Unit 1, Lesson 5
Scaling and Area
Integrated Math 3
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Here is a rectangle with a length of 5 units and a width of 2 units.
| scale factor | area of image in square units | factor by which the area changed |
|---|---|---|
| 0.5 | ||
| 1 | ||
| 2 | ||
| 2.5 | ||
| 3 | ||
| 4 |
Andre says, “We know that if a rectangle is scaled by a factor of
Jada says, “Here’s a shape that’s not a rectangle. Say its area is
Andre says, “These rectangles start to make a nice approximation of the blob. If we wanted to get closer, we could add even more rectangles. The sum of the areas of all the rectangles would add up to the area of the blob. I think we’re almost there!”
Here is a 5-unit-by-4-unit rectangle. Its area is 20 square units. What will the area be after it is dilated by a scale factor of 3?
The dilated rectangle’s dimensions can be written as
In general, when scaling a rectangle by a factor of
What about a shape that is not a rectangle? Consider the rounded shape called an ellipse in the image. We can approximate the area of an ellipse by filling it with many rectangles. The sum of the areas of the rectangles will be a little less than the area of the ellipse because they don’t fill it entirely. To get closer to the area of the ellipse, draw in more rectangles. Continuing the process infinitely would give the exact area of the ellipse.
When this ellipse is dilated using a scale factor of 4, each rectangle becomes 16 times larger. This means that the area of the ellipse increases by a factor of 16 as well. Any closed shape can be filled with rectangles that approximate its area. Because of this, when any shape is scaled by a factor of