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Walking through Paper

A shape can sometimes be deceiving! 

Topology is the mathematical study of shapes and spaces.

Topology involves looking at the shapes that result through stretching, transforming, deforming, folding and twisting.

This clever mathematical trick makes a giant loop out of a single piece of paper. All you need is a pair of scissors.

Objectives

  • Develop mathematical reasoning.

  • Make connections and solve problems.

  • Communicate and express their understanding.

  • Explain the importance of observation when doing math.

Materials

  • Per Student:
    one letter-sized paper
    scissors

Key Questions

  • Why do you need to cut from opposite sides of the paper?
  • What would happen if you repeated the activity by making fewer folds or more folds?

What To Do

  1. Fold the paper in half lengthwise.
  2. Create a series of crosswise folds about 1cm in width (an accordion fold) until the whole page is divided into 16 sections. Flatten each crease so you can see it clearly.
  3. Starting at one end, cut along the first crease from the central fold, stopping about 1.5cm short of the open edge. Cut the next crease from the opposing side, stopping about 1.5cm short of the folded edge. Continue cutting in this fashion along the full length of the paper.
  4. Now you have a rectangle made up of many smaller, near-rectangles. Avoiding the two end pieces, cut off the very ends of the rectangles with the folded edge.

Teacher Tip: Remind students to be careful when cutting as it is easy to miss a crease. You’ll want to cut along every fold to get the largest loop.

  1. Open up the page, and you’ll find a loop large enough to walk through!

Extensions

  • Can you predict how large of a loop you will make, knowing only the size of your paper and the number/size of cuts you will make?