Why do 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
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
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.
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?
The problem Covering
provides another context for investigating packing
Learners could design their own shapes which contain six
circles and challenge each other to estimate the areas.
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.