- Book: Alice in Pastaland: A Math Adventure by A. Wright.
- Topic: Combinations: Inspired by Alice in Pastaland, I asked the kids to figure out how many ways there are to choose two different kinds of pasta from ten different choices.
- Topics: Tangrams, Geometry: We did a number of tangrams from Tangrams: 330 Puzzles by R. Read.
How Did It Go?
We had four kids this week.
Alice in Pastaland
The kids were very excited to finish reading Alice in Pastaland. It doesn’t have all that much math content, but it does mention numbers frequently. It also inspired the next activity…
We’ve investigated this problem before, but this time I wanted them to come up with the general formula. When I gave them the problem, they had no idea how to proceed, and I’m sure they wouldn’t have gotten anywhere unless I got them started. I said that a good problem solving strategy is to look at simpler versions of the same problem.
First, I asked how many ways if there were two kinds of pasta; they quickly got one as the answer. Next I asked if there were three kinds. They were able to demonstrate all three, using the picture of plates of pasta I drew (above). Next I asked four kinds of pasta. After some wronger guesses they settled on five, by finding five different answers. I then wrote the table in the picture above: 4 columns of 3 rows each, AB/AC/AD, BA/BC/BD, CA/CB/CD, DA/DB/DC. The highlight of circle was that they noticed the duplicate pairs themselves: AB vs BA. They didn’t notice that every combination had one and only one match, but when I asked why there were the same number of originals as cross-outs, one of them realized they were in matched pairs. For 4, they simply counted to 6. Next we did 5, and one of the kids wrote out the entire table of 20 combinations (including duplicates), counted it, and divided by two. I asked if there was a faster way to do this, and they saw they could use multiplication. From here, with just a bit more help, one of the kids was able to answer the full problem. Out of the four kids, two were pretty involved and (I think) understood the answer at the end, the others probably not.
For a reward, they all got a (very small) prize at the end of circle.
Unlike previous circles, we just did the puzzles straight out of the book. This is quite a bit harder because they don’t have an outline to put their shapes in, and it’s quite a bit harder to understand the scale of the various parts.
The kids have varying abilities at Tangrams. One interesting difference is that some of them still have trouble copying a completed Tangram. They can get the shapes in the right general location but sometimes have problems with exactly orientation or positioning.
I started by giving each kid a different puzzle from the same page. It turns out that none of them are ready yet to solve a puzzle without help. So, I switched to having everyone work on the same puzzle. The thing I tried to teach them is that they should first look for where the two big triangles are. Some of the kids could solve some of the problems once they knew where the big triangles went. Working as a group was a pretty good way to teach them strategies for solving tangrams, but the disadvantage is that it’s now a direct race to finish, so one of the kids got frustrated when they were slower than the others. We ended up doing about six different puzzles as a group.