Activity: Drawing Squares

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

Let's discover how many squares you can make on different grids:

Dot grids labeled 1 by 1, 2 by 2, and 3 by 3

Note: "1 by 1" means how many sides (not how many dots).


So, let's try drawing in some squares and count how many:

1 by 1

Well, that's easy, there's just one:   1 by 1 dot grid with one square outlined in red

2 by 2

That seems to be easy too. There are four of them, aren't there?   2 by 2 dot grid showing four 1 by 1 squares highlighted in red
     
But wait, that's not the complete answer.
There's also this bigger one:		   2 by 2 dot grid showing one large 2 by 2 square highlighted in red

That makes five squares altogether - four 1 by 1 squares and one 2 by 2 square

Your Turn !

3 by 3

Over to you now. Here's the grid:

Empty 3 by 3 dot grid with 16 dots total

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?



Printable sheet of dot grids of various sizes


Now you can start to fill in a table:

  How Many
1 by 1 squares		 How Many
2 by 2 squares		 How Many
3 by 3 squares		 How Many
4 by 4 squares		 How Many
5 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:

and the totals are found by adding together square numbers.

Formula to The Rescue ... !

There's 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's another square too, this one:

squares9

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 one x by x square.

squares9

What's the value of x? Each side forms a right triangle where each leg is 1 unit long.

We can use Pythagoras' Theorem to find the length:

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.

Your Turn !

3 by 3

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:

Dot grids displaying tilted squares with side lengths root 2, root 5, and root 8

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?

Here's a table to help you:

  How Many
1 by 1		 How Many
2 by 2		 How Many
3 by 3		 How Many
4 by 4		 How Many
5 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:











Advanced

Can you find a formula to calculate the number of squares that have lengths that are square roots?

Can you then find a formula for the total number of squares in each case?

Are there any more squares lurking in there that we've missed?

Conclusion

What started off as seeming to be a simple exercise turned out to be quite complex. You really do need to use your brains to think this one through, but it's a challenging and rewarding exercise.

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