8.7: Transformations and Symmetry
Introduction
The Moving Wall
In one of the rooms of the museum, this tile completely covered two of the walls. The students walked inside and began moving automatically because of the pattern on the wall.
“This tile makes me dizzy,” Greg stated, sitting on a nearby bench.
“Yes, it seems to move,” Lane commented.
“I think it’s cool. Look at all of the symmetry and the transformations are everywhere,” Emma said smiling.
“Are you sure? You are beginning to sound like Mrs. Gilman,” Lane said jokingly.
“I am sure, and thank you for the compliment,” Emma said, nudging her best friend.
Do you think Emma can see the transformations? Well, they are there. While sometimes a pattern like this one can make you dizzy, there are many different transformations in pattern. Your task is to find them. Use what you learn in this lesson to revisit the problem and find the examples of transformations and symmetry in the tile pattern.
What You Will Learn
In this lesson you will learn how to demonstrate the following skills:
- Recognize transformations as the movement of a figure in a plane, classified as a translation (slide), a reflection (flip) and a rotation (turn).
- Identify and describe a translation.
- Identify and describe a reflection.
- Identify and describe a rotation.
- Distinguish between line symmetry and rotational symmetry.
Teaching Time
I. Recognize Transformations as the Movement of a Figure in a Plane
In this lesson we will examine different kinds of transformations. A transformation is the movement of a geometric figure.
There are three different kinds of transformations.
In a translation, also called a slide, the figure moves left, right, up, down, or a combination of these (which would be a diagonal motion).
In a reflection, the figure flips. A reflection is like a mirror image of the original figure.
Finally, in a rotation, the figure turns.
The key thing to remember is that in any transformation, the size and shape of the figure stay exactly the same, only its location changes or shifts.
Now let’s look at each type of translation in more detail.
II. Identify and Describe a Translation
The first type of transformation is called a translation. It is also known as a slide because the figure in question does exactly that. It moves up, down, to the left or to the right. Nothing about the figure changes except its location.
Here are some examples of translations.
Notice when you look at each example that all that changed for each figure was its location. There are different colors used to show you the actual translation, but other than the location, each looks exactly the same.
This is how you always know that you are working with a translation or a slide.
How do we perform a translation?
To perform a translation, we measure a distance and then redraw the figure. For example, let’s move this triangle 6 inches.
We measure 6 inches from each point of the triangle and make a new point. This way, every part of the triangle moves 6 inches.
In this way, we can translate any figure in any direction for any distance.
Write down the definition for a translation in your notebook as well as how to perform a translation given this example.
III. Identify and Describe a Reflection
You have heard the word “reflection” all the time. From the reflection in a mirror to the reflection in a pond, reflections are all around us.
How do we apply the term reflection to geometry?
A reflection is a different kind of transformation. In a reflection, the figure flips to make a mirror image of itself. Take a look at the reflection below.
The line in the middle acts like a mirror. We call this the line of symmetry. This is a vertical line of symmetry. Imagine standing in front of a mirror and holding up your left hand. Where is your hand in the mirror’s reflection? A reflected figure works the same way: when we flip it over the line, all of its points are reversed. When reflected, the figure above looks like a backwards \begin{align*}L\end{align*}. Notice that, on both sides of the line, the dot is closest to the line.
We can also reflect figures across a horizontal line of symmetry. Then our reflection would look like this.
In this case, the “top” of the figure becomes the “bottom” in the reflection! Notice, however, that in both cases the figures are symmetrical.
Write down the definition of a reflection and line of symmetry in your notebook.
IV. Identify and Describe a Rotation
Now let’s learn about the third kind of transformation. A rotation is a transformation that turns the figure in either a clockwise or counterclockwise direction.
How does the figure below change as it is rotated?
Imagine you could spin the figure around in a circle. It would not change, but might turn upside down. Figures can rotate as much as \begin{align*}360^{\circ}\end{align*}, a full circle. Let’s see how that might look.
When we rotate the figure a full \begin{align*}360^{\circ}\end{align*}, it ends up in the same place it began, unchanged!
8M. Lesson Exercises
Let’s identify some translations.
1.
2.
Check your answers with your neighbor then continue with the next section.
V. Distinguish Between Line Symmetry and Rotational Symmetry
We have mentioned that reflected figures are symmetrical, and that the line that acts as a mirror is called the line of symmetry.
Symmetry means that when we divide a figure in half, the halves are congruent. In other words, a figure is symmetric if its outlines mirror each other.
Look at the figure below. Imagine you can fold it in half. When you fold it, do the outlines of each half match? They do, so this figure has symmetry.
When we “unfold” the figure, we have two congruent halves. The “fold” line is the line of symmetry. It divides the figure into halves that the two haves are mirror images of each other! Every part on one half “mirrors” or corresponds to a part on the other half.
Now let’s try another figure. Can we fold it perfectly in half?
We can try to fold this figure a bunch of different ways, but it does not have a line of symmetry.
Rotational symmetry is a different kind of symmetry. It means that when we rotate a figure, the figure appears to stay the same. The outlines do not change even as the figure turns. Look at the figure below.
We can tell the figure has been rotated because the dot moves clockwise. However, the outlines of the figure have not changed. This figure has rotational symmetry because every time we turn it, one of the arms of the star always faces up.
The figure below, on the other hand, does not have rotational symmetry. Can you see why?
No matter how we work with this figure it will look different each time it is rotated. Therefore, we know that it does not have rotational symmetry.
8N. Lesson Exercises
Does each of the following figures have line symmetry, rotational symmetry, both, or neither?
Let’s look at each.
- Cross
- Arrow
- Quadrilateral
Take a few minutes to check your answers with a friend.
Real Life Example Completed
The Moving Wall
Here is the original problem once again. Reread it and then make a note of each of the types of transformations listed.
In one of the rooms of the museum, this tile completely covered two of the walls. The students walked inside and began moving automatically because of the pattern on the wall.
“This tile makes me dizzy,” Greg stated, sitting on a nearby bench.
“Yes, it seems to move,” Lane commented.
“I think it’s cool. Look at all of the symmetry and the transformations are everywhere,” Emma said, smiling.
“Are you sure? You are beginning to sound like Mrs. Gilman,” Lane said jokingly.
“I am sure, and thank you for the compliment,” Emma said, nudging her best friend.
In this tile pattern, make a note of a reflection. Make a note of a translation. Make a note of a figure that rotates or has rotational symmetry. Make a note of the line of symmetry.
When finished, compare your answers with a partner.
While you won’t find the exact answer here, there are many ways to explore transformations and symmetry by using this pattern. Work with a partner or with your whole class to figure out the math in this tile pattern.
Vocabulary
Here are the vocabulary words found in this lesson.
- Transformation
- when a figure moved on a plane. The figure doesn’t change, but its position does.
- Translation
- a slide. The figure moved up, down, right, left, or diagonally.
- Reflection
- a flip. The figure flips over a line of symmetry, much like a reflection in a mirror.
- Rotation
- a turn. The figure turns either clockwise or counterclockwise.
- Line of Symmetry
- the line that a figure reflects over. Also the line that divided a figure in half showing line symmetry.
- Rotational Symmetry
- the way a figure looks does not change no matter how you rotate it.
- Symmetry
- a figure that can be divided into two congruent halves is said to have symmetry.
Time to Practice
Directions: Identify the transformations shown below as translations, reflections, or rotations.
Directions: Tell whether the figures below have line symmetry, rotational symmetry, both, or neither.
Directions: Draw the second half of each figure.
Directions: Answer each of the following questions true or false.
16. A reflection has rotational symmetry.
17. A square has line symmetry and rotational symmetry.
18. If a figure is a transformation, then it always rotates clockwise.
19. A slide is also called a translation.
20. A flip has a line of symmetry because it is a reflection.
21. A rotation or turn always moves clockwise and never counterclockwise.