Proportion and scale: which relationship must stay fixed?

Where we are on the map

We move from equivalent ratios to finding an unknown quantity.

Begin with a familiar image

If 2 cm on a map represents 3 km, then 6 cm at the same scale represents 9 km.

Precise meaning

In direct proportion both quantities change by the same factor; in inverse proportion their product remains constant.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

what stays fixed? → direct or inverse? → units → calculate → check meaning

Worked example

2/3 = 6/9 because both values tripled; 2 × 9 = 3 × 6 provides a check.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Saying twice as many equally productive workers need twice as much time is wrong: workers × time stays constant.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Three litres of concentrate mix with 12 litres of water. Find each component in 35 litres of the same solution.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

The checkpoint combines fractions, ratio, percent, and proportion in unfamiliar settings.

Course contents