Package 3-2: Inside Seven Squares

Your child will work through a task to solve a problem about area.

Week 4 - Package 2 - Year 5 and 6 Mathematics - Inside Seven Squares

Adapted from NRICH maths.

Things you need

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


Back up

  • Pen or texta

  • Grid paper such as that in your child’s mathematics book.

Before you start

This is a reasoning activity adapted from nrich.maths and is expected to take some time. Try to encourage your child to solve the problem without stepping in to help at the first hurdle.

Your child will need to be able to draw lines accurately with a ruler using the dotty grid paper or ordinary grid paper as a guide. Drawing lines accurately is an important mathematics skill but is not necessary for the reasoning part of the activity. Your child will also need to know the basic properties of squares and right-angled triangles so you may want to do some research before you get started. In addition, your child will need to understand that area is the amount of space inside the boundary of a two-dimensional shape and can be measured in square units.

What your child needs to know and do

The problem

Seven squares are set inside each other. The centre point of each side of the outer square is joined to make a smaller square inside. This is repeated each time a square is drawn until there are 7 squares.

7 concentric squares rotated at 90 degrees from each other. Each square is half the area of the one outside it.
Largest square is coloured red.

The centre square has an area of 1 square unit.

What is the total area of the four outside right-angled triangles (the ones in red)?

What to do next

A useful way to get your child thinking about this problem is to make the beginning of the design by folding a square piece of paper. Using concrete materials is an important mathematical skill at all levels of learning. Remind your child that this is a tool for learning.

An A4 piece of paper can be made to represent a square as follows:

Image of A4 paper Fold one corner over to the opposite edge to make a right angled triangle.Cut or tear off the excess.

The following folding activity will help your child consider where the problem came from. It is really helpful to ask what they notice at each step of the process.

  1. Once you have your square piece of paper fold it along both diagonals to find the centre.

square paper folded along diagonals

  1. Fold each right angle carefully into the centre.

Each right angle folded to the centre of the square.

  1. Open out and use the folds to draw the next square.

Square unfolded and next square drawn along the folds.

  1. Fold in again and then fold the next four right angles into the centre of the new square. At this point the paper is going to start looking like a Chatterbox. Making a Chatterbox would be a lovely fun activity to do after the maths! It is a great way to practise accurate folding.

Paper is refolded and right angles of smaller square are folded to the centre.

  1. Open out again and draw the next square.

Paper opened out with next square drawn on new folds.

  1. Your child can continue this activity until it becomes too hard to fold the paper. I wonder whether 7 times really is the limit for folding a piece of paper? The folds will become less and less accurate as the paper gets thicker. (You might even like to view an old episode of Myth Busters about this!)

Folded one more time Page opened out again with smaller square drawn along folds.

Next you will need the dotty paper or squared paper from your child’s maths book, a pencil and ruler.

It would be wonderful if you could do the activity alongside your child or if they could work with a sibling to share ideas. Make sure you give your child a chance to think about and discuss where on the paper they will start, what size square to start with, and whether to start with the largest or the smallest square. There may be a few trial runs.

Once all of the squares have been drawn it is important to remember that the problem asks you to find the combined area of the four outer triangles (the red ones in the diagram). There will be a number of ways to explain this. All strategies are good strategies and it is worth brainstorming as many ways as possible.

Options for your child

Activity too hard?

If the drawing activity is too difficult there is still a lot of learning in the folding activity. Ask your child how much bigger the area of the original square is than the square created by the folds. Ask them how they know.

Activity too easy?

Ask your child to determine how many times bigger the area of the outside square than each of the successive squares.

Ask if they can predict the area of the square outside the largest one they have drawn. Can they create a table that shows the areas of each square so they can predict the areas of both smaller and larger squares?

Can they add perimeter to their table or length of each side to discover the relationship between the perimeters of the squares and their areas?

Extension/additional activity

A lovely extension activity that everyone can try also involves manipulating squares. The activity is called Fitted. From NRICH maths.

The problem

Nine squares with side lengths 1,4,7,8,9,10,14,15 and 18 cm can be fitted together with no gaps and no overlaps, to form a rectangle.

What are the dimensions of the rectangle?

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