5 mistakes learners make with proportion and scale, each one named and explained.
In direct proportion both quantities scale by the same multiplier; in inverse proportion one goes up as the other comes down, and the product stays fixed.
3 builders take 8 days, so 4 builders are given 9 days, because one more builder means one more day.
Adding is the first relationship anybody learns, and it fits the numbers often enough to survive.
Question. 3 builders take 8 days, so 4 builders take how long
A common answer. 9 days, one more for one more builder
The answer. 6 days
Why. The work is 24 builder days however many people turn up, so four of them take 6 days each.
Proportion is about multiplying, not about adding on the difference.
A scale of 1:50000 with 8 cm on the map gives 40 km or 0.4 km instead of 4 km.
The scaling is done correctly and then centimetres have to become kilometres, which is two conversions in a row.
Question. 8 cm at a scale of 1:50000
A common answer. 40 km
The answer. 4 km
Why. 8 x 50000 = 400000 cm, and there are 100000 cm in a kilometre, so that is 4 km.
Do the scaling first, then the units, and write the unit at every step so a factor of ten cannot hide.
More builders are given more days, so 4 builders taking 9 days becomes 6 builders taking 13.5.
Almost every proportion question so far has scaled both quantities the same way, so the hand does it before the head asks.
Question. 4 builders take 9 days, so 6 builders take how long
A common answer. 13.5 days
The answer. 6 days
Why. More builders means fewer days, and the total work of 36 builder days does not change.
Ask whether the answer should go up or down before you calculate. Inverse means the product stays fixed.
The scaled amount of flour is given where the butter was asked for, or the reverse.
Both quantities are scaled in the same working and both are correct answers to something.
Question. 180 g flour and 90 g butter for 12 biscuits, how much butter for 18
A common answer. 270 g, which is the flour
The answer. 135 g
Why. The butter starts at 90 g and scales by the same 1.5.
Circle the ingredient the question names before you start scaling anything.
y is 18 when x is 4, and y when x is 10 is given as 40, which is 4 times 10.
Direct proportion does mean multiply, and the question is full of numbers that look like they could be the one to multiply by.
Question. y = 18 when x = 4, find y when x = 10
A common answer. 40, which is 4 x 10
The answer. 45
Why. The multiplier is 18 divided by 4, which is 4.5, and 4.5 x 10 is 45.
Work the multiplier out from a pair you were given. Never borrow a number that happens to be lying about.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.