Separate Length Scaling from Area Scaling

Supplied materials

• Paper or a note app.
• Supplied rectangle: 4 by 7 units, enlarged by length factor 3. Perimeter = 2(width + length); area = width × length.

Learn the idea

Perimeter adds lengths. Multiplying each side by three multiplies the entire boundary length by three. The original perimeter is 2(4 + 7) = 22, and the enlarged sides 12 and 21 give perimeter 66.

Area multiplies two lengths. Scaling both by three introduces two factors of three, so area grows by nine. Keep square units for area and ordinary units for perimeter. A length factor cannot be applied once to area because both dimensions changed.

1. Compute the original perimeter 22 and area 28.
2. Scale both side lengths to 12 and 21.
3. Calculate the new perimeter and area; compare factors 3 and 9, then try the rectangle below.

Check your reasoning

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Take it further

Shrink the original 4-by-7 rectangle by factor 1/2. Its sides are 2 and 3.5; perimeter is 11 units and area 7 square units, half the perimeter but one quarter the area.

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