The Sieve of Brandon

We are taking a break from the Fraction a Day posts so we can bring you "The Brandon Method"
My nephew needed help to prepare for his Algebra test (which was only two days away) so I was fighting the competing goods of helping him succeed on the test and actually having him learn math.  Along the way the need to teach math won out for a while (well most of the time). We were doing prime factorization and greatest common divisors and I asked him what the primes were, then I brought out my handy dandy number grid and here is what he created:

I "colored in" 0 and 1, then asked him what was the first prime #, he said two, I then asked him to color every multiple of 2 (except 2 of course), then asked him what was the next prime #, (he said three) then asked him to color in every multiple of three and repeat.  The "Brandon technique" was to change colors each time, so he started with green for multiples of 2, then went to red for multiples of 3, etc.  After he got up to 8 I asked him to stop because we were running short on time, and we had a lot of material to cover, but his response was "but I'm having fun, I want to finish".  He persevered and stuck with it (a great habit, so why kill it just to pass a test).    I told him that was the first time I had seen anyone use that technique and that there may be one right answer, but there are multiple ways to get there and the important thing was to find the ways that worked for him (and understand different routes as well).  He was actually having fun doing math (we spent four hours doing nothing but Algebra and I finally had to tell him I was not going to teach any more and he had to go to bed.  This from a kid who usually does not like to do a lot of homework and tries to finish as quickly as possible.

Another teaching idea that came up was a better way for them to discover interesting things while doing factor trees, but we will save that for another post. Hint: think visually and you can move around parts of trees you already created, which leads from this will take forever (when he was asked to list all the prime factors for numbers up to 75) to an aha moment and a big smile.