Directions for teachers:
To engage students before reading the article, have students answer the “Before Reading” questions as a warmup in class or for homework. Now ask students to read the online Science News article “An enduring Möbius strip mystery has finally been solved.” Afterward, have them answer the “During Reading” questions. As an optional extension for deeper analysis on proportions and relative values, have students discuss the “After Reading” questions. This article also appears in the December 2 issue of Science News. Science News Explores offers another version of the same article written at a middle-school reading level.
Directions for students:
Read the online Science News article “An enduring Möbius strip mystery has finally been solved” and answer the following questions as directed by your teacher.
Before Reading Activity
1. With scissors, cut three long strips from a sheet of paper. Make sure every strip has the same width and the same length. Using a ruler, measure the length and the width of each strip in centimeters.
Student answers will vary.
2. Take one strip and make a loop by taping one end of the paper strip to the other. Take the second strip and twist the paper halfway around once before taping the two ends. If you need a visual, check out this video. The normal loop has two surfaces, an inside and outside. How many surfaces does the half-twist loop have? You can figure this out by running your finger along the loop until you arrive back at the point you started.
The half-twist loop has only one surface. If you start drawing a continuous line any point on the loop, you will eventually end back at your starting point.
3. Take the third strip and create another loop with a twist. Before taping, try to make the loop as small as you possibly can. What happens to the loop if you make it too small?
It turns into a flat triangle.
1. How does the number of surfaces differ between a strip of paper curled into a loop vs. a Möbius strip?
The Möbius strip has one continuous surface, whereas the simple paper loop has two.
2. What problem occurs if you attempt to make a Möbius strip from too short of a strip of paper?
If the strip of paper is too short, the Möbius strip will fold flat at the corners, resulting in triangles rather than a smooth bend.
3. Explain the meaning of the following symbol: √3
The symbol √3 means the square root of 3, which is equal to a value of about 1.73
4. In 1977, what did mathematicians hypothesize regarding the limitations of the Möbius strip?
In 1977, mathematicians hypothesized that the “triangular” Möbius strip was the shortest possible.
5. In 1977, mathematicians showed that a Möbius strip’s ratio of length and width must exceed a particular value. What is that value?
The ratio between a Möbius strip’s length and width must exceed about 1.57.
6. What assumption did Schwartz make regarding the shape of a sliced opened and flattened Möbius strip? What did he discover the shape to be?
Schwartz assumed that the shape formed would be a parallelogram. In fact, it’s a trapezoid.
7. What did Schwartz prove after correcting his previous mistake?
Schwartz proved that the length of the Möbius strip must be greater than √3 times its width.
1. What does this story tell us about the value of trying multiple approaches in addressing challenging problems?
This story suggests that when struggling with complex problems, approaching the problem with a different technique might reveal invisible flaws in our thinking. Playing around and trying new things can lead to new discoveries!
2. What was the proportion of your Möbius strip’s length to width? Did it fall within the proportion proved by Schwartz? Explain.
Students should find their strip’s proportion by dividing the length of the strip by its width. They should all find that the proportion falls within the ratio that Schwartz proved (the length of the Möbius strip must be greater than √3 times its width).
3. Come up with one other potential real-world application for a Möbius strip and describe it in one sentence. What is it about the Möbius strip that makes it so useful in this application? Feel free to be creative!
Student answers will vary for other potential applications. One possible use: In the bets used in vehicles or conveyor belts. The reason: The Möbius strip wears evenly on all sides because both sides of the belt get used equally. Students may expand upon the possible use as belts or get creative and suggest a new type of pasta that boasts a more even sauce distribution.