Difficult and not difficult

As I wrote this morning on twitter:

In 4 years of teaching my kids math, I've developed no ability at all to tell ahead of time if a problem will be hard or easy for them.

— Mike Lawler (@mikeandallie) November 26, 2014

The outcome isn’t really surprising at all since:

(1) I’ve never taught young kids before,
(2) I’ve never taught elementary math before,
(3) The experience with one kid probably doesn’t translate in any way to the other kid,

and . . . well, I could probably get to (100) without much difficultly.

Still, despite maybe not being intellectually surprising, it still surprises me. All. the. time.

We had friends staying overnight and got started with school a little late this morning. My younger son was starting a new section in his number theory book today – “Units Digits.” We went through a few examples of finding units digits. The problems we did were mainly introductory problems like: find the units digit of 4*23, 214*23, and 492*5137. Then we did the same exercise for a few powers of 4.

With those examples out of the way I wanted to explain how looking at patterns can help you find the last digit of large powers. I thought the large powers would make for a fun movie project so I started off our movie by asking him to find the last digit of . He worked through the problem as if we’d been studying this subject for weeks. I expected the transition from finding the pattern to evaluating what the pattern would be at the 1000th step to be much more difficult. The approach that I walked through in the 2nd half of the video is what I was expecting to be talking about (in pieces) for the entire discussion:

While I was working with my younger son, my older son was working on a few old math contest problems. The one linked here gave him quite a bit of trouble:

2003 AMC 10B Problem 5

Here’s the problem w/o the link:

“Moe uses a mower to cut his rectangular 90-foot by 150-foot lawn. The swath he cuts is 28 inches wide, but he overlaps each cut by 4 inches to make sure that no grass is missed. He walks at the rate of 5000 feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn: (a) 0.75 (b) 0.8 (c) 1.35 (d) 1.5 (e) 3.”

Contributing to my surprise was that first the first time ever last weekend he helped me mow the lawn. Ha!

 

If it wasn’t 35 degrees and raining this morning, I would have gone out in the back yard with him and mowed for a bit just for context! More to the math points, though, I was genuinely surprised at how much difficulty this problem gave him. Two of the larger struggles came from:

(1) The interaction of the 28 inch wide cut and the 4 inch wide overlap.

And, yes, this part of the problem is gimmicky, no question. However, even drawing a picture of what was going on in this problem was hard for him. We probably talked about it for 15 minutes before he had the “aha” moment of realizing you were essentially moving across the lawn in 2 ft steps.

(2) The transition from knowing you were moving in 2 ft steps to figuring out how long it would take to mow the lawn.

We had a picture on our whiteboard of a rectangle chopped up into a bunch of thin (90 foot long) strips. Maybe the long talk about the first half of the problem sort of used up all of his mental energy, who knows, but this part took a lot longer to work through than I would have guessed. Even when we got to the last bit and just needed to evaluate (75 strips) * (90 ft / strip ) / ( 5000 ft / hour ) he told me that he was worried that we hadn’t solved the problem correctly because the units were wrong.

I don’t mind struggles – in fact I want struggles. I’m just surprised (constantly) when I don’t see them coming. In fact, this morning when I looked at the problems he was going to be working on I was sure this was the one that we’d be talking about:

2003 AMC 10B Problem 7

I wonder how many years it will be until I get good at seeing these struggles ahead of time?