Fractions, Decimals and Percentages

Status: Mandatory

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Concepts

This study delivers 4 primary concepts and 0 secondary concepts.

Primary concept: Simplifying and Comparing Fractions (MA-Y6-C008)

Type: Skill | Teaching weight: 3/6

Mastery of simplifying fractions means pupils can divide the numerator and denominator of any fraction by their highest common factor to express it in its simplest form, and can compare and order fractions — including improper fractions and mixed numbers — by converting to a common denominator. A fully secure pupil understands that simplification does not change the value of the fraction, only its representation, and chooses to simplify at appropriate points in calculations rather than only at the end.

Teaching guidance: 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. Key vocabulary: simplify, simplest form, lowest terms, common factor, equivalent fraction, common denominator, improper fraction, mixed number, compare, order Common misconceptions: 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.

Differentiation

Model response (Entry): 8/12 = 4/6 = 2/3 (dividing by 2 twice) or 8/12 = 2/3 (dividing by 4, the HCF).

Model response (Developing): LCM of 8 and 12 is 24. 5/8 = 15/24. 7/12 = 14/24. 15/24 > 14/24, so 5/8 is larger.

Model response (Expected): Convert all to decimals: 3/4 = 0.75, 0.7, 2/3 = 0.666..., 0.72. Order: 2/3, 0.7, 0.72, 3/4.

Representation stages (CPA)

Primary concept: Addition and Subtraction of Fractions with Different Denominators (MA-Y6-C009)

Type: Skill | Teaching weight: 3/6

Mastery means pupils can add and subtract any pair of fractions — including those with different denominators and mixed numbers — by converting to equivalent fractions with a common denominator, and can simplify the result. A fully secure pupil applies this fluently in both calculation and problem-solving contexts, chooses the lowest common multiple of the denominators as the common denominator for efficiency, and manages the whole-number and fractional parts of mixed numbers correctly during subtraction.

Teaching guidance: 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. Key vocabulary: common denominator, equivalent fraction, lowest common multiple, mixed number, improper fraction, simplify, denominator, numerator Common misconceptions: 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.

Differentiation

Model response (Entry): 2 + 1 = 3. 3/7 + 5/7 = 8/7 = 1 1/7. Total: 3 + 1 1/7 = 4 1/7.

Model response (Developing): LCM of 6 and 9 is 18. 5/6 = 15/18. 2/9 = 4/18. 15/18 – 4/18 = 11/18.

Model response (Expected): 2 1/3 – 1 3/4. Convert: 2 4/12 – 1 9/12. Borrow 1 from 2: 1 16/12 – 1 9/12 = 7/12 cup more flour.

Representation stages (CPA)

Primary concept: Multiplication and Division of Fractions (MA-Y6-C010)

Type: Skill | Teaching weight: 3/6

Mastery of fraction multiplication means pupils can multiply a pair of proper fractions by multiplying the numerators and the denominators, simplify the result, and understand the result is smaller than either factor (when both are proper fractions). For division of a fraction by a whole number, mastery means pupils understand that dividing a fraction by n is the same as multiplying by 1/n (or equivalently, multiplying the denominator by n) and can apply this in context.

Teaching guidance: 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. Key vocabulary: multiply fractions, divide fractions, product, proper fraction, simplify, cross-cancel, reciprocal Common misconceptions: 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.

Differentiation

Model response (Entry): 2 × 4 = 8. 3 × 5 = 15. Answer: 8/15.

Model response (Developing): 3/4 ÷ 2 = 3/(4×2) = 3/8.

Model response (Expected): Cross-cancel: 3 and 9 share factor 3 (3→1, 9→3). 4 and 8 share factor 4 (4→1, 8→2). So 1/2 × 1/3 = 1/6. The answer is smaller because you are taking a fraction of a fraction — a part of a part is smaller than either part.

Representation stages (CPA)

Primary concept: Fraction-Decimal-Percentage Equivalences (MA-Y6-C011)

Type: Knowledge | Teaching weight: 2/6

Mastery means pupils can fluently convert between fractions, decimals and percentages for a wide range of values, including less common equivalences such as 1/8 = 0.125 = 12.5%, and can select the most appropriate representation for a given context without prompting. A fully secure pupil understands that the three representations are different ways of expressing the same proportion and can use this understanding to compare quantities expressed in different forms and to solve proportion problems.

Teaching guidance: 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. Key vocabulary: fraction, decimal, percentage, equivalent, convert, proportion, per cent, decimal point, recurring decimal Common misconceptions: 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.

Differentiation

Model response (Entry): 3/4 = 0.75 = 75%.

Model response (Developing): 3/8 = 0.375. 0.2 = 1/5. 2/5 = 40%.

Model response (Expected): 1/3 = 33.3...% ≈ 33.3%. This is more than 30%. So a 1/3 discount is better for the customer.

Model response (Greater Depth): Convert to decimals: 0.375, 2/5 = 0.4, 37% = 0.37, 3/8 = 0.375. Order: 37% (0.37), then 0.375 and 3/8 (equal, both 0.375), then 2/5 (0.4). I used decimals because they are easiest to compare digit by digit. Fractions would need a common denominator of 200, which is harder.

Representation stages (CPA)

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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: Fraction-decimal-percentage equivalence is fundamentally about representing the same proportion in three different notations — fluent conversion relies on understanding that 3/4, 0.75 and 75% all describe the same part-whole relationship. Question stems for KS2:

How many times bigger is this than that?

What fraction of the whole is this part?

Which unit of measurement fits best here? Why?

If we doubled the amount, what would change?

Secondary lens: Patterns — Multiplying fractions follows a clean pattern (multiply numerators, multiply denominators) and dividing by a fraction follows the keep-change-flip pattern — these regular algorithmic rules are the operational backbone of the cluster.

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Session structure: Pattern Seeking + Worked Example Set

This study uses 2 vehicle templates:

Pattern Seeking (main structure)

Enquiry focused on identifying relationships and regularities in data. Pupils pose questions about possible correlations, gather data through observation or measurement, organise and represent data graphically, identify patterns, and attempt to explain the underlying relationship.

question → data_gathering → graphing → pattern_identification → explanation Assessment: Data presentation with appropriate graph or chart, written description of the pattern found, and explanation of the possible reasons for the pattern, including evaluation of the strength of evidence. Teacher note: Use the PATTERN SEEKING template: pose a question that pupils investigate by collecting data and looking for relationships. Guide them to gather data systematically, present it in tables or graphs, and describe any patterns they find. Encourage them to suggest explanations for the patterns and consider whether the pattern always holds true. KS2 question stems:

What data do we need to collect to answer this question?

What does the graph or table show? Can you describe the pattern?

Does this pattern always happen, or are there exceptions?

What might explain the pattern you have found?

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: activate prior knowledge and address common misconceptions. Guide pupils through the concrete-pictorial-abstract progression, modelling each step with clear mathematical language. Provide varied practice that builds fluency, then extend with reasoning problems that require pupils to explain, justify, or spot errors. Use bar models and diagrams to build conceptual understanding. KS2 question stems:

What do you already know that could help you here?

Can you draw a bar model or diagram to represent this problem?

Where has this gone wrong, and how would you correct it?

Can you explain why this method works, not just how?

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Mathematical reasoning skills (KS2)

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.

Problem solving with unfamiliar and complex structures — Formulate and solve problems that require choosing from a wide range of mathematical knowledge, devising strategies for problems with no immediately obvious method, and persevering through multi-stage solutions in unfamiliar contexts.

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.

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 reasoning and generalisation — Express generalisations symbolically using algebraic notation, reason about the properties of unknown quantities, and use algebra to prove or disprove conjectures about numbers and geometric relationships.

Estimation, checking and reasonableness — Use rounding, inverse operations and known facts to estimate answers before calculating, check the reasonableness of results in context, and identify errors in worked examples by comparing expected and actual outcomes.

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Vocabulary word mat

Prior knowledge (retrieval plan)

Pupils should already know the following from earlier units:

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Assessment alignment (KS2)

KS2 test framework content domain codes assessed by this study:

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Scaffolding and inclusion (Y6)

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Knowledge organiser

Core facts (expected standard):

Simplifying and Comparing Fractions: Ordering a set of fractions, decimals and mixed numbers by converting to a common form, and simplifying complex fractions.

Addition and Subtraction of Fractions with Different Denominators: Adding and subtracting mixed numbers with different denominators, simplifying results, in problem contexts.

Multiplication and Division of Fractions: Multiplying proper fractions with simplification (cross-cancellation), dividing fractions by whole numbers, and understanding that multiplying proper fractions gives a smaller result.

Fraction-Decimal-Percentage Equivalences: Converting any fraction, decimal or percentage and selecting the most useful form for a given context.

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Graph context

Node type: MathsTopicSuggestion | Study ID: MTS-Y6-003 Concept IDs:

MA-Y6-C008: Simplifying and Comparing Fractions (primary)

MA-Y6-C009: Addition and Subtraction of Fractions with Different Denominators (primary)

MA-Y6-C010: Multiplication and Division of Fractions (primary)

MA-Y6-C011: Fraction-Decimal-Percentage Equivalences (primary)

Cypher query:

``cypher

MATCH (ts:MathsTopicSuggestion {suggestion_id: 'MTS-Y6-003'})

-[: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.