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Do Unto Caesar


Diminishing Returns


Why do this problem

The problem helps to extend learners' familiarity with fractions, but has the potential to prompt discussion about limits and infinity. Learners have the chance to develop convincing arguments and to share with others different ways of seeing the images. 

Possible approach

Start by asking learners to sketch copies of the first image and ask them to work out what proportion of the image is shaded. Share methods of working it out.
Now ask learners to imagine that the pattern continues into the centre of the page (show them the second image in the problem if they need help to visualise this). Ask them to discuss with their partner (without writing anything else down) what fraction of the image would be shaded if the pattern continued forever.
Bring the class together and share ideas. The discussion should bring out the concept of an infinite process approaching a limit in an informal way. Mathematicians are often trying to avoid lots of laborious work, so draw attention to any elegant ways learners have of seeing what the limit must be.
Compare the answers for the two different questions above - check that the first answer is a reasonable approximation to the second.
Now give pairs the chance to repeat this activity with some more patterns. There are three worksheets available to download and print (Colour, Black/White). Challenge them to come up with elegant ways of showing what the limit should be, and remind them to check that their answer for the partial sum is a close estimate to the limit.
Finish the session by sharing elegant ways of thinking about the limit.

Key questions

What fraction of the square is blue?
Can you convince someone else that your calculations are correct? 
How can you visualise the pattern to help you to find the limit?

Possible extension

Can you continue the patterns outwards? Can you use this to find a way of working out successive approximations to the limit?

Learners could investigate the perimeters of successive shapes in the sequences.

The patterns could be used as a brief introduction to the idea of self-similarity in fractal patterns.

Possible support

Challenge learners to make the patterns out of squares of paper in contrasting colours. By folding paper to create each section, learners will be able to spot relationships between the area of consecutive pieces of the pattern. Learners could work in pairs to create two copies of the pattern from two squares of contrasting colour by swapping pieces.

The article Zooming in on the squares discusses the idea of starting the process off and looking at successive estimates for the area.