Date: Sep 29, 2012 12:52 AM
Author: Louis Talman
Subject: Re: An Algebra 2 Test

On Fri, Sep 28, 2012 at 9:36 PM, Robert Hansen <bob@rsccore.com> wrote:

a^2 - (a-1)^2 = a^2 - (a^2 - 2a + 1) = 2a - 1 = 13.04822...
>
> This is a very good point, work the algebra FIRST. I wonder how many
> algebra teachers are even capable of concocting such a problem? Working
> backwards and making sure that 2*a doesn't require any more than single
> digit math (no digit greater than 4). I have to think about this. This
> works well with large integers as well, with the same condition on the
> digits.
>
> Lou, what do you say to problems like this with regards to our prior art
> discussion?
>
> Ha, calculators are allowed. Calculators with 70 digits of precision I
> suppose.:)
>
>

I get
65241132301442003123012230341430201/5000000000000000000000000000000000.
(Yes---it's in lowest terms.)

Though I would accept your decimal answer---which you arrived at the
semi-hard way. (You even made the semi-hard way harder than it needed to
be: 2 a - 1 = a + (a - 1) = a + b.)

It's easier to write a^2 - b^2 = (a - b)(a + b). In this case, a - b = 1,
so the answer is simply the sum of the two numbers given. Your restrictions
are therefore unnecessary: Kids should be able to do multi-digit addition
where it's sometimes necessary to carry a one. Even in their heads, given
that they can write down the digits of the answer as they produce them.

What do I think of it? It does require that the student know at least one
important algebraic identity, and the form a^2 - b^2 gives it away
immediately to the cognoscenti. The number of digits involved should warn
students that they are intended to solve it without their calculators.
Those are the only good things I can say about it. As David said, it's
pretty silly.

The pluses are outweighed by a very big minus: Questions like this convince
too many kids that the heart of mathematics consists of knowing the hidden
trick---and that, therefore, no real thought is ever needed. That's
absolutely the wrong message.

And the point isn't that one should "work the algebra FIRST". It's that
algebra is for making life easier.

--Louis A. Talman
Department of Mathematical and Computer Sciences
Metropolitan State College of Denver

<http://rowdy.mscd.edu/%7Etalmanl>