Grade by Grade
B. Number · Number Sense: Big Numbers, Square Roots and Rational Numbers

Perfect Squares

B1.2 A1About 40 minutes
Learning goals
  • I can explain what a perfect square is.
  • I can show a perfect square as a square array or the area of a square.
  • I can find the square of a whole number, like 12² = 144.
  • I can decide whether a number is a perfect square.

Words to know

perfect square
A number you get when you multiply a whole number by itself, like 36 = 6 × 6. Also called a square number.
square (a number)
Multiply a number by itself. 6 squared is written 6² and means 6 × 6.
array
Objects arranged in equal rows and columns.
area
The amount of surface a shape covers, measured in square units like cm² or m².
prime factorization
A number written as a product of prime numbers, like 36 = 2 × 2 × 3 × 3.

Remember

In earlier grades you found the area of a rectangle by multiplying its length by its width. In Grade 6 you broke numbers into prime factors with factor trees. You will use both ideas today.

A square patio

Amir's family has 36 square patio stones. Can they lay them in a perfect square, with the same number of stones on each side?

Yes! 6 rows of 6 stones makes a square, because 6 × 6 = 36.

Now they buy 4 more stones. Can 40 stones make a square? Try 6 × 6 = 36 (too few) and 7 × 7 = 49 (too many). No whole number times itself makes 40, so 40 stones cannot make a square.

A square patio of grey concrete stones, 6 rows of 6, surrounded by green grass, seen from above.
Count the stones: 6 rows of 6 makes 36. Because 36 stones can form a square, 36 is a perfect square. AI-generated image

What is a perfect square?

A perfect square is the product of a whole number multiplied by itself.

  • 36 = 6 × 6, so 36 is a perfect square.
  • 40 is not a perfect square.

We write 6 × 6 as 6² and say "six squared". The small 2 is an exponent: it means two 6s are multiplied. It is called squared because 6 × 6 is the area of a square with sides of 6.

1×112×243×394×4165×5256×636
The first six perfect squares: 1, 4, 9, 16, 25 and 36. Each one makes a square array of tiles.
nn²nn²nn²nn²
116361112116256
247491214417289
398641316918324
4169811419619361
525101001522520400
The squares of 1 to 20. Try to learn the squares up to 15² = 225.
Worked example

Example 1: Is it a perfect square?

Is 64 a perfect square? Is 50?

  1. Look for a whole number that times itself makes 64. 8 × 8 = 64, so 64 is a perfect square.
  2. For 50: 7 × 7 = 49 is too small and 8 × 8 = 64 is too big. There is no whole number between 7 and 8, so 50 is not a perfect square.

64 = 8², so it is a perfect square. 50 is not.

Worked example

Example 2: Area of a square

A square community garden has sides of 15 m. What is its area?

  1. Area of a square = side × side.
  2. 15 × 15 = 225. (Split it: 15 × 10 = 150 and 15 × 5 = 75. 150 + 75 = 225.)

The area is 225 m². So 225 is a perfect square: 225 = 15².

Worked example

Example 3: Use prime factors

Is 144 a perfect square? Use its prime factorization.

  1. Make a factor tree: 144 = 12 × 12 = (2 × 2 × 3) × (2 × 2 × 3).
  2. So 144 = 2 × 2 × 2 × 2 × 3 × 3.
  3. Can you split the factors into two identical groups? Yes: (2 × 2 × 3) and (2 × 2 × 3). Each group is 12.

Yes. 144 = 12 × 12 = 12². A number is a perfect square when its prime factors can be split into two identical groups (every prime factor appears an even number of times).

144 =2×2×2×2×3×32×2×3group = 122×2×3group = 12×Two identical groups, so 144 = 12 × 12 = 12²
The prime factors of 144 split into two identical groups of 2 × 2 × 3. Each group is 12, so 144 is a perfect square.

Last digit check

Perfect squares can only end in 0, 1, 4, 5, 6 or 9. A whole number that ends in 2, 3, 7 or 8 is never a perfect square. (Check it: square each digit 0 to 9 and look at the last digit of the answer.) But be careful: ending in 1, 4, 5, 6 or 9 doesn't prove a number is a perfect square. 14 ends in 4 and is not one.

Hands-on activity

Build the squares

You need: about 50 small objects that are the same, like coins, square crackers, buttons or building bricks, and paper.

  1. Build a square array with 1 object, then 4 (2 by 2), then 9 (3 by 3). Keep going as far as your objects allow.
  2. Write how many objects each square uses.
  3. Look at how many objects you add each time to make the next square: 1 → 4 (add 3), 4 → 9 (add 5), 9 → 16 (add ...?). What pattern do you see?
  4. Count out 20 objects. Can you make a square? Try 25. Try 30.

What to look for: you add 3, 5, 7, 9, ... each time (the odd numbers), because you add an L-shape around two sides of the square. 20 and 30 can't make squares, but 25 can (5 by 5).

Math talk: "How do you know 30 can't make a square?"

Just for fun

Sneaky pattern alert: 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16. Add the odd numbers in order and you get a perfect square every time. Squares have been secretly collecting odd numbers all along!

Your turn

1

What is 9²?

2

Is 49 a perfect square? Explain.

3

Is 60 a perfect square? Explain.

4

A square tile has sides of 11 cm. What is its area?

5

List all the perfect squares between 100 and 200.

6

What is 20²?

7

Which of these are perfect squares: 81, 90, 100, 125?

More practice: warm-up

1

What is 4²?

2

What is 1²?

3

What is 10²?

4

Is 25 a perfect square?

5

Is 30 a perfect square?

More practice: on your own

1

Can 72 square tiles be arranged into one square with no tiles left over?

2

Is 196 a perfect square? Explain.

3

Can a whole number ending in 7 be a perfect square? Explain.

4

225 = 3 × 3 × 5 × 5. Is 225 a perfect square? Use the prime factors.

5

What is the smallest number you can add to 90 to make a perfect square?

Challenge

1

72 = 2 × 2 × 2 × 3 × 3. Is 72 a perfect square? What is the smallest whole number you can multiply 72 by to make a perfect square?

2

Find 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 without adding them one by one.

3

List all the factors of 36. How many are there? Why do perfect squares have an odd number of factors?

Check your understanding

Quiz

1.

Which number is a perfect square?

2.

What is 13²?

3.

Which product shows that 49 is a perfect square?

4.

Just by looking at the last digit, which number cannot be a perfect square?

5.

How many perfect squares are between 1 and 50 (not counting 1 or 50)?

6.

A square patio has 12 stones along each side. How many stones are in the whole patio?

For parents and teachers

Big idea: Grade 7 expectation B1.2 asks students to identify and represent perfect squares, and determine their square roots, in various contexts. This lesson is the first half: what perfect squares are and how to show them with square arrays and areas. The next lesson covers square roots.

Common difficulties:

  • Thinking 6² means 6 × 2. Build or draw the square to show 6 rows of 6.
  • Assuming any number ending in 1, 4, 5, 6 or 9 is a perfect square. The last-digit test can only rule numbers out.
  • Not knowing the squares. Knowing squares to 15² by heart makes the next lesson (and later grades) much easier.

How to help: Do the build-the-squares activity, and quiz each other on squares in the car or at dinner. This lesson builds A1 skills: noticing patterns, making and testing conjectures ("Do all squares have an odd number of factors?"), and persevering.