michalchik@[EMAIL PROTECTED]
wrote:
> You may be right about at least some kids getting
> mislead by this technique. My main problems is that
> wwhat you suggest is really not practical. For example
> a common mistake kids make is accidently or mistakenly
> cancelling part of a numerator or denominator as in
> a^2/(a+1) --> a/1 --> a. There is no reversable step
> that they can use to test. Or sometimes the reverse
> step is just way too hard for the kid such as in
> (2x^2 + x + 2) (x^2-2x+3) = 2*x^4-3*x^3+6*x^2-x+6.
> Like I said, going over the probelm again is not
> goos since people are prone to make the same mistake
> twice. If you can come up with a better alternative
> i would be grateful.
I don't like your -3 test for two reasons. One reason
is that the computations involved with plugging in -3
are often not immediate, especially for 8th and 9th
grade students. Another reason is that you're missing
an op****tunity to reinforce some im****tant concepts
(e.g. dividing causes exponents to subtract, etc.)
What you're looking at are things I called "safety
checks" in my cl*****. The most im****tant two safety
checks are plugging in 0 and looking at leading terms.
Of course, you can't always use 0, so use 1 or -1,
although now the process isn't so immediate.
For example, I know (2x^2 + x + 2)(x^2 - 2x + 3)
has a leading term of (2x^2)(x^2) = 2x^4 and a
constant term of (2)(3) = 6, the latter obtained
either by plugging in x = 0 or by knowing the
lowest order term is obtained by appropriately
combining the lowest order terms in the inputs.
Also, I know (x^3 + 3x^2 - 4x - 12)/(x - 2)
can't be x + 3 for two reasons. One is that
the 'x' in x + 3 is not equal to x^3/x, and
the other is that the '3' in x + 3 is not
(-12)/(-2).
A few years ago I taught 4 or 5 sections of (college)
intermediate algebra one year, and I'd bet that 90%
of the incorrect algebraic manipulation problems I
found on tests could be seen immediately as incorrect
by just applying the two safety checks above.
Now these aren't going to catch everything, but that's
not the goal. It is not the goal to independently work
the problem (especially when many of the students, and
most of those in your target audience, can't work the
problem correctly in just one way), but rather to have
available 2-second "sight checks" that one can use as
safety checks.
I note that one of your examples, a^2/(a+1) --> a/1 --> a,
p***** both of my safety checks. However, this computation
relies on two separate errors occurring together. The first
is cancelling an 'a' without factoring an 'a' out in the
denominator. The second error is in replacing the cancelled
'a' with 0 instead of 1. Now I don't doubt that this can
happen (and I've almost certainly seen it, given how many
papers I've graded in nearly 30 years), but as I've already
indicated, the vast majority of algebraic errors can be
detected with the two 2-second "sight checks" I mentioned.
Dave L. Renfro
|
