Logistics: We had kindergarten through third graders in our room. This year, we required the children this age to bring a parent, where one parent for a multiple children is fine. We learned from experience that the younger children need an adult to help guide them through the activities. Requiring parents to attend helped make the experience better for everyone. We also had everyone sign a role, including their children's names and an email address where we could contact parents to send announcements or let them know when classes would be cancelled.
Preparation: I printed a handout, available to download from my website, here:
http://math.byu.edu/~jpurcell/mathcircles/Mobius-activities.docx
I also brought (1) scissors, about four pairs per table of six kids, (2) clear tape, about four rolls per table, (3) two boxes of markers, so each child could have a couple of colors and share with neighbors, and (4) strips of paper. I used fifteen papers in three colors, cut into strips. One set of strips were shorter, cut on the long side of a standard sheet of letter paper, roughly 8.5 inches by 2 inches each. The other set of strips were longer, roughly 11 inches by 2 inches.
Since this was the first activity, I had no idea how many kids would show up, so I printed 20 handouts and kept a master copy handy in case I had to run upstairs and print more. (Luckily, our classroom is near the math department office so I have that option.)
Arrival: I was in the classroom about five minutes before we began. As the children arrived, I asked them their names and what grade they were in. I then repeated the names I had learned as new children came in, so that I could call on each child by name. (I had to get help a couple of times.)
We had 12 kids show up, and we seated them in three tables. One table filled up before I arrived, with children only and a father on the side. The other two tables had three children and three parents. We also had the help of two undergraduate volunteers. These were a couple of friendly guys who were happy to help, but after the first 15 minutes I realized they didn't really know where they were needed. Once I figured that out, I had them sit at the table of six, one on either side, and instructed them to just get on the level of the children and talk to them about who they are and what they were doing, and help where needed.
Beginning: A little after 9:00, after I had learned most of the names, I welcomed everyone and asked for a show of hands who was at their first math circle. More than 2/3 of the class raised their hands. We then went over our Math Circle Rules, pirated from Tyler Jarvis:
Rules:
1. Make mistakes.
(It's important to make mistakes, because that means you're trying and learning.)2. Ask questions.
(After you make a mistake, or when you get stuck, ask for help. Or if we did something one way, but you wonder why we didn't try it a different way, ask! Questions lead to more fun.)3. Have fun.
(One of the most important rules for K-3 math circles.)4. Help others have fun.
(If your neighbor doesn't look like they are having fun, see if you can help them. Children this age really don't seem to be that great at helping their friends, but this rule also encompasses general behavior rules.)
The main activity:
I then handed out the worksheet, paper, tape, and markers (not the scissors yet) and showed students how to build a cylinder with paper and tape: Take a shorter strip of paper, bring the ends together, and tape them in place.
First question: How many edges does a cylinder have? The edge of the cylinder is the side of the paper. I asked them to color it with their markers. Even before coloring, they realized that a cylinder has two edges, one on the top and one on the bottom. But they had fun coloring anyway. The technical term for "edge" is "boundary". The cylinder has two boundary components.
I then asked how many sides a cylinder has, and pointed to the middle of the folded strip of paper. I can draw a line around a side. How many sides? One of the girls who was in math circle last year (and had done this exercise), raised her hand and said there were two sides. I agreed. There was the outside, and what else? At that point, all the children knew there was also the inside.
While they were still coloring their cylinders, I moved on to talk about the Mobius band. I showed them that to create a Mobius band, you twist the paper first, then tape the sides together like the cylinder. I asked the same questions: How many edges does a Mobius band have (boundary components)? How many sides?
Several of the children needed help putting together their Mobius bands, but the parents and undergrad helpers were able to help most of them before they got bored or frustrated. I wandered around the tables for several minutes asking students to show me their Mobius bands. One of the girls was coloring her cylinder in multiple colors, making a pretty bracelet. She wanted me to see, and I told her it looked very nice. Then helped her build a Mobius band and asked what would happen if she tried to color that like her pretty bracelet? (And she seemed to be back on task.)
One of the boys had finished coloring the edges of his Mobius band. I asked him how many edges he had colored. "Two". He said immediately. (This was not correct.) So I backed up. "I see you colored an edge brown," I said. "You colored the edge brown until you had finished with that edge. So if there are two edges, you need another color to color the second one, right?" He nodded and reached for another marker. But then I asked where he was going to start with the other color. He looked over the edges of his Mobius band, and his eyes went a little wider as he realized it was all colored.
"So how many edges?" I asked.
"One!" he said.
"Very good!" I said. "And how many sides are there?"
"Two!"
Again I told him that I could see he had colored one side already. If there were two sides, he needed to color the other side. He looked again and realized there was only one side!
Soon after that, I brought the class together and asked all of them how many edges and how many sides there were on a Mobius band. I asked, how many people were surprised it had just one? Who thought it would have two? About half the class, including many parents, raised their hands. It was fun to see their delight at the unexpected.
The next activity was cutting Mobius bands in half. I asked what would happen if I cut my cylinder in half? One child said it would fall into a strip. I agreed and showed them what would happen if I cut the cylinder along the side. But I didn't want to cut along the side -- I wanted to cut it down the center. I showed them how to cut it in half down the center (hint: pinch it in a small crease first, making a tiny cut, then put the scissors through that hole and continue cutting all around). Cutting down the center made two pieces: two more cylinders. I then passed out scissors, and asked the children what would happen if they cut their Mobius band in half?
OK. So I've done cutting of Mobius bands activity before with all ages, and the result is universally fun. The children get excited when they realize that after cutting it in half it's still in just one piece. This time, however, it worked a little differently. The youngest children had a hard time getting the scissors to cut the Mobius band down the middle. The kindergarteners really needed one on one help to cut it -- which was ok. We had the help. One boy kept slicing the side, and starting over. (He was making mistakes, which was one of our rules, I explained, so he was doing great.) Finally, I held the Mobius band for him while he made very tiny cuts, and then I twisted it so he could make the next cut, and so on, until he had gone all the way around.
By now, the third graders were done and getting bored, but a couple of kindergarteners still needed more help with scissors. I pointed out a few things that third graders had been doing to try to keep them working on their own. One girl had twisted a paper strip twice instead of once, and when she cut it she got two linked circles. I pointed that out and told others to try it. Another girl had twisted more than three times. One person was coloring sides and edges again, which I recommended. And then I went back to helping the kindergarteners with their cutting.
One of the questions I asked was what happened if you glued two Mobius bands together? A couple of the parents started trying, but they quickly got stuck. I pulled the class together and pointed out that one of the dads couldn't finish. They thought that was funny. I explained that the reason why this dad couldn't finish is that I had asked a trick question. You can't actually tape together two Mobius bands in 3 dimensions. We had a very brief discussion about dimension then. I think the older kids got it, but it was probably way too fast for the younger ones.
At this point, although there were more activities on the worksheet and 15 minutes left on the clock (we had been going for 45 minutes), I felt we were at a good stopping point. Some of the kids were losing interest. I broke out the cookies and told everyone they could stay and finish their coloring and cutting, or leave when they were ready. Cookies took about five minutes. I double checked with the parents that everyone had signed the role, and answered a few parent questions. The undergrads cleaned up for me while I talked to the parents (which was super nice!), and we were all done by 9:55.
Final thoughts:
I knew going into this that there would be a big difference between kindergarteners (age 5) and 3rd graders (age 8). But in the past, we've only worked with 1st through 3rd grades, and the difference in ability wasn't quite so pronounced. I didn't realize how much extra help the kindergarteners would need.
For next time: I need to have even more open ended activities that the older kids can be exploring while I'm helping the younger ones. Or, perhaps I need to line up the youngest with a parent or undergraduate helper buddy, and have them keep up with what they can while I speak more toward the older kids.
But in all, the activity was fun. The kids all said they had fun when we went back over the rules at the end of class. (And most of them also made a mistake and asked a question, too.) We will call it a success!
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