Paper With a Twist
By Michael Naylor

Start your year with exciting activities that feature simple loops of paper. Paper chains can help students count days or study patterns. Older kids can do experiments with Moebius strips to delve into a modern branch of mathematics called "topology." These activities not only hone three-dimensional thinking skills, but they're also full of surprises that capture the magic of math.

Younger kids can begin by alternating two colors to make a paper chain, as shown in the top half of this drawing. Older kids can make more complex patterns, like the ones shown in the bottom half.

A Link a Day (Grades K-2)
Keep track of this school year with a calendar chain. Start each school day by adding a link to a paper chain. Make every 10 links all one color, then switch to a different color for the next 10 links. As the chain grows day by day, your students will be able to count the links by tens and by ones. Consider having a "One Hundred Day" when the chain reaches 100 links long.

Paper Chain Patterns (Grades K-2)
Provide strips of colored paper or have your students cut their own, then have them decide on a color pattern. You can start younger kids with a simple two-color alternating pattern. Older kids can make more complicated patterns; be sure their pattern has a basic repeating unit that's repeated at least twice.

Once the paper chains are complete, compare the patterns – how are they different and, more importantly, how are they alike? For example, how is a pattern that repeats red-red-yellow similar to a pattern that repeats green-green-blue? The chains give your students an opportunity to look at the structure of patterns. They also make great classroom decorations!

Moebius Strip Variations (Grades 3-8)
Moebius strips are just about the closest thing we have to real magic in the world. To think about and understand the tricks these strips do is intrinsically satisfying – mental play at its best. Here are instructions for making many kinds of Moebius strips, along with questions to get your students thinking.

Moebius Strip Half-Twist:

1. Cut a strip of paper about two cm wide and as long as a sheet of paper.
2. Hold the ends of the paper together to make a ring, as if you're making a link in a paper chain.

3. Give one of the ends of the strip a half-turn and hold it to the other end of the strip. You'll see the strip is now a bit twisted, as shown right.
4. Tape the ends together to make a twisted loop.
5. Ask your students, "How many sides does the strip have?" Pretend you're an ant. Draw a dotted line along the center of the strip to show the path of the ant. Can you walk everywhere on the surface?
6. Ask your students, "How many edges does the strip have?" Try sliding a finger around the entire edge without lifting it off.
7. Ask your students to predict what will happen if the strip is cut along the dotted line, then try it and see if their predictions were correct. The kids will be amazed when they see that the cut produces not two pieces but one double-length loop.
8. Why does it do that? It's easier to see what's happening if you color along the edge of the Moebius strip with a highlighter. Cut the highlighted strip down the middle and see what happens to the edge.

Moebius Strip Full Twist:

1. Make another Moebius strip but, this time, give one end of the strip a full twist before taping it to the other end. It should look like the figure shown at above right.
2. How many sides and edges does this strip have? Run a finger around the edge. Do you have to lift your finger and put it down elsewhere to be able to touch both sides?

3. Cut the strip in half. You should get two linked, twisted loops.
4. Why does it do that? To find out, make another strip of this type and color each edge with a different marker. Notice that the strip's edges twist around each other, linking in three-dimensional space. Even though you've cut the edges apart, they've managed to stay linked!

Moebius Strip in Thirds:

1. Take a strip of paper and sketch lines lengthwise to divide the strip into thirds, as shown right.
2. Make a half-twist Moebius strip out of this strip of paper.
3. Ask your students to predict what will happen when the strip is cut into thirds. Write down all predictions so the students are committed to them.
4. Cut along the guidelines.
5. Surprise! Few people guess this correctly; don't worry if you missed it.
6. Why did you get this result? Make another of these strips and, this time, color the outside edges one color and the middle a different color. Cut it apart. Does this help you to see what's happening?

Moebius Links:

1. Make one loop that is not twisted.

2. Make a second loop that is not twisted and link it to the first loop as though you're making a paper chain, as shown right.
3. Turn the loops so they're perpendicular and tape them together on both sides of the connection.
4. Ask your students to predict what will happen if each ring is cut in half.

5. Start cutting! Hold the pieces together while making the second cut. Were any predictions correct?
6. Try these variations:
   
   • Make one link a Moebius strip.
   
   
   • Make both links Moebius strips.
   
   
   • Do you get a different result if one strip is twisted to the right and the other is twisted to the left?
   
   
   • Do you get a different result if both strips are twisted the same way?

Connections
M.C. Escher is known for his mathematical artwork. Search the Internet for a picture of his "Moebius II."

Invite students to write a never-ending story on a Moebius strip, such as, "It was a dark and stormy night. The crew had gathered below deck to tell stories. The captain went first, saying, ‘It was a dark and stormy night. The crew had gathered below deck to tell stories. The captain...'"

For more classroom activities and Moebius strip Internet links, go to www.wwu.edu/~mnaylor/moebius

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

August/September 2003, Vol.34, No.1