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This lesson will define reflection, rotation, and translation as they relate to math.

It will also show you an example of each one so that you can perform these transformations on your own.

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Transformations in Math

The next time you go in front of a mirror, I want you to notice your reflection and really think about what it is. Then I want you to turn to your side. Then I want you to slide to the left or to the right. You’ve just seen a reflection and performed a rotation and translation all in your own home. It’s not that hard! Let’s see what these really are in the world of math.

What is Reflection?

Reflections are all around us. You see your reflection in the mirror every morning. If you’re out on a calm lake, you may see the reflections of trees and mountains in the water. A reflection in math is the flipping of a point or figure over a line of reflection (the mirror line).

reflection of a triangle

The key word here is flipped. A reflection flips an image.

Look at the triangle on the screen on the top. It has been flipped over the horizontal line of reflection. Notice that every point will have the same distance from this central line, and a reflected image will always be the same size as the original image.You can create a reflection yourself really easily.

Again, let’s use a triangle. Draw one in any triangle shape you want. Then add a line of reflection next to it. Take each point of the triangle, and measure the distance from it to the line of reflection while making sure the mirror line is at a right angle to the line you are drawing.

Then, take this distance, measure it from the mirror line, and place it on the other side of the mirror line as a dot. Connect the dots up, and you have a reflected triangle!

What is Rotation?

Okay, that wasn’t too hard, was it? This next one is even easier! If you stand in one spot and turn, you rotate. Rotation is the turning of a figure or object around a fixed point.Once again, we turn the shape; we don’t flip it as we did during a reflection. When we turn the shape around a fixed point, a point called the center of rotation, then every single point will make a circle around this fixed point.We describe the amount of rotation in degrees.

If you turn directly to your left or right, that’s a 90-degree turn. If you turn to the side where your back is right now, that’s a 180-degree turn. If you turn to where your right side is but you do it by turning counterclockwise, you’ve turned 270 degrees. And finally, if you make a full circle as you turn, you’ve turned 360 degrees.

When you turn something counterclockwise, the degrees are positive, and if you turn something clockwise, the degrees are negative. Like in a reflection, the size and shape of the image doesn’t change during a rotation.Let’s perform a rotation together. Let’s take a red triangle and rotate around the coordinates (0, 0) by 90 degrees. Notice how every point is rotated the same amount, and because the degrees are positive, we rotate everything counterclockwise.

What is Translation?

And, even easier still than a reflection and rotation is a translation, and it has nothing to do with changing one language into another! Translation in math is a scenario where every point in a figure is moved the exact same distance and in the same exact direction, without being rotated, reflected, or resized.

In short, we simply slide the image in one direction or another. That’s it.Let’s perform a translation together to demonstrate how this works. Let’s move our triangle 40 units from its current location. We need to take every point and move it the exact same distance in order to translate the shape. See, that was easy, wasn’t it?

Lesson Summary

Let’s quickly sum everything up.A reflection is the flipping of a point or figure over a line of reflection (the mirror line).A rotation is the turning of a figure or object around a fixed point.And a translation is a scenario where every point in a figure is moved the exact same distance and in the same exact direction, without being rotated, reflected, or resized.

Reflection, Rotation ; Translation Overview

What is Reflection?
reflectionmath
Keywords Definitions
Reflection In math, a reflection is the flipping of a point or figure over a line of reflection (the mirror line).
Rotation the turning of a figure or object around a fixed point
Translation a scenario in which every point in a figure is moved the exact same distance and in the same exact direction without being rotated, reflected, or resized

Learning Outcomes

Complete the lesson with the intention of realizing these goals:

  • Recognize a reflection in mathematics
  • Describe a rotation in degrees
  • Perform a translation

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