Two weeks ago, I started my middle school group on Dave Marain’s Unsummables activity. 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

Thanks again, Dave, for a cool investigation!

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