Skip to content
Brainsladder

Approximate Irrational Numbers on a Number Line

Students use successive squaring to narrow down the location of irrational square roots between consecutive integers, then between consecutive tenths, to place them accurately on a number line. They understand that these approximations get arbitrarily close to the true value.

Try three of them

Real questions from this skill, graded here. No account, and nothing is saved.

Approximate Irrational Numbers on a Number Line

Grade 8 · The Number System · question 1 of 3

0 of 20

20 rungs. A star at the top.

√12 is between and 4

How it is taught

0:00 of 3:31

An irrational square root sits between consecutive squares.

A rational approximation is a decimal close to that root.

Read it: Square roots between whole numbers

√20 is the number that times itself makes 20. 4 × 4 = 16 and 5 × 5 = 25, and 20 is between them, so √20 is between 4 and 5.

  1. Find the squares just below and just above the number.
  2. The root is between the numbers that make those squares.
  3. It sits nearer the whole number whose square is nearer.

A square root undoes squaring: 4 × 4 = 16, so √16 = 4, and 5 × 5 = 25, so √25 = 5.

20 is between 16 and 25, so √20 is between √16 and √25: between 4 and 5.

20 is not a square, so √20 is irrational: its decimal never ends and never repeats.

√n is between the whole numbers whose squares are just below and just above n.

Halving the number: 20 ÷ 2 = 10.

A root times itself makes the number: 4 × 4 = 16 and 5 × 5 = 25, so √20 is between 4 and 5.

√86 is between ___ and 10 — 9. 9 × 9 = 81 and 10 × 10 = 100, so √86 is between 9 and 10.

Place √2 on a number line

Where does the square root of two live on a number line? It's irrational — its decimal never ends. Let's pin it down anyway.

First question: between which two whole numbers does √2 sit?

Square the integers around it. One squared is one, and two squared is four. Two falls between one and four, so √2 falls between one and two.

Great — √2 lives somewhere between 1 and 2 on the line.

Where does √2 live?√2 = 1.41421356…An irrational number — let's approximate it.Approximating √2Between which two integers?1² = 12² = 41 < 2 < 4so 1 < √2 < 20312√2Zoom in to the nearest tenth1.4 × 1.4 = 1.961.96 < 2 → too small1.5 × 1.5 = 2.252.25 > 2 → too big1.4 < √2 < 1.52 − 1.96 = 0.042.25 − 2 = 0.25much closer to 1.41.02.01.41.5√2√2 ≈1.4to the nearest tenth
Grade 8 · Squeezing √2 between integers, then between tenths

Now let's zoom in between 1 and 2 and find √2 to the nearest tenth.

Try one point four. One point four times one point four is one point nine six. Less than two — so √2 is bigger than 1.4.

Try one point five. One point five times one point five is two point two five. Bigger than two — so √2 is smaller than 1.5.

Squeezed it: √2 sits between 1.4 and 1.5.

Which tenth is it closer to? Two minus 1.96 is just 0.04. But 2.25 minus 2 is 0.25. Way closer to 1.4.

So we plot √2 just past 1.4 — that's where it lives.

To the nearest tenth, √2 is about 1.4. Keep zooming and you'd find 1.41, then 1.414 — closer and closer, but never exact.

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

More of the library

Every skill has a page

Practise this skill free

14 days of every skill, practice packets, Fluency sheets and up to four students. No card.

Start free