Unit 5, Lesson 4
Scaling and Area
Geometry
Lesson Synthesis
Ask students to share their thinking about what happens to the area of the blob when it is dilated by a scale factor of . Display this image for all to see.
Label several of the rectangles in the original blob . If students are not familiar with subscripts, explain that they are a way to label variables to make it easier to keep track of them. That is, is the area of the first rectangle in square units, is the area of the second rectangle, and so on.
Ask students how many rectangles there are in total (9) and how they can write an expression for the area of the original blob ().
Now ask students to write expressions for the areas of the corresponding rectangles in the dilated blob (), and so on. Ask students how they could write the area of the dilated blob().
We can use the distributive property to rewrite the expression for the blob’s area as , which is times the sum of all the original rectangles. That means the area of the dilated figure is times the area of the original figure. The takeaway is that the area of any shape, no matter how irregular, gets multiplied by when the shape is dilated by a scale factor of .
Most importantly, invite students to share strategies for how they solved the last problem. Ask them how the number 10 in the previous activity relates to this problem (the 10 was the area of the original rectangle, but for the blob, that number is 20).
Student Lesson Summary
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 and . Substitute these values into the area expression to get . Rearrange the numbers to get . This can be rewritten or . So the area of the dilated rectangle is 180 square inches, or 9 times the original. The diagram confirms that 9 of the original rectangles fit into the dilated rectangle.
In general, when scaling a rectangle by a factor of , the length and the width are both multiplied by , so the area is multiplied by .
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 , its area is multiplied by a factor of .