Proportional Reasoning-1 · Lesson 1 of 8
Observing Similarity in Change
“Discover proportional change by comparing resized images and distinguishing a common scale factor from equal additions or subtractions.”
• Compare the width and height of resized images. • Recognise a common multiplying factor. • Explain why equal subtraction may distort a picture. • Predict a missing dimension when resizing.
Have you ever dragged the edge of a photograph and made a person look unusually tall or wide? The picture may still fit inside a rectangle, but something about its shape has changed. A smaller photograph can look perfectly natural, while a larger one can look distorted. The important question is not simply how much space the image occupies. It is whether its width and height change together in the right way.
You will compare measurements before learning a new notation. Keep asking two questions: what happened to the width, and did exactly the same multiplying change happen to the height?
7.1 Observing Similarity in Change
Imagine five versions of the same tiger image. A, C and D retain the tiger’s proportions. B looks elongated, while E looks compressed and wider. Being rectangular is not enough to explain the difference: A and B are both rectangles.
Measure the horizontal width and vertical height consistently. Use the table to compare corresponding dimensions. The entries describe the same original image resized without cropping; matching the frame ratio would not by itself guarantee matching content if the picture had been cropped.
| Image | Width (mm) | Height (mm) |
|---|---|---|
| A | 60 | 40 |
| B | 40 | 20 |
| C | 30 | 20 |
| D | 90 | 60 |
| E | 60 | 60 |
From A to C, the width changes from 60 mm to 30 mm: it is multiplied by one half. The height changes from 40 mm to 20 mm: it is also multiplied by one half. Every horizontal and vertical distance shrinks consistently, preserving the image’s proportions.
From A to D, the width changes by 90 ÷ 60 = 3/2. The height changes by 60 ÷ 40 = 3/2 as well. A common factor can be greater than 1, less than 1 or fractional; the key is that both dimensions use the same factor.
For the positive quantities considered here, corresponding quantities change proportionally when they are multiplied by the same factor.
Problem
A picture is 60 mm wide and 40 mm high. Both dimensions are halved. Find its new size.
Problem
Change the 60 mm width of image A to 90 mm. What should its height become?
Now compare A and B. Both dimensions have decreased by 20 mm. However, the width factor is 40/60 = 2/3, while the height factor is 20/40 = 1/2. Those factors differ. Equal subtraction therefore does not preserve these proportions.
For E, the width remains 60 mm, so its factor is 1. Its height becomes 60 mm, so that factor is 3/2. Again the factors differ. A square frame is not a suitable proportional resize of this particular 60-by-40 image.
Problem
A 12 cm by 8 cm picture becomes 18 cm by 14 cm. Does it preserve proportions?
The instruction “make both sides 2 cm longer” is generally different from “make both sides twice as long.” Proportional resizing uses a common multiplying factor. Special cases such as squares do not make equal addition a general resizing rule.
Quiz
Which resize preserves the proportions of a 60 mm by 40 mm picture?
A width changes from 60 mm to 30 mm. What factor must also be applied to the height?
Why does subtracting 20 from both 60 and 40 distort this image?
A 10 cm by 6 cm image is enlarged to width 15 cm. What height preserves its proportions?
What should you compare when checking proportional resizing?
Practice Problems
- A 12 cm by 8 cm image is doubled. Find its new dimensions.
- A 15 cm by 10 cm image is reduced to one third of each dimension. Find the new size.
- A 20 cm by 12 cm image has its width changed to 35 cm. Find the proportional height.
- Does changing 10 cm by 6 cm to 14 cm by 10 cm preserve proportions? Explain.
- Give a smaller and a larger proportional version of an 18 cm by 12 cm rectangle. Explain your choices.
1. Multiply both dimensions by 2. 2. The new size is 24 cm by 16 cm.
1. Width = 15 ÷ 3 = 5 cm. 2. Height = 10 ÷ 3 cm, or 3⅓ cm. The fractional height is allowed.
1. Factor = 35/20 = 7/4. 2. Height = 12 × 7/4 = 21 cm.
1. Width factor = 14/10 = 7/5; height factor = 10/6 = 5/3. 2. They are unequal, so adding 4 cm to both does not preserve proportions.
1. Using factor 1/2 gives 9 cm by 6 cm. 2. Using factor 3/2 gives 27 cm by 18 cm. 3. Each version uses a common factor for both dimensions; many other correct pairs are possible.
Key Takeaways
• Proportional resizing multiplies corresponding dimensions by the same factor. • A factor greater than 1 enlarges; a positive factor below 1 reduces. • Equal additions or subtractions generally do not preserve proportions. • A rectangular frame can still contain a distorted image. • Find the factor from one known dimension and apply it to the other.
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Ratios and Their Simplest Form