B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

Estimating Square Roots

B1.3 A1About 40 minutes

Learning goals

I can find the two whole numbers a square root is between.
I can estimate a square root to one decimal place and check by squaring.
I can use a calculator to find a square root and round it sensibly.
I can explain whether my estimate is too high or too low.

Words to know

perfect square
A whole number that is a whole number times itself, like 49 = 7 × 7.

square root
A number that, multiplied by itself, gives the original number. √49 = 7 because 7 × 7 = 49. The √ sign means the positive square root.

estimate
A careful guess that is close to the exact value.

guess and check
Try a value, square it, and adjust up or down until you are close enough.

Remember from Grade 7

In Grade 7 you found square roots of perfect squares, like √64 = 8, and you said that √50 is between 7 and 8. You were told you would learn to estimate square roots like this in Grade 8. Now is the time!

A square of floor space

Amira wants to tape a square on the floor for a game. She has room for 50 square tiles' worth of area (50 square units). How long should each side be?

She needs √50. It is not a whole number, and (as you saw last lesson) it is irrational. But she can still find a very good estimate.

Four steps for estimating a square root

Find the perfect squares on either side. 49 < 50 < 64, so 7 < √50 < 8.
Decide which one it is closer to. 50 is only 1 away from 49 but 14 away from 64, so √50 is much closer to 7.
Guess and check to one decimal place. Square your guesses:

7.1² = 50.41 (a bit too big)
7.0² = 49 (too small)

So √50 is between 7.0 and 7.1. 50 is closer to 50.41 than to 49, so √50 ≈ 7.1 to one decimal place.

Check with a calculator if you have one: √50 = 7.071 067 8… ≈ 7.07.

A quick shortcut for step 2: 50 is 1 of the 15 steps from 49 to 64. 1/15 ≈ 0.07, so √50 ≈ 7 + 0.07 = 7.07. This "fraction of the way" estimate is very close, but always a tiny bit too low.

Top: 50 is just past 49 on the way to 64. Bottom: in the same way, √50 is just past 7 on the way to 8. The dashed lines join each number to its square root.

Guess	Guess squared	Compared with 50
7	49	too small
7.1	50.41	a little too big
7.05	49.702 5	too small
7.07	49.984 9	very close, a tiny bit small
7.08	50.126 4	too big

Guess and check for √50. Each guess gets closer: √50 is between 7.07 and 7.08.

A 7 × 7 square has area 49. To reach 50, add two thin strips of width x, each about 7x. Since 14x ≈ 1, x ≈ 0.07 and √50 ≈ 7.07. (Not to scale.)

Worked example

Example 1: Estimate √90

Estimate √90 to one decimal place.

Perfect squares: 81 < 90 < 100, so 9 < √90 < 10.
90 is 9 steps past 81, out of 19 steps to 100. 9/19 is about 0.47, so try about 9.5.
Check: 9.5² = 90.25 (a little too big). 9.4² = 88.36 (too small).
90 is much closer to 90.25 than to 88.36, so √90 is closer to 9.5. Calculator: √90 = 9.486 8…

√90 ≈ 9.5

Worked example

Example 2: A bigger number

Estimate √200 to one decimal place.

Which perfect squares are near 200? 14² = 196 and 15² = 225. So 14 < √200 < 15, very close to 14.
Try 14.1² = 198.81 (too small) and 14.2² = 201.64 (too big).
200 − 198.81 = 1.19 and 201.64 − 200 = 1.64. 200 is a little closer to 198.81.
Calculator: √200 = 14.142 1…

√200 ≈ 14.1

Worked example

Example 3: Using a calculator well

A square rink has an area of 30 m². Find the side length to the nearest tenth of a metre.

Estimate first: 25 < 30 < 36, so the side is between 5 m and 6 m, about halfway or a bit less.
Calculator: √30 = 5.477 2…
Round to the nearest tenth: the hundredths digit is 7, so round up to 5.5. This matches the estimate. ✓

Each side is about 5.5 m.

Watch out!

√50 is not 25. Taking a square root is not the same as dividing by 2.
When you guess and check, always square your guess. Don't multiply it by 2.
Estimate before you press the calculator's √ key, so you will notice if you pressed the wrong key.

Just for fun: the oldest square-root homework

About 3 700 years ago, a student or scribe in ancient Babylon pressed an estimate of √2 into a small clay tablet. In our decimals it works out to about 1.414 213, which is off by less than one millionth. No calculator, no guess-and-check app, just clay and patience.

Hands-on activity

Tape a 2 m² square

You need: a measuring tape or metre stick, painter's tape (or chalk outdoors, with permission), a calculator.

Mark a square on the floor with an area of exactly 1 m². (What is each side?)
Now plan a square with an area of 2 m². Estimate its side length using perfect squares, then guess and check: 1.4² = ?, 1.5² = ?
Tape out your 2 m² square using your best estimate, to the nearest centimetre.
Stand inside each square. Does the 2 m² square look like twice as much floor? Does it have twice the side length?

What to look for: the 1 m² square has sides of 1 m. For 2 m², 1.4² = 1.96 and 1.5² = 2.25, so the side is just over 1.4 m. A calculator gives √2 = 1.414…, so about 1.41 m (141 cm). Doubling the area does not double the side length: 1.41 m is much less than 2 m.

Math talk: "How long would the sides be for a square with 4 times the area of the first one?" (2 m.)

Your turn

Between which two whole numbers is √30?

Between which two whole numbers is √75? Which is it closer to? Estimate it to one decimal place.

Estimate √20 to one decimal place.

Estimate √10 to one decimal place.

Without a calculator: is √99 closer to 9 or to 10?

Use a calculator to find √150 to the nearest hundredth.

A square rink has an area of 30 m². About how long is each side?

More practice: warm-up

Between which two whole numbers is √40?

Between which two whole numbers is √5?

Between which two whole numbers is √85?

Is √3 closer to 1 or to 2?

Between which two whole numbers is √120?

More practice: on your own

Estimate √60 to one decimal place.

Estimate √45 to one decimal place.

Estimate √130 to one decimal place.

Use the "fraction of the way" shortcut to estimate √28. Then check with a calculator.

A square picture has an area of 500 cm². Find its side length to the nearest tenth of a centimetre.

Challenge

Estimate √1 000 to the nearest whole number.

A square has an area of 2 m². A second square has sides twice as long. What is the area of the second square?

Sam says √70 ≈ 8.5. Without a calculator, check his estimate and improve it to one decimal place.

Check your understanding

Quiz

√55 is between which two whole numbers?

Which is the best estimate of √26?

4.8² = 23.04 and 4.9² = 24.01. So √24 is:

Which is closest to √99?

A square has an area of 50 m². About how long is each side?

Which of these is a perfect square?

For parents and teachers

Big idea: B1.3 asks Grade 8 students to estimate and calculate square roots, in various contexts. In Grade 7 they found square roots of perfect squares only. Now they estimate square roots of any number, check by squaring, and use a calculator sensibly. Knowing the perfect squares up to 15² = 225 by heart (part of the Operations unit, B2.2) makes this much easier.

Common difficulties:

Halving instead of square-rooting (√50 = 25).
Checking a guess by doubling instead of squaring.
Rounding before comparing. Encourage writing out the squares of both neighbouring tenths.

How to help: Practise perfect squares in the car or at dinner ("What's 13 squared?"). Do the taping activity: seeing that a 2 m² square is not twice as wide is a powerful "aha" moment. Encourage your child to estimate first and say whether a calculator answer makes sense; that habit is part of the A1 processes.