Where it goes wrong
Traditional teaching
Set up a proportion, cross-multiply, solve for x. Cross-multiplication works — and hides the reasoning so completely that most students can't say why. Ratio becomes a trick to survive rather than the single most useful piece of everyday math.
The reteach
The Alice Method
Start in the kitchen. A recipe for 4 people must feed 6: everything scales together, or the pancakes taste wrong. Build ratio tables (2 cups : 3 eggs → 4 : 6 → 6 : 9) until scaling feels physical. Then introduce the unit rate — 'how much for one?' — as the universal decoder. Cross-multiplication arrives last, as a shortcut the child can already justify.
Mental model
The recipe
A ratio is a recipe: the amounts can change, but the relationship can't. Doubling everything keeps the taste; doubling only the flour ruins it. 'What's being kept constant?' is the question — and the answer (the unit rate) is a number you can find.
Transfer
Where this shows up for the rest of their life
Unit thinking runs the adult world: price per ounce, miles per gallon, revenue per user, risk per dose. And proportional reasoning is the gateway to science — speed, density, and concentration are all ratios. A child fluent in 'per' reads the world differently.
Watch for these
Common misconceptions
The misconception
“Additive thinking: 2:3 scales to 4:5 because 'both went up by 2.'”
The repair
Ratios scale by multiplying, not adding. The recipe test exposes it instantly: add 2 of everything to a recipe and taste what happens. Multiplication preserves the relationship; addition destroys it.
The misconception
“Cross-multiplication is understanding.”
The repair
It's compression. Useful once the reasoning exists, dangerous before — students who only cross-multiply can't tell when a situation isn't proportional at all (and many aren't).
Seen, not said
The visual explanation
A ratio table growing rightward — 2:3, 4:6, 6:9, 20:30 — with the unit rate highlighted as the column where one quantity equals 1. Scaling becomes a pattern you can see marching across the page.
Try it
Interactive example
Percent is ratio's most famous costume: a rate 'per hundred.' Play with the bar and the whole.
Percent, decimal, and fraction are one number wearing three outfits
Slide the shading. All three names update together — because they were never different things.
percent
30%
decimal
0.30
fraction
30/100
30% of 80 = 24