Number - Fractions
KS2MA-Y3-D004
Understanding tenths, unit and non-unit fractions with small denominators, equivalent fractions, adding and subtracting fractions with the same denominator, and comparing and ordering fractions.
National Curriculum context
Year 3 is the point at which pupils extend their understanding of fractions significantly beyond the simple halves and quarters of KS1, encountering a wider range of unit and non-unit fractions and the fundamental concept of equivalent fractions. The non-statutory guidance explains that pupils should count up and down in tenths on a number line and connect tenths to place value — understanding that one-tenth is the same as dividing by 10 — which lays essential groundwork for decimal notation in Year 4. Pupils learn to add and subtract fractions with the same denominator and to compare and order fractions, developing the understanding that fractions are numbers that can be placed on a number line. This domain provides the conceptual foundations for the substantially more complex fraction and decimal work of Years 4, 5 and 6, and its careful development is essential for future algebraic and proportional reasoning.
6
Concepts
2
Clusters
0
Prerequisites
6
With difficulty levels
Lesson Clusters
Understand unit and non-unit fractions with small denominators
introduction CuratedTenths (bridging fractions and place value), unit fractions and non-unit fractions together establish the conceptual foundation of the fraction number system in Year 3. All three mutually co-teach.
Compare, order and add fractions with the same denominator
practice CuratedEquivalent fractions, adding/subtracting with the same denominator and comparing/ordering fractions are the procedural and relational skills built on the conceptual foundation. All three are mutually co-taught.
Teaching Suggestions (1)
Study units and activities that deliver concepts in this domain.
Understanding Fractions as Parts of a Whole
Mathematics Worked Example SetPedagogical rationale
Y3 is where fractions shift from simple halving and quartering to a genuine number concept. Children must understand that a fraction describes equal parts of a whole and that the denominator tells you how many equal parts. Fraction tiles and fraction walls make the equal-parts requirement visible and prevent the common misconception that any two pieces are halves. Introducing tenths connects fractions to place value and lays groundwork for decimals in Y4.
Concepts (6)
Tenths as fractions and in place value
knowledge AI FacilitatedMA-Y3-C024
A tenth arises when one whole is divided into 10 equal parts; each part is one tenth (1/10). This concept bridges fractions and decimals because one tenth (1/10) is the first decimal place (0.1). Counting in tenths connects to the number line and to measurement (e.g. 1/10 of a metre = 10 cm). Mastery means pupils understand tenths as both fractions and as parts of decimal numbers, can count in tenths on a number line, and recognise that dividing a one-digit number by 10 gives tenths.
Teaching guidance
Use a metre stick divided into 10 equal parts (each 10 cm = 1/10 of a metre) as a concrete tool. Fraction strips or circles divided into 10 equal sectors provide visual representations. Count in tenths on a number line from 0 to 2: 0/10, 1/10, 2/10... 10/10, 11/10... showing tenths beyond 1. Connect to division: 6 ÷ 10 = 6/10 (six tenths). Note that the formal decimal notation (0.1) is not introduced until Year 4, but the fractional concept (1/10) is established here.
Common misconceptions
Pupils may think 1/10 is a large fraction because 10 is a large number — not understanding that larger denominators mean smaller parts. They may count tenths as: one tenth, two tenths... ten tenths, eleven tenths instead of one whole and one tenth. The connection between 1/10 and 0.1 is not made until Year 4 so pupils at this stage work purely in fraction notation.
Difficulty levels
Dividing a concrete whole into 10 equal parts to discover tenths using fraction strips or a metre stick.
Example task
Fold a paper strip into 10 equal parts. Colour 1 part. What fraction have you coloured?
Model response: I coloured 1 out of 10 equal parts. That is one tenth, or 1/10.
Counting in tenths on a number line and recognising that 10/10 = 1 whole.
Example task
Count in tenths from 0 to 1 on a number line. What is 7/10? Where does it go?
Model response: 0/10, 1/10, 2/10, 3/10, 4/10, 5/10, 6/10, 7/10, 8/10, 9/10, 10/10. 7/10 is seven tenths, just past halfway (5/10). 10/10 = 1.
Connecting tenths to division by 10 and counting in tenths beyond 1 whole.
Example task
What is 3 divided by 10? Count in tenths from 8/10 to 14/10.
Model response: 3 / 10 = 3/10 (three tenths). Counting: 8/10, 9/10, 10/10 (= 1), 11/10 (= 1 and 1/10), 12/10, 13/10, 14/10 (= 1 and 4/10).
Connecting tenths to measurement contexts and reasoning about their size relative to other fractions.
Example task
A metre stick is divided into 10 equal parts. How long is each part in centimetres? Is 1/10 of a metre greater or less than 1/4 of a metre? Explain.
Model response: Each part is 100 cm / 10 = 10 cm. 1/10 of a metre = 10 cm. 1/4 of a metre = 25 cm. So 1/10 < 1/4 because dividing into more parts makes each part smaller.
CPA Stages
concrete
Dividing a metre stick into 10 equal sections using masking tape, handling fraction strips cut into tenths, and counting physical tenth-pieces to build whole quantities
Transition: Child counts in tenths fluently past 1 whole and states that 10/10 = 1 without needing the physical strips
pictorial
Drawing number lines marked in tenths, shading fraction bars divided into 10 equal parts, and recording tenths as fractions on paper
Transition: Child places any number of tenths on a number line (including beyond 1) and connects tenths to division by 10 without the drawn bar
abstract
Working with tenths mentally, connecting division by 10 to tenths, and counting in tenths across whole number boundaries
Transition: Child converts between tenths and wholes mentally, stating division-by-10 results as tenths without hesitation
Delivery rationale
Primary maths (Y3) with concrete stage requiring physical manipulatives (metre stick with masking tape markers, fraction strips (tenths)). AI delivers instruction; facilitator sets up materials.
Unit fractions with small denominators
Keystone knowledge AI FacilitatedMA-Y3-C025
Unit fractions have a numerator of 1 (e.g. 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10). In Year 3, pupils extend their KS1 understanding of 1/2 and 1/4 to a wider range of unit fractions. They find unit fractions of discrete sets of objects and numbers, and place unit fractions on a number line. Mastery means pupils understand that the denominator tells how many equal parts the whole is divided into, the numerator tells how many parts are being counted, and can compare unit fractions (understanding that 1/3 > 1/4 > 1/5 because larger denominators mean smaller parts).
Teaching guidance
Use fraction walls (visual representations showing fractions of a whole, with bars divided into halves, thirds, quarters, fifths etc.) as the primary pictorial tool. Fraction circles and folded paper provide concrete experience. Emphasise the equal-parts definition — 1/3 means one of THREE EQUAL PARTS, and all three parts must be the same size. Place unit fractions on a number line between 0 and 1. Compare: '1/4 < 1/3 because splitting into 4 parts makes each part smaller than splitting into 3 parts.'
Common misconceptions
Pupils commonly believe larger denominators mean larger fractions (thinking 1/8 > 1/3 because 8 > 3). They may also accept non-equal partitions as valid fractions. When finding 1/3 of 9 objects, pupils may distribute them one at a time without checking the equal-parts condition.
Difficulty levels
Finding unit fractions of sets of objects by equal sharing using concrete manipulatives.
Example task
Find 1/3 of 12 counters by sharing them equally into 3 groups.
Model response: 12 counters shared into 3 groups = 4 counters in each group. 1/3 of 12 = 4.
Finding unit fractions of quantities using division, and placing unit fractions on a fraction wall or number line.
Example task
Find 1/5 of 20. Place 1/5 on a number line between 0 and 1.
Model response: 1/5 of 20 = 20 / 5 = 4. On the number line, 1/5 is between 0 and 1/2, closer to 0.
Comparing unit fractions and explaining that larger denominators make smaller parts.
Example task
Put these fractions in order from smallest to largest: 1/2, 1/8, 1/4, 1/3.
Model response: 1/8, 1/4, 1/3, 1/2. The larger the denominator, the smaller the fraction, because you are splitting the whole into more parts.
CPA Stages
concrete
Finding unit fractions of sets by physically sharing counters into equal groups, and using fraction circles and fraction strips to show one equal part of a whole
Transition: Child finds unit fractions of sets by dividing and compares unit fraction pieces, saying 'Bigger denominator means smaller pieces'
pictorial
Using a fraction wall to compare unit fractions, placing unit fractions on a number line between 0 and 1, and drawing fraction bars to find unit fractions of quantities
Transition: Child orders any set of unit fractions from smallest to largest and finds unit fractions of quantities using division without the fraction wall
abstract
Comparing and ordering unit fractions mentally, finding unit fractions of quantities by division, and explaining why larger denominators give smaller fractions
Transition: Child instantly orders unit fractions and explains the inverse relationship between denominator size and fraction size
Delivery rationale
Primary maths (Y3) with concrete stage requiring physical manipulatives (counters, sorting trays (3, 4, 5, 6, 8 compartments)). AI delivers instruction; facilitator sets up materials.
Non-unit fractions with small denominators
knowledge AI FacilitatedMA-Y3-C026
Non-unit fractions have a numerator greater than 1 (e.g. 2/3, 3/4, 5/8). They represent multiple parts of a whole divided equally. In Year 3, pupils learn to recognise, find and use non-unit fractions, building on their secure knowledge of unit fractions. Mastery means pupils can find a non-unit fraction of a set (e.g. 3/4 of 20 = 15) by first finding the unit fraction and multiplying, and can place non-unit fractions on a number line.
Teaching guidance
Build from unit fractions: 'We know 1/4 of 20 = 5, so 3/4 of 20 = 3 × 5 = 15.' Use fraction bars and circles to show multiple parts coloured. Place non-unit fractions on a number line — this shows that 3/4 is closer to 1 than to 0, and that fractions get larger as the numerator increases (when denominator is fixed). Connect to the concept that a non-unit fraction is several equal parts combined.
Common misconceptions
Pupils sometimes find the unit fraction correctly but forget to multiply by the numerator (finding 3/4 of 20 as just 5 rather than 15). Some pupils treat the numerator and denominator as independent whole numbers (saying 3/4 of 20 = 3 out of 4 of 20, then not knowing how to proceed). The connection between finding fractions of quantities and division is not always clear.
Difficulty levels
Finding non-unit fractions of sets using concrete grouping: first finding the unit fraction, then taking multiple groups.
Example task
Find 2/5 of 15 counters. First find 1/5, then take 2 groups.
Model response: 1/5 of 15 = 3. So 2/5 of 15 = 2 x 3 = 6.
Finding non-unit fractions of quantities using the divide-then-multiply strategy, with pictorial fraction bars.
Example task
Find 3/4 of 20. Show this on a fraction bar.
Model response: 1/4 of 20 = 5. 3/4 of 20 = 3 x 5 = 15. On the fraction bar: 20 split into 4 equal parts (5 each), 3 parts shaded = 15.
Finding non-unit fractions of quantities abstractly and placing non-unit fractions on a number line.
Example task
Find 5/8 of 40. Where would 5/8 sit on a number line between 0 and 1?
Model response: 1/8 of 40 = 5. 5/8 of 40 = 5 x 5 = 25. On the number line, 5/8 is between 1/2 (4/8) and 3/4 (6/8), just past halfway.
Solving problems requiring finding non-unit fractions and reasoning about what fraction remains.
Example task
There are 24 sweets. Priya eats 3/8 of them. How many are left? What fraction is left?
Model response: 3/8 of 24 = 3 x 3 = 9 sweets eaten. 24 - 9 = 15 left. Fraction left: 8/8 - 3/8 = 5/8.
CPA Stages
concrete
Finding non-unit fractions of sets by first sharing counters into equal groups (unit fraction) then taking multiple groups, using fraction circles to show several parts shaded
Transition: Child consistently finds the unit fraction first, then multiplies by the numerator, verbalising: 'I divide by the bottom, then multiply by the top'
pictorial
Drawing fraction bars to find non-unit fractions of quantities, shading the correct number of parts, and placing non-unit fractions on a number line
Transition: Child draws fraction bars and calculates non-unit fractions without needing counters, and places non-unit fractions accurately on number lines
abstract
Finding non-unit fractions of quantities mentally using the divide-then-multiply strategy, and reasoning about what fraction remains
Transition: Child calculates non-unit fractions of quantities within 5 seconds and reasons about complementary fractions without visual support
Delivery rationale
Primary maths (Y3) with concrete stage requiring physical manipulatives (counters, sorting trays). AI delivers instruction; facilitator sets up materials.
Equivalent fractions
knowledge AI FacilitatedMA-Y3-C027
Equivalent fractions are different fractions that represent the same value (e.g. 1/2 = 2/4 = 3/6 = 4/8). Pupils in Year 3 use diagrams and visual representations to recognise and show equivalent fractions with small denominators. Mastery means pupils can identify and generate equivalent fractions, understand that multiplying both the numerator and denominator by the same number produces an equivalent fraction, and use equivalence to compare and order fractions.
Teaching guidance
Fraction walls are the most powerful tool: line up bars showing halves, quarters and eighths to see that 1/2 = 2/4 = 4/8 visually. Fraction circles also demonstrate this well. Folding paper: fold a strip in half (showing 1/2), then fold again to show quarters (2/4), then again (4/8). Use diagrams to show the pattern: multiplying top and bottom by 2 doubles both. At Year 3, the formal rule (multiply by n/n) is introduced pictorially rather than algebraically.
Common misconceptions
Pupils who have not developed the pictorial understanding may not believe that 1/2 = 2/4 because 'the numbers are different.' They may think that multiplying both numerator and denominator by the same number changes the value of the fraction. Some pupils confuse equivalent fractions (same value) with equal fractions (identical notation).
Difficulty levels
Using fraction walls or folded paper to see that two fractions can represent the same amount.
Example task
Fold a paper strip in half. Now fold it in half again to make quarters. How many quarters equal one half?
Model response: 2 quarters equal 1 half. 1/2 = 2/4.
Identifying equivalent fractions from a fraction wall diagram by comparing bar lengths.
Example task
Using a fraction wall, find a fraction equivalent to 1/3.
Model response: 2/6 = 1/3. Two sixths bars line up exactly with one third bar on the fraction wall.
Generating equivalent fractions by multiplying the numerator and denominator by the same number, and using diagrams to verify.
Example task
Find two fractions equivalent to 2/4. Show one on a diagram.
Model response: 2/4 = 1/2 (dividing both by 2) and 2/4 = 4/8 (multiplying both by 2). Diagram: a bar split into 4 parts with 2 shaded = a bar split into 8 parts with 4 shaded.
Using equivalent fractions to solve problems, including finding missing numerators or denominators.
Example task
3/? = 6/8. What is the missing denominator? Explain how you know.
Model response: The numerator doubled from 3 to 6, so the denominator must also double. 3/4 = 6/8. The missing denominator is 4.
CPA Stages
concrete
Folding paper strips to discover equivalent fractions physically: folding in half then folding again to see that 1/2 = 2/4, using fraction circles and fraction strips to overlay and compare
Transition: Child finds equivalent fractions by folding or overlaying and states the equivalence without being prompted (e.g. 'That's the same as 2/4')
pictorial
Using a fraction wall to identify equivalent fractions by lining up bars, and drawing diagrams to show that multiplying numerator and denominator by the same number produces equal fractions
Transition: Child generates equivalent fractions by multiplying top and bottom by the same number and checks using a diagram
abstract
Generating equivalent fractions by multiplying or dividing numerator and denominator by the same number, and recognising equivalence without diagrams
Transition: Child generates and simplifies equivalent fractions mentally, explaining the multiplicative relationship between numerator and denominator
Delivery rationale
Primary maths (Y3) with concrete stage requiring physical manipulatives (paper strips, fraction circles (halves, quarters, eighths)). AI delivers instruction; facilitator sets up materials.
Adding and subtracting fractions with the same denominator
skill AI FacilitatedMA-Y3-C028
When fractions have the same denominator, they can be added or subtracted by operating on the numerators only, keeping the denominator constant (e.g. 3/7 + 2/7 = 5/7; 5/8 – 2/8 = 3/8). The constraint 'within one whole' means results remain between 0 and 1. Mastery means pupils can reliably add and subtract same-denominator fractions, understand why only the numerators change, and connect this to counting in fractions on a number line.
Teaching guidance
Use a fraction number line to show addition as movement along the line: start at 3/7, move 2/7 further to reach 5/7. Fraction bars provide a concrete/pictorial approach: shade 3 parts of a 7-part bar, then shade 2 more — 5 parts shaded in total. Emphasise: 'The denominator tells us the size of each piece; the pieces don't change when we add, we just count more of them.' Connect to whole-number addition: 3 sevenths + 2 sevenths = 5 sevenths, just as 3 apples + 2 apples = 5 apples.
Common misconceptions
The most common error is adding or subtracting both numerators and denominators: 3/7 + 2/7 = 5/14 (wrong). Pupils who make this error do not understand that the denominator represents the type of unit being counted, which does not change when you combine same-type units. Subtracting to get a zero numerator (3/5 – 3/5 = 0/5 = 0) also confuses some pupils.
Difficulty levels
Adding same-denominator fractions using a fraction bar: shading additional parts to see the total.
Example task
A bar is divided into 5 equal parts. Shade 2 parts red, then shade 1 more part blue. What fraction is shaded altogether?
Model response: 2/5 + 1/5 = 3/5. Three out of five parts are shaded.
Adding and subtracting same-denominator fractions using fraction number lines or bars, with answers remaining within one whole.
Example task
Start at 3/7 on a number line. Move forward 2/7. Where do you land? Then subtract 4/7. Where do you land?
Model response: 3/7 + 2/7 = 5/7. Then 5/7 - 4/7 = 1/7.
Adding and subtracting same-denominator fractions abstractly, explaining that only numerators change.
Example task
Calculate 5/8 + 2/8 and 6/9 - 4/9. Explain why you only add or subtract the numerators.
Model response: 5/8 + 2/8 = 7/8. 6/9 - 4/9 = 2/9. Only the numerators change because the denominator tells us the size of the pieces, which does not change. 5 eighths + 2 eighths = 7 eighths, like 5 apples + 2 apples = 7 apples.
CPA Stages
concrete
Adding and subtracting same-denominator fractions using fraction strips: physically combining and removing fraction pieces, counting the resulting parts
Transition: Child adds and subtracts same-denominator fractions with strips, consistently keeping the denominator the same and saying: 'The pieces don't change size, I just count more or fewer'
pictorial
Using fraction bars and number lines to show same-denominator addition and subtraction, recording the calculation alongside the diagram
Transition: Child records same-denominator fraction calculations correctly alongside diagrams and explains why only numerators are added or subtracted
abstract
Adding and subtracting same-denominator fractions mentally, explaining the reasoning: 'Sevenths plus sevenths equals sevenths — like apples plus apples equals apples'
Transition: Child calculates same-denominator fraction additions and subtractions instantly, never adding the denominators, and explains the reasoning using the 'counting units' analogy
Delivery rationale
Primary maths (Y3) with concrete stage requiring physical manipulatives (fraction strips (fifths, sevenths, eighths), fraction circles). AI delivers instruction; facilitator sets up materials.
Comparing and ordering fractions
skill AI FacilitatedMA-Y3-C029
In Year 3, pupils compare and order unit fractions (understanding that 1/2 > 1/3 > 1/4 > 1/5 because larger denominators make smaller parts) and compare fractions with the same denominator (where the larger numerator indicates the larger fraction, e.g. 5/8 > 3/8). Mastery means pupils can compare any two fractions encountered at this stage with justification, and order a set of unit fractions or same-denominator fractions correctly.
Teaching guidance
Use fraction walls to compare unit fractions visually — the bar divided into halves is longer per part than the bar divided into quarters. For same-denominator fractions, colour fraction bars and compare the shaded areas. Use a shared number line to place multiple fractions and compare their positions. Introduce the language 'greater than', 'less than' and 'equal to' explicitly in fraction contexts. Avoid the symbol approach until pupils have strong conceptual understanding.
Common misconceptions
The most persistent misconception is 'larger denominator = larger fraction' for unit fractions (thinking 1/8 > 1/3). For same-denominator fractions, pupils may compare denominators rather than numerators. Some pupils are confused by equivalent fractions appearing at different positions in an ordered list.
Difficulty levels
Comparing unit fractions using a fraction wall to see which piece is larger.
Example task
Look at a fraction wall. Which is bigger, 1/3 or 1/5? How do you know?
Model response: 1/3 is bigger than 1/5. On the fraction wall, the third bar is longer than the fifth bar because splitting into fewer parts makes each part bigger.
Ordering unit fractions from smallest to largest and comparing fractions with the same denominator.
Example task
Order from smallest to largest: 1/4, 1/2, 1/6, 1/3. Then order: 2/7, 5/7, 1/7, 6/7.
Model response: Unit fractions: 1/6, 1/4, 1/3, 1/2 (larger denominator = smaller fraction). Same denominator: 1/7, 2/7, 5/7, 6/7 (larger numerator = larger fraction).
Comparing and ordering fractions with justification, using both the unit fraction rule and same-denominator rule.
Example task
Which is greater, 3/8 or 5/8? Which is greater, 1/3 or 1/8? Explain each answer.
Model response: 5/8 > 3/8 because they have the same denominator and 5 > 3 (more eighths). 1/3 > 1/8 because both are unit fractions and thirds are bigger pieces than eighths (fewer parts = bigger pieces).
Comparing fractions that require reasoning beyond the basic rules, using benchmarks like 1/2.
Example task
Without drawing, which is greater: 3/5 or 2/7? Explain your reasoning.
Model response: 3/5 is greater than 1/2 (because 3 is more than half of 5). 2/7 is less than 1/2 (because 2 is less than half of 7). So 3/5 > 2/7.
CPA Stages
concrete
Comparing fractions by physically holding up fraction pieces side by side: unit fraction pieces to see which is bigger, and same-denominator fraction groups to see which has more
Transition: Child compares fractions using pieces and states the rule: 'For unit fractions, smaller denominator means bigger piece; for same denominators, bigger numerator means bigger fraction'
pictorial
Using a fraction wall and number lines to compare and order fractions, drawing arrows or markers to show relative positions
Transition: Child places fractions on a number line and orders them correctly without checking against the fraction wall
abstract
Comparing and ordering fractions mentally, selecting the correct comparison strategy (unit fraction rule vs same-denominator rule) and justifying each comparison
Transition: Child selects the correct comparison strategy without prompting, orders any set of unit fractions or same-denominator fractions instantly, and uses benchmark fractions like 1/2 to compare
Delivery rationale
Primary maths (Y3) with concrete stage requiring physical manipulatives (fraction circles, fraction strips). AI delivers instruction; facilitator sets up materials.