## Monday, March 30, 2009

### The Perfect Challenge

I am not a huge fan of making kids memorize things for its own sake. But memorization does have its place. For example, for a student to be able to efficiently simplify a radical, it really, really helps if she recognizes a perfect square when she sees one. For example, $\sqrt{338}$ looks like it would be complicated to simplify, but by the time I'm done with my freshmen, they can do this without a calculator. They notice it's even, so they write it as $\sqrt{169}\sqrt{2}$, and then they recognize that 169 is the square of 13, so they write $13\sqrt{2}$. Done. But recognizing the 169 is the lynchpin.

I actually have them memorize all the perfect squares from 1 to 400. The first 10 they kind of know already, so it's really a matter of learning the square of 11 through the square of 20. How do I accomplish this feat? Bribes. This little exercise is a testment to the lengths a 15 year old will go for a Lemonhead. (Also lemme just say, I'm pretty sure I stole this idea from someone, but I have no idea who or when. I'm not trying to claim credit, just get the word out.)

Once the Perfect Challenge starts, we begin every class period with a short, timed quiz, and they know exactly what will be on it. I just jumble up the questions. If a student can complete it perfectly in the allotted time (60 - 90 seconds, depending on my mood), she gets a piece of candy. We do it every day until every student can do it perfectly. Some years one or two students can do it the first day. It has never taken more than 7 tries.

Here is my Excel Document with 7 quizzes, 2 per page.

Jim McClain said...

Kate, I have my eighth graders memorize the perfect squares up to 26 by using an alphanumeric code, where 1 = A, 4 = B, 9 = C, etc.

The first day I do this, I don't even tell them what the code is. I put a complex message on the board and simply ask them to decode it. It's really not that big a challenge to decode, but once they have the key, they really think they're clever.

Once they figure out the code, I leave them secret messages on the board each day while I'm in that chapter, and they enjoy decoding the messages. The trick is to use each letter often enough that no perfect square goes unused. You may have to create some pretty bizarre messages to get the variation you need. They get lots of repetition by the time the chapter is over, and it's more natural (and more fun!)than just telling them to memorize them.

CassyT said...

My son can figure perfect squares in his head faster than I can do them on a calculator. He learned how in 6th grade by understanding place value. Of course it helped that he'd been taught with Singapore curriculum since 1st grade, by teachers that understood mathematics well.

He was up to 27 * 27 last fall, when his Algebra teacher said "enough". Here's how his brain works the numbers.
27 * 20 = 27 * 10 * 2 = 540
27 * 7 = 140 + 49
then he adds 540 + 140 + 49=729
faster than I can type it.

Kate said...

Hi Jim - Sounds fun! But can you explain it in a little more detail? Would you write like 64 25 144 144 225 for "hello"?

Kate said...

Hi Cassy - My goal isn't so much to have them compute squares, but to recognize them when they see them and know the root.

But, I like your son's way! I teach something similar based on (a+b)^2 = a^2 + 2ab +b^2. Using your example, 27^2 = (20 + 7)^2, which I do in my head in this order:
20*7*2 = 280
20^2 = 400
7^2 = 49
sum 280 + 449 = 729

Jim McClain said...

That's exactly right, Kate. There isn't anything more to it than that.

Nick said...

The stars were aligned as I happened to read this post, and we're studying factoring, so I began the perfect squares challenge. Thanks for the suggestion. I do a similar thing with having seventh graders memorize fraction, decimal, percent conversions. for halves through tenths. It's maddening when kids don't recognize simple fractions in decimal form.

jd2718 said...

I encourage some more square knowledge, without setting a requirement, or quizzing/rewarding.

On the other hand, I've been known to ask for instantaneous recall of powers of 2 up to 12...

When did you (adults writing here) memorize perfect squares past 13, and how far do you know them without hesitation?

(Me? extended through 20 in young adulthood, then pushed to 33 when I started teaching, but I always pause for 28.)

Kate said...

I had to memorize up to 20 at some point in school. I didn't really know them until probably student teaching. I don't know too many above 20.

As an update, school psychologist friend pointed out that this quiz incorporates both instantaneous reward (candy) and negative reinforcement (we can stop as soon as everyone can do it).

Ann said...

Kate,

So, do you require the students who get the quiz correct to keep requizzing until one day everyone in the class gets them all right?

Kate said...

Yes, I do.

I don't believe that doing it once means they really know it. They could have just crammed for 10 minutes before class.

They don't mind, because they get a good little grade and a piece of candy every time.

Kate said...

Oh yeah...and there's also the mild peer pressure on the kids who haven't done it yet. I've seen kids who have been successful helping the others get ready - quizzing them in study hall and the like. In my classroom environment it's a positive pressure. I don't think anyone is made to feel bad.

Ann said...

Ok. So you give them a quiz grade, or however you count it in the gradebook and a piece of candy. I'm assuming you just grade them during class? I like this idea. I'm getting ready to start a trigonometry unit in my geometry class. I'm excited to try this! Thanks for the tips!

Kate said...

Actually I just have them trade and mark each others' papers. I read through the correct answers quickly, and they are instructed to mark a slash through any ones missed, and write how many missed at the top of the paper.

You're welcome! I hope it works out for you.