Fractions, Decimals, and Percentages

The big idea

Fractions, decimals, and percentages are not three separate topics. They are different representations of the same numbers and the same relationships. A pupil who understands this has a fundamentally richer and more flexible number sense than one who treats each form as an isolated skill to learn.

Why this matters for secondary maths

Fractions, decimals, and percentages appear throughout secondary mathematics. Ratio, probability, proportion, algebraic fractions, trigonometric values, percentage change, compound interest, and the interpretation of data all depend on pupils being able to move fluently between these forms and to operate confidently within each one.

The conceptual difficulty is that a fraction is not one thing. It is simultaneously:

A part of a whole: $\frac{3}{4}$ means 3 parts out of 4 equal parts.

A point on a number line: $\frac{3}{4}$ sits three-quarters of the way from 0 to 1.

A division: 3 ÷ 4 = $\frac{3}{4}$ . This is one of the most important connections in the framework, established in Concept Section 8 (Division).

A ratio: $\frac{3}{4}$ expresses the relationship ‘3 for every 4’.

An operator: $\frac{3}{4}$ of 20 means ‘three-quarters of twenty’. A multiplicative scaling, connecting to Concept Section 6 (Additive and Multiplicative Reasoning).

Pupils who understand only the ‘part of a whole’ meaning will struggle when fractions appear in other roles. For example, interpreting $\frac{7}{4}$ as ‘seven parts out of four’ makes no intuitive sense in a part–whole model, but makes perfect sense as a point on the number line or as a division (7 ÷ 4 = 1.75).

This section depends on six earlier concept sections, more than any other section in the framework:

Place Value and Unitising (Concept Section 1) provides decimal place value.
The Laws of Arithmetic (Concept Section 2) govern fraction operations.
Zero and One as Identities (Concept Section 4) explains why equivalent fractions work: multiplying numerator and denominator by the same number is multiplying the fraction by 1 in the form $\frac{n}{n}$ .
Additive and Multiplicative Reasoning (Concept Section 6) provides the conceptual shift to proportional thinking that fractions demand.
The Area Model (Concept Section 7) extends to fraction multiplication.
Division (Concept Section 8) establishes the fraction-as-division connection.

This density of dependencies reflects the genuine complexity of fraction understanding and is the reason fractions are one of the most challenging areas to teach well.

Key representations

Fraction walls and bars

Rectangular bars divided into equal parts, stacked to show relationships between fractions with different denominators. A fraction wall makes visible that $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}$ , because the shaded regions align.

Strengths: Excellent for comparing fractions, finding equivalences, and seeing why common denominators are needed for addition. Makes the idea of ‘same-sized pieces’ concrete.

Limitation: Becomes cluttered with large or dissimilar denominators. Less useful for fraction multiplication or for fractions greater than 1.

Number lines

Fractions, decimals, and percentages all placed on the same number line. This is the single most important representation for connecting the three forms, because it makes the equivalence $\frac{1}{2}$ = 0.5 = 50% visible as the same position.

Strengths: Shows fractions as numbers with definite positions, not just ‘parts of shapes’. Supports ordering, comparison, and the density of fractions (there is always another fraction between any two fractions). Handles improper fractions and mixed numbers naturally.

Limitation: Does not, on its own, explain fraction operations. Needs to be complemented by area models for multiplication and bar models for addition.

Area models for fraction multiplication

A unit square partitioned both horizontally and vertically to show the product of two fractions. For example, $\frac{1}{2} \times \frac{1}{3}$ : divide the square into 2 rows and 3 columns, creating 6 equal parts. The overlap of the shaded $\frac{1}{2}$ and the shaded $\frac{1}{3}$ covers 1 of the 6 parts, giving $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ .

Strengths: Makes the ‘multiply numerators, multiply denominators’ rule visible and meaningful. Connects directly to the area model for whole-number multiplication (Concept Section 7). Explains why the product of two proper fractions is smaller than either factor.

Limitation: Requires careful drawing. Can become complex with larger numerators or denominators.

Bar models for fraction-of-an-amount

A single bar representing the whole, divided into equal parts. To find $\frac{3}{4}$ of 20, divide the bar into 4 equal parts (each worth 5), then shade 3 parts (giving 15). This connects the operator meaning of fractions to a proportional model and supports the transition to ratio problems.

Hundred squares and grids for percentages

A 10 × 10 grid where each cell represents 1%. Shading 25 cells shows 25% = $\frac{1}{4}$ = 0.25. This representation anchors percentages in a concrete model and supports conversion between forms.

Worked examples

Example 1: Equivalent fractions through multiplying by 1

Show that $\frac{3}{5}$ is equivalent to $\frac{6}{10}$ .

$\frac{3}{5} \times \frac{2}{2} = \frac{6}{10}$ . Since $\frac{2}{2} = 1$ , and multiplying by 1 does not change a number’s value (as established in Concept Section 4, Zero and One as Identities), the two fractions are equal. This is not a ‘trick’ or a procedure; it is a consequence of the multiplicative identity.

Example 2: Addition of fractions using common denominators

Calculate $\frac{2}{3} + \frac{1}{4}$ .

We need pieces of the same size. Rewrite both fractions with a common denominator:

$\frac{2}{3} = \frac{8}{12}$ (multiplying by $\frac{4}{4} = 1$ ) and $\frac{1}{4} = \frac{3}{12}$ (multiplying by $\frac{3}{3} = 1$ ).

So $\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$ .

The key structural idea: we can only add fractions when the units are the same size. This is the same principle as unitising (Concept Section 1): 3 tens + 4 tens = 7 tens works because the units match, just as 8 twelfths + 3 twelfths = 11 twelfths.

Example 3: Fraction multiplication using the area model

Calculate $\frac{2}{3} \times \frac{3}{4}$ .

Draw a unit square. Divide one side into 3 equal parts and shade 2 of them (representing $\frac{2}{3}$ ). Divide the other side into 4 equal parts and shade 3 of them (representing $\frac{3}{4}$ ). The square is now divided into 12 equal parts. The doubly shaded region covers 6 of the 12 parts.

So $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$ .

Notice: 2 × 3 = 6 (numerators multiplied) and 3 × 4 = 12 (denominators multiplied). The area model makes visible why the procedure works. This extends the same area model reasoning pupils used for whole-number multiplication in Concept Section 7.

Example 4: Connecting fractions, decimals, and percentages

Express $\frac{3}{8}$ as a decimal and as a percentage.

$\frac{3}{8}$ means 3 ÷ 8 (the fraction-as-division connection from Concept Section 8). Calculating: 3 ÷ 8 = 0.375.

To convert to a percentage: 0.375 × 100 = 37.5%.

So $\frac{3}{8}$ = 0.375 = 37.5%. These are three names for the same number, the same position on the number line.

Example 5: Finding a fraction of an amount

Find $\frac{3}{5}$ of 40.

Divide 40 into 5 equal parts: 40 ÷ 5 = 8. Each fifth is worth 8. Three-fifths: 3 × 8 = 24.

This combines division (Concept Section 8) and multiplication. The bar model makes this visible: a bar of length 40 divided into 5 parts, with 3 parts shaded.

Notice that ‘finding $\frac{3}{5}$ of 40’ means $\frac{3}{5}$ × 40 = 24. The word ‘of’ signals multiplication. This is the operator meaning of a fraction, rooted in multiplicative reasoning (Concept Section 6).

The bridge to algebra

The arithmetic of fractions is the direct foundation for algebraic fractions. Every operation pupils learn with numerical fractions reappears in identical structural form with algebraic fractions.

Equivalent fractions → Equivalent algebraic fractions:

Arithmetic: $\frac{3}{5} = \frac{6}{10}$ (multiplying by $\frac{2}{2} = 1$ ).

Algebra: $\frac{x}{3} = \frac{2 x}{6}$ (multiplying by $\frac{2}{2} = 1$ ). The same multiplicative identity principle is at work.

Adding fractions with common denominators → Adding algebraic fractions:

Arithmetic: $\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$ (same denominator, so combine numerators).

Algebra: $\frac{2}{x} + \frac{3}{x} = \frac{5}{x}$ (same denominator, so combine numerators). The unitising principle is identical: we are combining like units.

Adding fractions with different denominators → Adding algebraic fractions:

Arithmetic: $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$ .

Algebra: $\frac{1}{x} + \frac{1}{y} = \frac{y}{x y} + \frac{x}{x y} = \frac{x + y}{x y}$ . The structural reasoning is the same: find a common denominator, convert, combine.

Simplifying fractions → Simplifying algebraic fractions:

Arithmetic: $\frac{12}{18} = \frac{2}{3}$ (dividing numerator and denominator by their common factor, 6).

Algebra: $\frac{6 x}{9 x} = \frac{2}{3}$ (dividing by the common factor $3 x$ ). Simplifying algebraic fractions uses exactly the same reasoning as simplifying numerical fractions, which is why the factor structure of numbers (Concept Section 13) matters.

Fractions as expressions of proportion:

The statement ‘ $\frac{3}{4}$ of the class chose option A’ is a proportional relationship. In algebra, proportional relationships are expressed as $y = k x$ , where $k$ is the constant of proportionality. Fractions are the language of proportionality, and fluency with fractions is fluency with the algebraic expression of proportion.

The key message for teachers: Algebraic fractions are not a new topic introduced in Key Stage 4. They are the same operations pupils have been practising since Year 7, with letters in place of specific numbers. If numerical fraction work is structural and well understood, the algebraic extension is a natural continuation.

Key vocabulary

Term	Definition
Fraction	A number expressed as $\frac{a}{b}$ , where $a$ is the numerator and $b$ is the denominator. Represents a division, a point on the number line, a part of a whole, a ratio, or a scaling operation.
Numerator	The top number in a fraction. Tells how many parts (or how many of the unit) we have.
Denominator	The bottom number in a fraction. Tells the size of the parts (or the unit we are counting in).
Equivalent fractions	Fractions that represent the same value: $\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$ . Generated by multiplying or dividing numerator and denominator by the same non-zero number.
Proper fraction	A fraction where the numerator is less than the denominator (value between 0 and 1).
Improper fraction	A fraction where the numerator is greater than or equal to the denominator (value of 1 or greater).
Mixed number	A whole number and a proper fraction combined: $1 \frac{3}{4}$ .
Common denominator	A shared denominator used when adding or subtracting fractions with different denominators.
Simplify / cancel	To express a fraction in its simplest form by dividing numerator and denominator by their highest common factor.
Percentage	A fraction with a denominator of 100. Literally ‘per hundred’: 45% = $\frac{45}{100}$ = 0.45.
Decimal	A number expressed using the base-ten place value system extended to the right of the ones column: tenths, hundredths, thousandths.
Reciprocal	The multiplicative inverse of a number: the reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ . A number multiplied by its reciprocal gives 1.

What we don’t say

Avoid	Why	Say instead
‘Multiply the top and bottom by the same number’ (without explanation)	This describes the procedure but not the reason. Pupils who do not know why this works cannot judge when it is appropriate. The reason is the multiplicative identity: multiplying by $\frac{n}{n} = 1$ does not change the value.	‘Multiply by 1 in the form $\frac{n}{n}$ . Since $\frac{n}{n} = 1$ the value does not change.’ This gives pupils the structural reason, rooted in Concept Section 4 (Zero and One as Identities).
‘To divide fractions, flip and multiply’ (without explanation)	This is a procedural shortcut that hides the mathematical reasoning. Pupils who flip and multiply without understanding why will not be able to adapt the reasoning or check their answers. Division by a fraction means ‘how many of this fraction fit into the other number?’	‘Dividing by $\frac{a}{b}$ is the same as multiplying by its reciprocal, $\frac{b}{a}$ . We can understand this through the area model or by asking ‘how many groups of $\frac{a}{b}$ fit into the number?’’
‘You can’t compare fractions with different denominators’	You can compare them on a number line, by converting to decimals, or by cross-multiplying. The issue is not that comparison is impossible but that addition requires common denominators. Conflating these creates confusion.	‘To add or subtract fractions, we need the same denominator so the pieces are the same size. To compare fractions, we can also use the number line or convert to decimals.’
‘Percent means out of a hundred’ (as the full explanation)	While etymologically correct, this definition does not explain percentages greater than 100% or the use of percentages as operators (e.g. a 15% increase). It reduces percentages to a static part–whole model.	‘Percent means per hundred. 45% means 45 for every 100. This works as a proportion, not just as a count. A percentage can be more than 100% when the quantity exceeds the original whole.’
‘A fraction is a part of a shape’ (as the sole meaning)	Fractions are also numbers on a number line, divisions, ratios, and operators. If pupils only see fractions as shaded regions in circles or rectangles, they will struggle with improper fractions, fraction operations, and algebraic fractions.	‘A fraction is a number. It can represent a part of a whole, a point on a number line, a division, a ratio, or a scaling.’

Common misconceptions and how to surface them

Misconception 1: Larger denominator means larger fraction.

Pupils may think $\frac{1}{8}$ is bigger than $\frac{1}{3}$ because 8 is bigger than 3. This reveals a failure to understand that the denominator tells you the size of the parts. More parts means smaller parts. Surface this by asking: ‘Would you rather have $\frac{1}{3}$ of a pizza or $\frac{1}{8}$ of the same pizza? Which piece is bigger?’ Then confirm on a number line.

Misconception 2: Adding fractions by adding numerators and denominators separately.

Pupils write $\frac{1}{3} + \frac{1}{4} = \frac{2}{7}$ . This is a natural but incorrect generalisation from whole-number addition. The error reveals that the pupil does not understand that fractions can only be added when the units are the same size. Use a fraction wall: shade $\frac{1}{3}$ and $\frac{1}{4}$ and show that the total is not $\frac{2}{7}$ (which is less than $\frac{1}{3}$ ). Then show the common-denominator method and connect it to the unitising principle (Concept Section 1).

Misconception 3: Treating the fraction bar as a separator rather than a division.

Pupils who see $\frac{3}{4}$ as ‘a 3 and a 4 with a line between them’ rather than as ‘3 divided by 4’ will struggle to convert fractions to decimals and will not understand improper fractions. Ask: ‘What does $\frac{3}{4}$ mean as a calculation?’ A pupil who cannot say ‘3 divided by 4’ has not made the fraction-as-division connection.

Misconception 4: Believing that multiplication always makes numbers bigger.

When multiplying by a fraction less than 1, the result is smaller than the starting number: $\frac{1}{2}$ × 8 = 4. Pupils who expect multiplication to enlarge are applying whole-number intuition. The area model helps: a $\frac{1}{2}$ -by-8 rectangle has area 4, which is smaller than 8. This connects to the broader shift from additive to multiplicative reasoning (Concept Section 6).

Misconception 5: Treating fractions, decimals, and percentages as separate topics.

Pupils who can calculate with decimals but freeze when they see fractions, or who can find 50% but not $\frac{1}{2}$ of an amount, have compartmentalised their knowledge. Ask: ‘Give me three different ways to write the same number.’ A pupil who can move between $\frac{3}{4}$ = 0.75 = 75% understands the connected system; one who cannot is treating each form in isolation.

Component 9: Diagnostic questions

Question 1: Put these numbers in order from smallest to largest: 0.35, $\frac{3}{8}$ , 40%, $\frac{1}{3}$ .

What this reveals: To compare these, the pupil must convert between forms. $\frac{1}{3}$ ≈ 0.333, 0.35, $\frac{3}{8}$ = 0.375, 40% = 0.4. Correct order: $\frac{1}{3}$ , 0.35, $\frac{3}{8}$ , 40%. A pupil who can do this fluently understands the connected system. A pupil who cannot convert between forms has not yet internalised the equivalences.

Question 2: True or false: $\frac{1}{3} + \frac{1}{3} = \frac{2}{6}$ . Explain your reasoning.

What this reveals: False, $\frac{1}{3} + \frac{1}{3} = \frac{2}{3}$ , not $\frac{2}{6}$ . A pupil who answers ‘true’ is adding both numerators and denominators, revealing the fundamental misconception about fraction addition. A pupil who answers ‘false’ and explains that the denominator stays the same because we are adding thirds (same-sized pieces) demonstrates structural understanding. Note also that $\frac{2}{6} = \frac{1}{3}$ , which is less than each addend, a sense-checking argument (Concept Section 17) that answer must be wrong.

Progression spine

Stage	Key ideas	Notes
Primary (Y1–Y6)	Fractions of shapes. Fractions on number lines. Equivalences (halves, quarters, thirds). Simple addition and subtraction with same denominators. Fractions of amounts. Introduction to decimals and percentages.	Pupils typically arrive in Year 7 with the part–whole model dominant. The number line and fraction-as-division meanings may be underdeveloped.
Year 7	All five meanings of a fraction developed explicitly. Equivalent fractions through multiplying by 1. Addition, subtraction, multiplication, and division of fractions. Fraction–decimal–percentage conversions. Area model for fraction multiplication. Fractions on the number line.	The emphasis is on connecting the forms and developing all meanings, not just procedural fluency with one form.
Years 8–11	Algebraic fractions (simplifying, adding, multiplying). Percentage change and compound interest. Probability as fractions and decimals. Ratio and proportion expressed as fractions. Recurring decimals as fractions. Trigonometric values as fractions.	Algebraic fractions in KS4 use exactly the same operations as numerical fractions. Pupils whose numerical fraction work is structural will find the algebraic extension natural.