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'Brush Loads' printed from http://nrich.maths.org/
Why do this problem?
is very practical, and encourages visualisation. It can help pupils' development of the concepts associated with area - surface area. It is an investigation that can be extended very far indeed, and some pupils may succeed beyond your expectation.
It is worthwhile for pupils to work in pairs or threes and to have cubes in front of them so as to enable good discussions to take place. The youngsters usually find that keeping shapes flat produces smaller numbers and making the tallest towers produces the largest, but challenge them to explain why this is the case. Look out for children moving one cube about from one place to another -
you can get into good quality conversations about how many square faces are being covered by the new position and how many new square faces are added.
Some younger children find it hard to count the faces that are visible, and they recount some or miss some out - don't we all? The children have to invent ways of being more and more accurate. It's good to ask them about their ways of counting and you may very well get these kinds of replies: "I look at each cube in turn and count the faces that are visible" or "Well I go all around the edge
then count the ones on top". So for example with these $8$ cubes:
they'd count "$3$ plus $3$ plus $3$ plus $3$ plus $8$". It then becomes very interesting that with $9$ cubes there will also be $4$ threes but adding $9$ at the end. Moving to $10$ cubes it would be $3 + 4 + 3 + 4 + 10$. The discussions can go on and on as the confidence grows.
How are you counting the faces?
Are you able to make a shape with more/fewer faces?
Can you explain why or why not?
The more mature youngsters who have no difficulties in counting will probably move on to tabulating their results. You might want to use standard units (cm$^2$) with older children. If you use multilink, $1$BL might be equivalent to $4$ cm$^2$. Here is a spreadsheet of some findings related to this investigation that you may find useful. But, please don't just hand it out to the
Probably the best way to support learners who are struggling is to make sure they have a supply of cubes and encourage them to discuss what they are doing.