Proof in Algebra I?
Do you teach proof in Algebra I?
I do. All the tools are there for some basic stuff. We start with = as an equivalence relation (reflexive, symmetric, transitive). We (real) are closed for + and x, (though we only rest on this implicitly).
For addition we have an identity and inverses (opposites), for multiplication we have an identity and inverses (reciprocals) except for 0, and we have a special multiplication property for 0. We have an associative property for +, and for x, too. We have a commutative property for +, and for x, too. We have a distributive property of x over +.
We define subtraction as the addition of the opposite, division as multiplication by the reciprocal. We obtain equal quantities when we add or multiply equal quantities by equal quantities. I use substitution – I think we can prove that it is ok… not sure though.
(more, and a sample, below the fold —>)
Anyway, we have all this stuff, why not use it? Yesterday kids showed that (a + b) + c = (b + c) + a (nope, that’s not just the commutative property), and today we played more. I think proving that -(a – b) = b – a was the highlight, since so many kids did most of it on their own.
-(a – b) –> -(a + -b) –> -a + -(-b) –> -a + b –> b + -a –> b – a
Although I write it up two column style, with reasons. A girl kept recognizing the definition of subtraction (that was new for her today, I think, the really getting it part). Out of her enthusiasm, I think we agreed to parenthetically refer to the definition as That Girl’s Property (GsP). Another girl hung around after class, wanting to know if we would name the distributive property after her….
As a note, if you do very little of this, the easiest for the kids are the ones where the reasons are supplied, and they fill in the steps. This is the reverse of what most text books use for their crutch. And it’s not what I usually do, but I have fairly good buy in and a fairly high level of tolerance for frustrating tasks.
JD2718 
I teach proofs like that before I start proofs in geometry class. It is a good way to show the kids how to write a proof, using what they know.
When I taught at a large high school, we used logic proofs up front like that for Course II.
At my little school we teach Algebra, Geometry, Trig (not I, II, III or A and B) and our geometry text does not support logic proofs well. One year I stuck them in, all the same, and it went ok. But we are not having major proof problems, and I am the only big logic supporter, so it didn’t catch on.
But we like the proof in algebra. It’s a little early, but it’s a nice preview.
We also (thanks to the fantastic younger teacher I share planning with) have the kids do a Sudoku project at the very start of the year. They justify the first ten steps. We have statements, reasons, a definite sequence… Works quite nicely.
Jonathan
I like your Algebra proofs. I don’t think I’ve seen that done around here.
I’d like to know more about the Sudoku project as well!
The link to which your figure leads is also quite interesting in the set of problems there. I did not find any reference to that in the post, though.
Clueless,
I was just looking for something that looked algebra-proofy. I did notice that it seemed to be part of some well-constructed work sheet or test, but didn’t dig. What else did you find?
Interesting, it was actually the problem set for Chapter 2 of the Geometry course.
More inforamtion at http://marian.creighton.edu/~dkath/index.htm
they don’t allow us time in the syllabus
in the typical “high school math for college students” courses
to do this kind of thing;
furthermore there’s nowhere near enough
exercises in the texts; they just want to *pretend*
to’ve introduced the field axioms and let the student
take on faith that they’re important for *something*.
it’s one of the most frustrating things in my working life
(and i’m frustrated about *something* or other
several times a day on average …).
sometimes i’ve *made* time and created
exercises (of the i-give-steps, tell-me-reasons type;
if i ever do it again, i’ll try the opposite style, too.)
the buy-in typically isn’t great. students at this level
generally don’t trust me very well
(& whenever i take issue with the way
things are done in the text, it costs me
a little more of that already-scarce resource).
on a more pleasant note:
there was a problem in graph theory once years ago
that was known (to a certain math-for-poets class)
as “lamont’s theorem” (because the guy who first knew
the answer … and i *mean* first: before me) was lamont.
i’ve very seldom seen a student so radiantly pleased
with anything i’ve said as when i just dropped that
into the lecture …
hmm. misplaced closing paren.
it should have been an ellipsis
(followed two words later by
a *properly* placed closing paren).
but i expect my meaning was clear.
& it’s the thought that counts.
right? it had better be.
(otherwise, my failure to have learned how
[after several decades of trying pretty hard]
to cobble together a simple declarative sentence
would be, well, not to put too fine a point on it,
kind of depressing.)
in other, equally useless & uninteresting news:
wordpress ate my other recent comment
(on the “last left over” problem). blag.
TGIF.
The last left over comment may have gone into the spam filter. I check before dumping it, but if it is very full, I can miss things.
And yes, doing the proof is a luxury, but I would argue that all of us in high school have time to do at least some demos… I teach college algebra; I know there is no time at all.
please make some more sense for the clueless
who the hell are the clueless
clueless,
Wazzup, dont understand proofs me neither
I have a coming article on high school proofs. Anyway, I have written two blogs that discuss a proof why do we have to flip the inequality sign when multiplying inequalities by a negative number. You may want to check it out.
http://math4allages.wordpress.com/2009/12/03/triangle-angle-sum/
http://math4allages.wordpress.com/tag/flip-inequality/