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Why do this problem?
involves many aspects of mathematics, both content and processes, and it is a good one to introduce investigative approaches. Children will be counting and developing spatial awareness, and you can use it to highlight working systematically, representing and making conjectures.
In order for children to begin to make conjectures about a mathematical situation, they need to be very familiar with that situation. In this case, learners must be very comfortable with the making of shapes according to the rules. So, this activity would be best introduced as practically as possible with cubes, going through the rules very carefully so that everyone really understands what
is and what is not allowed. Ask pupils to show examples of what is allowed, and when one example is shown that is not allowed, invite children to explain the reasons. You could also use the pictures of the 'two on two' cubes as a stimulus for this.
As learners begin to work on the challenge, perhaps in pairs, it is a good idea to encourage them to record the shapes they make. Emphasise that this recording is for them to remember what they have done and need not (in this particular case) be understood by any adults. They just need to be able to look back and see whether they have made that shape before. You could stop the group after
five minutes or so to share some ways of representing the models that pairs have developed. (If a large number of cubes is available and there are fewer pupils then they can keep each shape that is made and so there is no need to record them.)
Once all the different ways of having two on three have been found, ask children to talk in pairs about their predictions for two on four. Invite them to write up their ideas, with reasons, on a sheet of A4 to come back to later. As the class works on making the different two on four models, you can bring them together at various stages to talk about how they know they will be able to make
them all and whether they want to alter their conjectures based on what they have done so far.
Tell me about the way that you are making these shapes.
How do you know you will have them all?
You look as if you have a method for making more and more shapes - tell me about it.
Is this shape the same or different compared with that one? Why?
So, what will be the number of shapes you can make when you have more red cubes? Why?
Encourage the pupils who ask "I wonder what would happen if I ...?" - these could be different numbers of green and/or red cubes, altering the rules or maybe introducing another coloured cube.
For the highest-attaining
Go to Four Layers for some extension work for the most able of pupils.
If cubes are in short supply and each pair can only have five cubes then pooling results in a group may help.