math/, not teaching; be it university, high school, primary school, or something else. Similarly, for one aiming to program a spacecraft, there's much more to know about spaceflight, than about programming per se.

And yes, I've taken courses on (university's) teaching, even though I've never managed to get my certificate. And, I can ask a family member on the matter; he's got his, in school teaching.

well, homeschooled kids in the US tend to give off really broken vibes. There's something weird about it most of the time.

subjects?

I've followed this thread for a while and I'm struck by your strident support for teachers and teaching.

You seem to suggest that we mere mortals can't understand teaching despite being educated in grade school, high school, college, and graduate school. And, despite having children and grandchildren go through the same process.

Consider these facts about teaching and teachers:

• As a group, teaching majors have the lowest SAT scores
• Schools of education are regarded as the dregs of universities' schools
• If it required special training to teach, why is it that a

• University professors--people who teach--aren't required to have teacher training or teaching credentials
• Teaching majors learn such things as teaching methods, teaching tools, etc. The one thing they don't learn--specifically, elementary school teachers--is something to teach; they are no better educated in basic subjects than they were when they left high school.

When I was in elementary school, several teachers told our classes, for example, that "I was never very good at math," or "I was never very good at spelling." But they taught those subjects nevertheless.

Public schools are worse now than they were 50 years ago, and they continue to get worse.

When I was in grade school, we had smart kids and we had dumb kids, but ALL could read. That's not true today, even for high school kids.

Your vaunted teachers have failed us. Jim Thompson's examples of how his granddaughter was given a problem for which she was not equipped, and the teacher jumping into simultaneous equations with no prior preparation in single-variable algebra are only two examples. The examples of misguided teachers and teaching methods are endless.

The best advice I can give - study basics of algebra (solving for x) and then maybe polynomial functions.

Then learn a computer programming language. You'll use some of the algebra you used.

And in addition learn number systems to base x. Binary and Hexadecimal for starters.

I've met a number of teachers - some of whom shouldn't even be in the classroom. One of the more eye opening things I ever did was do program reviews in area public high schools.

One of the classes was teaching the kids how do use the MS Office suite. The day we were reviewing they were doing a payroll spreadsheet and I noted they all had a cheat sheet for the tax on earnings.

I asked the teacher if they intended to teach the kids how to use VBA. With VBA you could just write code to get it to calculate the tax.

The teachers response was that you needed advanced math to be able to program a computer.

So on my review notes I said that the most math you might need is the first few weeks of Algebra I. Apparently this caused great consternation in the school. Good - anything that rocks them out of their little comfort zone is a good thing.

Of course it also serves as a lesson, never invite an I.T. type to do program reviews.

hey! stop changing the rules!

a-b , a+b satisfies those two sets of results

but c,c satisfies them too, and can trivially be proven correct, and if a function is all that's required why make more effort?

my (and possibly your) conjecture is

(a+b)² + (a-b)² = 2c²

multipying out the parenteses

a²+2ab+b² + a²-2ab+b² = 2c² combining like terms 2a² + 2b² = 2c² dividing by constant 2 a² + b² = c² which was given at the start. looks it's a correct conjecture.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

I didn't change the rules. Though I left the question open at first.

I am actually not going to post the answer here and would ask that anyone who does work it out do likewise. It is obvious once you see it.

I thought by now this one would be too well known but apparently not. Google doesn't find a suitable crib for this one.

you said sqrt(a^2+b^2) AKA "c" was an integer in your first post:

What was c^2 for? is it an attempt at misdirection? the problem with that is it makes finding a valid answer easier.

the pattern is obviuos, I had to derive 2a^2 + 2b^2 before I believed it. I posted it before I read this.

No. Me misremembering an old maths interview algebra puzzle.

On reflection. It almost certainly was = c and = 2c.

I guessed that. Please don't put it onto Wiki though.

I too have been following this thread. Some of the comment have put a smile on my face. I have been working in and around schools and teachers for many years. I even had a few opportunities to actually teach the teaching staff in some local elementary schools a few years back how to use their newly installed alarm system. The results of those classes proved that my efforts fell on deaf ears since most did not learn much of anything with all the problems that happened after. Either I am not a good teacher or the students were not good learners. :-)

On a different circumstance I was asked to repair a classroom phone that was not ringing in Room 224 at a particular elementary school. Upon arrival I checked in at the office and said I was going up to Rm 224 to take a look. Entered the room and the teacher was at her desk on the other side of the room. I identified myself and said why I was there. She started into a rant about the phone not working for over 2 weeks bla bla bla.. I explained that I just received the work order a couple of hours before........................... I looked at the phone and flipped up the lever that was marked "ringer on/\ - ringer off \/" I called the office and asked them to call Rm 224. Of course the phone rang and the teacher jumped out of her chair. "WHAT did you do??" She asked. I motioned for her to come over to the phone and pointed to the lever and said; "This was in the OFF position" She said; "Is THAT what THAT is for??" I sent the school district a invoice for an hour of my time.

The point of these little stories................. from my point of view teachers that are in the teaching field are only as good as their desires. Most do not have common sense. Some are better than others. All will defend what they do, some are right and some are not. And lastly, it is a proven fact that 50% of all certified teachers graduated in the bottom half of their class.

Have a nice day. I will now retreat to my lurking position.

Les

Exactly! It was on the 'approved' syllabus, so that is how it MUST be taught! Since the teacher doesn' really know what it means, anyway, it must be right!

Or, she could be the opposite problem, the teacher is a wiz at math, who just 'does' it, and never really thought about the process involved. I had one teacher in grad school like that. For him, it was just obvious and he couldn't understand that his explanation made absolutely non sense!

Charlie