Tangrams and math at the market

who: J2 and J1
when: all day (sick kids at home)
what did we use: tangrams

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A quick-start activity and conversation from our kids' recent home-sick days.

Tangrams

(Note: people like pictures, but I don't like spoilers. I've included some of our tangram pictures at the bottom of this post)

We got a book of tangram puzzles from the grandparents when we were visiting. It was a good catalyst for getting out the nice tangram set that came with our RightStart math kit. While the book gave us some good ideas, the best one was a simple progression we (J2 and I) came up with on our own: make isosceles right triangles with 1, 2, 3, 4, 5, 6, and 7 pieces.

When we did attempt puzzles from the book, we quickly noticed that we never came up with the same solution that the book had. Admittedly, we only did about half a dozen puzzles, but this led to the natural question of how many solutions we could find for our triangle progression. That naturally opens a really interesting discussion about when you should consider two solutions to be the same (rotations, reflections)?

Another path to follow is related to dissections: we had a sense that some solutions are more satisfying than others because they can't be broken into "typical" sub-shapes. Making this idea more precise is difficult, but worth pursuing.

One last path for the triangle progression is to see what solutions are possible simultaneously. There are several ways to specify this, but here is a specific challenge for you:

Let P be a set of positive integers summing to 7. Using the 7 traditional tangram pieces at one time, make isosceles right triangles so that, for each p in P, there is exactly one triangle with p pieces.

Can you find a set P that works?

Math at the market

Someone was nice enough to buy me a bag of passion fruit. For some reason, the price came up: 80 baht for 1 kg (we weighed it, just to confirm). I recalled another market where I had purchased 800 grams for 100 baht. Of course, that leads to instant discussion:

• Which seller has a cheaper price? How do you know?
• How much cheaper is one price than the other? What are sensible ways to compare?
• Why might the prices be different? Different place and time are obvious ones.
• If the two sellers were next to each other in the market at the same time, would people only buy from the cheaper source? Why/why not? What factors complicate this?

Also, if you were paying attention, you will realize that, yes, this is how I thank someone for giving me a gift: lead them along a mathematical conversation!

Some pictures

Avoid this section if you don't want hints about some tangram configurations.

We thought our approach to making a letter "L" shape was better than the one suggested by the book. Both have an annoying triangle tip poking out. Our version otherwise has a common and consistent width on the two legs which the book didn't have.

Simple rectangle. This is an example of something that comes quickly once you figure out the classic 7-piece square.

One of the members of our triangle family and a cousin of the class square. This gives away solutions for 1, 2, 5, and 7 piece triangles, so sorry about that.