Comments on: In Defense Of Busy Work

By: Touzel

Touzel — Wed, 24 Feb 2010 05:41:18 +0000

I counted the rows a little differently and came up with a pattern I’ve never used before in problems like this.

I expressed each row, not as the total number of circles on that row like:
1 + 2 + 3 + … + 17 + 18

But rather as a sum of two numbers, the first being the row number and the second being whatever is leftover:
(1 + 0) + (2 + 1) + (3 + 2) + … + (17 + 16) + (18 + 17)

This large sum can be re-written into two separate sums: the sum of the first number of each group and the sum of the second. The former is simply a summation of the row number, and the latter is almost identical, missing only number from the previous sum (e.g. the second sum would go up to 17, but not 18. Obviously, the sum of the two summations works out to be a perfect square.

This seems obvious to me now, but, for some reason, seemed unique five minutes ago.

By: daniel

daniel — Sun, 07 Feb 2010 23:44:41 +0000

Not directly related to this example, but shouldn’t some math operations become ‘automatic’ so that we can focus on the deeper issues? i.e. I don’t want to be thinking (or worrying) about adding, subtracting, multiplying, and dividing integers when I’m learning to solve systems of equations. In my experience, students who are not solid in the fundamentals quickly become overwhelmed when faced with more abstract concepts. It would seem that ‘busy’ work has its place.

By: phanson

phanson — Sun, 07 Feb 2010 05:09:03 +0000

My 5th graders would love something like this. Why? Because of the different ways it can be solved. If you aren’t confident mathematically, you count the circles. But you realize (quickly) that there needs to be a better way. They are savvy enough to notice the number pattern, and could solve it that way as well. Personally I solved it geometrically, seeing the 2 right triangles and simply multiplying LxW (or squaring 18).

It certainly illustrates how math problems can be solved in a myriad of ways… and that knowing your math facts can save you some time (I’m picturing some counting the circles, as that is where they’re at).

By: sam b

sam b — Sun, 07 Feb 2010 00:21:03 +0000

hi,

gonna give this to my 14/15 yr olds (in britain) on monday and see what they make of it. i really really like this because there are two methods to solve it (one which you haven’t mentioned) and i will be giving it to them on a print out…

1) algebraic – notice sums of rows total square numbers – see 18 rows and hence answer is 18^2

2) geometric – cut out one of the 17 high right angled triangles on the side (the ones your students tried to double) and stick it onto the other 18 triangle to make an 18×18 square

any other methods? i love the link between algebraic result and shape here and it’s going to be the perfect intro to the algebra work we’re about to start

By: John

John — Fri, 05 Feb 2010 22:16:35 +0000

I agree with the comment above, teaching students to see “the trick” to this is indeed the issue. Not this particular problem but all like it.

I have worked with people that will just start counting. You come back 3 days later and they tell you it will be 10 more days. A good engineer will either find the pattern, write a program or make a machine to solve the problem.

One last note though, there are others that will always “automate” the process even for super simple issues. This is taking it to the other extreme. Common sense is really hard to teach.

By: Kevin

Kevin — Fri, 05 Feb 2010 14:47:17 +0000

My appologies, that is a very valid point, there I was considering the elegant way you could program something to actually factorise a quadratic when in actuallity just using the formula would get the same result.

By: Kevin

Kevin — Fri, 05 Feb 2010 14:24:47 +0000

On the subject of the programming a calculator, I probably agree that doing 30 questions when you know how to do them is tedious, but I would have asked for the student to explain the program and how could he take that further.

By: Craig

Craig — Fri, 05 Feb 2010 12:00:43 +0000

I should proofread more often. I meant to say the Locker Problem comes to mind.

By: Craig

Craig — Fri, 05 Feb 2010 11:59:55 +0000

Why is it that students always have a hard time recognizing perfect squares? The problem comes to mind as well.

By: Matt E

Matt E — Fri, 05 Feb 2010 11:53:01 +0000

I’m torn about the “Program my calculator to do it” solution, though. While, yes, it does require some solid understanding to be able to write such a program successfully, the program does not care about tedium. “Brute force” will always work, but it requires the least from of the programmer. If we want to talk about efficiency, though… well, I feel as though that is a) worthwhile, but b) a topic for a Comp Sci class.