Dilation - Enlargement and Reduction
comparing quantities (ratio and proportion)
Through the study of dilations and projections students explore the concepts of similarity and scale factor as they relate to two-dimensional and three-dimensional objects. We begin by exploring and constructing enlargements and reductions of polygons using the concept of ratios to model the relationships between similar shapes.
For example, if we start with a given square 'A', we can construct other squares using ratio and proportion concepts.
...build square 'B' so that the ratio of the area of square 'B' to the area of square 'A' is 9 to 1...
If we arrange 9 copies of square 'A' into another square, we'd have the required 9-times-greater area.It's dimensions would be 3 times the dimensions of square 'A', or, the ratio of the dimensions of square 'B' to the dimensions of square 'A' would be 3 to 1.
... build square 'C' so that the ratio of the edge of square 'B' to the edge of square 'C' is 1 to 2...
That means the dimensions of square 'C' must be twice the dimensions of square 'B'.It's area is four times the area of square 'B', or, ratio of the area of square 'B' to the area of square 'C' is 1 to 4.
... build square 'D' so the ratio of the perimeter of square 'D' to the perimeter of square 'A' is 5 to 1...
The ratio of the perimeter is equal to the ratio of the dimensions, since they are both measured in linear units, therefore, the dimensions of square 'D' are five times as long as the dimensions of square 'A'.It's area is twenty-five times that of Square 'A', or, the ratio of the area of square 'D' to the area of square 'A' is 25 to 1.
Here are a few other situations for you to think about on your own:
...the ratio of the area of square 'D' to the area of square 'E' is 1 to 4......the ratio of the perimeter of square 'F' to the perimeter of square 'B' is 2 to 3...
...the ratio of the area of square 'B' to the area of square 'G' is 1 to 100...
In the above examples, square 'B' is called an enlargement of square 'A' by a Scale Factor of 3. At the same time, square 'A' is a reduction of square 'B' by a Scale Factor (SF) of 1/3. Both enlargements and reductions are called dilations. In the same way, square 'D' is an enlargement of square 'A' by a Scale Factor of 5, and square 'A' is a reduction of square 'D' by a Scale Factor 1/5.
These same concepts are explored with other polygons and shapes as the concept of similarity is defined precisely as a relationship in which the shapes' corresponding angles are congruent, and the corresponding sides of each shape are in direct proportion (SF) to each other. So, in similar shapes, size varies with respect to SF, but shape remains unchanged.
Similarity and Scaling can also be explored through projections, in which projection lines passing through projection points (also called centers of dilation) can be used to create similar shapes by applying Scale Factors to distances measured from the center of dilation. Here are some examples - in each one, the location of the center of dilation has a different effect on the relationships of the similar shapes (Center of Dilation inside shapes, All shapes on the same side of the Center of Dilation, Center of Dilation between shapes, Center of Dilation on the sides of a shape.)
Center of Dilation inside shapes (left):
'O' is the center of dilation, or projection pointOB is 1/2 of OB', so ABCDE is a reduction of A'B'C'D'E' by an Scale Factor of 1/2.
OB' is twice OB, so A'B'C'D'E' is an enlargement of ABCDE by a Scale Factor of 2.
About what Scale Factors could be used between A'B'C'D'E' , ABCDE and the outer pentagon?
All shapes on the same side of the Center of Dilation (below):
OA' is 2/3 as long as OA'', so A'B'C'D' is a reduction of A''B''C''D'' by a Scale Factor of 2/3OA'' is three times as long as OA, so A''B''C''D'' is an enlargement of ABCD by a Scale Factor of 3
Center of Dilation between shapes* (below):
OA is equal to OA', A'B'C' is a dilation of ABC by a scale factor of -1.OB'' is twice OB, A''B''C'' is an enlargement of ABC by a Scale Factor of 2. Also, *OA'' os twice OA', so figure A'' is an enlargement of figure A' by a Scale Factor of -2.
*Some shapes are inverted with respect to others; when the Center of Dilation is between the projected shapes they create negative dilations.
Center of Dilation on the sides of a shape (below):
All the shapes have a corresponding colinear side, the one the Center of Dilation is on.
OA' is three times OA'', so figure A' is an enlargement of figure A'' by a Scale Factor of 3.
OA' is 3/2 the length of OA, so figure A' os an enlargement of A by a scale factor of 3/2.
The same relationships hold true for three-dimensional dimension and area as well. Imagine the effect of doubling the dimension (Scale Factor = 2) of a cube on the surface area and volume:
Imagine enlarging or shrinking the following cube solids; use different scale factors like 3, 1/2, 4, 2/3, .9, etc... Do you see a pattern in the effects of the Scale Factors on Surface Area? How about the Volume?
In class, we explore a variety of solids, including rectangular and triangular prisms, and discuss and generalize the relationships between Scale Factor, proportion, similarity, and changing the unit. As well as helping students further develop the vocabulary and characteristics of two-dimensional and threee-dimensional shapes, the activity provides students the opportunity to develop deeper intuitive understanding of plane and solid geometry, which will provide the inductive foundation for later work in Euclidean Constructions and deductive proofs in Course III of Math Alive!.
