Number - Fractions (including decimals and percentages)

Develop simplification as a natural extension of equivalent fraction knowledge from Year 5: rather than multiplying up to find equivalents, pupils now divide down to find simpler equivalents. Explicitly connect to the HCF work in the calculation domain. Use fraction walls and number lines to verify that simplified fractions are equivalent to the original. For comparing fractions, progress from using benchmark fractions (less than ½, equal to ½, greater than ½) to converting to a common denominator. Include improper fractions and mixed numbers in comparison activities.

When simplifying, pupils often divide by 2 (or another common factor) rather than the highest common factor, leaving fractions in a partially simplified state (e.g., simplifying 12/18 to 6/9 rather than 2/3). Some pupils believe a fraction with a larger numerator and denominator is always greater than one with smaller numbers. When comparing fractions with different denominators, pupils sometimes compare numerators or denominators in isolation rather than converting to a common denominator.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.

Use fraction diagrams and number lines to demonstrate why fractions must have the same denominator before they can be added or subtracted. Progress carefully through three stages: same denominators (already secured), one denominator is a multiple of the other (simpler case), and denominators that share only the factor 1 (requiring multiplication of both). For mixed numbers, teach both the 'convert to improper fraction' method and the 'deal with whole and fractional parts separately' method, allowing pupils to choose. Subtraction of mixed numbers where regrouping is required (e.g., 3 1/4 - 1 3/4) needs careful attention.

The classic and persistent misconception is adding numerators and denominators separately (e.g., 1/3 + 1/4 = 2/7). Use diagrams extensively to show why this is incorrect. When subtracting mixed numbers, pupils often subtract the smaller fraction from the larger regardless of which is the subtrahend (e.g., computing 3 1/4 - 1 3/4 as 3 + (3/4 - 1/4) = 3 2/4). Failing to simplify the final answer is also common; build simplification into the routine.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.

For multiplication, begin with the concrete context of 'a fraction of a fraction' (e.g., ½ of ¾ of a pizza). Diagrams showing a rectangle divided first into quarters horizontally and then into halves vertically make the multiplication of fractions visually clear. Establish the rule (multiply numerators, multiply denominators) and connect it to the diagram. Cross-cancellation (simplifying diagonally before multiplying) can be introduced to keep numbers manageable. For division by a whole number, use sharing contexts: if ¾ of a bar of chocolate is shared equally between 3 people, each person gets ¼. Generalise to the rule of multiplying the denominator.

Pupils apply addition/subtraction strategies to multiplication (finding a common denominator before multiplying), which is unnecessary and leads to errors. When dividing a fraction by a whole number, pupils sometimes divide the numerator rather than multiply the denominator (e.g., computing 3/4 ÷ 3 as 1/4, which coincidentally gives the right answer, but using incorrect reasoning that fails for other examples like 3/4 ÷ 2). Build conceptual understanding alongside procedural competence.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.

Build on the fraction-decimal-percentage equivalences established in Years 4 and 5 by extending to eighths, thirds and other fractions that produce recurring or multi-place decimals. Organised practice — completing equivalence tables, using sorting activities, and playing matching games — builds fluency. Percentage bar models support problem-solving: drawing a bar divided into 100 equal parts and shading the relevant percentage helps pupils visualise the relationship. Connect explicitly to the ratio domain by using percentages as a common scale for comparison.

Pupils sometimes treat fractions, decimals and percentages as unrelated rather than as equivalent representations, failing to convert between them when it would be helpful. Common specific errors: believing 25% = ¼ but not knowing that 75% = ¾; confusing 0.5% with 50%; writing 0.8 as 0.8% rather than 80%. Regular work with real-world contexts (sale discounts, test scores, nutritional information) builds meaningful understanding alongside the procedural fluency.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.