Why do this
problem?
This investigation provides pupils with an engaging context
in which to explore the factors of $36$ through rectangular
boxes. It will also give them a chance to pursue their own
questions and take ownership of the investigation themselves.
Possible approach
You could present this investigation orally to begin with by
asking learners to sketch one arrangement of the $36$ sweets on
their mini-whiteboards or on paper. Encourage them to share
some of their ideas in pairs and then with the whole group. You
might find you need to discuss limitations such as whether the
sweets can be different shapes or different sizes. Read the
question together to make sure the task is understood. Decide
on various limitations such as whether the sweets can be
different shapes or different sizes. Then, with learners
working in pairs, start on the designs.
Children could then work in their pairs to find other designs.
It would be useful to have coloured pencils and squared paper
available, and (if possible) isometric paper for those who
explore triangular boxes. Having counters on hand to represent
the sweets would also be helpful, particularly when it comes to
looking at their arrangements in the boxes.
This investigation would work well as an extended activity with
space on the wall dedicated to displaying what has been found
out so far. Once the children have contributed a range of ideas
to the wall, take time to bring them together and look at what
they have found out. Are there other questions they would like
to ask as a result?
Key questions
How could you arrange $36$ sweets in a
rectangular box?
Are there any other ways?
What can you say about the number $36$? What are its
factors?
How does that help?
How do you know you have got all the rectangular
designs?
How are you going to colour your arrangement so that no
sweets of the same colour are adjacent to each
other?
Will you always be able to colour the sweets like
this?
Possible extension
Some learners might like to use triangular or hexagonal cells,
rather than squares, for the sweets. They could also explore
the 'four colour map problem' (or four colour theorem). Can
they draw a 'map' for which it is necessary to have more than
four colours so that no two 'countries' which share a border
are the same colour?
Possible support
Using counters in four colours to
represent the sweets will help children to try out their ideas in
a less intimidating way than committing something to paper. You
could use the interactivity in the
Growing Garlic problem instead of counters by reducing the
number to $36$.