Why do this problem?
This problem allows learners to see the value of estimating
before making accurate calculations, and to see that sometimes
an estimate is all that is needed. The problem also offers an
opportunity to practise calculating areas and working out
lengths accurately using trigonometry.
Possible approach
Show the image of the six boxes and explain that we're
interested in comparing the areas of different boxes made to
hold six circular discs. Ask learners to compare the areas of
A and B, and allow everyone time to consider and then discuss
in pairs which is bigger and why.
In discussing A and B, key ideas to consider include
comparing the parts of the shapes which are the same, or
comparing the size of the gaps between circles. Next, allow
the class some time to discuss in pairs or small groups how
to order all six shapes. Stress that the importance is not so
much in the order the learners come up with but in their
reasons for placing them in that order.
After allowing some time to work on estimating, suggest to
the learners that some calculations may help them to put the
shapes in order. Learners may decide they do not need to work
out every area to be certain of the correct order - for
example, if they are certain their estimate has identified
the biggest or the smallest area they may choose not to
calculate that one. They could then work in small groups to
create a poster or presentation showing the correct order for
the shapes and the justification and calculations they used
to find it. The hint contains two diagrams which suggest an
approach for working out the areas using trigonometry, which
could be shared with the class if appropriate.
Key questions
Which shape do you think has the largest area and why?
What angles can we work out? What lengths do we know?
For which shapes do you think you need to work out the areas in
order to be certain you had ordered the shapes correctly?
Possible extension
The problem
Covering Cups provides another context for investigating
packing circular shapes.
Learners could design their own shapes which contain six
circles and challenge each other to estimate the areas.
Possible support
Some of the calculations for the exact areas require knowledge
of trigonometry, but the problem could be tackled instead by
constructing accurate diagrams of the boxes and measuring in
order to calculate their areas.