Comparing/ordering | Year 6

01 / LEARN

Understand the idea.

Compare the same-sized parts

Fractions with the same denominator count equal-sized parts, so compare their numerators. With different denominators, rename both fractions using a common denominator. This makes the piece sizes match while keeping the original amounts unchanged.

Choose a denominator both original denominators divide into. For quarters and sixths, twelfths work because $12$ is divisible by both $4$ and $6$ . Remember to scale each numerator by its own matching factor.

Rename to compare the same whole
Fraction	Equivalent twelfths
3/4	9/12
5/6	10/12

Use benchmarks to check

A benchmark can make a comparison quicker. To compare with $\frac{1}{2}$ , ask whether the numerator is smaller or larger than half the denominator. This can separate fractions that lie on opposite sides of one half.

After ordering renamed fractions, report the original fractions in that order. For real quantities, also check that the wholes are the same size: a greater fraction of a small bottle need not contain more water than a smaller fraction of a large bottle.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Which is greater: $\frac{3}{4}$ or $\frac{5}{6}$ ?

Worked solution

1
Rename the quarters as twelfths.
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
Splitting each quarter into three makes twelfths.
2
Rename the sixths as twelfths.
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
Splitting each sixth into two gives the same-sized parts.
3
Compare the equal-sized parts.
$\frac{9}{12} < \frac{10}{12} \Rightarrow \frac{3}{4} < \frac{5}{6}$
Ten twelfths exceeds nine twelfths, so five sixths is greater.

Answer

$\frac{5}{6}$ .

EXAMPLE 02

Worked example 2

Order $\frac{2}{3}$ , $\frac{1}{4}$ and $\frac{3}{8}$ from least to greatest.

Worked solution

1
Rename all three fractions in twenty-fourths.
$\begin{aligned} \frac{2}{3} & = \frac{16}{24} \\ \frac{1}{4} & = \frac{6}{24} \\ \frac{3}{8} & = \frac{9}{24} \end{aligned}$
Twenty-four is divisible by 3, 4 and 8.
2
Order the new numerators.
$6 < 9 < 16$
All the pieces are twenty-fourths, so their counts determine the order.
3
Return to the original fraction names.
$\frac{1}{4} < \frac{3}{8} < \frac{2}{3}$
These fractions correspond to six, nine and sixteen twenty-fourths respectively.

Answer

$\frac{1}{4}, \frac{3}{8}, \frac{2}{3}$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Which is greater: $\frac{7}{12}$ or $\frac{1}{2}$ ?

Show answer & reasoning

$\frac{7}{12}$ .

$\frac{1}{2} = \frac{6}{12}$ . Since seven twelfths exceeds six twelfths, $\frac{7}{12}$ is greater.

Keep practising

6 worksheet questions · 10 marks

Open worksheet Worked solutions

Read the questions online

Compare $\frac{3}{8}$ and $\frac{5}{8}$ using $<$ or $>$ .
1 mark
Which is greater: $\frac{2}{3}$ or $\frac{3}{5}$ ?
2 marks
Order $\frac{1}{2}$ , $\frac{3}{4}$ and $\frac{2}{3}$ from least to greatest.
2 marks
Is $\frac{4}{9}$ below or above one half?
1 mark
Give a fraction strictly between $\frac{1}{4}$ and $\frac{3}{4}$ .
2 marks
Identical bottles are filled to $\frac{5}{6}$ and $\frac{7}{8}$ . Which contains more water?
2 marks

Read the worked solutions

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

Compare $\frac{3}{8}$ and $\frac{5}{8}$ using $<$ or $>$ .
1 mark
The pieces have the same size, so compare the numerators.
$\frac{3}{8} < \frac{5}{8}$ .
Which is greater: $\frac{2}{3}$ or $\frac{3}{5}$ ?
2 marks
$\frac{2}{3} = \frac{10}{15}$ and $\frac{3}{5} = \frac{9}{15}$ .
$\frac{2}{3}$ .
Order $\frac{1}{2}$ , $\frac{3}{4}$ and $\frac{2}{3}$ from least to greatest.
2 marks
Use twelfths: $\frac{6}{12}$ , $\frac{9}{12}$ and $\frac{8}{12}$ .
$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}$ .
Is $\frac{4}{9}$ below or above one half?
1 mark
Half of 9 is 4.5; 4 is less than 4.5.
Below.
Give a fraction strictly between $\frac{1}{4}$ and $\frac{3}{4}$ .
2 marks
$\frac{1}{2} = \frac{2}{4}$ lies between them. Other correct answers are possible.
For example, $\frac{1}{2}$ .
Identical bottles are filled to $\frac{5}{6}$ and $\frac{7}{8}$ . Which contains more water?
2 marks
$\frac{5}{6} = \frac{20}{24}$ and $\frac{7}{8} = \frac{21}{24}$ .
The bottle filled to $\frac{7}{8}$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Equivalence; comparing/ordering; addition/subtraction; practical quantities.

Official curriculum source ↗

In this lesson

Understand the idea.

Compare the same-sized parts

Use benchmarks to check

See the method in action.

Worked example 1

Rename the quarters as twelfths.

Rename the sixths as twelfths.

Compare the equal-sized parts.

Worked example 2

Rename all three fractions in twenty-fourths.

Order the new numerators.

Return to the original fraction names.

Try it yourself.

Check your understanding

Keep practising