Squares and Square Roots

The square of a number?

When we multiply an integer by itself we call the product the square of the number.
For example: 4 x 4 = 16
The square of 4 is 16
The pictures above show why we call these products squares.

4 by 4 grid in outline form 16 units arranged 4 by 4

Squared Numbers

The examples below show how squares can be found.

Number Multiplied by itself Square
1 1 x 1 1
2 2 x 2 4
3 3 x 3 9
4 4 x 4 16
5 5 x 5 25
6 6 x 6 36
7 7 x 7 49
8 8 x 8 64
Number Multiplied by itself Square
9 9 x 9 81
10 10 x 10 100
11 11 x 11 121
12 12 x 12 144
13 13 x 13 169
14 14 x 14 196
15 15 x 15 225
 12 x 12 multiplication grid with squares highlighted

Squared Words!

You'll hear different words used when people talk about squares. e.g.

  • Square of a number: 25 is the square of 5
  • Squaring a number: multiplying the number by itself
  • A squared number: 100 is a square number
  • 3 squared: 3 squared is 9
  • What's the square of 9? The square of 9 is 81
  • Perfect square: Perfect square is another term for square number

Make sure your children know the short way of finding a square on a calculator. e.g. to find 6 x 6, enter 6, x, =

Writing squares

Squares are powers of two.

We write squares using the same notation that we use with other powers.
e.g. for 3 squared (or 3 x 3)
we write 32

We could talk about 3 to the power of 2, or the second power of three but we don't usually do so; we say 3 squared. We write 32

Square Roots

You can think of finding square roots as the opposite of finding squares. You find the square root of a number (let's call it number A) by finding the number that, when multiplied by itself produces number A. The examples below show this:

Number Square Root
1 x 1 =
1 1
2 x 2 =
4 2
3 x 3 =
9 3
4 x 4 =
16 4
5 x 5 =
25 5
6 x 6 =
36 6
7 x 7 =
49 7
8 x 8 =
64 8
  Number Square Root
9 x 9 =
81 9
10 x 10 =
100 10
11 x 11 =
121 11
12 x 12 =
144 12
13 x 13 =
169 13
14 x 14 =
196 14
15 x 15 =
225 15
16 x 16 =
256 16

Note: Think of these examples: 4 x 4 = 16 and - 4 x - 4 = 16.
Positive numbers have two square roots; a positive (called the principal square root) and a negative. Unless you are asked for the negative square root, you can just give the principal square root.

When exploring squares and square roots with your children, emphasize that each process is the inverse of the other. In other words, one undoes the other.

Writing Square Roots

illustrated example of the square root symbol

Make sure that your children know the square root symbol and that they can find and use it on a calculator.

Calculating and Estimating Square Roots

Most calculators have a square root button that quickly calculates square roots. There are other ways of calculating square roots but they aren't quick. If you need an approximate value for a square root you can use a method like the one below.

What is the square root of 42?
What two squares does 42 come between? 36 and 49
What are the square roots of 36 and 49 6 and 7
So the square root of 42 is between 6 and 7.
Let's use some trial and error to get an approximate answer.
Let's try 6.5 42 ÷ 6.5 = 6.46
We're looking to get the number we divide by to be as close to the answer we get as possible. In this case we have 6.5 and 6.46
Close, but we can get closer.
Let's take the average of 6.5 and 6.46 and try that. 42 ÷ 6.48 = 6.48
So we have an square root that is accurate to two decimal places. You can repeat these steps to get as accurate an answer as you want.

Squares and Square Roots Chart

chart showing squares and square roots (to 10)

This printable version of the squares and roots chart above will help visualize the relationship between squares and square roots.

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