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