In article
<234fa1d2-5150-4bea-83c3-7af69cc16a8a@[EMAIL PROTECTED]
>,
michalchik@[EMAIL PROTECTED]
<michalchik@[EMAIL PROTECTED]
> wrote:
>On May 4, 11:14=A0am, magi...@[EMAIL PROTECTED]
(Arturo Magidin) wrote:
>> In article
<67bf4bbd-9ae0-4e9c-9e93-5763534a4...@[EMAIL PROTECTED]
>com>,
>>
>> michalc...@[EMAIL PROTECTED]
<michalc...@[EMAIL PROTECTED]
> wrote:
>> >On May 3, 11:42=3DA0pm, Arturo Magidin <magi...@[EMAIL PROTECTED]
> wrote:
>> >> > 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
reliabl=
>e
>> >> > 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"?
>>
>> You did not address this.
[...]
>> It is misleading to encourage students to test a universal proposition
>> by checking a single test case. It is misleading to encourage students
>> to test a universal proposition that is to hold for ALL numbers by
>> testing INTEGERS. It puts them in the rut of thinking that the only
>> "numbers" are the integers; they never bother with fractions, or real
>> numbers. I cannot tell you how many students I have to flunk because
>> they apparently don't know that there are a few numbers strictly
>> between 0 and 1. In my humble opinion, your test is both misleading
>> and likely to create false impressions on the students. If you
>> encourage them to multiply out to test factoring, then it should be
>> just as hard/easy for them to test "simplifications" in a similar
>> manner: through long division or through multiplication.
>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.
But "multiplying out" ->is<- a practical way of checking factoring?
How so?
> 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.
But this arises because they are not cancelling identical FACTORS in
fractions, or identical SUMMANDS in sums. That is indeed a categorical
error.
>Or sometimes the reverse step is just way too hard for the kid
>such as in (2x^2 + x + 2) (x^2-2x+3) =3D 2*x^4-3*x^3+6*x^2-x+6.
What is the difficulty here that they find so hard, exactly? They
presumably did not go to this equality directly, but rather went
halfway through by first multiplying out and then adding terms with
identical powers of x. The step to check is only the last one.
>Like I
>said, going over the probelm again is not goos since people are prone
>to make the same mistake twice.
And if they don't know how to add or multiply correctly, which is
usually the case with mistakes like the above, how is plugging in -3
going to save them?
Look: the major problem I have with this is that you are presenting it
on the same level playing field as categorically different
"checks". "Plug in an answer to check if it solves the equation" is
simply a different category of test than "test the identity by
plugging in -3" (or any other number, or any finite number of inputs,
for that matter). As I mentioned elsewhere, this 'check' is
more akin to casting out 9s in sums and products than it is to 'plug
back in to check solution'. By presenting it in the same playing
field, you are in effect implicitly telling them they are
equivalent. This is a real problem. This is more, as was said, a
"sanity check" then a "test of the simplification", much like casting
out nines, or checking the order of magnitude of an answer to see if
it is in the right ballpark.
To further complicate things, encouraging them to always use the same
INTEGER for testing these kinds of expressions is also liable to
reinforce mistaken impressions, such as that "number" means "integer".
>If you can come up with a better
>alternative i would be grateful.
Ah, but every "alternative" is just not practical. Apparently....
First, I would not present this in the same breath as "plug in" or
"multiply out" for the reasons I've mentioned many times. Present it
as a parallel to casting out 9s, perhaps, especially if they are
familiar with the latter and realize the problems with it. Second,
rather than suggest a particular integer all the time, I would suggest
trying 'pseudo-random' numbers; sometimes -3, sometimes 2, sometimes
-1, sometimes 0, sometimes 1/2, sometimes -1/3, etc (and push
fractions, at least simple egyptian ones, from time to time).
Thirdly, present it as a test of ->equality<-, not of "simplify";
after all, this "test" works equally well for "factor": just plug in
something to both sides and see if you come up with the same thing. If
two expressions are equal, then they are equal for any particular
number. Repeat this latter mantra over and over and over again, so
they understand that this is what is behind this check. This would
make it (i) more useful, since it can be used in situations other than
"simplify", and (iii) by repeating the mantra, you are driving home
what is behind it, rather than making it a pseudo-magical property of
the number -3.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
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