# 6.7: Translations, Rotations and Tessellations

**At Grade**Created by: CK-12

## Introduction

*The Dome*

Dylan came storming in the door after a busy day at school. He slammed his books down on the kitchen table.

“What is the matter?” his Mom asked sitting down at the table.

“Well, I made this great geodesic dome. It is finished and doing great, but Mrs. Patterson wants me to investigate other shapes that you could use to make a dome. I don’t want to do it. I feel like my project is finished,” Dylan explained.

“Maybe Mrs. Patterson just wanted to give you an added challenge.”

“Maybe, but what other shapes can be used to form a dome? The triangle makes the most sense,” Dylan said.

“Yes, but to figure this out, you need to know what other shapes tessellate,” Mom explained.

“What does it mean to tessellate? And how can I figure that out?”

**These are both good questions that will be answered in this chapter. Use the lesson to learn about tessellations. At the end of the lesson, you can help Dylan to answer these questions.**

*What You Will Learn*

In this lesson you will learn how to do the following skills.

- Recognize translation transformations (slides) and rotation transformations (turns).
- Use coordinate notation to describe translations of given figures and their resulting images.
- Use coordinate notation to describe rotations around the origin of given figures and their resulting images, \begin{align*}90^\circ\end{align*} clockwise, \begin{align*}90^\circ\end{align*} counterclockwise and \begin{align*}180^\circ\end{align*}.
- Identify rotational symmetry and the least angle of rotation in given images.
- Use any combination of transformations (flips, slides, turns) to create tessellations, and identify polygons that do not tessellate.

*Teaching Time*

I. **Recognize Translation Transformations (Slides) and Rotation Transformations (Turns)**

In the last lesson, you learned all about a ** transformation** called a

*reflection***. A**

*transformation***is the movement of a geometric figure on the coordinate plane. Remember that a**

*reflection***is the flip or mirror image of a geometric figure over the \begin{align*}x\end{align*} or \begin{align*}y-\end{align*}axis.**There are other kinds of transformations too.

**Two other types of transformations are** *translations***and** *rotations***. A** *translation***is when a geometric figure slides up, down, left or right on the coordinate plane. The figure moves its location, but doesn’t change its position. A** *rotation***is a turn. A figure can be turned clockwise or counterclockwise on the coordinate plane. In both transformations the size and shape of the figure stays exactly the same.**

Now let’s look at these two types of transformations in more detail.

II. **Use Coordinate Notation to Describe Translations of Given Figures and Their Resulting Images**

**When we perform translations, we slide a figure left or right, up or down. This means that, in the coordinate plane, the coordinates for the vertices of the figure will change.**

Take a look at the example below.

We can represent this triangle by using coordinate notation. Coordinate notation is when we write ordered pairs to represent each of the vertices of a geometric figure that has been graphed on the coordinate plane.

(-1, 5)

(-1, 2)

(-5, 2)

These are the coordinates of the vertices of the triangle.

**If we slide this triangle 3 places down, all of its vertices will shift 3 places down the \begin{align*}y-\end{align*}axis. That means that the ordered pairs for the new vertices will change. Specifically, the \begin{align*}y-\end{align*}coordinate in each pair will decrease by 3.**

Look at this example.

**Now the \begin{align*}y-\end{align*}coordinate of each ordered pair decreased by three units. We can see how the ordered pairs changed from the first image to the next image.**

\begin{align*}(-1, 5) & \rightarrow (-1, 2)\\ (-1, 2) & \rightarrow (-1, -1)\\ (-5, 2) & \rightarrow (-5, -1)\end{align*}

**The \begin{align*}y-\end{align*}coordinate changed from 5 to 2, from -1 to 2 and from 2 to -1. As we move down, the value of the coordinate also moved down.**

**If we were to move the image up three units on the \begin{align*}y-\end{align*}axis, then we would increase the value of the \begin{align*}y-\end{align*}coordinate by three units.**

**If we were to move the image to the right then we would increase the \begin{align*}x-\end{align*}coordinate. If we were to move it to the left, then we would decrease the \begin{align*}x-\end{align*}coordinate.**

**We can translate figures in other ways too. We can move figures diagonally by changing both their \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}coordinates. One way to recognize translations, then, is to compare their points. The \begin{align*}x-\end{align*}coordinates will all change the same way, and the \begin{align*}y-\end{align*}coordinates will all change the same way.**

**To graph a translation, we perform the same change for each point. Let’s try graphing a translation.**

Example

*Graph the following translation five places to the right.*

**Now you can see by looking at this square that there are four vertices, so there are four sets of ordered pairs to represent these vertices. Here are the ordered pairs.**

**(-4, 3)**

**(-1, 3)**

**(-1, 6)**

**(-4, 6)**

**The translation is to move the square five places to the right. That means that we are going to change the \begin{align*}x-\end{align*}coordinate and not the \begin{align*}y-\end{align*}coordinate.**

\begin{align*}(-4, 3) + 5 &= (1, 3)\\ (-1, 3) + 5 &= (4, 3)\\ (-1, 6) + 5 &= (4, 6)\\ (-4, 6) + 5 &= (1, 6)\end{align*}

**Now let’s look at the graph of the translation.**

**Notice that while it is helpful to graph the square both first and then as a translation, it isn’t necessary to do so to figure out the coordinate notation. If you know the vertices of the figure that you are translating and you know how you are moving it, then you can figure out the new coordinates of the vertices.**

Example

*A triangle with the coordinates (0, 2), (2, 2) and (2, 5) is graphed on the coordinate grid. Find the coordinates of a translation moved three units down.*

**To work through this one, first notice the direction of the translation. It is to move the triangle three units down. Down means that we will be subtracting three and down also means that we will be changing the \begin{align*}y-\end{align*}coordinate since up and down involves the \begin{align*}y-\end{align*}axis. Here is our action.**

\begin{align*}(0, 2) - 3 &= (0, -1)\\ (2, 2) - 3 &= (2, -1)\\ (2, 5) - 3 &= (2, 2)\end{align*}

**If we were to graph the translation, here is what we would see.**

III. **Use Coordinate Notation to Describe Rotations Around the Origin of Given Figures and their Resulting Images, \begin{align*}\underline{90^\circ}\end{align*} Clockwise, \begin{align*}\underline{90^\circ}\end{align*} Counterclockwise and \begin{align*}\underline{180^\circ}\end{align*}**

Now let’s look at another kind of transformation: rotations. **A** *rotation***is a transformation that turns the figure in either a clockwise or counterclockwise direction.** We can turn a figure \begin{align*}90^\circ\end{align*}, a quarter turn, either clockwise or counterclockwise. When we spin the figure exactly halfway, we have rotated it \begin{align*}180^\circ\end{align*}. Turning it all the way around rotates the figure \begin{align*}360^\circ\end{align*}.

Now if you look at these two triangles, you can see that one has been turned a quarter turn clockwise. If we talk about that turn or rotation in mathematical language, we can describe the turn as \begin{align*}90^\circ\end{align*} clockwise. We could also turn it \begin{align*}180^\circ\end{align*}, which would show a triangle completed upside down.

**Next, let’s look at rotating figures on the coordinate plane.**

Example

*Rotate this figure \begin{align*}90^\circ\end{align*} clockwise on the coordinate plane.*

**First, let’s write down the coordinates for each of the points of this pentagon.**

\begin{align*}& A (-3, 5)\\ & B (-4, 4)\\ & C (-3, 3)\\ & D (-1, 2)\\ & E (-1, 4)\end{align*}

**Now we have the points. The easiest way to think about rotating any figure is to think about it moving around a fixed point. In the case of graphing figures on the coordinate plane, we will be rotating the figures around the center point or origin.**

**If we rotate a figure clockwise \begin{align*}90^\circ\end{align*}, then we are going to be shifting the whole figure along the \begin{align*}x-\end{align*}axis. To figure out the coordinates of the new rotated figure, we switch the coordinates and then, we need to multiply the second coordinate by -1. This will make perfect sense given that the entire figure is going to shift.**

**Let’s apply this to the coordinates above.**

\begin{align*}A (-3, 5) &= A^\prime (5, -3) = (5, 3)\\ B (-4, 4) &= B^\prime (4, -4) = (4, 4)\\ C (-3, 3) &= C^\prime (3, -3) = (3, 3)\\ D (-1, 2) &= D^\prime (2, -1) = (2, 1)\\ E (-1, 4) &= E^\prime (4, -1) = (4, 1)\end{align*}

**Now we can graph this rotated figure. Notice that we use \begin{align*}A^\prime\end{align*} to represent the rotated figure. Here is the graph of this rotation.**

**That is a great question. Let’s think about what would happen to the figure if we were to rotate it counterclockwise. To do this, the figure would move across the \begin{align*}y-\end{align*}***axis***in fact, the \begin{align*}x-\end{align*}coordinates would change completely. In actuality, we would switch the original coordinates around. The \begin{align*}x-\end{align*}coordinate would become the \begin{align*}y-\end{align*}coordinate and the \begin{align*}y-\end{align*}coordinate would become the \begin{align*}x-\end{align*}coordinate. Then, we need to multiply the new \begin{align*}x-\end{align*}coordinate by -1.**

Here is what that would look like.

\begin{align*}& A (-3, 5) \rightarrow A^\prime (-5, -3)\\ & B (-4, 4) \rightarrow B^\prime (-4, -4)\\ & C (-3, 3) \rightarrow C^\prime (-3, 3)\\ & D (-1, 2) \rightarrow D^\prime (-2, -1)\\ & E (-1, 4) \rightarrow E^\prime (-4, -1)\end{align*}

**Now we can graph this new rotation.**

**We can also graph figures that have been rotated \begin{align*}180^\circ\end{align*} too. To do this, we multiply both coordinates of the original figure by -1.**

**Let’s see what this looks like.**

\begin{align*}A (-3, 5) &= A^\prime (3, -5)\\ B(-4, 4) &= B^\prime (4, -4)\\ C (-3, 3) &= C^\prime (3, -3)\\ D (-1, 2) &= D^\prime (1, -2)\\ E (-1, 4) &= E^\prime (1, -4)\end{align*}

**Now we can graph this image.**

*Write the three ways to figure out new coordinates for rotating \begin{align*}90^\circ\end{align*} clockwise and counterclockwise and for rotating a figure \begin{align*}180^\circ\end{align*}. Put these notes in your notebook.*

IV. **Identify Rotational Symmetry and the Least Angle of Rotation in Given Images**

In the last section you learned how to rotate a figure. Now let’s think about symmetry and rotations. **We can call this** *rotational symmetry***. A figure has rotational symmetry if, when we rotate it, the figure appears to stay the same. The outlines do not change even as the figure turns.** Look at the figure below.

**Look at this image. The star will look the same even if we rotate it. We could turn it \begin{align*}72^\circ\end{align*} or \begin{align*}144^\circ\end{align*} clockwise or counterclockwise. It won’t matter. The star will still appear the same.**

Example

*Does this figure have rotational symmetry?*

**While the outline of this image has rotational symmetry, the design inside prevents it from having rotational symmetry. If we turn the circle, then the design inside will change. Therefore this image does not have rotational symmetry.**

**Look at this hexagon. It has rotational symmetry. You can see that because we can rotate it \begin{align*}90^\circ\end{align*} and \begin{align*}180^\circ\end{align*} and it will still look exactly the same.**

**Now look carefully at the hexagon. We could rotate it less than \begin{align*}90^\circ\end{align*} too and it still has rotational symmetry. We can also look at the angles to determine rotational symmetry. Each time we turn the figure, it has two parallel sides on the top and bottom and four other sides at the same angles. It has rotational symmetry.**

V. **Use any Combination of Transformations (flips, slides, turns) to Create Tessellations, and Identify Polygons that Do Not Tessellate**

We can use translations and reflections to make patterns with geometric figures called *tessellations.***A tessellation is a pattern in which geometric figures repeat without any gaps between them.** In other words, the repeated figures fit perfectly together. They form a pattern that can stretch in every direction on the coordinate plane. Take a look at the tessellations below.

This tessellation could go on and on.

We can create tessellations by moving a single geometric figure. We can perform translations such as translations and rotations to move the figure so that the original and the new figure fit together.

**How do we know that a figure will tessellate?**

**If the figure is the same on all sides, it will fit together when it is repeated. Figures that tessellate tend to be regular polygons. Regular polygons have straight sides that are all congruent. When we rotate or slide a regular polygon, the side of the original figure and the side of its translation will match. Not all geometric figures can tessellate, however. When we translate or rotate them, their sides do not fit together.**

*Remember this rule and you will know whether a figure will tessellate or not! Think about whether or not there will be gaps in the pattern as you move a figure.*

**Sure. To make a tessellation, as we have said, we can translate some figures and rotate others.**

Example

*Create a tessellation by repeating the following figure.*

First, trace the figure on a piece of stiff paper and then cut it out. This will let you perform translations easily so you can see how best to repeat the figure to make a tessellation.

This figure is exactly the same on all sides, so we do not need to rotate it to make the pieces fit together. Instead, let’s try translating it. Trace the figure. Then slide the cutout so that one edge of it lines up perfectly with one edge of the figure you drew. Trace the cutout again. Now line the cutout up with another side of the original figure and trace it. As you add figures to the pattern, the hexagons will start making themselves!

Check to make sure that there are no gaps in your pattern. All of the edges should fit perfectly together. You should be able to go on sliding and tracing the hexagon forever in all directions. You have made a tessellation!

## Real-Life Example Completed

*The Dome*

**Here is the original problem once again. Reread it and then answer the questions at the end that Dylan asks his Mom.**

Dylan came storming in the door after a busy day at school. He slammed his books down on the kitchen table.

“What is the matter?” his Mom asked sitting down at the table.

“Well, I made this great geodesic dome. It is finished and doing great, but Mrs. Patterson wants me to investigate other shapes that you could use to make a dome. I don’t want to do it. I feel like my project is finished,” Dylan explained.

“Maybe Mrs. Patterson just wanted to give you an added challenge.”

“Maybe, but what other shapes can be used to form a dome? The triangle makes the most sense,” Dylan said.

“Yes, but to figure this out, you need to know what other shapes tessellate,” Mom explained.

“What does it mean to tessellate? And how can I figure that out?”

**Now answer the two questions. What does it mean to tessellate? How can you figure out which figures will tessellate and which ones won’t.**

*Solution to Real – Life Example*

**First, let’s answer the question about tessellations. What does it mean to tessellate?**

To tessellate means that congruent figures are put together to create a pattern where there aren’t any gaps or spaces in the pattern. Figures can be put side by side and/or upside down to create the pattern. The pattern is called a tessellation.

**How do you figure out which figures will tessellate and which ones won’t?**

Figure that will tessellate are congruent figures. They have to be exactly the same length on all sides. They also have to be able to fit together. A circle will not tessellate because there aren’t sides to fit together. A hexagon, on the other hand, will tessellate as long as the same hexagon is used to create the pattern.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

- Transformation
- moving a geometric figure on the coordinate plane.

- Coordinate Notation
- using ordered pairs to represent the vertices of a figure that has been graphed on the coordinate plane.

- Reflection
- A flip of a figure on the coordinate plane.

- Translation
- A slide – when a figure moves up, down, left or right on the coordinate plane, but does not change position.

- Rotation
- A turn – when a figure is turned \begin{align*}90^\circ, 180^\circ\end{align*} on the coordinate plane.

- Rotational Symmetry
- when a figure can be rotated but appears exactly the same no matter how you rotate it.

- Tessellation
- a pattern made by using different transformations of geometric figures. A figure will tessellate if it is a regular geometric figure and if the sides all fit together perfectly with no gaps.

## Time to Practice

Directions: Answer the following questions about rotations, translations and tessellations.

- What is a translation?
- What is a rotation?
- What is a tessellation?
- True or false. A figure can be translated up or down only.
- True or false. A figure can be translated \begin{align*}180^\circ\end{align*}.
- True or false. A figure can be rotated \begin{align*}90^\circ\end{align*} clockwise or counterclockwise.
- True or false. A figure can’t be translated \begin{align*}180^\circ\end{align*}.
- When rotating a figure \begin{align*}90^\circ\end{align*} counterclockwise, we switch the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} coordinates and multiply which one by -1?
- When rotating a figure \begin{align*}90^\circ\end{align*} clockwise, we multiply which coordinate by -1?
- True or false. When rotating a figure \begin{align*}180^\circ\end{align*}, we multiply both coordinates by -1.

Directions: Write the new coordinates for each rotation given the directions.

A Triangle with the coordinates (-4, 4) (-4, 2) and (-1, 1)

- Rotate the figure \begin{align*}90^\circ\end{align*} clockwise
- Rotate the figure \begin{align*}90^\circ\end{align*} counterclockwise
- Rotate the figure \begin{align*}180^\circ\end{align*}

A Triangle with the coordinates (1, 3) (5, 1) (5, 3)

- Rotate the figure clockwise \begin{align*}90^\circ\end{align*}
- Rotate the figure counterclockwise \begin{align*}90^\circ\end{align*}
- Rotate the figure \begin{align*}180^\circ\end{align*}

A Triangle with the coordinates (1, -1) (3, -4) (5, -1)

- Rotate the figure clockwise \begin{align*}90^\circ\end{align*}
- Rotate the figure counterclockwise \begin{align*}90^\circ\end{align*}
- Rotate the figure \begin{align*}180^\circ\end{align*}

Directions: Answer the following questions.

- Will a regular pentagon tessellate?
- What characteristics does a figure have to tessellate?
- Can a tessellation have any gaps?

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