The Number Makers


Last Thursday, in my evening math circle (I now have a daytime Thursday circle, but more on this in a separate post), we are continuing to discuss what is math.

E. brought a calculator and decided to use it even for simpler problems. That’s perfectly fine with me. A. said that he doesn’t need a calculator because his brain is calculator-fast. To demonstrate, he calculated an answer to the following warm-up problem (took him less than a second):

It takes an elevator 3 seconds to get from the first floor of a building to the third floor. How long does the elevator take to go from the first floor to the ninth? (from Anna Burago’s Mathematical Circle Diaries, Year 1: Grades 5 to 7)

A’s answer was 9. See the problem? I explained that calculators aren’t all that smart. They are fast and accurate, but not smart. So enjoying mental math is terrific, but the results must be checked against the original problem – does my answer make sense. Perhaps it’s better to have a brain that is like, well, a brain, and not like a calculator.

It was K who pointed out that 9 wouldn’t work and explained why not. After some more work – drawing elevator shafts, counting floors and spaces between, discussing whether the size of an elevator or where it starts matters, they did arrive at the answer.

Then we moved on to the main activity – making numbers 1-15 out of five 2’s (here’s a link to the Google Doc with the puzzle). Interestingly, they immediately thought back to the Four Four’s problem, but somehow forgot that they could concatenate the twos. There is still a lot of confusion about the order of operations and I ended up helping everyone to get their parenthesis right. Perhaps that’s what we should do next time – explore the order of operations.

Everyone got stuck on odd numbers larger than 10. That’s when I asked them to think about one operation that they used in Four Fours, but haven’t used here yet. Eventually they re-discovered concatenation, yet 11 was proving to be difficult to crack until K mentioned that 11 was half of 22. So I thought after that, 13 and 15 would be a breeze. M did 15 (he was the only one working on the list out of order) and shared his discovery.

And yet 13 was unyielding. They were stuck on it for at least 5 minutes. Granted, that was almost the end of the meeting and they were tired. But M, K and A all said they wanted to continue with the problem and solve all 15 numbers no matter what (that’s a new and exciting development in itself!) I gave them a hint – to think about what do 11, 15 and 13 have in common and how can their solutions for the first two help with the last one. Almost right away, working together (!!!) they saw it and finished the entire 1-15 list.

After a round of high-fives, they said they were too tired to do anything else today (understandably). And for the last 15 minutes of the circle they just jumped on the trampoline.


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