Middle Schoolers and the Unsummables
They had created much of the chart during that original class time two weeks ago. So on Wednesday, I asked them to take them out and finish up with them. I asked those who believed they were finished if they were confident that they had found all the ways for every number. They were not, so I asked them how they might approach the problem so they could be confident. One young lady suggested that they start with the sequences that start with 1 (1+2, 1+2+3, …) then those that start with 2 (2+3, 2+3+4, …) and so on until each list got larger than 35, the number they were going up to. Hooray! I asked the group what we might call a strategy like that, and hinted that they might like to look at the list of strategies they had all written in the front of their notebooks on their fist day with me. After more guesses than I’d have expected, they did come up with “Make an Organized List”. After they spent some more time finishing and/or checking their lists, I went around the room and asked how many ways they had for each number. Where there was disagreement, they listed the ways and sorted it out. Because I wasn’t convinced that they had all gotten all the “ways” down accurately or readably, and because I thought the organization of my chart might make some of the patterns pop out a bit better, I handed out my own chart, which I had organized according to the very strategy we had just discussed:
You can download my whole chart up to 36 (though I squashed down the columns on the right to fit it onto one page) here: unsummables.JPG
I asked them, as homework, to come up with at least 5 conjectures. We talked about what a conjecture was. I put the following “starters” on the board, and told them they could use mine or come up with their own:
All the odd numbers _____________
All the numbers that can be expressed as the sum of three consecutive numbers _________
All the unsummable numbers _____________
So today I collected them and I was pretty impressed that a lot of the kids seemed to really “get it” and came up with great conjectures. Some of the younger kids (I have 10- to 13-year-olds in this group) came up with mostly observations about the structure of the chart, but I still think that looking at those kinds of patterns was also a useful exercise, even if they didn’t really come up with many conjectures. All but one student came up with a reasonable conjecture about the unsummables.
Here are some of the conjectures they came up with about the unsummable numbers:
- if you double four, then double it again, and then again, they are all unsummable
- all numbers that are unsummable are multiples of four
- starting with one, every time the number doubles it is an unsummable. Example 1, 2, 4, 8, 16, 32, 64, etc.
- all unsummables are factor of 4
- all the unsummables are sum of 4
- If there is a number that can’t be solved, then it times 2 can’t be either, such as 4, 8, 16, 32, 64
- all of the unsummables are divisible by 8 and 4
So… most of them were on the right track, but none of them had the vocabulary to express the pattern as “the powers of two”. And we have a little vocabulary confusion (factor, sum). But all great observations.
Some of our other conjectures (some true, and some not):
- every number that is divisible by 3 (except 3) has a number combination that involves three numbers
- there are no numbers with more than three different number combinations
- there is no number that is divisible by four, that has more than one number combination
- all odd numbers have a consecutive addition problem that only has 2 numbers, such as 21 = 10+11
- all the numbers that can be expressed as 3 consecutive numbers are a multiple of 3
- evens other than multiples of 4 are summable
- triangle numbers are summable
- there are no more than 4 ways [to sum any number]
- all the numbers that are divisible by 6 have at least 3 consecutive numbers
- there is about a 50% chance that a number will have only 1 way
Some statements that are more observations about the table than conjectures:
- in each column the spaces between consecutive expressions keeps getting larger by one — the first has 2 spaces, then 3, etc.
- all the diagonal lines going from right down to left end in the same number
- the diagonal lines are getting steeper
We went around the room and had each student state his or her favorite conjecture. Then we talked about what they had noticed about the unsummables. We got to talk about one-way implication when we discussed the observation/conjecture that “all numbers that are unsummable are multiples of four,” since someone else pointed out that not all multiples of four are unsummable. When we looked at the conjectures from the kids who had found the “doubling” patterns for the unsummables, we quickly got to the point where we noticed that those numbers were all “two to the something” and introduced the phrase “powers of two”. I asked the kids to think about why the powers of two might be unsummable, but they weren’t really ready to get very far with that. I did decide to walk them through a proof (as I presented in this comments thread). They did seem to mostly follow it, though I think only a couple really appreciated it. But perhaps I laid some groundwork for them to be more comfortable with mathematical proofs in the future.
Thanks again, Dave, for a cool investigation!