Larry Zimmerman strikes again

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 x² – y² = 2 has no solutions in integers

Translate this into words.

Say this in words: x² + y²

(x + y)²

x² + 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?”