Fractions: Halves, Quarters, and Thirds

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Concepts

This study delivers 1 primary concept and 1 secondary concept.

Primary concept: Fractions as 'fractions of': 1/3, 1/4, 2/4, 3/4 (MA-Y2-C012)

Type: Knowledge | Teaching weight: 3/6

Year 2 extends the set of fractions pupils can find and name from halves and quarters (Year 1) to include thirds and three-quarters. Pupils use fractions to describe a part of a discrete quantity (e.g. 1/3 of 12 = 4) or a continuous quantity (e.g. 1/3 of a length). A quarter of a quantity is one of four equal parts; two quarters (2/4) is two of those four equal parts; three quarters (3/4) is three of those four equal parts. Three-quarters is explicitly the first non-unit fraction pupils encounter. Mastery means pupils can find any of these fractions of given quantities and lengths, and write the fraction notation correctly.

Teaching guidance: Use sharing as the primary concrete approach: to find 1/3 of 12, share 12 equally among 3 (each person gets 4). To find 3/4 of 12, first find 1/4 (share among 4 = 3), then multiply by 3 (three-quarters = 3 groups of the unit fraction = 9). Use fraction strips and number lines to show fractions of continuous quantities. The curriculum specifies that pupils should count in fractions up to 10 on the number line (1 1/4, 1 2/4, 1 3/4, 2...) to understand fractions as numbers in their own right. Write fraction notation clearly: the vinculum (fraction bar) separates numerator and denominator. Key vocabulary: fraction, third, quarter, three-quarters, numerator, denominator, unit fraction, non-unit fraction, equal parts, whole Common misconceptions: Pupils frequently confuse the denominator with the number of shaded parts, rather than the total number of equal parts. When finding fractions of quantities by sharing, pupils may share unequally and not recognise the error. For three-quarters, pupils often do not connect this to 3 × (1/4) — they see it as a new fraction to memorise rather than as three unit fractions combined. Writing fraction notation is confused: some pupils write the numerator below the line.

Differentiation

Model response (Entry): Half of 10 is 5 (each plate gets 5). A quarter of 8 is 2 (each plate gets 2).

Model response (Developing): First find 1/4 of 12 = 3. Then 3/4 = 3 × 3 = 9. So 3/4 of 12 is 9.

Model response (Expected): 0, 1/4, 2/4, 3/4, 1, 1 1/4, 1 2/4, 1 3/4, 2. One third of 18 is 6.

Representation stages (CPA)

Secondary concept: Fraction equivalence: 2/4 = 1/2 (MA-Y2-C013)

Type: Knowledge | Teaching weight: 3/6

The recognition that 2/4 and 1/2 name the same quantity — the same point on the number line, the same part of a whole — is the first exposure to fraction equivalence in the national curriculum. This is conceptually significant: two fractions with different numerators and denominators can have the same value. Mastery means pupils understand why 2/4 = 1/2 (because two of four equal parts is the same as one of two equal parts), and can identify this equivalence in different contexts.

Differentiation

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Thinking lens: Scale, Proportion and Quantity (primary)

Key question: How big, how many, or how much — and how does that change how we think about it? Why this lens fits: Finding fractions of shapes and sets is directly about proportional part-whole reasoning — a half is always the same proportion regardless of the size of the whole, which is why 2/4 = 1/2. Question stems for KS1:

Which one is bigger?

Which group has more?

How could we check which is heavier?

Is this a lot or a little?

Secondary lens: Patterns — Equivalent fractions reveal a pattern: multiplying or dividing numerator and denominator by the same number preserves the proportion — pupils begin to see this regularity when they spot that 2/4 and 1/2 shade the same amount.

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Session structure: Practical Application + Worked Example Set

This study uses 2 vehicle templates:

Practical Application (main structure)

A hands-on sequence where pupils apply knowledge and skills to solve a practical problem or create a functional outcome. Begins with a real-world context, builds skills through rehearsal, guides design or planning, supports making or problem-solving, and concludes with evaluation against success criteria.

context → skill_rehearsal → design → make_or_solve → evaluate Assessment: Practical outcome (solution, product, program) evaluated against defined success criteria, with written or verbal explanation of the process and decisions made.

Worked Example Set

A mastery-oriented mathematics sequence moving through the concrete-pictorial-abstract progression with activation and reasoning extension phases. Begins by activating prior knowledge, introduces new concepts with physical manipulatives, transitions to pictorial representations, develops abstract fluency, applies in context, and extends through reasoning challenges.

activation → concrete → pictorial → abstract → application → reasoning_extension Assessment: Graduated practice set moving from guided examples to independent application, with reasoning task requiring explanation of method and justification of answers. Teacher note: Use the WORKED EXAMPLE SET template: begin by activating what children already know using a quick warm-up. Introduce new concepts using physical objects they can touch and move. Move to pictures and drawings that represent the same idea. Then show how to record it using numbers and symbols. Let children practise with similar examples and talk about their thinking. KS1 question stems:

Can you show me with the objects?

Can you draw a picture to help you work it out?

What number sentence matches what you did?

Can you explain how you got your answer?

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Why this study matters

Y2 extends fraction understanding from halves and quarters to include thirds, and from unit fractions to non-unit fractions (2/4, 3/4). The critical new idea is that 2/4 is equivalent to 1/2 -- this is the first encounter with fraction equivalence. Fractions of quantities (1/3 of 12) connect fractions to division. Concrete folding and sharing remain essential, but pupils now also work with fraction notation and begin to reason about the relationship between fractions.

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Pitfalls to avoid

Thinking fractions only apply to shapes, not to quantities -- practise fractions of sets of objects alongside shapes

Not understanding that 2/4 and 1/2 are the same amount -- use fraction walls and overlapping fraction strips

Finding 1/3 of a quantity by splitting into unequal groups -- teach systematic sharing one at a time

Reading 3/4 as 'three fours' rather than 'three quarters' -- model correct fraction language consistently

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Mathematical reasoning skills (KS1)

These disciplinary skills should be woven through teaching, not taught in isolation:

Checking and verifying results — Use inverse operations, estimation or an alternative method to check whether a result is reasonable, and adjust working when an answer does not make sense in context.

Mathematical proof — Understand and apply the concept of mathematical proof, distinguishing between evidence, conjecture and proof, constructing simple proofs by exhaustion or direct argument, and recognising why a finite number of examples cannot prove a universal statement.

Identifying and describing patterns — Spot numerical and spatial patterns, describe the rule that generates a sequence, and use the rule to predict further terms, providing the foundation for algebraic generalisation.

Critical evaluation and error analysis — Critically evaluate the validity of mathematical arguments and solutions presented by others, identifying errors in reasoning or calculation, explaining why a result is or is not correct, and constructing counter-examples to disprove false claims.

Algebraic and procedural fluency — Manipulate algebraic expressions, formulae and equations accurately and efficiently, applying learned procedures to a wide range of numerical and symbolic contexts, including working with negative numbers, surds, indices and standard form.

Problem solving in varied and unfamiliar contexts — Apply mathematics to solve multi-step problems presented in a range of contexts, breaking problems into manageable parts, selecting appropriate representations and methods, and interpreting results in relation to the original problem.

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Vocabulary word mat

Prior knowledge (retrieval plan)

Pupils should already know the following from earlier units:

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Scaffolding and inclusion (Y2)

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Access and Inclusion

Likely barriers

This study has high demands on: Vocabulary Novelty (Fraction vocabulary introduces 'numerator', 'denominator', 'third', 'quarter', 'equal parts', and 'fraction of' — all new mathematical terms that describe relationships rather than objects.), Abstractness Without Concrete Anchor (Fractions as 'fractions of' (1/3, 1/4, 2/4, 3/4) require understanding part-whole relationships with multiple different denominators. Each denominator represents a different equal-sharing scenario. Without extensive concrete partitioning experience, the notation is meaningless.).

Universal supports

Apply by default for all learners:

Vocabulary Pre-Teaching — Explicitly teaching key vocabulary before the main lesson begins, so that unfamiliar terms do not block access to the concept. Pre-teaching uses the define-show-use-check pattern: define the word simply, show it in context with visual support, use it in a sentence, then check the child can use it themselves. Typically targets 2-4 key words per session.

Visual Supports — Providing visual representations alongside or instead of verbal/written information: icons, diagrams, picture cues, symbol-supported text, visual timetables, and graphic organisers. Visual supports make abstract information concrete and persistent (the child can refer back to them), reducing reliance on auditory processing and transient memory.

Targeted options

Simplified Language Wrapper — Rewriting task instructions, questions, and explanations using simpler sentence structures, shorter sentences, and more common vocabulary — while preserving the full complexity of the underlying concept. The mathematical, scientific, or literary idea is not simplified; only the language surrounding it is made more accessible. This requires careful judgement about which words are domain-essential (keep) versus incidental complexity (simplify). (targets: Vocabulary Novelty)

Word Bank — Providing a curated set of words the child may need during a writing or response task, displayed persistently on screen. This offloads spelling from working memory, allowing the child to focus on content, sentence structure, and ideas. The word bank contains domain-specific vocabulary, connectives, and high-frequency words the child is known to struggle with. (targets: Vocabulary Novelty)

Adaptive Difficulty Stepping — Using the DifficultyLevel data to present tasks at a level matched to the child's current attainment, stepping up only when the child demonstrates readiness. For a child working at 'entry' level while peers are at 'expected', this means presenting entry-level tasks with the option to progress — never assuming the child should start where their year group expects. The DifficultyLevel descriptions, example_tasks, and common_errors drive the adaptive presentation. (targets: Abstractness Without Concrete Anchor)

Worked Example First — Showing a fully worked example of the type of task the child will be asked to complete before they attempt their own. The worked example is annotated to show the thinking process, not just the answer. This reduces the cognitive load of figuring out both WHAT to do and HOW to do it simultaneously. Particularly effective for procedural tasks in maths and structured writing in English. (targets: Abstractness Without Concrete Anchor)

Concrete Manipulatives (Extended) — Maintaining access to physical or on-screen manipulatives beyond the point where the curriculum typically moves to pictorial or abstract representation. Some children with dyscalculia or learning difficulties need to remain at the concrete stage significantly longer than their peers. This is a pedagogically valid position — concrete understanding IS mathematical understanding, not a lesser version of it. (targets: Abstractness Without Concrete Anchor)

Use with caution

Simplified Language Wrapper — construct risk: conditional. Unsafe when assessing: language_load

Word Bank — construct risk: conditional. Unsafe when assessing: vocabulary_novelty

Concrete Manipulatives (Extended) — construct risk: conditional. Unsafe when assessing: abstractness_without_concrete_anchor

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Knowledge organiser

Core facts (expected standard):

Fractions as 'fractions of': 1/3, 1/4, 2/4, 3/4: Finding unit and non-unit fractions of quantities and lengths, and counting in fractions on a number line.

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Graph context

Node type: MathsTopicSuggestion | Study ID: MTS-KS1-011 Concept IDs:

MA-Y2-C012: Fractions as 'fractions of': 1/3, 1/4, 2/4, 3/4 (primary)

MA-Y2-C013: Fraction equivalence: 2/4 = 1/2

Cypher query:

``cypher

MATCH (ts:MathsTopicSuggestion {suggestion_id: 'MTS-KS1-011'})

-[:DELIVERS_VIA]->(c:Concept)

-[:HAS_DIFFICULTY_LEVEL]->(dl)

RETURN c.name, dl.label, dl.description

``

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Generated from the UK Curriculum Knowledge Graph — zero LLM generation.