Now that we've had a chance to work through HW3 and part of HW4 in our house, I wanted to share a few things we ran into that might be stumbling blocks for other kids.
1) In the answer key to HW3, I suggest that kids might try to figure out the pattern for how many socks Grufftina needs to pull out depending on the number of colors. This part is pretty straightforward: She needs to grab a sock for each color in the drawer and then one extra.
The last part of this question suggests that maybe we could come up with a rule even if we don't know the actual number. In the text, I wrote this down as Grufftina having "n" colors in the drawer, and I was looking for kids to come up with the idea that whatever "n" is, "n+1" socks will do the job here. This turned out to be trickier than I thought it might be, mostly because my daughter was trying to sort out what "n" might stand for (maybe "nine?" maybe "ninety?")
My wife had an excellent way of breaking through this logjam, though, that I want to share here: She said that the sock monsters had some words for numbers that weren't the same as ours, so Grufftina might tell you that she has "Blorg" different colors in the drawer, but we don't know what "Blorg" means. Can you still tell her how many to pull out?
This turned out to totally do the trick, so definitely feel free to cross out my very boring "n" and do something like this instead if it helps.
2) OK, this one we haven't really run into yet, but as I was writing the answer key to HW4 it struck me that Q2 and Q3 are probably just a little too hard because of how many colors you need to keep track of. I was trying to get away from needing to talk about how you might multiply the numbers together to get to the right answer, but with 5 colors and 3 feet to think about, the tables/diagrams just get kind of big and probably annoying for young kids to deal with.
My suggestion here is that reducing the number of colors to 4 probably makes this all MUCH easier to think about. In particular, for Q2, it makes it easy to do "leave one out" reasoning about the number of different mismatched 3-sock sets with all colors different: If there are 4 colors and 3 feet that have to be all different, there's always one color left out! So leaving out each color in turn tells you that there are 4 different mismatched sets (as long as we don't care about what foot the sock goes on).
I may come up with an alternate version of the Boofernaut problem that's written this way and replaces Q3 with a question about finding all the different ways to rearrange a mismatched set on 3 feet. Stay tuned for an alternate draft in the next week or so.
If you happen to try either of these problems (or any others) and have thoughts about how to improve or simplify the questions for kids, please let me know! This is all a work in progress, so I'm very much still figuring out how to present some of these fairly tough questions so that young kids can work on them and have fun.