All Activities U Möbius Madness
for parents
for teachers

Möbius Madness

Cut, twist, and tape a strip of paper to create this famous and surprising mathematical artifact.

Ages: 4+ (with help)
Players: 1
Time: 10-20 Minutes
Type: curiosities
Location: tabletop
Share This Activity
Ages: 4+ (with help)
Players: 1
Time: 10-20 Minutes
Type: curiosities
Location: tabletop

Instructions

Have you ever played with a one-sided piece of paper? What's that you say? Impossible?! Get ready to be confounded and surprised by your very own Möbius Strip.

First, cut a strip of paper.

Strip being cut from a piece of paper

Now take your strip of paper and form a loop, then give one end of the strip of paper a half twist. In other words, hold each end of the strip in a separate hand. Then turn over one end so that what was the bottom side is now the top side. It should look like this:

Strip of paper with ends connected together in a loop
Loop with no twist
Strip of paper with ends connected together with a half twist
Loop with half twist

Now, connect the ends with a piece of tape. As you do, be sure and preserve the half-twist you made.

Mobius strip with tape and scissors

There it is, your own Möbius Strip! Not so amazing at first. To appreciate it, you've got to play with it a bit. First, draw a line down the middle of one side of the piece of paper. As you do, you'll eventually notice that you are, in fact, drawing your line on the reverse of the side you started on, and then at some point you'll meet back up with the starting point of your line!

Drawing a line down the center of a mobius strip

This is why a Möbius Strip is, mathematically, a "one-sided" shape. One way to approach this realization with your kids is to first ask them to pick two colors, one for each side. As they start coloring the Möbius Strip, they'll discover that they will actually only use one color to cover the entire strip.

You can do something similar with the edge of your Möbius Strip. Use a marker to color one edge of your strip. Again, you'll find yourself coloring what you had thought was the "other" edge and eventually reaching the point where you began. So, a Möbius Strip is a shape with one side and one edge!

Mobius strip with edge partially colored
Mobius strip with edge fully colored

Now try cutting up your Möbius Strip. First, try splitting the strip in two by cutting it in half down the middle, almost as if it were a road and you were cutting it down the center divider between the lanes.

Cutting down the center of a mobius strip

When you reach the point where you began, try to pull your strips apart. You'll find that they are, in fact, still just a single strip!

A single long loop of paper

Is this a new, longer but skinnier Möbius Strip? To find out, try drawing a line down one side of it (or coloring one side). Remarkably, this new strip is not a Möbius Strip! It has two distinct sides (and two edges too, if you'd like to check).

Make yourself another Möbius Strip to try this next cut. Draw small marks to split the width of your strip into thirds. Then cut lengthwise along one of these marks.

Mobius strip with two dashes drawn on it
Mobius strip being cut into thirds

You end up with two interlocking strips!

Two interlocking strips of paper

Are these both one-sided Möbius Strips? Or is just one of them? Or neither?

This is a great activity to work alongside your child (or with them if they need help cutting the paper). As you play, ask questions. It's okay if you don't know the answers yourself! The simple act of sharing your own sense of curiosity and wonder helps your child learn to be curious as well. See Variations below for other questions to ask and other things to try when playing with a Möbius Strip.

Don't forget: it's Beast Academy Playground, not Beast Academy Study Hall. Change the rules, be silly, make mistakes, and try again. The Variations and Learning Notes are here for you if you want to dive deeper, but not all of them apply to learners of every age. The most important thing is to have fun.

Variations d

Twistier Möbius Strips:

What happens if, instead of one half-twist, you make two half twists (aka a full twist), or 3, or 4? If this many twists gets difficult, you can help yourself by using an even longer strip of paper. Try cutting these different Möbius Strips in half. You'll be surprised what happens!

Interconnected Möbius Strips:

Once you've played around a bit with different loops and Möbius Strips, try connecting two strips together. First, tape two strips together like this:

Two strips of paper taped together in a plus sign shape

Connect opposite ends of this plus sign with tape, giving one or two or no twists to the paper before taping. Then cut each Möbius Strip apart and see what happens. For example, try giving one strip a half-twist, and the other strip a half-twist in the opposite direction as your first strip.

One of the strips connected in a loop with a clockwise twist
Clockwise twist
The other strip connected in a loop with a counter-clockwise twist
Counter-clockwise twist

(Don't worry if you get the twists wrong. The results of cutting them up are surprising and interesting no matter how you twist!)

Then cut each of the two interlocking Möbius Strips lengthwise down the middle. Here's what you get!

Two interlocking hearts
We ♡ math!

Classroom Tips d

Playing with a Möbius Strip is a great quiet task for after a test or quiz, or use this as an activity between units or the day before a break.

Discussion Questions

  • After you cut your Möbius Strip down the middle, do you still have a Möbius Strip? How is it the same? How is it different?
  • When you cut one-third of the way into your mobius strip, what do you get? More Möbius Strips? Regular loops? One of each?
  • What would happen if you made 2 twists? 3 twists? 4 twists? Is there a pattern? (see Variations)

Alignment with Beast Academy Curriculum

  • Level 2, Chapter 12: Problem Solving
  • (And Book 5D has a Möbius Strip roller coaster on the cover!)

See Variations and Learning Notes for more ideas on how to adapt this activity and incorporate it into your classroom.

Learning Notes d

Topology:

The questions we asked about "sides" and "edges" in playing with the Möbius Strip are typical of the field of mathematics called "topology." In topology, we think about what characteristics a shape or object has, characteristics that don't change even if the shape is twisted or stretched or otherwise "squished around" in some way.

A classic mathy joke about topologists is that they can't tell the difference between a coffee mug and a donut. The joke is that, topologically, both of these objects are the same. A coffee mug and a donut both have a single hole in them through which you could poke your finger. For the coffee mug, this hole is the handle. It's true that a donut is a uniform circular shape around its hole, whereas a coffee mug has more ceramic in the "cup" part of the mug than in the handle, not to mention a big indent where the coffee goes. But you could imagine taking a lump of clay shaped like a donut and stretching and molding it to form a coffee mug with a handle, and in doing so not needing to poke another hole, or tear the clay apart anywhere. Hence, these two shapes are topologically the same! Whew! That made me hungry! Pass me that coffee mug!

What do you think of this activity?

We're always looking to improve. Submit your feedback to us below.

Share This Activity
Materials
  • paper
  • scissors
  • tape
  • pencil
Learning Goals
  • wonder
Common Core Standards
Image of Ms. Q

Ready to level up?

Keep problem solving with Beast Academy’s full math curriculum for students ages 7–13. Check out our captivating comic book series and immersive online platform.

LEARN MORE

Bring problem-solving to your classroom

Keep your entire class engaged with a full book and online math curriculum, for students ages 7–13. 98% of teachers say they’re satisfied with Beast Academy.

LEARN MORE
Image of a BA book

Recommended Videos

Image of Ms. Q

Ready to level up?

Keep problem solving with Beast Academy’s full math curriculum for students ages 7–13. Check out our captivating comic book series and immersive online platform.

LEARN MORE

Bring problem-solving to your classroom

Keep your entire class engaged with a full book and online math curriculum, for students ages 7–13. 98% of teachers say they’re satisfied with Beast Academy.

LEARN MORE
Image of a BA book