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?”