Diagonally Square

Why do this problem?

This task is designed to encourage learners to generate lots of examples, look at similarities and differences, and to make, and refine, a conjecture based on their noticings. They are then invited to create a mathematical argument in order to prove, or disprove, their conjecture. As they put together their reasoning, they will be drawing on knowledge of geometrical properties and relationships.

Possible approach

Begin the task by showing the image and asking children what they see and what they notice. Give them time to think on their own, then talk with a partner, before inviting pairs to share ideas with the whole class. As you facilitate the discussion, encourage other members of the class to comment on and question the contributions, rather than you doing the validation. Depending on the responses, and the experience of your class, you may wish to draw attention to particular noticings.

For the purposes of this task, focus on the diagonals of the square meeting at right angles. Will this always be the case? Invite learners to explore examples for themselves. After a short time, you may need to bring them together for a mini plenary to talk about how they are generating examples. What do they know about squares? They may be drawing round squares that are available in the classroom; they may be drawing their own squares using a ruler and protractor; they may be using squared paper. How are they checking the angle at which the diagonals meet? 

As they work, circulate around the room, observing and listening. When you bring everyone together again, you may wish to highlight particular ways of working that you feel the whole class would benefit from hearing about. How are pairs choosing their examples? Some might suggest that it is important to create some very small squares and some much bigger squares to try.

Explain that mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true. The class' challenge is to provide an argument that would convince mathematicians that the diagonals of a square always meet at right angles. Give them time to work in pairs and, as they discuss their ideas, circulate round the room again, listening out for watertight chains of reasoning to share with the whole group. 

If learners have not had much experience of proof, you may wish to share the proof sorter with them, which will help them get a feel for what a proof 'looks like'. Don't be afraid of using the word 'proof' with primary learners - being able to prove is an essential part of being a mathematician.

Key questions

What do you notice about the picture?
Will your noticing always be true, for any square? How could you find out?
Can you create a convincing mathematical argument?

Possible support

Printing this sheet of many different sized squares will help learners investigate more examples.

Possible extension

Encourage learners to consider the diagonals of a rectangle. A square is a rectangle so it would be easy to assume that the diagonals of a rectangle meet at right angles too. Do they? Why or why not?