Activity: Drawing Squares

For this activity all you need is a grid of dots, a pencil and your brain.

Let us discover how many squares you can make on different grids: Note: "1 by 1" means how many sides (not how many dots).

1 by 1

 Well, that's easy, there's just one: 2 by 2

 That seems to be easy too. There are four of them, aren't there? But wait, that's not the complete answer. There's also this bigger one: That makes five squares altogether - four 1 by 1 squares and one 2 by 2 square

3 by 3

 Over to you now. Here's the grid:  Hint: For the 3 by 3 case, you will expect to get 1 by 1 squares, 2 by 2 squares and 3 by 3 squares. How many of each?

Now you can start to fill in a table:

 How Many 1 by 1 squares How Many2 by 2 squares How Many 3 by 3 squares How Many4 by 4 squares How Many5 by 5 squares Total 1 by 1 Grid: 1 1 2 by 2 Grid: 4 1 5 3 by 3 Grid: 4 by 4 Grid: 5 by 5 Grid:

Did you notice anything about the numbers in the table?

They are all square numbers:

• 12 = 1,
• 22 = 4,
• 32 = 9,
• etc ...

and the totals are found by adding together square numbers.

Formula to The Rescue ... !

There is actually a formula for adding the first n square numbers:

Sn = n(n+1)(2n+1) / 6

Example: The number of squares in the 5 by 5 case

Try substituting n = 5 into the formula:

Sn = n(n+1)(2n+1) / 6
S5 = 5 × (5+1) × (2×5+1) / 6
S5 = 5 × 6 × 11 / 6
S5 = 55

So, we seem to have solved the question. Yipee!

But wait ... there's more!

I said you would need to use your brains. Let's go back to the 2 by 2 case:

2 by 2

There is another square too, this one: Why is it a square? It has four equal sides and four right angles, so that's a square.

So, that makes six squares altogether.

Four 1 by 1 squares, one 2 by 2 square and onex by x square.

What is the value of x? We can use Pythagoras' Theorem to find it:

x2 = 12 + 12 = 1 + 1 = 2
So x =  √2

So, we have four 1 by 1 squares, one 2 by 2 square and one √2 by √2 square.

3 by 3

• Are there any more squares?

YES! Can you find them?

4 by 4 and 5 by 5

Also try the 4 by 4 grid, and the 5 by 5 grid

As you proceed, you will find squares like these: What are the lengths of the sides of these squares?

You can use Pythagoras' Theorem to work that out yourself

In each case, how many do you get of each one?

 How Many1 by 1 How Many2 by 2 How Many3 by 3 How Many4 by 4 How Many5 by 5 How Many√2 by √2 How Many√5 by √5 How Many√8 by √8 How Many√10 by √10 How Many√13 by √13 How Many√17 by √17 Total 1 by 1 Grid: 1 1 2 by 2 Grid: 4 1 1 6 3 by 3 Grid: 4 by 4 Grid: 5 by 5 Grid: