AC9M7N01 · Year 7 · Number

Square numbers and square roots

ACARA v9 CONTENT DESCRIPTION “describe the relationship between perfect square numbers and square roots, and use squares of numbers and square roots of perfect square numbers to solve problems”

Some numbers have a special shape. If you take 9 small tiles, you can arrange them into a perfect 3 by 3 square with none left over. The same works for 4 tiles, for 16, for 25. These are the square numbers, and seeing them as actual squares is the key to understanding both squaring and its reverse, the square root.

This year you learn to describe the relationship between perfect squares and square roots, and to use both to solve problems. The ideas sound abstract until you picture them as areas and side lengths, at which point they become some of the most intuitive in all of number.

Square numbers as squares

Squaring a number means multiplying it by itself. We write 3 squared as a small raised 2, and it equals 3 times 3, which is 9. The name comes straight from geometry: a square with sides of 3 units has an area of 9 square units. Every perfect square is the area of some square grid, so 4, 9, 16 and 25 are square numbers because they each fill a tidy square with no gaps.

Square numbers are squares
Each square number counts the unit squares in a square grid of that side.
A perfect square is what you get by multiplying a whole number by itself. Drawn as a grid, 3 times 3 fills a 3 by 3 square with 9 small squares, which is why 9 is called a square number.

This picture explains why squaring grows so quickly. Add one to the side and you add a whole new row and column of tiles, so the jump from 4 squared to 5 squared is not one step but nine extra tiles. Keeping the area picture in mind stops the common mistake of thinking that squaring just doubles a number, when in fact it multiplies the number by itself.

The square root reverses it

A square root runs the operation backwards. Where squaring turns a side length into an area, the square root turns an area back into its side length. The square root of 16 is 4, because a square of area 16 has sides of length 4. Squaring and rooting are inverse operations, two directions of the same relationship, which is why one undoes the other.

Square root is the reverse
Squaring takes a side to an area; the square root takes the area back to the side.
Squaring and taking a square root are opposite moves. Squaring 4 gives 16; the square root of 16 brings you back to 4. The root answers the question, what side length makes a square of this area.

Once you see roots this way, you can estimate ones that are not exact. The square root of 20 is not a whole number, but since 20 lies between the perfect squares 16 and 25, its root must lie between 4 and 5. This kind of reasoning, anchoring an unknown root between two squares you do know, turns square roots from a mystery into a sensible estimate, and it sets up much of the work you will meet later with areas and right-angled triangles.

Teaching tip: physical square tiles make this topic click. Have the student build squares of side 1, 2, 3, 4 and 5, counting the tiles each time to discover the sequence 1, 4, 9, 16, 25. Noticing that the gaps between them grow by the odd numbers, 3 then 5 then 7, is a small piece of mathematical delight that makes the pattern memorable.

The most common confusion is treating squaring as doubling. Returning to the grid image fixes it instantly: doubling 4 gives a 2 by 4 rectangle of 8, while squaring 4 gives a 4 by 4 square of 16. The shapes are visibly different, and so are the answers.

Builds on: Properties of prime, composite and square numbers (AC9M6N02). That unit introduced square numbers as a property of whole numbers; this unit adds the square root and the perfect-square to root relationship.
Quick self-check
1. What is a perfect square?
2. What is the value of 6 squared?
3. What is the square root of 49?
4. Squaring and taking a square root are best described as
5. Between which two whole numbers does the square root of 20 lie?