Ratios and Proportions: Comparing by Division, Not Difference

A ratio compares two quantities by division, not by subtraction — and that one sentence is the whole topic. If a recipe uses 2 cups of flour to 3 cups of milk, the ratio 2:3 stays true for 4:6, 20:30 and 1:1.5. What stays constant is not the gap, it is the multiplier.

Why children get this wrong. Asked "if 2 pencils cost 60¢, what do 5 cost?", many children add: 60 + 3 = 63. They are using difference where they need scaling. The fix is to make them find the one first: one pencil is 30¢, so five are 150¢. Unit rate first, always.

A proportion is two equal ratios. 2/3 = 8/12. Solve it by asking "what did the top get multiplied by?" — not by cross-multiplying, which works but teaches nothing at this age.

Percentages are ratios out of 100, which is why they are so useful for comparison: 17 out of 20 and 43 out of 50 are impossible to compare until you make them 85% and 86%. Start with 10% (divide by ten) and 50% (halve), then build the rest: 30% is three lots of 10%, 25% is half of half.

Where ratios really live. Recipes scaled up for guests, map scales, mixing paint, gears, speed, exchange rates and every price-per-unit label in a supermarket. Ratio is the most-used maths topic in adult life and one of the least practised in primary school.

Continue with division facts (unit rate is division), math word problems, customary units and games for 5th graders.