This problem will allow students to engage with calculations of
areas of disks, often used as an approximation to real objects in
science. They will also practice visualisation skills.
To find the best answer students will need to resort to
trigonometry, although this is not necessary to make a satisfactory
attempt at the question.
Possible approach
Whilst students might be familiar with the concept of the area of a
disk they might be unable immediately to see how to apply this to
the question in hand. Some discussion might facilitate this. Note
that there are two 'obvious' different ways to pack disks. Which
will work best? Discuss this. When they do the packing calculation,
be sure to note that students must be clear as to exactly how may
rows and columns of disks will fit into the grid. They can do this
exactly (using trigonometry) or by drawing an accurate diagram and
taking a measurement. Or, perhaps, they will produce some other
valid method.
Key questions
Describe your method of packing the disks clearly in
words.
How many different sensible packing methods might you try?
Can you be sure that each disk is completely contained within
the confines of the square?
What order of magnitude checks could you make to test that your
answer is sensible?
Possible extension
Can students repeat the question by filling an equilateral triangle
with side 1m?
Possible support
Provide coins or counters for the students physically to work
with.