Ratio Notation and Relationships

Activities to try

1. 1

   One ratio, three costumes — and what breaks under 'add 4'

   Why it works: A ratio is a single multiplicative idea wearing three costumes (ratio a:b, fraction a/b, straight line y=kx through the origin); its signature is a constant multiple, not a constant difference. When the quantities scale, the difference changes but the multiple — and the colour, the ratio and the gradient — stay the same. An additive rule breaks the line-through-origin and changes the shade, which is what separates it from a ratio.

   1. Work with the ratio 1:3 (e.g. red:yellow paint) and extend a ratio table: 1:3, 2:6, 3:9, 4:12, scaling both quantities together.
   2. For each new row, plot the point (x,y) on a coordinate graph and note that the points fall on a straight line passing through the origin (0,0).
   3. Observe a perceptible constant-ratio context alongside: in an idealised, thoroughly mixed model, a paint swatch mixed in that ratio keeps the same orange shade as the batch grows.
   4. Name the same relationship in three forms on this one example: ratio 1:3, fraction/multiple (y is 3 times x; x is 1/3 of y), and linear function y=3x with gradient 3 (the gradient equals y÷x).
   5. Switch the rule from multiplicative (y=3x) to additive (y=x+4) and watch what changes: the graph's line no longer passes through the origin, the plotted points leave the straight proportional line, and the swatch shade shifts row by row.
2. 2

   Read the ratio table both ways

   Why it works: A ratio table encodes both a scalar multiplier (down columns) and a functional multiplier (across, the y=kx rule); the multiple can be used directly to predict any value. A double number line aligned at zero shows equal ratios lining up. Part-to-part and part-to-whole are different comparisons.

   1. Take a ratio table and read DOWN the columns to find the scalar multiplier (how each row scales from the previous).
   2. Read ACROSS the rows to find the constant functional relationship a:b / the y=kx rule between the two quantities.
   3. Use a double number line aligned at 0 so that equal ratios line up, and check that both lines start at aligned zeros.
   4. Given x=10 for a 1:3 relationship, work out y directly using the multiplier (×3) rather than extending the table row by row.
   5. Distinguish 'x is 1/3 OF y' (part-to-part, the correct multiplicative reading) from 'x is a fraction of the whole x+y' (part-to-whole); for example, with 2 red : 5 blue beads, red is 2/5 of the blue count (part-to-part) but 2 out of 7 total beads (part-to-whole), anchoring each reading to its own picture.
3. 3

   Sort and match: is it a ratio relationship?

   Why it works: Sorting and matching across forms, with deliberate additive distractors, can check whether a learner can translate among ratio, fraction and linear form and can reject an additive table as not proportional. A straight-line graph only shows proportionality when it passes through the origin.

   1. Sort several tables of values into 'ratio relationship' vs 'not a ratio relationship', justifying each by whether y÷x is constant.
   2. Match equivalent forms: connect 1:3 with y=3x and with a straight line through the origin.
   3. Predict y for a new x in a given table, with an additive answer offered as a tempting but wrong option, and explain why the additive choice fails.
   4. Given two graphs, identify which shows proportionality and explain that it must pass through the origin.
   5. Test a tricky table — x=2,y=5 and x=4,y=7 — and decide whether it is a ratio relationship (it is additive: +3 each time, but 5÷2 ≠ 7÷4, so no constant multiple).