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Impossible Picture?

Age 14 to 16
Challenge Level

Why do this problem:

In this curious problem the diagram shows a seemingly impossible situation but by showing some perseverence and when provided with appropriate scaffolding, students can make sense of what's going on. This problem forces us to explore our assumptions when working with diagrams. Diagrams can be misleading, and in investigating this problem students will need to question all our assumptions carefully and justify their strategies. 

Possible approach :

Project the first diagram on the board. "Here is a square that has been cut up into 6 pieces. the 6 pieces rearrange to give 3 squares." Show the second picture. 

Give each student a few squares of paper to experiment with and challenge them to cut their square up as shown in the picture. It is unlikely that anyone will manage to make 3 exact squares from the pieces, and even if they do, the correct dimensions may not be obvious.

After students have had a a chance to explore bring the class together and discuss what they have found out. You may wish to show this interactive to see how the pieces change. Invite students to think about what constraints are needed to make the third shape a square. 

Once students have discussed the necessary conditions for the third shape to be a square, invite them to try to find appropriate dimensions. Some students may use Pythagoras' Theorem; others might try to get as close to a square as they can using trial and improvement.

Finally you could ask your students to find the ratios of the areas of the 3 squares. 

Key questions :

What assumptions have you made?
Can you describe what properties the diagram has when it is possible to make 3 squares?
How would you convince someone else?

Possible extension :

Students could create their own version of the interactive in GeoGebra

Possible support :

Square Coordinates and Tilted Squares might provide a useful starting point for thinking about some of the ideas raised in this task.