Package 2-2: Reasoning with squares

In this task your child will develop reasoning skills through paper folding activities.

Week 3 - Package 2 - Year 5 and 6 Mathematics - Reasoning with squares


Things you need

Have these things available so your child can complete this task.


  • 2 pieces of A4 paper – each a different colour

  • Pair of scissors

  • Spare paper and pencil

Back up

  • 2 pieces of A4 paper – same colour

  • 2 pieces of A3 paper

  • 2 pieces of A5 paper

  • Tearing along the fold works for this activity.

Before you start

This is a reasoning activity. It looks very simple but can take a long time. It is important to give your child time to think about the task before stepping in with a solution.

If your child is in Year 5 or Year 6 they should be very familiar with squares and their properties. Younger children can also join in the activity, however, as children have been recognising and describing squares since Kindergarten. They will just use different language.

Before students start the activity, you may need to show them how to create the two squares that they need.

  1. Fold one of the pieces of A4 paper in half and either tear along the fold or cut along the fold. This will give you two pieces of A5 paper. Put one of them aside.

Image of step 1
2  A4 papers side by side

  1. Take one of the A5 pieces of paper and fold to create a square.

Image of step 2
Fold to create a square.

  1. Cut or fold and tear the spare strip of paper.

Image for Step 3
Remove extra strip

  1. Make another square in the same way but with the A4 piece of paper so you have a large square and a small square.

What your child needs to know and do

The first thing you need to do to check your child’s understanding of squares is to ask them what the difference is between the two ‘squares’ they have in front of them and real squares. They should know that you cannot pick up real squares because they only have two dimensions – length and width. The paper squares in front of them have depth, even though it is very small. If they need more clarity you could get them to draw around the shapes they have made onto another piece of paper or take a photograph of them. They cannot pick up real squares because they only have two dimensions. These paper squares are just learning tools that are going to help us with an activity.

The next step is to ask your child to brainstorm everything they know about squares. They can write their ideas on a separate piece of paper or onto the squares themselves, or they can record themselves talking about squares. Here is a cheat sheet with some of the things they may know:

  • Squares have 4 equal sides

  • They have four equal angles that are right angles

  • They have 2 equal diagonals

  • They have 4 lines of symmetry

  • They have two pairs of parallel lines

  • The diagonals create right angles where they cross

  • They have rotational symmetry around their central point

  • They are a special kind of rectangle

  • They are a quadrilateral

Your child may be able to tell you all of these things and more, or they may know just some of them.

Once you have established what your child knows, ask them to use one of the paper squares to prove what they have said what they know. For example, how can you use only the paper square to prove that all four angles are equal?

Hint: your child needs to manipulate the paper square by folding, but give them a little time to think this out.

The final and most important part of this activity uses both squares. Using only the two squares (no rulers) prove that the small square is exactly half the size of the big one. Remember to give your child time to think this through for themselves. It is not as easy as it looks. You may like to have a go yourself. There are some hints at the end of the document based on ideas from Year 5 and 6 students, but try not to look unless you have to.

What to do next

If your child manages to prove that the small square is exactly half the size of the large square see if they can explain what they have done to someone else, either someone in their family or one of their friends using a platform such as Skype or Zoom. Teaching someone else is the best way to embed learning.

Your child could also continue making smaller and smaller squares by halving the spare A5 paper and then halving the A6 paper and so on. They can then arrange them into a pattern. Two colours of paper really help here.

Options for your child

Activity too hard?

You may need to help your child to make the squares or make them yourself.

Remember that your child may use more everyday language to describe a square.

Activity too easy?

Ask your child to use two different methods to prove that one square is half the size of the other.

Extension/Additional activity

This activity can also be done with other shapes. Here are the instructions for making a rhombus:

Start with the two pieces of A4 paper as with the square and make one of them into two A5 pieces. Discard the spare. Fold each piece of paper as follows. This is tricky and you may need to help your child.

  1. Fold into quarters to create parallel folds.

Image of step 1
Fold paper into parallel quarters.

  1. Fold in the bottom right, right angle so that it is pointing along the first fold from the right.

Image of step 2

  1. Fold in the bottom left right angle in the same way to make the folds symmetrical.

Image of step 3

  1. Fold the top right, right angle - angle over so that the right side of the piece of paper is directly over the fold you made in the last step. Make sure you have created a really sharp angle.

Image of step 4

  1. Do the same with the left side.

Image of step 5
Complete polygon

Once you and your child have made the two rhombuses you can do the activity in the same way you did with the two squares. However, there are many interesting things you can both do with the rhombus.

  • How many folds do you need to make an equilateral triangle?

  • How many folds do you need to make a right angled triangle?

  • How many folds do you need to make a scalene triangle?

  • How many folds do you need to make an isosceles triangle?

  • How many folds do you need to make a trapezium?

  • How many folds do you need to make a regular hexagon?

If you make a number of these rhombuses in different colours you can use them to make a variety of tessellating patterns and you can do the same thing with all of the other shapes you have made.

Hints for the Square activity as promised.

1. Try folding each of the squares into triangles.

2. Try putting the smaller square in the centre of the large one and rotating it 45 degrees so that each angle is touching the sides of the large square.

Return to top of page Back to top