Proportion

Activities to try

1. 1

   Proportion lab: drive one quantity, watch the other

   Why it works: Quantity B's response to driving quantity A reveals the structure: in direct proportion the ratio holds (and the line goes through the origin), in inverse proportion the product holds (one goes down as the other goes up), and in linear-but-not-proportional situations a fixed charge means the line misses the origin so doubling fails. Proportionality is a property to test, not a default to assume.

   1. Work with three switchable scenarios: pens & cost (4 pens £6 → 10 pens £15, directly proportional); taps filling a bucket or workers digging a hole (inversely proportional); and a phone plan of £15/month + £0.20/min (linear but NOT proportional).
   2. Before changing anything, PREDICT where the next point will land on the graph and what the paired value should be — write or say the prediction and your reason first.
   3. Change quantity A (e.g. with a slider or by choosing a new value) and watch quantity B recompute, with the point plotting live on a graph while a ratio table fills in alongside.
   4. In the phone-plan scenario, double the minutes while predicting the cost will also double; observe that the plotted point lands below your prediction and the traced line clearly does NOT pass through the origin.
   5. In the inverse scenario, halve the number of taps and observe the fill time double, with the curve bending away from the straight line you expected.
2. 2

   Sort situations into direct, inverse, or neither

   Why it works: Classifying situations and matching them to graphs forces the discrimination that defines the topic: not every straight line is proportional (it must pass through the origin), and additive scaling is not the same as multiplicative scaling.

   1. Sort six real-world situations into three bins: directly proportional, inversely proportional, or neither.
   2. For each, justify the placement with a test: for direct proportion, check that the ratio stays constant (so doubling one doubles the other, and zero gives zero); for inverse, check that the product stays constant (so doubling one halves the other). 'Zero gives zero' alone is not enough: a square's side and its area both start at zero, but they are not proportional.
   3. Choose which of several graphs shows direct proportion, deliberately comparing a straight line through the origin against a straight line that misses the origin.
   4. Complete a ratio table for a directly proportional situation by multiplying (scaling columns), and check whether an answer found by adding a fixed difference gives the same result.
3. 3

   Reference triptych: three graphs, three tests

   Why it works: For straight-line graphs, origin behaviour is a quick diagnostic: through the origin means direct proportion, and missing the origin means linear but not proportional. A decreasing curve should prompt a check of whether the product stays constant, because not every decreasing curve shows inverse proportion.

   1. Study a labelled set of three graphs side by side: a straight line through the origin (direct), a smooth decreasing curve (inverse), and a straight line that misses the origin (linear but not proportional).
   2. Reveal, for each graph, its defining property — 'does the line start at (0,0)?' for the lines and 'does the product stay the same?' for the curve.
   3. Use the triptych as a consolidation reference while classifying new situations, checking origin behaviour as a fast first diagnostic.