XYZ * 1000 + XYZ = XYZ( 1000 + 1) = 1001 * XYZ.

Those who did try the problem either found this relationship after thinking about it for a while or found part of the answer (11 was a common response if the full result wasn't found.) That one's a good answer by itself since it notices all numbers of the form XYZXYZ satisfy the divisibility rule for 11.

We then broke into group, after I encouraged folks to pair with someone different than their normal partners. Here's the set I wrote up: https://drive.google.com/open?id=0B6oYedIeLTUKeUdWdDdCUHJ6dXM

I'm going to mostly talk about the first one which I found via MathForLove.com. This asks the students to try adding a digit to itself 3 times i.e. 6 + 6 + 6 and then multiplying it by 37. The goal is to observe the pattern and come up with a reason why its true. To start I wasn't certain how easy or hard this would be but I suspected it would at least be a good warm up. It turns out, this was quite challenging for the kids. Assuming they multiplied correctly and there was a few times I pointed out some careless errors, everyone noticed the pattern right away. (D + D + D) * 37 = DDD. But it was really hard without scaffolding for lots of the room to come up with reasons. I ended up prompting a bit to take a look at the 37 and think about using the commutative and associate properties for those who were stuck for longer periods of time. Eventually everyone found the key 3 * 37 = 111 but once again there is need here for more practice/play with regrouping. Like the MOEMS problems I really want everyone to sit back and consider a simple computation and how it can be rearranged. I'm going to keep looking for exercises that require the same principal and see if I can introduce them in future weeks. Also because I was stretched thin, I didn't have as much time as I wanted to observe the kids sharing results back and forth. So I'm looking forward to trying this again with some more help and focusing more narrowly on this aspect.

For the next problem of the week, inspired by mikesmathblog I gave out an old AMC problem (although minus the multiple choices):

Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?

This should pair really well with the fourth graders' current rate and ratios unit.

For the problem of the week, once the kids saw 1001 was a factor, could they explain why it has to be the largest factor?

ReplyDeleteThe idea of the 37 trick was behind a short Square One TV segment my kids really liked: SQ1TV.

Unfortunately, when I checked, the video wasn't available anymore.

I didn't stress that part of the process as much with the kids but I did hear some fuzzy logic along those lines. Definitely a good point for next time to explore. I'm still busy brainstorming other activities to encourage more exploration of number rearrangement.

ReplyDeleteIs the magic 1089 trick on the middle of this page number rearrangement? Just checking that I know what you mean.

ReplyDeleteFWIW, I remember the teacher I got that trick from made a really elaborate show: without showing her their interim or final results, she had the kids go through the steps, then write their result on a piece of paper. She then took the class outside, burned the paper and rubbed the ashes on her forearm, revealing the number 1089.

Yes anything where the you might use some combination of commutative, associative transforms.

DeleteI just found an interesting decoding riff on this:

ReplyDeleteYE * ME = TTT. Note: TTT must be a multiple of 37.