Overhanging Squares

This island is of particular use early on in children’s understanding of area. There are opportunities to emphasise being systematic when trying out possibilities. Since there are many examples to find, it might be a good idea for them to try and come up with them in groups.

Initially, I’d get them to just make a shape with an area of 3 squares, then 4 squares. Then, rather than introduce it all at once, show them the below examples and get them to create two shapes without overhanging squares and two with them to ensure they understand the principle. A square is ‘overhanging’ if it isn’t being supported below it. We are trying to make shapes without overhanging squares.

How many possible ways are there when using an area of 3 squares?

For 3 squares, there are 4 possible ways. I am not counting the first two as the same even though they are the same when rotated. That might be something you explore later but to make it accessible, I keep it as this first.

What if we used an area of 4 squares?

This is a logical next step. I get them to predict how many they think first of all. Hopefully, they should be able to realise there will be a greater number of possibilities!

In fact, there are 8.

It leads to an interesting conversation about the link between the number of possibilities for areas of 3 and 4 squares. We can expect conjectures of there being 12 or 16 possibilities for an area of 5 depending on whether we believe we are adding four or doubling between areas of 3 and 4. Of course, other possibilities could be happening!

What if we used an area of 5 squares?

Search inquiry reasoning skill in maths

This is where it is really useful to have a conversation about systematic thinking and using cubes to help with it. Cubes allow the principle of moving one or two squares to make a minor adjustment to create a new example. You could also try and find the possibilities with a base of one, a base of two, etc. This is using the Combing skill to search for new possibilities.

The knowledge that there are likely to be 12 or 16 possibilities can drive them on to keep finding more examples.

What if we found the number of possibilities for examples that had to have overhanging squares?

Whilst not necessary, this is a possible extension.

Children’s possible use of reasoning skills (click on each heading):

Search
Search inquiry reasoning skill in maths

This island focuses on the use of the Combing skill. The children won’t necessarily naturally be systematic but in trying to find all the examples, you can highlight how easy it is to miss out possibilities if they aren’t systematic.

Organise

Using a table which shows the number of possibilities for each area size will allow them to notice a pattern.

However, equally, you might want to consider how the possibilities are arranged. By showing similar examples together it encourages them to be systematic.

Discover

This is an island that focuses on finding all the possibilities rather than making lots of conjectures. The obvious opportunity for a conjecture is how many there are for each area. However, they might also try and predict subsets for each area – how many there are with a base of two for example.

Investigate
Investigate inquiry reasoning skill in maths

Once conjectures have been made about the number of possibilities for a specific area, you can refer to these when trying to find the number of possibilities. They are validating the conjecture and also using it to know how likely there are further examples.

Argue

With the examples above, the children might consider why the greatest number of possibilities are there for a base of three with an area of five.

Explore

This is a fairly standard investigation that is about exhausting all the possibilities rather than going off in different directions. But with enough experience, I’m sure the children could come up with a new direction!

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