Grade by Grade
B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional Reasoning

Square Numbers and Square Roots to Know by Heart

B2.2 A1About 40 minutes
Learning goals
  • I can recall the squares of whole numbers from 1 to 15 (and 20) and their square roots quickly.
  • I can use these facts to find squares and square roots of multiples of 10, decimals and fractions.
  • I can explain why squaring and taking a square root undo each other.
  • I can use square numbers to solve problems about square areas.

Words to know

perfect square
A number that is a whole number multiplied by itself, like 144 = 12 × 12 = 12².
square root
A number that, multiplied by itself, gives the original number. √144 = 12 because 12 × 12 = 144. In this unit √ means the positive square root.
squaring
Multiplying a number by itself. Squaring 0.7 gives 0.7² = 0.49.
inverse operations
Operations that undo each other. Squaring and taking the square root are inverse operations.

Remember from Grade 7

In Grade 7 you learned what a perfect square is, listed the squares from 1² to 20², and found square roots of perfect squares, like √64 = 8. In Grade 8 the goal is to know the common ones by heart so you can use them instantly, and to stretch them to decimals, fractions and multiples of 10. (Estimating square roots of numbers that are not perfect squares, like √50, is taught in this grade's Number Sense unit.)

Let's start in the community garden

A community garden has a square plot with an area of 144 m². How much fencing goes around it?

You need a side length that, multiplied by itself, makes 144. If you know 12² = 144 by heart, the side is 12 m right away. The fence is 4 × 12 = 48 m.

Knowing square facts by heart saves time in geometry, in measurement and in algebra, and next year in Grade 9 math.

A fenced community garden plot with raised beds of tomatoes, beans and lettuce.
A fenced garden plot like this one needs a fence all the way around. If a square plot has an area of 144 m², knowing 12² = 144 tells you each side is 12 m. AI-generated image
nn²nn²nn²nn²
116361112116256
247491214417289
398641316918324
4169811419619361
525101001522520400
Know at least 1² to 15² and 20² by heart. Read the table backwards for square roots: √196 = 14.

Stretching the facts

Once you know 3² = 9, you also know:

  • Multiples of 10: 30² = 30 × 30 = 900, so √900 = 30. And √1 600 = 40 because 40² = 1 600.
  • Decimals: 0.3² = 0.3 × 0.3 = 0.09, so √0.09 = 0.3. Also 1.2² = 1.44, so √1.44 = 1.2.
  • Fractions: (3/4)² = 9/16, so √(9/16) = 3/4. Take the square root of the top and the bottom: √(25/49) = 5/7.

Squaring and square roots undo each other. √(7²) = √49 = 7, and (√49)² = 7² = 49.

Whole square = 1Each small square = 0.01Shaded: 0.3 × 0.3 = 0.09So √0.09 = 0.3
A 10 by 10 grid is one whole, so each small square is 0.01. The 3 by 3 shaded corner is 0.3 × 0.3 = 9 hundredths = 0.09. That is why 0.3² = 0.09 and √0.09 = 0.3.

Decimal place check

When you square a decimal, the answer has twice as many decimal places: 0.3 (one place) → 0.09 (two places); 0.12 (two places) → 0.0144 (four places).

So for a square root, halve the decimal places: √0.0144 = 0.12. Watch out: √0.9 is not 0.3, because 0.3² = 0.09.

Worked example

Example 1: A square root from memory

Find √196.

  1. Think: what number times itself is 196?
  2. 14 × 14 = 196.

√196 = 14

Worked example

Example 2: A decimal

Find √0.64.

  1. 64 = 8², so try 0.8.
  2. 0.8 × 0.8 = 0.64. ✓ (One decimal place doubles to two.)

√0.64 = 0.8

Worked example

Example 3: A fraction

Find √(36/121).

  1. √36 = 6 and √121 = 11.
  2. Check: (6/11)² = 36/121. ✓

√(36/121) = 6/11

Worked example

Example 4: A square poster

A square poster has an area of 2 500 cm². Find its side length and its perimeter.

  1. 2 500 = 25 × 100, and 50 × 50 = 2 500. So the side is 50 cm.
  2. Perimeter = 4 × 50 = 200 cm.

Side 50 cm; perimeter 200 cm

Hands-on activity

Build the squares and make flashcards

You need: square crackers, sticky notes, square tiles or graph paper; 15 index cards (or paper squares); a pencil.

  1. Build squares that are 1 by 1, 2 by 2, 3 by 3, up to 6 by 6. Count the pieces in each.
  2. Write the totals in a row: 1, 4, 9, ... What do you add each time to get the next square?
  3. Make flashcards: write 13² on the front and 169 on the back, and so on for 1² to 15² and 20². On the back also write the root fact: √169 = 13.
  4. Quiz yourself (or a family member) every day this week. Put the cards you know instantly in one pile and the slow ones in another. Practise the slow pile.

What to look for: squares 1, 4, 9, 16, 25, 36. You add 3, 5, 7, 9, 11, the odd numbers, because each bigger square adds a new row, a new column and one corner piece.

Each bigger square adds the next odd number of tiles1² = 12² = 43² = 94² = 165² = 25+3+5+7+9
Each new square adds a new row, a new column and one corner tile. The jumps are the odd numbers 3, 5, 7, 9, so 16 + 9 = 25 = 5².
Just for fun

Why are square roots such good gardeners?

They always get to the root of the problem, and every plot comes out perfectly square.

Your turn

1

13² = ?

2

√121 = ?

3

0.7² = ?

4

√2 500 = ?

5

√(16/81) = ?

6

Is 200 a perfect square? How do you know?

7

A square table top has an area of 2.25 m². How long is each side?

More practice: warm-up

1

9² = ?

2

√64 = ?

3

20² = ?

4

√1 = ?

5

√0.25 = ?

More practice: on your own

1

√1.21 = ?

2

15² − 14² = ?

3

Write √(49/100) as a decimal.

4

Which is greater: √169 or 12.5? Explain.

5

A square quilt is made of 196 equal square patches. How many patches are along each side?

Challenge

1

Find two perfect squares next to each other in the list (like 9 and 16) whose difference is 25.

2

Square tiles have sides of 0.4 m. How many tiles cover a square floor that is 3.6 m by 3.6 m?

3

Find √0.0036.

Check your understanding

Quiz

1.

What is √144?

2.

What is 0.5²?

3.

Which number is a perfect square?

4.

What is √(9/25)?

5.

A square garden has an area of 400 m². How long is each side?

6.

What is √0.81?

For parents and teachers

Big idea: Ontario's Grade 8 expectation B2.2 asks students to understand and recall commonly used square numbers and their square roots. "Recall" means quick, without counting or a calculator, the way multiplication facts are recalled. Grade 7 introduced perfect squares; this year is about fluency and stretching the facts to decimals, fractions and multiples of 10.

Common difficulties:

  • Halving instead of rooting (√144 = 72, or √400 = 200).
  • Decimal places: 0.3² written as 0.9, or √0.9 written as 0.3.
  • Confusing 2 × n with n² (7² = 14).

How to help: Use the flashcards for a few minutes a day, a little at a time. Point out square things at home (floor tiles, a chessboard: 8² = 64 squares) and ask for the side length when you know the area.