Ordering rational numbers on number lines | Year 7

01 / LEARN

Understand the idea.

Compare the same-sized parts

To compare fractions, give their parts the same size. With equal positive denominators, the greater numerator counts more of those equal parts. Otherwise use equivalent fractions with a common denominator or convert to exact decimals.

A larger numerator alone does not guarantee a larger fraction. Quarters are larger than eighths, so three quarters must be renamed before comparing it with five eighths. Both fractions must refer to the same-sized whole.

Positions on a number line

Number-line values increase to the right. Check the labels and count equal intervals, not tick marks, to find each interval's size. A fraction or decimal represents one value and therefore one position on the line.

Trailing zeros help align decimal places without changing the number: $0.4 = 0.40$ . Compare corresponding tenths, then hundredths and thousandths as needed. This explains why a longer decimal can still lie to the left of a shorter one.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Find common parts

Compare $\frac{3}{4}$ and $\frac{5}{8}$ .

Worked solution

1
Rename quarters as eighths.
$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$
Eighths provide a common-sized part for both fractions.
2
Compare the counts of eighths.
$\frac{6}{8} > \frac{5}{8}$
Six parts are more than five when each part has the same size.
3
State the comparison in the original forms.
$\frac{3}{4} > \frac{5}{8}$
Equivalent fractions occupy the same positions on the number line.

Answer

$\frac{3}{4} > \frac{5}{8}$

EXAMPLE 02

Mix different forms

Order $0.6$ , $\frac{5}{8}$ and $0.58$ from smallest to largest.

Worked solution

1
Convert the fraction exactly.
$\frac{5}{8} = \frac{625}{1000} = 0.625$
Multiplying numerator and denominator by 125 gives thousandths.
2
Align all three values to thousandths.
$0.6 = 0.600, 0.58 = 0.580$
Trailing zeros preserve the values while making place-value comparison clear.
3
Order and restore the original forms.
$0.580 < 0.600 < 0.625, 0.58 < 0.6 < \frac{5}{8}$
The increasing counts of thousandths determine the increasing values.

Answer

$0.58 < 0.6 < \frac{5}{8}$

03 / YOUR TURN

Try it yourself.

Check your understanding

Which is further right: $\frac{2}{5}$ or $0.45$ ?

Show answer & reasoning

$0.45$

Convert $\frac{2}{5}$ to $0.40$ . Since $0.40 < 0.45$ , the value $0.45$ is further right on the number line.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Compare $\frac{5}{6}$ and $\frac{3}{4}$ using equal parts.
2 marks
Order 0.7, 0.67 and 0.705 from smallest to largest.
2 marks
On a number line from 0 to 1, give the left-to-right order of $\frac{1}{4}$ , $\frac{3}{4}$ , $\frac{1}{2}$ .
2 marks
Find a fraction strictly between $\frac{1}{3}$ and $\frac{1}{2}$ .
2 marks
Which is closer to 1: $\frac{7}{8}$ or 0.8?
3 marks
Order race times $12.4$ s, $12 \frac{3}{8}$ s and $12.45$ s from fastest to slowest.
3 marks

Read the worked solutions

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

Compare $\frac{5}{6}$ and $\frac{3}{4}$ using equal parts.
2 marks
$\frac{5}{6} = \frac{10}{12}$ and $\frac{3}{4} = \frac{9}{12}$ .
$\frac{5}{6} > \frac{3}{4}$
Order 0.7, 0.67 and 0.705 from smallest to largest.
2 marks
Compare 0.670, 0.700 and 0.705.
0.67, 0.7, 0.705
On a number line from 0 to 1, give the left-to-right order of $\frac{1}{4}$ , $\frac{3}{4}$ , $\frac{1}{2}$ .
2 marks
Divide the unit into four equal parts; these occupy the first, second and third marks.
$\frac{1}{4}, \frac{1}{2}, \frac{3}{4}$
Find a fraction strictly between $\frac{1}{3}$ and $\frac{1}{2}$ .
2 marks
$\frac{1}{3} = \frac{4}{12}$ and $\frac{1}{2} = \frac{6}{12}$ .
For example, $\frac{5}{12}$ .
Which is closer to 1: $\frac{7}{8}$ or 0.8?
3 marks
The distances are $\frac{1}{8} = 0.125$ and $0.2$ .
0.125 is smaller.
$\frac{7}{8}$
Order race times $12.4$ s, $12 \frac{3}{8}$ s and $12.45$ s from fastest to slowest.
3 marks
$\frac{3}{8} = 0.375$ .
A shorter time is faster; compare 12.375, 12.400 and 12.450.
$12 \frac{3}{8}$ s, $12.4$ s, $12.45$ s

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Equivalent fractions and mixed numbers; ordering rational numbers on number lines; adding/subtracting fractions; multiplying/dividing fractions; four operations with decimals; converting between fractions and decimals; multi-step problems and checking estimates.

Official curriculum source ↗

In this lesson

Understand the idea.

Compare the same-sized parts

Positions on a number line

See the method in action.

Find common parts

Rename quarters as eighths.

Compare the counts of eighths.

State the comparison in the original forms.

Mix different forms

Convert the fraction exactly.

Align all three values to thousandths.

Order and restore the original forms.

Try it yourself.

Check your understanding

Keep practising