Placing real numbers on number lines | Year 9

01 / LEARN

The idea, step by step

Locate the value before drawing it

Equal distances on a number line represent equal differences in value. Start with neighbouring integers that trap a root. Because one squared is one and two squared is four, the square root of two lies between one and two. Its position is closer to one.

Fractions and roots belong on the same line. Convert to a useful decimal when needed, or compare exact bounds. Keep the exact label beside the point even if a decimal estimate helps locate it.

The direction of negative values

Values increase from left to right. A larger positive magnitude is farther right; a larger negative magnitude is farther left. Since the square root of five is a little above two, its negative is a little left of negative two.

When two estimates look almost equal, keep more digits or compare their squares if both are nonnegative. Squaring negative values does not preserve their order. Compare their positive magnitudes first, then account for the negative signs.

A common trap

$- \sqrt{11}$ lies to the right of $- 3$ because $11 > 9$ .

Instead: A larger negative magnitude is farther left. Locate the positive root, then reflect its position in zero.

02 / WORKED EXAMPLES

See the thinking

Example 1

Three representations

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

$1^{2} < 2 < {1.5}^{2}$ , so $\sqrt{2} < 1.5$ .
$3 / 2 = 1.5 < 1.6$ .

\sqrt{2} < 3 / 2 < 1.6

Example 2

A negative root

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

$3^{2} < 11 < 4^{2}$ , so $3 < \sqrt{11} < 4$ .
Negate and reverse the order.

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

Check your understanding

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

Show answer

$- 2$ .

$\sqrt{5} > 2$ , so $- \sqrt{5} < - 2$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Give consecutive integer bounds for $\sqrt{19}$ .
2 marks
Order $1.4$ , $\sqrt{2}$ and $3 / 2$ .
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 $- 1.6$ , $- \sqrt{2}$ , $- 3 / 2$ 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 $3 / 2$ .
2 marks
${1.4}^{2} = 1.96 < 2 < 2.25 = {1.5}^{2}$ .
$1.4 < \sqrt{2} < 3 / 2$ .
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 $- 1.6$ , $- \sqrt{2}$ , $- 3 / 2$ and $0$ .
3 marks
$3 / 2 = 1.5$ and $1.4 < \sqrt{2} < 1.5$ .
Reverse the positive order for the negative magnitudes.
$- 1.6 < - 3 / 2 < - \sqrt{2} < 0$ .
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 ↗

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