Week 2 Group 3: Tiling Puzzles

I had a group of 5 girls (one age 7, one 8, and three 9) and 2 other parents. One was new to math circles, two had been to one before, and two were math circle veterans. About halfway through the circle, I got a call from my older daughter’s school that she was sick and I had to go pick her up immediately. The kids and parents in my living room said “Go! Go! We’re fine here.” They were correct – when I returned 40 minutes later, they were right where I had left them, doing yet another tiling activity. Here are some details from the different activities:

Escher Tilings – we spent a lot of time looking at these. We also looked at some architectural tilings – the Alhambra, and the Dome of the Rock – and some from nature. We talked a bit about Escher, but mostly focused on the question of why I might have printed out all these seemingly unrelated pictures, and, to quote our youngest participant, “What DO these pictures have to do with math?” I left those questions for the kids to ponder and discuss, and they came up with numerous insightful answers. I also asked why Escher would want to do this kind of art. Some kids said “because they are beautiful,” but another said, “That’s not all. I think he wanted to get people curious.”

The kids then spent some time (at my request) creating some tilings of their own with pencil and paper. The surrounding conversation gave rise to such mathematical concepts as vertex/vertices, whether there’s a difference between tilings and tessellations, periodic and non-periodic tilings, and more.

Next I brought out the pentominoes. Most of the kids had seen them before. We talked about their name, their properties, and their role in the children’s book Chasing Vermeer. I mentioned to the families that the Chasing Vermeer website has an interactive pentominoes puzzle (http://www.scholastic.com/blueballiett/games/pentominoes_game.htm). We talked about how some people have named the pieces according to their resemblance to alphabet letters, and discussed whether having a universal naming convention was a good idea. We also wondered how the character in the book who carried pentominoes around in his pocket actually fit them in there. One of the parents remembered that his were paper.

Pentomino Rhino – I had enough pentomino sets for everyone, but only one picture of the rhino pattern, which I had drawn on graph paper, to a scale of 4. One child grabbed it and started laying her tiles directly on the picture. I hadn’t realized that my template fit the pentominoes exactly. Two other kids joined her, while two others were totally left out. I explained that the challenge of pentominoes is to do it without a template. They didn’t care; they all wanted templates.

Not everyone was able to share easily, so I quickly held the one template we had up to the window and made tracings for everyone. Two kids wanted to work together, and the other three worked independently. “That was easy!” they all announced very soon. I asked them to see if they could find other arrangements that produce the rhino, and they came up with two others. “Give us a real challenge,” demanded one child (age 8). I quickly looked online for something to challenge them in the same vein as the rhino puzzle, but couldn’t seem to get the google search worded well. Instead, I gave them the instructions for the Pentomino City, so they got to work on that. It was at that point when I had to leave.

If I were to do this activity again, I would have shown only one set of pentominoes, and a tiling pattern that was not to the same scale as the actual tiles. I actually had a nice wooden set of pentominoes on hand, not just the plastic ones. In retrospect, I would have had all 5 kids share that wooden set and see what happened before getting out the other (plastic) sets.

Pentomino City – I was away while the kids did this. The parents photographed all the houses the kids made for me, though. They reported that it went well, and the kids enjoyed it. I don’t have any feedback beyond this.

Guess my Shape – I also missed this activity, unfortunately. When I did a debriefing with the kids at the end of the circle, a few mentioned that this was their least favorite activity, although everyone reported liking everything to some degree. The parents reported that the kids had a lot of frustration describing the shapes. Especially difficult was left-right orientation, about which the kids weren’t confident. We discussed the importance of communication ability in mathematics.

Esher Tesselations – this was not in the original packet, but I had printed out an article from the National University of Singapore describing how to make your own. While I was out, one of the parents showed the kids how to do this. When I returned, everyone was delightfully engaged in making their own Escheresque creations. While kids worked, I very loudly explained to the parents that it was good that I had left, that a math circle leader is doing her job if she talks less, or maybe even not at all.

(Here’s the URL for the site that we used: http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/Understanding_Escher.pdf . For some reason I can’t get that site to load tonight. I do have a printout, though, so if you want this info I could scan in the page with illustrated details. It’s page 9 of the document, which is a 17-pager full of fascinating math at a level kids can understand.)

Small paper, giant hole – I was surprised that some of the kids had seen this activity before. Two had done it in a local program “Science in the Summer” in the Philadelphia libraries, and one had read the instructions for it in an American Girl magazine. But no one knew how to do it. Four of the kids got right to it, but one was too caught up in tessellations to move on. Kids used each other’s work to build upon strategies, and got very close. But it was frustrating too. I told them of how I had been frustrated last night, late last night, very late, at the kitchen table for a very long time making mistake after mistake after mistake. It turns out that I hadn’t been doing the last step, so I had a bag full of unfinished attempts. The kids asked to have my unfinished attempts to get a better idea of how to do it. This didn’t help much, however. Then I mentioned that I had finally gotten one to work last night. “Can we use that to figure out how to do it?” asked the youngest. I handed it to her, and also finished a few of those unfinished attempts so other kids could try to work backwards too. I explained the value of the strategy of working backwards. The idea that mathematicians actually sometimes do this put a big smile on the face of the girl who had suggested it. Unfortunately, this still didn’t generate any aha moments.

Some of the kids had moved on to cutting paper snowflakes when the rest of the kids (and the 2 parents) asked me to show them how to do the big hole. People had given up. I handed everyone new paper, and set forth to model as they cut. Now I was in the hot seat in terms of communicating math accurately. This was hard! Things got better, though, when we started describing the cuts, shapes, and negative space in real-world terms (door, window, tongue, etc.) For the endgame, we were all using mouth-vocabulary. I instructed them to do the final step as a cut from the tongue crease to the back of the throat, separating the tonsils. Now everyone understood. An added bonus to the mouth-vocabulary (besides the obvious success) was that this language had recaptured the interest of the kids who had faded.

Chinese checkers – We were just about out of time, with one parent on her way to pick up, and the others talking about leaving soon. I showed the non-snowflake kids how to play. We played for a few minutes while lightly discussing the math of the board. Then I showed them a pentagonal board. Two of the kids got very interested in designing a new set of rules to go along with this board. I felt like I was privy to the R&D department of a strategic game company. We never did get into the mathematical differences in the two types of boards. The kids were ready to play with the new rules when time was really up and people had to go home.

I asked for feedback from the kids on which activities they liked and didn’t like. They generally liked everything, especially the Escher tessellations and the snowflakes. Things individual kids mentioned as not-as-favorites were the hole activity and guess the shape. The house activity was both a favorite and a least favorite. I can tell, though, that they really did enjoy it all because no one wanted to stop and no one wanted to leave after nearly two and a half hours.

(I have some photos, but their file size is too large. If I can figure out how to make them smaller, I will post them.)

Our math circle is tomorrow, so I wanted to get in my "before" ideas now! I'm really excited to introduce them to the Escher tilings. I think he is an amazing artist and I think the kids will be amazed too! I love the idea of drawing a tiling picture, but I'm wondering if the kids will be able to do this. The age range is from 7-11. If they are struggling too much I was thinking of letting them use stamps to create their tiling picture.

I'm looking forward to seeing how they do with the rhino picture. My kids struggle a lot more with pentomino pictures then tangrams or pattern blocks which have become easy for them. I think this will be a good challenge for them. I don't expect to have any distraction problems, but I do expect for some of them to complain that they just want to make their own picture ( the ones that shy away from a challenge).

This brings me to the pentomino city. I expect that everyone will love this project and the only problem will be getting them to move on in a reasonable amount of time. I'll have to really get them excited about the following activity so they'll be more motivated to move on.

I think the guess my shape activity will be a real eye opener for most of them as to how important is is to be specific when communicating. I've done this activity before with tangrams and the kids really had trouble communicating clearly enough on their own to recreate the picture well. I'm wondering now if it's better to completely let them do it on their own and to see that the two pictures are completely different and work it out themselves instead of helping them to get the picture right the first time.

I can't wait to see their faces when I show them the solution for small paper giant hole. I'm positive none of them will figure it out, but I expect they'll be amazed when I show them how to do it.

Finally, I always enjoyed a game of Chinese checkers when I was a kid and I think it's a great way to end the class on a fun relaxing note. I think the kids will love it!

It think my son will like this pentominoes blocks but resist doing the assignments and want to create his own forms. He loves Escher and we have already tried Tesselating patterns. We have a puzzle of the lizards and painted them on paper and tennis shoes last year. I adapt most lessons to deal with my son's autism. We will do some tiling with hexagons on the drive way. I think he will like this.

What happened" My son did not make the rino but did make several patterns on his own and did a couple of the cards that I bought with the set of pentominoes manipulatives. We also looked at a lego version of the Escher staircases but have not tackled that project yet. We also talked about the lack of negative space in the tessellation process which he can see with the Escher artwork. We did do some compass drawing with the circle and interconnecting circles in a tiling patterns of scared geometry. He did the intersecting flower of life pattern and a version of Metatron's cube. Here is the chart I showed him. Sacred Geometry Chart

Here is his 1st success. He did it very fast and was thrilled. Very empowering for him.

Reply 1

Esher Tilings - I think that kids will want to spend a lot of time with these. I think I will read up on Esher to anticipate questions like "what WAS he thinking?"

Pentomino Rhino - I expect some frustration but then some cooperation

Pentomino City - I think kids will want to spend more time. I also think they may combine their work into something larger.

Pentomino Fossil - THE DETAILS ON THIS ACTIVITY ARE NOT IN THE PDF, SO I DON'T KNOW WHAT IT IS.

Guess my shape - this sounds incredible - I can't wait! I think kids will have a hard time describing their shapes but feel such a sense of triumph when their partner guesses correctly

Small paper, giant hole - I DON'T UNDERSTAND THE TASK OF THIS PROBLEM! CAN YOU GIVE MORE DETAIL IN THE INSTRUCTIONS? Is the task to create a hole on an intact piece of paper without cutting through the edge?

Chinese checkers - looks like fun - my guess is kids will want much more time but that we'll be running out of time/attention by then.

Looking forward to doing all of these. I've always wanted to do tilings/tesselations with kids but didn't really know where to start - thank you!