Diagonally square
Problem
Take a look at this image:
What do you see?
What do you notice?
Ayah has a conjecture. She thinks that the diagonals of the square meet at right angles.
Do you agree? How could you find out?
Draw another square and its diagonals.
Do the diagonals of your new square meet at right angles?
What happens if you draw another square and its diagonals?
Do the diagonals of a square always meet at right angles?
Mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true.
Can you create an argument that would convince mathematicians?
Once you have had a good think about it, you might like to look at this proof that has been scrambled up.
Can you rearrange the steps into the correct order?
If you would prefer to work away from a screen, you could print off, cut up and rearrange the statements on this sheet (it includes two copies of each statement).
Toby has another conjecture. He says, "I know that a square is a special type of rectangle. So I think that the diagonals of rectangles meet at right angles too."
What do you think?
Do the diagonals of a rectangle always meet at right angles?
Getting Started
Try drawing lots of different squares and testing each one. You may find it helpful to print off this sheet, which contains a few squares of different sizes.
Student Solutions
Thank you to everybody who sent in their ideas about this activity.
Saanvi from Newcastle under Lyme Prep School in the UK sent us this explanation:
For this problem, I think it is easiest to imagine it as a square piece of paper. If you fold the paper in half diagonally, then there is a 180° angle in the centre. Fold it in half again (dividing the page into quarters), and there is a 90° angle in the middle. This will work on every square.
This is a good idea, Saanvi - sometimes making a model out of paper makes it much easier to see what is happening.
Ananya from Halstead Prep School in England agreed that the lines would always meet at 90°. Ananya gave this reasoning:
We know that every angle in a square is 90° and there are only 2 diagonal lines which can fit in the square (making 4 separate triangles). When you draw the diagonal line, the corners would now be split into 45° each. We know that a triangle adds up to 180° and two of the angles are 45° each, hence the third angle (where the diagonal lines are intersecting) is 90° (180°-[45°+45°]). Now the Mathematician will know that the diagonal lines will always meet up to 90°.
Well done for using the facts you know about angles in a triangle to justify why the third angle must be 90 degrees!
Haleema and Fern from Pierrepont Gamston Primary School sent in similar solutions. Fern thought about the angles made by the diagonals of a rectangle, and she sent in this picture:
Good ideas! Fern says that the diagonals of a rectangle will make two acute angles and two obtuse angles. I wonder if we can notice anything interesting about those angles from Fern's picture?
Prayaan from King's College Junior School in the UK also began to think about the angles made by the diagonals of a rectangle. Have a look at Prayaan's full solution to see his ideas. I particularly like the diagram showing which shapes are special types of which other shapes, although I think triangles should be in the polygon section.
In Prayaan's investigation, he looks at rectangles that are 1 by 2, 1 by 4, 1 by 6 and 1 by 8 to see what angles the diagonals meet at. I wonder what Prayaan would find if he measured the angles for a 1 by 100 rectangle? Or 1 by 1000?
We also had similar solutions sent in from Dhruv at St. Anne's RC Primary School in the UK, from Ci Hui Minh Ngoc at Kong Hwa School in Singapore and from the children at The Sterling Hall School. Well done to everybody who had a go at this activity!
Teachers' Resources
Diagonally Square
Take a look at this image:
What do you see?
What do you notice?
Ayah has a conjecture. She thinks that the diagonals of the square meet at right angles.
Do you agree? How could you find out?
Draw another square and its diagonals.
Do the diagonals of your new square meet at right angles?
What happens if you draw another square and its diagonals?
Do the diagonals of a square always meet at right angles?
Mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true.
Can you create an argument that would convince mathematicians?
Once you have had a good think about it, you might like to look at this proof that has been scrambled up.
Can you rearrange the steps into the correct order?
Toby has another conjecture. He says, "I know that a square is a special type of rectangle. So I think that the diagonals of rectangles meet at right angles too."
What do you think?
Do the diagonals of a rectangle always meet at right angles?
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?