Dilation for my old eyes...

     When I saw we were learning dilation, my mind first went to my annual eye exam and having my pupils dilated. I was hoping the math version would be more fun than the eye exam version, and I am still on the fence with my decision. They both leave me with a bit of a headache. A dilation in geometry is a translation of an image that creates an image similar to the original, but bigger. Occasionally, the resulting image will be smaller than the original, and in that case we refer to it as a reduction. In order to perform a dilation, you must have a scale and a center of dilation. For the purposes of illustrating a dilation, I found this image here.

     As you can see, the two shapes are similar in that their corresponding angles are all congruent, but they are different in that their corresponding sides are different lengths. The image gives us the center of dilation, but it does not give us the scale. To find the scale, we must look at two corresponding sides. In this example, we have a 6 as the original image and a 12 as the dilated image. Because our original image is smaller than the dilation, we know the scale factor will be greater than 1. (If we reversed the roles and our larger image was our original, our resulting image would be smaller, a reduction, and have a scale of less than 1.) To find our scale we can use the equation 6*X=12 and solve for X. This dilation must have a scale factor of 2. This means that the dilated image sides should all be twice the length of the original image. If we were looking for the center of dilation, we would need to know the scale. Since our scale is 2, that means that the distance from our center of dilation to our dilated image needs to be twice as far as the length from the center of dilation to our original image. 

     I think typing out the rules for dilation helped me to understand them a little better, maybe reading them will have the same affect for you! If not, there is an awesome video that takes dilation and explains it in its simplest form and why we need to understand it here.

In the mirror

         All those old 9th grade frustrations are resurfacing again. I did the math today, and it's been 23 years since I've had to think about any of these geometry concepts. I was able to do them okay then and now, but I have never felt confidence in explaining them, or even confidence that I fully understand them. Anyways, back to the mirror. Reflections are a funny thing. When I look in the mirror, I think I see myself, and don't even consciously consider that what I am actually seeing is a reflection. I had to alter my thoughts for this lesson.
          Within the realm of math and geometry, a reflection is thought of as a flipping or folding of a shape over the line of reflection. That means that when I am asked to perform a reflection, the most important things for me to consider are the line of reflection (or mirror line), and where the shape's vertices fall on the grid in respect to the line of reflection. This means that if I have a shape with vertices at (-5,2), (-3,-3) and (-1,2) and I want to do a reflection of this triangle across the y-axis, I'm going to need to look closely at those points. A basic rule for reflecting over the y-axis will be that each point's mirror will be as follows: (x,y) mirrored to (-x,y). That means for the triangle above, my reflection vertices would be (5,2), (3,-3), and (1,2). If I was reflecting across the x-axis, the rule would be (x,y) mirrored to (x,-y).  I found this awesome chart to help with a few of the other common mirror lines and the rules for finding new coordinates here.

Outside of these common lines of reflection, a person would have to see where the image sits in relation to the mirror line in order to find the mirror image across a different line of reflection. If all else fails, students could try folding their paper along the line of reflection to see where placement for the mirror image should be. Good luck! If you need me I'll be staring at my reflection in the mirror, admiring all my new grey hairs thanks to geometry!