A square number is literally a square

Take 4 counters and you can arrange them in a 2 by 2 square. Take 9 and you can make a 3 by 3 square. Take 10 and you cannot make a square at all. That picture is where the word comes from.

Squaring and square rooting

Operation	What it does	Example
Squaring	Multiply a number by itself	7² = 7 × 7 = 49
Square rooting	Find the number that was squared	√49 = 7

They are opposites, like adding and subtracting

Squaring takes 7 to 49, and square rooting takes 49 back to 7. If you are given the side of a square and want the area, you square. If you are given the area and want the side, you take the root.

The square numbers, and their patterns

The first twelve square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144. Learning them is worth the effort, because they turn up constantly.

Pattern one: the gaps are the odd numbers

The differences are exactly the odd numbers in order. To turn a 3 by 3 square into a 4 by 4 square you add a strip of 3 down the side, a strip of 3 along the top and one more in the corner, which is 7 counters.

Pattern two: the last digit gives a quick test

Every square number ends in 0, 1, 4, 5, 6 or 9, and none ever ends in 2, 3, 7 or 8. So 4327 cannot be a perfect square. But ending in 6 does not prove a number is square, since 26 ends in 6 and is not. The test can rule out but never rule in.

Finding a square root by prime factorisation

Break the number into prime factors. Because a square is a number multiplied by itself, every prime factor appears an even number of times. Take one factor from each pair, multiply them, and that is the root.

The method tells you when a number is not square

If any prime factor is left over without a partner, the number is not a perfect square. That is useful information, not a failure. Multiplying or dividing by that lonely factor produces the nearest perfect square.

Estimating a root that is not whole

To estimate the square root of 50, notice that 49 is 7 squared and 64 is 8 squared, so the root lies between 7 and 8, and much closer to 7 because 50 is only just past 49.

Two common slips

The first is confusing squaring with doubling: 7 squared is 49, not 14. The second is forgetting that the units of an area are square units: a square of side 7 cm has area 49 square centimetres.

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Worked Example 1

Find the square root of 576 by prime factorisation.

Step 1: break it into primes576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3

Step 2: make pairs(2 × 2), (2 × 2), (2 × 2), (3 × 3). Every factor has a partner.

Step 3: take one from each pair2 × 2 × 2 × 3 = 24

Check24 × 24 = 576. And 20² = 400 while 25² = 625, so an answer between 20 and 25 was expected.

Worked Example 2

What is the smallest number 720 must be multiplied by to make a perfect square?

Factorise720 = 2 × 2 × 2 × 2 × 3 × 3 × 5

Look for the lonely factorThe 2s and 3s pair up, but the 5 has no partner.

Supply the partnerMultiply by 5. 720 × 5 = 3600, and √3600 = 60.

The same idea backwardsIf asked to DIVIDE instead, the answer is also 5: removing the lonely factor leaves 144, which is 12².

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Activity 12.1

Build the square numbers with counters and find the pattern in the gaps

What you needAbout sixty small counters: pebbles, seeds, bottle tops, or squared paper.

MethodBuild a 1 by 1 square, then 2 by 2, then 3 by 3 up to 7 by 7. Each time ADD to the square you already have, and count how many you had to add.

Where to lookWatch the SHAPE of what you add: a strip down one side, a strip along the top, and one counter in the corner.

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Squaring means multiplying a number by itself: 7² = 7 × 7 = 49.

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1What is a square number, and why is it called that?

2Explain how to find the square root of 576 using prime factorisation.

3Why can no square number end in 2, 3, 7 or 8, and what are the limits of that test?

4What is the smallest number by which 720 must be multiplied to make a perfect square? Explain your reasoning.

5Between which two whole numbers does the square root of 90 lie, and how would you decide which it is nearer to?

Question 1 of 10

1What is 8 squared?

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Unit 12 · Squares and Square Roots

Squares and Square Roots

Mathematics · Grade 7

What You Will Be Able To Do

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Square any number, and take the root of any perfect square

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Tell at a glance that some numbers cannot be perfect squares

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Find a root of a large number by prime factorisation, not by guessing

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Trap a root that is not whole between two whole numbers

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Take 4 counters and you can arrange them in a 2 by 2 square. Take 10 and you cannot make a square at all, however you push them about.

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Presenter notes: Start with counters rather than notation. Hand each pair about twenty pebbles or bottle tops and ask them to arrange exactly ten into a square. They will push them around for a while and fail, and somebody will say it cannot be done. Then ask for nine, and it works immediately. That contrast is the definition of a square number, discovered rather than announced, and the word square now means something they have handled.