Number - Fractions (including Decimals and Percentages)
KS2MA-Y5-D004
Comparing and ordering fractions; adding and subtracting fractions with different denominators; multiplying proper fractions; mixed numbers and improper fractions; decimals to three places; percentages as fractions and decimals.
National Curriculum context
In Year 5, fractions become significantly more complex: pupils add and subtract fractions with different denominators using equivalent fractions (requiring LCM knowledge), multiply proper fractions and mixed numbers by whole numbers, and convert between mixed numbers and improper fractions. The non-statutory guidance explains that pupils connect fractions to division (a/b = a ÷ b) and use this connection to find equivalences. Percentages are introduced as a special form of fraction (per hundred) with decimal and fraction equivalences. Decimal place value is extended to three decimal places (thousandths). This domain is one of the most demanding in Year 5 and requires confident fluency with all prerequisite fraction and multiplication knowledge from Years 3 and 4.
4
Concepts
2
Clusters
5
Prerequisites
4
With difficulty levels
Lesson Clusters
Convert between mixed numbers and improper fractions and add fractions with different denominators
introduction CuratedMixed numbers/improper fractions and adding with different denominators are the procedural fraction targets that build on Year 3-4 equivalence work. Together they represent the Year 5 fraction arithmetic cluster.
Extend decimal place value to thousandths and understand percentages
practice CuratedDecimal place value to three decimal places and the concept of percentage as parts per hundred are both representational knowledge concepts that expand the number system. Together they set up the fraction-decimal-percentage equivalence work.
Teaching Suggestions (1)
Study units and activities that deliver concepts in this domain.
Fractions, Decimals and Percentages
Mathematics Pattern SeekingPrerequisites
Concepts from other domains that pupils should know before this domain.
Concepts (4)
Adding and subtracting fractions with different denominators
skill AI DirectMA-Y5-C008
To add or subtract fractions with different denominators, both fractions must be expressed with a common denominator using equivalent fractions. The lowest common multiple of the denominators gives the most efficient common denominator. For example, 1/3 + 1/4: LCM of 3 and 4 is 12; 1/3 = 4/12, 1/4 = 3/12; 4/12 + 3/12 = 7/12. Mastery means pupils can find a common denominator, convert both fractions, add or subtract, and simplify if possible.
Teaching guidance
Begin with denominators where one is a multiple of the other (1/4 + 1/8: 1/4 = 2/8, so 2/8 + 1/8 = 3/8 — only one fraction needs converting). Extend to cases requiring LCM (1/3 + 1/4 as above). Use fraction bars to show equivalence visually before computing abstractly. Connect to LCM knowledge from number theory. When results exceed 1 whole, convert to mixed number (7/5 = 1 2/5).
Common misconceptions
The classic error is adding numerators and denominators independently (1/3 + 1/4 = 2/7). Pupils who do not find a common denominator first will always make this error. When only one denominator needs converting, some pupils convert both unnecessarily and make errors. Forgetting to simplify the answer (leaving 6/9 instead of 2/3) is also common.
Difficulty levels
Adding fractions with the same denominator and with related denominators where one is a multiple of the other.
Example task
Work out 1/4 + 3/8.
Model response: Convert 1/4 to 2/8. Then 2/8 + 3/8 = 5/8.
Adding and subtracting fractions where neither denominator is a multiple of the other, finding a common denominator using LCM.
Example task
Work out 2/3 + 1/4.
Model response: LCM of 3 and 4 is 12. 2/3 = 8/12. 1/4 = 3/12. 8/12 + 3/12 = 11/12.
Adding and subtracting fractions with any denominators, simplifying results and converting improper fractions to mixed numbers.
Example task
Work out 5/6 + 3/4. Give your answer as a mixed number in its simplest form.
Model response: LCM of 6 and 4 is 12. 5/6 = 10/12. 3/4 = 9/12. 10/12 + 9/12 = 19/12 = 1 7/12.
CPA Stages
concrete
Using fraction strips and fraction walls to physically demonstrate why a common denominator is needed, and overlaying strips to convert fractions to the same-sized pieces before combining
Transition: Child explains why fractions need a common denominator before adding and finds the equivalent fractions using strips
pictorial
Drawing fraction bars to show conversion to a common denominator, recording the equivalent fraction steps on paper, and using LCM to find the most efficient denominator
Transition: Child finds common denominators and adds/subtracts fractions on paper without fraction strips, simplifying the answer
abstract
Adding and subtracting fractions with different denominators mentally or with minimal jottings, converting to mixed numbers when results exceed 1, and simplifying
Transition: Child adds and subtracts fractions with different denominators fluently, converting improper fractions to mixed numbers and simplifying automatically
Delivery rationale
Upper primary maths (Y5) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.
Mixed numbers and improper fractions
skill AI DirectMA-Y5-C009
An improper fraction has a numerator greater than or equal to its denominator (e.g. 7/4). A mixed number expresses the same value as a whole number and a proper fraction (e.g. 1 3/4). Converting between them: 7/4 = 4/4 + 3/4 = 1 + 3/4 = 1 3/4; and 2 3/5 = (2 × 5 + 3)/5 = 13/5. Mastery means pupils convert fluently in both directions, understand that the two forms represent identical values, and can position both on a number line.
Teaching guidance
Use a number line with fractions and whole numbers marked: 0, 1/4, 2/4 (=1/2), 3/4, 4/4 (=1), 5/4 (=1 1/4)... This makes the continuity of fractions beyond 1 visible. For converting improper to mixed: divide numerator by denominator (the quotient is the whole number; the remainder is the new numerator over the same denominator). For mixed to improper: multiply whole number by denominator and add numerator.
Common misconceptions
When converting mixed to improper, pupils forget to add the numerator after multiplying (computing 2 × 5 = 10 for 2 3/5 and writing 10/5 rather than 13/5). When converting improper to mixed, pupils who use the division method may not write the remainder as a fraction (writing quotient as the answer without a remainder fraction). They may also think improper fractions are 'wrong' rather than a valid form.
Difficulty levels
Identifying improper fractions and mixed numbers on a number line, understanding that 5/4 means 'more than one whole'.
Example task
Place 7/4 on the number line. Is it more or less than 2?
Model response: 7/4 = 1 3/4. It is less than 2. [Places it between 1 and 2, three-quarters of the way along]
Converting between improper fractions and mixed numbers using division.
Example task
Convert 17/5 to a mixed number. Convert 3 2/7 to an improper fraction.
Model response: 17 ÷ 5 = 3 remainder 2, so 17/5 = 3 2/5. 3 2/7 = (3 × 7 + 2)/7 = 23/7.
Converting fluently in both directions and using the appropriate form for addition, subtraction and comparison.
Example task
Which is larger, 2 3/5 or 11/4? Show your working.
Model response: 2 3/5 = 13/5 = 52/20. 11/4 = 55/20. 55/20 > 52/20, so 11/4 is larger.
CPA Stages
concrete
Using fraction circles and fraction strips to build improper fractions beyond 1 whole and convert them to mixed numbers by grouping whole circles
Transition: Child converts between improper fractions and mixed numbers using pieces, explaining the grouping process verbally
pictorial
Drawing fraction bars and number lines to show the conversion between improper fractions and mixed numbers, recording the division method on paper
Transition: Child converts in both directions on paper using the division/multiplication method without drawing fraction bars
abstract
Converting between improper fractions and mixed numbers mentally, and using these conversions in fraction calculations
Transition: Child converts between forms instantly and uses these conversions fluently within fraction calculations
Delivery rationale
Upper primary maths (Y5) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.
Percentages as fractions and decimals
knowledge AI DirectMA-Y5-C010
Percent means 'per hundred' (from Latin per centum). A percentage is therefore a fraction with denominator 100: 45% = 45/100 = 0.45. Common equivalences to know: 50% = 1/2 = 0.5; 25% = 1/4 = 0.25; 10% = 1/10 = 0.1; 1% = 1/100 = 0.01; 20% = 1/5 = 0.2. Mastery means pupils can convert between percentage, fraction and decimal forms for all common equivalences and for any percentage, and can find a percentage of a quantity.
Teaching guidance
Use a hundredths grid (100 squares representing 100%): shade 45 squares to show 45%. Money provides a natural percentage context: 50% of £80 = £40 (half). Teach the three forms side by side: fraction, decimal, percentage. Building up from known benchmarks: 50% is a half, 25% is a quarter, 10% is a tenth, then 5% = half of 10%, 20% = double 10% etc. Finding a percentage: multiply by the decimal equivalent (35% of 60 = 0.35 × 60 = 21).
Common misconceptions
Pupils sometimes think 'per cent' and 'percentage points' are interchangeable in all contexts. They may write 5% as 0.5 (rather than 0.05) or as 5/10 (rather than 5/100). Converting 0.07 to a percentage: some pupils say 7% (correct) but others say 70% (multiplying by 10 rather than 100). The relationship between percentage and decimal requires multiplying or dividing by 100, not 10.
Difficulty levels
Shading a hundredths grid to show common percentages and connecting to the fraction form.
Example task
Shade 50% of the hundredths grid. What fraction is this?
Model response: 50 out of 100 squares shaded. 50% = 50/100 = 1/2.
Converting between percentages, fractions and decimals for common benchmarks: 50%, 25%, 10%, 1%, 20%, 75%.
Example task
Write 25% as a fraction and a decimal. Write 0.1 as a percentage.
Model response: 25% = 1/4 = 0.25. 0.1 = 10%.
Converting any percentage to a fraction and decimal, and finding a percentage of a quantity using known equivalences.
Example task
Find 15% of £80. Show your method.
Model response: 10% of £80 = £8. 5% = half of 10% = £4. 15% = £8 + £4 = £12.
CPA Stages
concrete
Using a 10×10 hundredths grid to shade percentages, connecting to money (£1 = 100p) and fraction equivalences physically
Transition: Child states the fraction and decimal equivalent of any common percentage (10%, 20%, 25%, 50%, 75%) without the grid
pictorial
Drawing bar models to find percentages of quantities, recording fraction-decimal-percentage conversion tables, and using number lines to show equivalences
Transition: Child converts between fractions, decimals and percentages on paper and finds percentages of quantities using the bar model method
abstract
Converting between fractions, decimals and percentages mentally, and finding percentages of quantities using efficient mental strategies
Transition: Child converts fluently between all three forms and calculates percentages of quantities mentally using benchmark strategies
Delivery rationale
Upper primary maths (Y5) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.
Thousandths and decimal place value to 3 decimal places
knowledge AI DirectMA-Y5-C013
A thousandth is 1/1000 = 0.001 — the third decimal place. Extending decimal place value to three places allows measurement to the nearest millimetre (1 mm = 0.001 m), weight to nearest gram (1 g = 0.001 kg), and volume to nearest millilitre. Mastery means pupils can read, write, order and compare numbers with up to three decimal places, identify the value of any digit, and connect decimal and fraction notation for thousandths.
Teaching guidance
Extend the place value chart to include tenths, hundredths and thousandths columns. Use a number line from 0 to 1 marked in 0.001 steps (or at key reference points) for ordering. Measurement contexts: a 5 km race recorded as 4.876 km = 4 km 876 m; a mass of 2.345 kg = 2 kg 345 g. Connect to division: 3 ÷ 1000 = 0.003 (three thousandths). Compare numbers with 3 decimal places column by column from left.
Common misconceptions
Pupils compare decimals with different numbers of decimal places incorrectly: thinking 0.4 > 0.35 might be true (correct) but thinking 0.035 > 0.4 because 35 > 4 (incorrect — confusing place values). The word 'thousandths' is confused with 'thousands', leading to errors when moving between large and small numbers. Writing 0.003 as 3/1000 is secure; writing 3/1000 as 0.003 is less so.
Difficulty levels
Reading and writing numbers with one decimal place in a measurement context (e.g. 3.7 m), identifying the value of the tenths digit.
Example task
A rope is 4.6 m long. What does the 6 represent?
Model response: The 6 represents 6 tenths of a metre, which is 60 cm.
Reading, writing and ordering numbers with up to three decimal places, identifying the value of each digit.
Example task
Order these: 0.345, 0.35, 0.4, 0.305. What is the value of the 5 in 0.345?
Model response: Order: 0.305, 0.345, 0.35, 0.4. The 5 in 0.345 is worth 5 thousandths (0.005).
Comparing and ordering decimals with different numbers of decimal places, connecting to fractions, and placing them on a number line.
Example task
Place 0.075, 0.7 and 0.507 on a number line from 0 to 1. Write 0.075 as a fraction.
Model response: Order: 0.075, 0.507, 0.7. 0.075 = 75/1000 = 3/40. [Places correctly on line with 0.075 near 0, 0.507 just past halfway, 0.7 at seven-tenths]
CPA Stages
concrete
Using place value counters (0.001, 0.01, 0.1, 1) on a decimal place value mat and connecting to metric measurement: 1 mm = 0.001 m, 1 g = 0.001 kg
Transition: Child reads the value of any digit in a three-decimal-place number and connects thousandths to mm and g measurement contexts
pictorial
Using place value charts with three decimal columns, placing decimals on zoomed-in number lines, and comparing numbers with different numbers of decimal places
Transition: Child compares and orders decimals to three places without a chart, appending trailing zeros mentally to equalise decimal places
abstract
Working with thousandths mentally: reading digit values, comparing and ordering, converting between fractions and decimals, and rounding to 1 or 2 decimal places
Transition: Child works with three-decimal-place numbers fluently, comparing, ordering, converting and rounding without visual aids
Delivery rationale
Upper primary maths (Y5) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.