Proportional Reasoning Vocabulary

Activities to try

1. 1

   Talk it through with a market-stall trader (or science reporter)

   Why it works: A term is pinned to the behaviour it names — you decide 'direct' vs 'inverse' by checking whether the quantities move together in a constant ratio (direct) or oppositely with a constant product, so doubling one halves the other (inverse), not by spotting the word 'more'. If neither holds, the relationship is not proportional. A rate is incomplete without its units.

   1. Respond in-role to a character who throws real situations at you and listens for the right word: '3 apples for £1 — what do we call this, and what are its units?' (target: it is a rate; its unit rate is 'pounds per apple', found by sharing £1 across 3 apples ≈ £0.33 per apple — or equivalently 3 apples per pound).
   2. When you label 'more workers, fewer hours' as direct, answer the follow-up: 'I add two workers and the time drops — are those two quantities moving the SAME way or opposite?' and revise to 'opposite — and if every worker works at the same rate, doubling the workers halves the time (workers × hours stays constant), so it is INVERSE proportion.'
   3. Describe at least one directly proportional situation, one inverse, and name a rate aloud WITH its units (e.g. '120 miles in 2 hours is 60 miles per hour — a rate/speed, miles per hour').
   4. Say a full sentence that explains WHY, e.g. 'as the petrol increases, the cost increases with it in a constant ratio, so it is direct proportion.'
2. 2

   Sort-and-tag: direct / inverse / neither, with unit tags

   Why it works: The boundary between direct, inverse, and non-proportional situations becomes visible through contrasting examples and non-examples; every rate carries units.

   1. Sort situation cards into three bins: direct proportion, inverse proportion, and neither (non-proportional).
   2. Attach the correct unit tag to each rate card (speed = mph or m/s; density = kg/m³).
   3. Complete sentence frames using the vocabulary, e.g. 'as … increases, … so this is … proportion.'
   4. Read the feedback that names the tell ('both increase together in a constant ratio → direct') and re-sort any you placed wrongly.
3. 3

   Frayer-style term map (reference and consolidation)

   Why it works: Each term has a fenced meaning with clear examples, non-examples, and (for rates) units; 'proportion' and 'scale factor' have specific mathematical senses distinct from loose everyday use.

   1. For each term (direct proportion, inverse proportion, ratio, rate, unit rate, compound unit, scale factor), fill a four-box map: definition, characteristics, an example, and a non-example.
   2. Add the units for each rate/compound measure (speed: mph or m/s; density: kg/m³).
   3. Explicitly contrast the everyday sense of 'proportion' (an amount) with the mathematical sense (a multiplicative part-to-whole relationship / constant ratio).
   4. Note that 'scale factor' is a MULTIPLIER, not an additive change.