Why do this problem?

This activity invites discussion amongst pupils which will encourage them to use vocabulary associated with position and transformations.

You may like to read our Let's Get Flexible with Geometry article to find out more about developing learners' mathematical flexibility through geometry.

Possible approach

It is important for children to have paper or card copies of the shapes to work on the activity, or to have access to a tablet/computer in pairs so they can use the interactivity. This sheet contains four copies of the square or you could make your own on squared paper (the line dividing the square in two is drawn from one corner to the midpoint of the opposite side). It may be a good idea to use paper which is coloured on one side only and talk about whether the new shape should be the same colour all over.

You could introduce the task using the interactivity projected onto the board, or by inviting learners to gather round some large printed pieces. You could begin by showing them the square made of the two pieces and simply ask what they see. As there are no right/wrong answers to this question, it is an accessible entry point for all children. Use the comments that learners make to introduce the task itself.

Encouraging learners to be systematic in their discovery of 'new shapes' is important if they are going to be asked how they 'know' they have found every solution. Look out for those children who have developed a system and ask them to share their method with the whole group. For example, they might keep one shape fixed and find all the ways of placing the second shape. (There are many ways to be systematic, so try not to assume that everyone will use the same way as you!)

If it doesn't come up naturally, it might be helpful to ask the group how they are keeping track of the shapes they have found. If they are using the interactivity, a screenshot could be useful, or they could sketch the shapes they have created whether they are working on screen or using card. It might help some children to have many copies of the pieces so they can create each new shape and not have to re-use the pieces to find others.

Key questions

How will you record your findings?
How do you know you have found all the shapes?

Possible extension

Invite children to make another cut so that they have three pieces. How many new shapes can they make now? What cuts make the 'best' new set of shapes?

Possible support

Having several copies of the square will mean that children can stick down each arrangement.