7.1 Observing Similarity in Change
Most of us resize photos all the time, pinch to zoom, drag a corner to enlarge, squeeze a banner to fit a frame. Sometimes the resized picture looks completely natural; other times it ends up oddly stretched or flattened.
Picture five posters of the same elephant, each printed at a different size. Three of them, call them A, C and D, look perfectly natural. The other two, B and E, look off: in B the elephant seems stretched tall and thin, and in E it looks squat and bloated.
What causes this? Let us behave like mathematicians, measure and search for a pattern.
| Poster | Width (mm) | Height (mm) |
|---|---|---|
| A | 80 | 60 |
| B | 60 | 40 |
| C | 40 | 30 |
| D | 120 | 90 |
| E | 80 | 80 |
Math Talk Poster E is a square (80 \times 80), so maybe that is why it stands out. But B is a rectangle, exactly like the three “good” posters, and it still looks wrong. So being square cannot be the whole story. What is really happening?
Compare poster A with poster C. The width drops from 80 to 40, half. The height drops from 60 to 30, also half. Both width and height shrank by the same factor, \tfrac{1}{2}. The proportions are kept, so C reads as a smaller copy of A.
Now compare A with B. The width falls by 20 mm (80 \to 60) and the height also falls by 20 mm (60 \to 40). The difference is identical, yet look at the factors: the width became \tfrac{60}{80} = \tfrac{3}{4}, while the height became \tfrac{40}{60} = \tfrac{2}{3}. Different factors, so the elephant gets distorted.
The Key Idea Two quantities change proportionally when they are multiplied by the same factor, not when the same amount is added to or subtracted from them. Posters A, C and D look alike because their widths and heights were scaled by the same factor. Comparison by multiplication (ratio) is the core of proportional reasoning; comparison by subtraction (difference) is something else entirely.
Check poster D yourself: width 80 \to 120 is a factor of \tfrac{120}{80} = \tfrac{3}{2}, and height 60 \to 90 is also \tfrac{90}{60} = \tfrac{3}{2}. Same factor, so D is another faithful copy of A.