Fractions: Equivalence, Comparison and Ordering

Equivalent fractions name the same amount, even though their numerators and denominators look different. To compare or order fractions, first ask whether the fractions can be expressed in equal-sized parts. A strip, bar or number line can make the relationship visible; multiplying or dividing the numerator and denominator by the same non-zero whole number gives the matching numerical form.

For Year 5, this is the heart of the topic: recognise equivalent forms, compare their sizes and place them in order. The national curriculum in England says pupils should compare and order fractions whose denominators are multiples of the same number, and identify equivalent fractions represented visually, including tenths and hundredths.[1]

Contents

What makes two fractions equivalent?

A visual route to equivalent fractions

How to generate an equivalent fraction

Three reliable ways to compare fractions

Worked example: order three fractions

How to use a number line

Mistakes that reveal the next teaching step

A short parent-led practice routine

Questions children should be able to explain

Next step

What makes two fractions equivalent?

In a fraction such as 3/5, the denominator tells us that a whole has been split into five equal parts. The numerator tells us that three of those parts are being considered.

Now imagine splitting each fifth into four equal smaller pieces. The same whole contains 20 of these smaller pieces, and the original three fifths cover 12 of them. Nothing has been added or removed, so:

3/5 = 12/20

The written numbers have changed because the size of each named part has changed. The point on the number line has not changed. That is what the equals sign means here: both fractions have the same value.

This gives a useful test for any model. If two fraction strips are meant to show equivalent fractions, they must represent the same-sized whole. Comparing three shaded pieces of a small bar with six shaded pieces of a much longer bar can be misleading because the wholes differ.

A visual route to equivalent fractions

The Education Endowment Foundation recommends purposeful use of manipulatives and representations. It says the representation should reveal mathematical structure, that pupils should connect the model to the mathematical idea, and that the support should not become something they depend on indefinitely.[2]

For equivalence, try this sequence:

Draw two bars of exactly the same length.

Split the first into three equal parts and shade two: 2/3.

Split each third in the second bar into two equal parts. There are now six equal parts.

Shade the same length. Four of the sixths are shaded: 4/6.

Write the relationship: 2/3 = 4/6.

Then ask, “What happened to every third?” Each third was split into two, so both the numerator and denominator were multiplied by 2. The diagram explains why that numerical rule preserves the value.

A representation is most useful when it leads to reasoning. The EEF notes that comparison and discussion of different representations can support conceptual understanding, while too many representations at once may cause confusion.[2] One clear bar model followed by the matching symbols is often more helpful than switching rapidly among several unrelated models.

How to generate an equivalent fraction

To generate an equivalent form, multiply or divide the numerator and denominator by the same non-zero whole number.

Example 1: enlarge both numbers

Find the missing numerator:

3/5 = ?/20

The denominator has been multiplied by 4 because 5 × 4 = 20. Apply the same change to the numerator:

3 × 4 = 12

Therefore:

3/5 = 12/20

A quick check is to simplify 12/20 by dividing both numbers by 4, which returns 3/5.

Example 2: simplify both numbers

Simplify 14/21.

Both 14 and 21 are divisible by 7:

14 ÷ 7 = 2 21 ÷ 7 = 3

Therefore:

14/21 = 2/3

Dividing only the numerator or only the denominator changes the fraction's value. The same operation must be applied to both.

Three reliable ways to compare fractions

A comparison should be justified, not guessed from which fraction contains the larger-looking numbers. Choose the simplest relationship available.

1. Same denominator: compare the numerators

Compare 5/8 and 3/8.

Both wholes are divided into eighths, so the pieces are the same size. Five eighths contain more of those pieces than three eighths:

5/8 > 3/8

2. Same numerator: compare the size of each part

Compare 3/5 and 3/8.

Each fraction contains three equal parts, but fifths are larger than eighths when the wholes are the same size. Therefore:

3/5 > 3/8

This can feel counter-intuitive because 8 is greater than 5. A fraction's denominator does not count how many parts have been taken; it tells us how finely the whole has been divided.

3. Related denominators: write equivalent forms

Compare 7/10 and 2/5.

Tenths and fifths are related because each fifth can be split into two tenths:

2/5 = 4/10

Now compare equal-sized parts:

7/10 > 4/10

So:

7/10 > 2/5

This method is especially useful when one denominator is a multiple of the other. It keeps the comparison tied to equivalence rather than turning it into a remembered trick.

Worked example: order three fractions

Put these fractions in ascending order:

1/3, 5/6, 2/9

The denominators 3, 6 and 9 are all factors of 18, so eighteenths give equal-sized parts for all three fractions.

Convert each fraction:

1/3 = 6/18 because both numbers are multiplied by 6.

5/6 = 15/18 because both numbers are multiplied by 3.

2/9 = 4/18 because both numbers are multiplied by 2.

Now order the numerators:

4/18 < 6/18 < 15/18

Therefore:

2/9 < 1/3 < 5/6

Check the result by estimating position: 2/9 is a little more than one fifth, 1/3 is one of three equal parts, and 5/6 is close to one whole. The estimate agrees with the equivalent-fraction calculation.

How to use a number line

A number line treats fractions as numbers rather than only as shaded pieces of shapes. The national curriculum's earlier fractions guidance describes pupils using fractions as numbers on the number line and deducing relationships such as size and equivalence.[1] The EEF also identifies number lines as a particularly effective representation across Key Stages 2 and 3.[2]

To compare 1/2 and 3/6:

Draw a line from 0 to 1.

Mark the halfway point and label it 1/2.

Divide the same interval into six equal steps.

Count three sixths from 0.

The label 3/6 lands on exactly the same point as 1/2, so 1/2 = 3/6.

For ordering, plot every fraction on the same line and read from left to right. The shared line matters: separate number lines with different lengths or scales can hide the comparison.

Mistakes that reveal the next teaching step

“The larger denominator makes the larger fraction”

A child says 3/8 > 3/5 because 8 is greater than 5. Return to two same-length bars. Split one into fifths and the other into eighths, then compare three parts from each. This makes the size of each part visible.

Multiplying only one number

A child writes 2/3 = 4/3. Ask what happened to the bar: splitting every third in two creates twice as many parts in the whole and twice as many shaded parts. Both numbers must change: 2/3 = 4/6.

Using unequal wholes

A drawing appears to show 1/2 < 2/4, but the second bar is longer. Redraw both fractions against the same whole. Equivalence is about the same value, so the reference whole must be consistent.

Finding a common denominator but not an equivalent numerator

A child changes 1/3 to 1/18. Ask, “How many eighteenths cover the same length as one third?” Since one third contains six eighteenths, the equivalent form is 6/18.

Reversing the order symbols

Read the complete statement aloud. For 2/9 < 1/3, say, “Two ninths is less than one third.” If the spoken sentence is false, revisit the symbol rather than recalculating everything.

A short parent-led practice routine

Use a small set of questions and keep the reason visible.

Show one equivalence. Draw equal bars for 3/4 and 6/8. Ask what stayed the same and what changed.

Complete a missing value. Try 4/7 = ?/21. Since 7 × 3 = 21, the answer is 12/21.

Compare a related pair. Try 3/4 and 5/8. Write 3/4 = 6/8, so 3/4 > 5/8.

Order a short set. Try 1/4, 7/8, 3/4. In eighths these are 2/8, 7/8, 6/8, so the order is 1/4 < 3/4 < 7/8.

Ask for an explanation. “Why did the value stay the same?” is more revealing than “What rule did you use?”

If the calculation is correct but the explanation is unclear, return to one diagram and connect every line of working to it. If the diagram is secure, gradually remove it and ask the child to picture or sketch only what they need. This follows the EEF's principle that representations should help pupils reach independent mathematical reasoning rather than become a permanent prop.[2]

The fraction reasoning in this guide can be practised with pencil and paper; the value comes from connecting each diagram to the matching numerical relationship.

Questions children should be able to explain

A secure answer includes the reason, not just the symbol.

Why are 3/5 and 12/20 equal?

Which is greater, 4/7 or 4/9, and why?

How can a bar model show that 2/3 = 6/9?

Why must the wholes be the same size when comparing two fraction diagrams?

How would you order 1/3, 5/6 and 2/9 without using decimals?

If two fractions occupy the same point on a number line, what does that tell you?

These prompts stay within equivalence, comparison and ordering. Fraction arithmetic and decimal conversion are different teaching steps.

Next step

Choose one pair of equivalent fractions and one three-fraction ordering question. Ask the child to solve each with a diagram, then with numbers, and explain how the two methods match.

Try a Maths lesson

Sources

[1] https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study — Department for Education, National curriculum in England: mathematics programmes of study.

[2] https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/maths-ks-2-3 — Education Endowment Foundation, Improving Mathematics in Key Stages 2 and 3.