I wanted to share another few excerpts from Larry. He gave a talk last Tuesday at the NY Math Circle's PD for Middle School Teachers, and I was impressed again by how he asks questions to extend thinking about a problem, offer more entry points, and deepens my interest. He does it so quickly, I can't help but get interested and then my mind is thinking about 10 different but connected things at once, and whatever I get into relates to the bigger picture, and it's like magic differentiated instruction and guided inquiry. It's really awesome. We did a bunch of problems, and here are three that I wrote down his questions for:

**Show that the sum of two odd numbers is an even number.**

What is an odd number?

What is an even number?

Is 4 even?

Is 0 even?

Are there any even primes?

What’s the next even prime?

Are all prime #s greater than 2 odd?

Are all #s greater than 2 prime?

Is -4 even?

Is -6 even? Why?

So now, what is an odd number?

Is every even a multiple of 4?

**Show that no perfect sq ends in 3.**

Do you believe it?

Do any end in 6?

Did you know that every sq ends in 6 iff the number’s 10s digit must be odd?

Would anyone like to see a proof of that? Good! Go do it.

(I was curious to observe here that every sq is

- a multiple of 5

- one less than a multiple of 5

- one more than a multiple of 5)

**Show that x ^{2} – y^{2} = 2 has no solutions in integers**

Translate this into words.

Say this in words: x^{2} + y^{2}

(x + y)^{2}

x^{2} + 5x + 6 = (x + 2)(x + 3) Find the value of x that makes this false.

Factor it using diff of squares: (x + y)(x – y) = 2

Note: remember given info

Integers factors must be 2 & 1

“I wanna savor that for a minute. To some people that sounds trivial, but to me it sounds profound. Never dismiss the obvious as being trivial. Even if it’s obvious it may be very very important.”

x + y = 2 and x – y = 1 (or other way around)

“Do something, do anything, and if it doesn’t work, SO WHAT?”

Mr. Zimmerman is quite a teacher. I had the honor to be one of his student. The day he retired, Brooklyn Technical High School lost one of its greatest teacher! It is good to hear that he is still involve with teaching.

ReplyDeleteI really enjoy your blog!! I was just looking over this question. Shouldn't you also check x+y=-2,x-y=-1 and vice versa. Given, they will not have a solution in the integers, but still..

ReplyDeleteThanks for the comment and addition! Sorry to take so long to reply. Of course, you're absolutely right that there are four possible pairs of factors! Beautiful.

ReplyDelete