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Reasoning about the number of matches needed to build squares that share their sides.

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Tangram Paradox

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!


Stage: 2, 3 and 4 Challenge Level: Challenge Level:1


Look at this image for a short while before turning away.
Can you:

  • Recreate the image?
  • Describe the image?
  • Say some mathematical things about what you notice?
  • Think of some mathematical questions you would like to ask about it?

Here is a pdf document containing the image.

Baravelle Spiral

This image is taken from the NRICH Mathematics Posters CD called "Exploring Squares" published by Virtual Image. More details of this and the "Exploring Circles" CD can be foundhere.


Why do this problem

The aim of using this image is to encourage discussion about the different ways of seeing, and to pose questions that can form the focus of further investigation.

Possible approach

You can find a pdf with the image here. Look at the image for about a minute and then "hide it". Then share ideas.

Follow up by selecting one or two of the ideas for the group to explore further, for example:
  • Recreating the image
  • Finding what fraction of the whole the areas is shaded red, blue and white.

Key questions

Can you continue the pattern inwards? Outwards?
Can you give a convincing argument for the fractions of the whole that you can see?

Possible Extension

Consider summing infinite series.
What are the ratios of the lengths of the sides of the triangles?

Possible support

Can you find the fractions and or recreate the shape by folding? Some learners might like to try the problem Inside Seven Squares.