MMT 405 - Geometry for Middle School Teachers
Doubling a Square (Hands on, Minds on)

Task:         Given a collection of equal sized squares and the “usual” sorts of materials (string, graph paper, scissors, tape, ...) but no rulers. Make a square that is twice the area of one of the original squares.

Central Question:         How do you know it has twice the area of the original square? Can you show that it must be twice the area or only that it looks as if it is?

Looking Back:         Are you satisfied with your explanation? What did you know about a square that you needed to use? Are there things you know about a square that you did not use?

Are you satisfied with your approach? Is there any other way you might have done it? Are there several other ways to do it? Which ways do you like best?

Extension:         Would the same techniques work for a rectangle? How about other shapes? What properties should a shape have to make your techniques work? Can you justify your procedures?

A Further Extension:         Using dynamic geometry software (Geometer’s Sketchpad or Geogebra) instead of a piece of paper, construct a square twice the size of a given square. Be sure that it can be dragged and dropped into a different location and/or a different orientation and still be a square with the correct size. Can you make it so that both squares  can be simultaneously dragged into a different sized squares?

Observation: The construction behaves differently depending on the object you drag around. Observe some of these differences. It also behaves differently depending on how it was built? How do you explain these differences?

Some Possible Solutions:

Solution 1:         (“Brute force” method) Using the graph paper, measure the length of the side of the square and calculate the area. Double this number and take the square root in order to find the length of an edge of the desired square. Then construct this square. (Of course, there are a number of ways to do this construction, some of which may not be obvious.)

Solution 2:         Draw the diagonals of one square and cut it into four pieces along these lines. Stick these four pieces around the outside of the square.

Solution 3:         Tape four squares together. Fold in the corners so the mid-points of adjacent sides are connected. Crease and unfold. Cut off the four triangles formed (which could be reassembled to make one of the original squares). The polygon which is left is a square and has twice the area of the original. How do you know this?

Solution 4:         Cut two squares along a diagonal. Reassemble them to form the desired square.

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This page last updated on May 26, 2010