Similarity and scale: one shape at different sizes

Where we are on the map

We move from identical figures to the same shape at different sizes.

Begin with a familiar image

A screen image and a poster keep one shape when width and height are multiplied by the same factor.

Precise meaning

Similar figures have equal corresponding angles and proportional sides. Under factor k, lengths scale by k, areas by k², and volumes by k³.

The lesson’s support signal

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

match vertices → equal angles → one k → lengths k → areas k² → volumes k³

Worked example

For 3,4,5 and 6,8,10, every ratio is 2; perimeter doubles and area quadruples.

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

Comparing noncorresponding sides is wrong. Establish the vertex order and verify one factor across every pair.

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. A 1:50 model door is 4.2 cm high. Find its real height and the area ratio of similar panels.

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

Area and volume explain why formulas count unit squares and cubes.

Course contents