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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.
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
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
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.
What fraction of the square
Can you convince someone else that your calculations are
How can you visualise the pattern to help you to find the
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
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
The article Zooming
in on the squares discusses the idea of starting the process
off and looking at successive estimates for the area.