Just a fun thought for a quick activity (ideally, it would be a routine, for example a weekly one) aimed at (a) cultivating an appreciation for numbers’ distinct personalities; (b) promoting the idea that a pattern holding for small cases doesn’t necessarily keep holding; and (c) giving kids an opportunity to be creative with math. It just occurred to me, and I don’t have a class to try it out on, so I can’t give you any implementation feedback. Here’s the idea:
n is the smallest number that…
Keep track of what number week it is since the beginning of the year. So this could be week 3. Then, each week, as like a bonus-challenge type of a thing, put the problem to all students to think of a property of this number not shared by any smaller natural number. Publicly compile what people find at the end of the week. This activity gets more interesting, and harder, the higher the number.
Here is a brainstorm, so you know what I mean; although many of these are not the kinds of things a kid would come up with.
3 – smallest odd prime; smallest nontrivial triangular number; smallest number of points needed to prevent somebody else from drawing a line thru all of them
4 – smallest composite; smallest nontrivial square; smallest nontrivial sum of triangular numbers; smallest number that is exceeded by its number of partitions
5 – smallest nontrivial sum of squares (4+1); a pentagon is the fewest-sided polygon in which you can inscribe a star
6 – smallest number with 2 distinct nontrivial factors; smallest composite with an even number of factors; smallest nontrivial perfect number
7 – smallest prime for which 2p+1 isn’t also prime (i.e. smallest prime that’s not a Sophie Germain prime); 1/7 is the first fraction in the sequence 1/n for which the cycle length of the decimal expansion actually reaches the theoretical maximum of n-1
8 – smallest nontrivial cube; smallest sum of distinct odd primes
9 – smallest odd composite
10 – smallest two digit number
11 – smallest nontrivial palindrome
12 – smallest number exceeded by the sum of its proper factors; smallest number that is not a prime power but is still divisible by a (nontrivial) square
…and I’m already having trouble thinking of something for 13.
Variations would be “n is the largest number that …” and “n is the only number that…” I’m imagining with high school, to keep it from getting too hard, that the challenge could be to find any one of the three. Another variant to make it easier would just be “n is the number of…” (the object filling in the blank can’t be defined in terms of n; so “3 is the number of vertices of a triangle” isn’t a legit answer, but “8 is the number of vertices of a cube” is).
Actually another way to make it more accessible would be to make it into a project rather than a routine: for as many of the numbers 1-100 as possible, find either a) an interesting property that it but no smaller natural number has; b) a property that it but no greater natural number has; c) a property that it alone has; or d) some interesting object or set counted by that number.
That way, in order to participate, you’re not forced to find something for every single number. Some people will find something for just 15 numbers, others for 40, maybe a handful of others for 80 and those kids will get really intent on the ones they haven’t found yet.
Another cool thing I could see coming of this is a real awakening of the idea of proof, as people try to decide if a number really is the greatest or only number with a given property. For example, while making the list above I noticed for the first time that 9 is the square of the number of factors it has, and wondered if there were any other numbers like this. Trivially, 1 as well; but I’ve just about convinced myself that that’s it; there are no more. How would a student justify this claim?