Wednesday, April 19, 2017

4/18 the series "Infinite Series"

Spring break really flew by and yesterday to my surprise Math Club was already resuming. Things started with small snafu, the door to our room was locked. While we were waiting in the hall for the custodian I went over some administrative items. I'm still looking for a few kids to round out the group going to the upcoming WSMC Olympiad, I wanted to acknowledge the high participation in the problem of the week and that I'd bring candy in next week. Finally, I also started laying the groundwork for the talk next month and asked the kids to start thinking about questions to ask our guest mathematician. 

If only there was a whiteboard in the hall I would have gone over the previous problems of the week but sadly we waited a few extra minutes instead.

For this week I wanted to try out the Infinite Series youtube webcasts with the kids. I thought the above video on proofs was a good first choice since one of my priorities is to emphasize understanding why things work and how it will become increasingly important (and computation less) for the kids as they move forward. In fact, I'm trying as much as possible to add in comments about the math progression whenever appropriate. This is one of those areas I feel is not well understood in 5th grade.  Most of the kids know they're working towards algebra, geometry and probably Calculus. They don't necessarily know what Calculus is about even in the most broadest sense and they don't often think what happens after they finish that sequence. I also think they take it for granted that Math topics are all a roughly linear sequence which is not truly the case beyond school math.

What's also nice about the video is it structured around several problems and even has breaks where you're supposed to try them out first.

I took full advantage of that format and stopped 3 times:

1. The chessboard / domino coverage question was the easiest and one of the boys came up with the standard reasoning in a few minutes.
2. Probability of sticks forming a triangle. I wasn't sure if the kids had been exposed to the triangle inequality so I played that part before pausing. Interestingly everyone said "Oh yeah" even if they didn't recognize it by name.   No one came up with he answer but there was a lot of good discussion before I resumed.
3. Sum of odds formula:  Again no-one fully came up with an answer but I was satisfied with the thinking along the way.

In general this was a bit of a balancing act on how long to let the kids grapple with each problem, knowing they would probably not crack them. I wanted enough time so that the explanations really resonated afterwards but still allowed me to finish the video. In the end I had about 10 minutes of the session left. I thought the quality of discussion was particularly good even though everyone reasoned at their group of tables. Perhaps this was a residue of our work on the whiteboards the last few weeks.

Finally, to round things out I brought two sample Sudoku puzzles and an older purple comet problem set:   I thought most kids would prefer the Sudoku but I was pleasantly surprised that many asked for both so they could try them out.  This represents a shift in my organizational thinking. I'm tactical about this but especially with new activities I'm not sure the length of, I'm jumping right in and saving my old warm-up ideas for the end instead.  I see more benefit from having a light weight activity for those whose focus is used up than a transitional one at the beginning and it means I'm shorting my main focus much less often. If the activity takes the whole time and everyone is engaged I'll just save the extra puzzle for another week.

I went with an infinite series conceptual riddle. My hope is to have a group debate next week.

You’re a venal king who’s considering bribes from two different courtiers.

  • Courtier A gives you an infinite number of envelopes. The first envelope contains 1 dollar, the second contains 2 dollars, the third contains 3, and so on: The nth envelope contains n dollars.
  • Courtier B also gives you an infinite number of envelopes. The first envelope contains 2 dollars, the second contains 4 dollars, the third contains 6, and so on: The nth envelope contains 2n dollars.

Now, who’s been more generous?

Courtier B argues that he’s given you twice as much as A — after all, for any n, B’s nth envelope contains twice as much money as A’s.

But Courtier A argues that he’s given you twice as much as B — A’s offerings include a gift of every integer size, but the odd dollar amounts are missing from B’s.

So who has given you more money?

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