Dilations on Coordinate Grids

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. 

• A dilation that creates a larger image is called an enlargement.

• A dilation that creates a smaller image is called a reduction.

• A dilation stretches or shrinks the original figure.

• A description of a dilation includes the scale factor (or ratio) and the center of the dilation.

• The center of dilation is a fixed point in the plane.

• If the scale factor is greater than 1, the image is an enlargement (a stretch).

• If the scale factor is between 0 and 1, the image is a reduction (a shrink).

• If the scale factor is 1, the figure and the image are congruent; there is no change.
• If the scale factor is 0, the figure disappears.

First we looked at a set of eyes, one a factor of three times larger than the other. An eye doctor puts drops in the eyes to make the pupil larger so the doctor can use a magnifying lens to see into the back of the eye to examine the retina, cornea, and blood vessels. Normally, when light is directed at an eye the pupil gets smaller to protect the eye. The dilation from the larger pupil to the smaller was 1/3, the reciprocal of the enlarged scale factor of 3.

I then showed the Mathletes a chess piece with a scale factor of 1 and two other dilations with scale factors of 1/2 and 2. We looked at the area relationships which are two dimensional squares (for example, the chess piece with a scale factor of 2 had a 2^2 or area 4 times the original piece. The older Mathletes were able to find the area ratios even if the scale factor was not a whole number. For example, if the scale factor was 3.5 or 7/2, the area ratio would be 49/4 or 12.25.

Then we moved to the coordinate grid where creating a dilation with a center about the origin, point (0,0). If we have a point (2,5) and we want to dilate that point to a scale factor of 2, just multiply each value by 2 (multiplication is just repeated addition), so we have (4,10). If we have point (10,8) and we want to dilate with a scale factor of 1/2, then we divide each value by 2, so (5,4).

We were able to dilate triangles to scale factors of 2 and 4, a chess knight to a scale factor of 2 and 1/2, a letter K to a scale factor of 3 and 1/3, and a letter M to a scale factor of 1/4, and several other pictures to different scale factors.

The older classes looked at scale factors of common fractions. These are challenging but a lot of fun. Seth asked a question about a scale factor of -3. So we looked at this and found that the dilation was still 3 times the original but had moved to a different location with a 180 degree rotation instead of being in line with the origin.

Of course, I gave the children blank coordinate grids so they can create their own picture and dilate to any scale factor they choose.