School and University Partnership for Educational Renewal in Mathematics An NSF-funded Graduate STEM Fellows in K–12 Education Project

University of Hawai‘ i Department of Mathematics

UHM Department of Mathematics superm@math.hawaii.edu

A cut of scissors 1 Discovering axes of symmetry

Description A regular polygon is drawn on a sheet of paper. How can you fold the sheet to be able to cut out the polygon with only one straight cut with a pair of scissors?

Grade levels 3rd to 5th

Materials The handout and a pair of scissors

Duration 1 – 2 periods

Proposed Implementation

Students begin by working either alone or in small groups of two or three, with a handout for each student. Discuss the outcome as a class when a majority of the students have successfully completed the handout.

Preliminary analysis of the activity

(steps students will predict and teacher

interventions)

Goals Students discover the axes of symmetry Possible approaches The student starts with “doubling up” the polygon. For polygons with an even number of sides, two choices are possible: along a diagonal or along a center line. The student continues to fold, always along an axis of symmetry of the original polygon (which is not necessarily an axis of symmetry of the folded shape). Once all of the sides of the polygon are on top of each other, the student can now make the cut. Notes Initially, a student may not understand the relationship between making one straight cut with scissors, and cutting out a polygon. A reformulation of the problem is: how can you fold the paper to put all sides of the polygon on top of each other (to align all sides). This connection can be useful for math journal entries. Class Discussion After many students have completed the activity, have them describe the result of folding, for example, a student can explain which sides are on top of each other after making the folds. Unfold the polygons that have been cut out and observe the creases

UHM Department of Mathematics superm@math.hawaii.edu

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made by folding the paper. The goal is to lead the students to create a definition of axes of symmetry of the polygon. The teacher can use the students’ thoughts and definitions to formalize the concept.

Mathematical Concepts

Axes of symmetry

Possible Developments

Formalize the idea of axes of symmetry

One can study if one method is more effective than another, in terms of the number of folds. For example, always folding in half along the diagonal is often the most efficient method (only two folds in the square, versus three or more otherwise).

UHM Department of Mathematics superm@math.hawaii.edu

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Other resources For a detailed solution of the square, isosceles triangle, and regular pentagon, see (in French, but there are lots of pictures):

[1] the detailed solution for regular polygons

http://icp.ge.ch/dip/maths/IMG/pdf/plus-polyreg.pdf and

[2] the presentation of the problem http://icp.ge.ch/dip/maths/IMG/pdf/presentationciseau.pdf This activity is a starting point for a formal definition/discussion of axes of symmetry. The folds are necessary because of the axes of symmetry of a regular polygon. In the case of polygons with an even number of sides (square, hexagon, octagon, …), these folds are also axes of symmetry of the folded form, which yields a little discovery. When the sheet is unfolded, the axes of symmetry are seen, for example by folding the sheet along the creases (see [1]).