Doubling A Rectangle's Sides
Doubling a rectangle sides explores how multiplying both the length and width of a rectangle by 2 quadruples the area in Grade 8 Saxon Math Course 3. Students discover that linear scale changes have a squared effect on area, and apply this understanding to real-world situations involving scaling. This concept reinforces the distinction between linear and quadratic growth.
Key Concepts
Property Doubling the length and width of a rectangle doubles the perimeter, but it makes the area four times as large.
Examples Room A (4x5 yd) has $P=18$ yd and $A=20$ sq yd. Room B (8x10 yd) has $P=36$ yd (doubled) and $A=80$ sq yd (quadrupled). A 3x4 rectangle has $P=14$ and $A=12$. A 6x8 rectangle has $P=28$ (doubled) and $A=48$ (quadrupled).
Explanation This is a cool geometric trick! When you double the sides, the perimeter just stretches out along the edge, so its total length doubles too. But the area? It grows in two directions, length AND width, so you are actually multiplying the original area by two twice ($2 \times 2$), which equals four!
Common Questions
What happens to the area when you double both sides of a rectangle?
The area increases by a factor of 4. If original area = l x w, and you double to get 2l x 2w, the new area = 4lw, which is four times the original.
Why does doubling sides quadruple the area and not just double it?
Area is the product of two dimensions. Doubling each dimension means multiplying the area by 2 x 2 = 4, not just 2.
What if you only double one side of the rectangle?
If only one side is doubled, the area doubles. For example, doubling only the length gives area = 2l x w = 2lw.
How does this apply to other shapes?
The same principle applies to all similar figures. Scaling all linear dimensions by a factor k scales the area by k squared, regardless of the shape.
How is doubling rectangle sides used in Saxon Math Course 3?
Saxon Math Course 3 uses this concept to illustrate the squared relationship between linear scale and area, helping students avoid the common error of assuming doubling dimensions doubles area.