Placing real numbers on number lines | Year 9

01 / LEARN

Understand the idea.

Locate the value before drawing it

A number line uses a consistent scale: equal numerical differences occupy equal distances. To locate $\sqrt{2}$ , compare squares first. Since $1^{2} < 2 < 2^{2}$ , its point lies between $1$ and $2$ , regardless of its decimal approximation.

Use a decimal estimate to position the point more precisely, but retain its exact label $\sqrt{2}$ . To compare a root with a fraction, write the fraction as a decimal or compare squares of the nonnegative values.

The direction of negative values

Values increase to the right. Negating reflects a point across zero, so it reverses order: if $a < b$ , then $- a > - b$ . A larger distance from zero makes a negative number smaller, rather than larger.

Compare the positive magnitudes of negative roots before reflecting them. Squaring both sides preserves order for nonnegative numbers, but not for an arbitrary pair of negative numbers. State that condition when using squares to compare values.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Three representations

Order $\sqrt{2}$ , $1.6$ and $3 / 2$ .

Worked solution

1
Express the fraction as a decimal.
$\frac{3}{2} = 1.5$
This gives a direct comparison with 1.6.
2
Compare the root with 1.5 using squares.
$2 < {1.5}^{2} = 2.25$
Both numbers being compared, the root and 1.5, are nonnegative.
3
Combine the comparisons.
$\sqrt{2} < \frac{3}{2} < 1.6$
The square comparison places the root below 1.5, and 1.5 is below 1.6.

Answer

$\sqrt{2} < 3 / 2 < 1.6$ .

EXAMPLE 02

A negative root

Find consecutive integer bounds for $- \sqrt{11}$ .

Worked solution

1
Trap 11 between consecutive square numbers.
$3^{2} = 9 < 11 < 16 = 4^{2}$
The positive square root must lie between the corresponding nonnegative integers.
2
Take positive square roots.
$3 < \sqrt{11} < 4$
Taking square roots preserves order for nonnegative values.
3
Reflect the interval across zero.
$- 4 < - \sqrt{11} < - 3$
Negating reverses both inequalities; -4 is the smaller endpoint.

Answer

$- 4 < - \sqrt{11} < - 3$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Which is farther right: $- \sqrt{5}$ or $- 2$ ?

Show answer & reasoning

$- 2$ .

Since $5 > 2^{2}$ , $\sqrt{5} > 2$ . Negating gives $- \sqrt{5} < - 2$ , so $- 2$ is farther right.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Give consecutive integer bounds for $\sqrt{19}$ .
2 marks
Order $1.4$ , $\sqrt{2}$ and $\frac{3}{2}$ .
2 marks
Give consecutive integer bounds for $- \sqrt{6}$ .
2 marks
Which is larger: $\sqrt{10}$ or $3.2$ ? Explain without rounding the root.
2 marks
Order $- 1.6$ , $- \sqrt{2}$ , $- \frac{3}{2}$ and $0$ .
3 marks
On the supplied number line mark $\sqrt{3}$ and $- \sqrt{3}$ to about one decimal place.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Give consecutive integer bounds for $\sqrt{19}$ .
2 marks
$16 < 19 < 25$ .
$4 < \sqrt{19} < 5$ .
Order $1.4$ , $\sqrt{2}$ and $\frac{3}{2}$ .
2 marks
${1.4}^{2} = 1.96 < 2 < 2.25 = {1.5}^{2}$ .
$1.4 < \sqrt{2} < \frac{3}{2}$ .
Give consecutive integer bounds for $- \sqrt{6}$ .
2 marks
$2 < \sqrt{6} < 3$ ; reflect in zero.
$- 3 < - \sqrt{6} < - 2$ .
Which is larger: $\sqrt{10}$ or $3.2$ ? Explain without rounding the root.
2 marks
Both are nonnegative and ${3.2}^{2} = 10.24 > 10$ .
$3.2$ .
Order $- 1.6$ , $- \sqrt{2}$ , $- \frac{3}{2}$ and $0$ .
3 marks
$\frac{3}{2} = 1.5$ and $1.4 < \sqrt{2} < 1.5$ .
Reverse the positive order for the negative magnitudes.
$- 1.6 < - \frac{3}{2} < - \sqrt{2} < 0$ .
On the supplied number line mark $\sqrt{3}$ and $- \sqrt{3}$ to about one decimal place.
3 marks
$\sqrt{3} = 1.732 \dots$ .
Near $1.7$ and $- 1.7$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 9 real-number classification and applications; root bounds and sensible digital approximation. Full surd manipulation is developed in Year 10.

Official curriculum source ↗

In this lesson

Understand the idea.

Locate the value before drawing it

The direction of negative values

See the method in action.

Three representations

Express the fraction as a decimal.

Compare the root with 1.5 using squares.

Combine the comparisons.

A negative root

Trap 11 between consecutive square numbers.

Take positive square roots.

Reflect the interval across zero.

Try it yourself.

Check your understanding

Keep practising