On May 3, 11:42=A0pm, Arturo Magidin <magi...@[EMAIL PROTECTED]
> wrote:
> On May 3, 7:57 pm, "michalc...@[EMAIL PROTECTED]
" <michalc...@[EMAIL PROTECTED]
> wrote:
>
> > I tutor a lot of algebra and as you know there are three basic types
> > of problems that algebra students are given. The ones that say
> > "Solve", "Factor" and the ones that say "Simplify".
>
> > Solve you check by plugging the answers back in.
>
> > Factor you check by multiplying things back together.
>
> > Simplify... Well neither my students nor I were ever taught a reliable
> > and simple check for simplified expressions. Example "Simplify (x^2
> > -1) / (x+1)" this becomes "x-1".
>
> And if "factor" you check by multiplying things back together, why is
> it that "simplify" cannot be checked similarly? You go from (x^2-1)/(x
> +1) to x-1 by either long division (which can be checked through
> multiplication), or by factoring and cancelling; the factoring can be
> checked, and the cancelling can be checked simply by multiplying and
> dividing again. So how is this different from 'factor" and "solve"?
>
>
>
> > The best check I have thought of I call the -3 check. You plug in the
> > number -3 into the unsimplfied expression and the simplified
> > expression and see if you get the same answer. This technique is not
> > perfect. Sometime you will get the same answer even if the expressions
> > are not equivalent such as -3x and x^2 and the test blows up for
> > expressions like 1/(x+3). Still these problems come up seldomly.
>
> In other words: you encourage them to check a universal proposition by
> making a single test case?
>
> > -3 is the smallest and easiest to use number that rarely gives a false
> > check.
>
> Why is -3 any better than, say, -2?
>
>
>
> > 0 fails in multiplication x=3D2x
> > 1 fails in exponentiation x=3Dx^2
> > 2 has some weird properties x+x =3D x*x =3D x^x
> > 3 has a problem with abs(x) =3D x and some other minor ones
>
> Every number has problems; why? Because you are trying to test a
> universal proposition via a single test case. Hardly what I would call
> pointing them in the right direction. Obviously, though, you think it
> is quite clever. Good for you. I shall be sure not to offer this test
> to my students, however. Encouraging them to check something that must
> hold for all x by plugging in a single number will only give them
> false confidence, if not downright the wrong idea.
>
> > -3 distinguishes all the above situations and is easier to computer
> > than larger numbers.
>
> > So, I want to get your opinion on this test and ask if you can think
> > of anything better. I encourage my students to review their
> > calculations but many times they will make the same mistake more than
> > once if they don't know that their answer is wrong. Remember, in
> > general I am teaching this to 8th and 9th graders. They are not very
> > sophisticated or abtract thinkers yet and thier ability to keep track
> > of multiple things is quite limited.
>
> So it's better to teach them that they can establish that "all cats
> are black" by looking at their neighbor's and checking?
>
> Arturo Magidin, sans .sig
I always teach them that there are cases when this won't work and your
would need to test with all numbers to be sure. The remultiplying
before cancelling allwos them to check one step in a simplification
but not the whole process. You might be right that -2 would be a good
choice but I will have to think about it. As for the ad hominum attack
that I must think I am really clever for thinking of this, I do not
and never pretended to be so. I posted here to hear if people had
better suggestions and get an idea of what reasonable objections
people had.
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