Ratio, Proportion and Rates of Change

Teach conversion as multiplication or division by powers of 10, using a conversion staircase or table showing the relationships between units. For length: mm → cm → m → km with factors of 10, 100 and 1000. Extend to area units (1 m² = 10,000 cm²) and volume units (1 m³ = 1,000,000 cm³) by reasoning about the number of dimensions. Include time conversions (which are not base-10) and conversions between metric and common imperial units. Use dimensional analysis for compound units.

Pupils commonly multiply when they should divide (or vice versa) — particularly converting from smaller to larger units. Area and volume conversions cause more difficulty because pupils apply linear conversion factors instead of squaring or cubing them (e.g., thinking 1 m² = 100 cm² instead of 10,000 cm²). Time conversions are error-prone because they are not base-10.

Secondary maths concept — abstract, procedural, and objectively assessable.

Use maps and architectural plans as concrete starting points: measure distances on the map and use the scale to find real distances. Introduce scale factor as a multiplier — a scale of 1:50000 means every 1 cm on the map represents 50000 cm (500 m) in reality. Progress to using scale factors to enlarge and reduce shapes. Practise creating scale drawings with a given scale factor. Connect to ratio work: a scale of 1:5 is the same as the ratio 1:5. Include contexts where pupils must choose an appropriate scale for a drawing.

Pupils often confuse which way the scale factor works — multiplying by the scale factor to go from real to model instead of the reverse. When a map scale is given as 1:50000, pupils may not convert between centimetres on the map and metres or kilometres in reality. Some pupils apply the scale factor to areas without squaring it, or to volumes without cubing it.

Secondary maths concept — abstract, procedural, and objectively assessable.

Use bar models and fraction walls to visualise one quantity as a fraction of another. Start with straightforward examples: 'What fraction of 20 is 5?' (5/20 = ¼). Progress to improper fractions: 'Express 30 as a fraction of 20' (30/20 = 3/2). Connect to ratio — expressing one quantity as a fraction of another is closely related to finding the ratio between them. Include decimal and contextual examples: 'A 750 ml bottle is what fraction of a 2-litre bottle?' Connect to percentage work: expressing as a fraction is the first step towards expressing as a percentage.

Pupils often put the quantities the wrong way round — dividing the whole by the part instead of the part by the whole. When the answer is an improper fraction, pupils may think they have made an error because 'fractions should be less than 1'. Some pupils do not simplify their answers, leaving fractions in non-simplified form.

Secondary maths concept — abstract, procedural, and objectively assessable.

Introduce ratio through practical sharing activities: 'Share 20 counters between two people in the ratio 3:2.' Use bar models as the primary representation — each part of the ratio corresponds to one bar, and the total is divided into equal parts. Teach simplifying ratios using HCF, connecting to fraction simplification. Practise converting between ratio forms (part:part and part:whole) and between ratios and fractions. Include contexts: recipes, mixing paint colours, map scales. Emphasise that a ratio shows a multiplicative relationship, not additive.

Pupils often treat ratio as additive — thinking that increasing a 2:3 recipe by 4 means adding 4 to each part (giving 6:7) rather than multiplying (giving 8:12). Some pupils confuse ratio and fraction, thinking 2:3 means ⅔ of the total rather than ⅖. When simplifying ratios, pupils may divide the parts by different numbers. Mixing up part:part and part:whole ratios is also very common.

Secondary maths concept — abstract, procedural, and objectively assessable.

Use bar models to represent the total quantity divided into parts according to the ratio. For part:part ratios, find the total number of parts first, then find the value of one part (total ÷ number of parts), then multiply to find each share. For part:whole ratios, the 'whole' bar is one of the given parts. Present multi-step problems: 'Orange squash is mixed in the ratio 1:4 concentrate to water. How much concentrate is needed for 2 litres of squash?' Progress to three-part ratios. Emphasise the unitary method — finding the value of one part first.

Pupils often divide the quantity by the ratio numbers directly (e.g., dividing £60 in ratio 2:3 as £60 ÷ 2 = £30 and £60 ÷ 3 = £20, giving £50 not £60). Some pupils add the ratio parts incorrectly. Others confuse part:part with part:whole — if told '¾ are girls', they may set up a ratio of 3:4 (girls:total) instead of 3:1 (girls:boys).

Secondary maths concept — abstract, procedural, and objectively assessable.

Contrast additive and multiplicative relationships using side-by-side examples: 'If I add 3 to the length, I add 3 to the width' (additive) versus 'If I double the length, I double the width' (multiplicative). Show that multiplicative relationships can be expressed as ratios (the ratio stays constant) or as fractions (one quantity is always the same fraction of the other). Use double number lines and ratio tables to represent these relationships. Connect to direct proportion — when two quantities have a constant multiplicative relationship, they are directly proportional.

The fundamental misconception is applying additive reasoning to multiplicative situations. For example, when told that 3 pencils cost 45p, pupils may calculate 5 pencils as 45p + 30p = 75p (adding 2 × 15p) rather than 5 × 15p = 75p. Some pupils cannot distinguish between situations requiring additive versus multiplicative reasoning without explicit scaffolding.

Secondary maths concept — abstract, procedural, and objectively assessable.

Use concrete examples to show the same relationship expressed as a ratio and as a fraction: 'The ratio of blue to red counters is 2:3' means '⅖ of the counters are blue and ⅗ are red'. Use conversion exercises where pupils move between the representations. Show that the relationship can also be expressed as a linear function: if b:r = 2:3, then b = ⅔r. Plot this function on a graph to show the proportional relationship visually. Include percentage conversions as a third representation of the same relationship.

Pupils often struggle to convert between ratio and fraction because they confuse part:part ratio with part:whole fraction. For a ratio 2:3, pupils may write the fraction as ⅔ (which gives the relationship between the parts) rather than ⅖ or ⅗ (which give the part-to-whole fractions). The connection to linear functions is new and abstract — pupils may not see why y = ⅔x represents the same relationship as 2:3.

Secondary maths concept — abstract, procedural, and objectively assessable.

Teach percentage increase and decrease using the multiplier method: a 15% increase means multiplying by 1.15, a 15% decrease means multiplying by 0.85. Use bar models to show why: the original amount is 100%, so increasing by 15% gives 115% = 1.15. For reverse percentage problems ('After a 20% discount, the price is £48. What was the original price?'), emphasise that £48 represents 80% of the original, so the original is £48 ÷ 0.80. Include simple interest calculations in financial contexts.

The most common error in reverse percentage problems is adding/subtracting the percentage from the new amount — for example, if a price after a 20% increase is £60, pupils calculate the original as £60 - 20% of £60 = £48, rather than £60 ÷ 1.20 = £50. Pupils also frequently confuse percentage increase with finding a percentage of a quantity.

Secondary maths concept — abstract, procedural, and objectively assessable.

Introduce through real-world contexts: currency exchange, recipe scaling, speed-distance-time. Use double number lines and ratio tables to organise proportional reasoning. Establish the unitary method: find the value for one unit, then scale to the required amount. Introduce the algebraic representation y = kx, where k is the constant of proportionality. Plot directly proportional relationships on a graph and note they always pass through the origin and are straight lines. Compare with examples that look similar but are not directly proportional.

Pupils often assume any relationship between two variables is directly proportional. They may not check whether the graph passes through the origin, which is a requirement for direct proportion. Some pupils confuse direct proportion with a linear relationship that has a non-zero y-intercept (y = mx + c where c ≠ 0). The unitary method is sometimes applied where it is not valid.

Secondary maths concept — abstract, procedural, and objectively assessable.

Use practical contexts: if 2 workers take 12 hours to build a wall, 4 workers take 6 hours (as workers double, time halves). Contrast with direct proportion to highlight the difference: in direct proportion, both quantities increase together; in inverse proportion, as one increases the other decreases. Introduce the algebraic representation y = k/x and show the characteristic curve on a graph (a reciprocal curve). Use tables of values to check whether a relationship is inversely proportional by showing that the product xy remains constant.

Pupils often confuse inverse proportion with negative correlation or with simply decreasing values. Some pupils think inverse proportion means 'opposite' — that y = -kx is inverse proportion. The key diagnostic is whether the product xy is constant (inverse proportion) versus the ratio y/x being constant (direct proportion). Pupils may also struggle with the graphical representation, expecting a straight line.

Secondary maths concept — abstract, procedural, and objectively assessable.

Introduce speed as the first compound unit: distance ÷ time. Use the formula triangle (distance, speed, time) to practise all three rearrangements. Progress to density (mass ÷ volume) and unit pricing (cost ÷ quantity). Emphasise units: speed in m/s or km/h, density in g/cm³ or kg/m³. Use real-world contexts: comparing prices per kilogram in a supermarket, calculating journey times. Practise converting between units of compound measures (e.g., km/h to m/s). Connect to ratio and proportion — compound units express rates.

Pupils commonly confuse which quantities to divide when calculating compound measures — for example, calculating density as volume ÷ mass instead of mass ÷ volume. Unit conversions within compound measures cause additional errors. Some pupils cannot reconcile different units of speed (mixing up m/s and km/h). The idea that a compound unit combines two different measurements into one is conceptually challenging.

Secondary maths concept — abstract, procedural, and objectively assessable.

Use real-world financial contexts: bank accounts, saving for a purchase, comparing deals, understanding bills and payslips. Teach simple interest using the formula I = PRT/100 and compare with compound interest conceptually. Include percentage increase and decrease in financial contexts: VAT, discounts, profit and loss. Practise 'best buy' problems comparing unit prices. Introduce budgeting activities where pupils must make financial decisions within constraints. Emphasise that financial mathematics is an application of percentage and ratio skills, not a separate topic.

Pupils often confuse simple interest (calculated on the original amount) with compound interest (calculated on the accumulated amount). Some pupils add VAT by calculating the percentage and then forgetting to add it to the original price. Others struggle with reverse percentage problems in financial contexts. The concept of percentage profit being calculated on the cost price (not the selling price) is frequently misunderstood.

Secondary maths concept — abstract, procedural, and objectively assessable.

Present paired problems where the same surface structure masks different underlying structures: 'Tom has 3 more sweets than Jill' (additive) versus 'Tom has 3 times as many sweets as Jill' (multiplicative). Use bar models to make the structural difference visible. Teach pupils to ask: 'Am I comparing by adding/subtracting or by multiplying/dividing?' Proportional reasoning extends multiplicative reasoning to situations involving two variables: if one doubles, does the other double (direct proportion) or halve (inverse proportion)? Include problems that require identifying the type of reasoning before calculating.

The most fundamental misconception at KS3 is applying additive reasoning to proportional situations. For example: 'If 3 workers finish in 12 days, 6 workers will finish in 9 days' (subtracting 3) rather than 6 days (halving). The bar model is the most effective tool for diagnosing and correcting this error. Some pupils also apply proportional reasoning to additive situations, multiplying when they should add.

Secondary maths process concept — problem-solving benefits from structured AI delivery with facilitator for extended reasoning.