# 11.7: Transformations

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

## Introduction

*The Clubhouse*

Cody and his Dad are working on building a clubhouse in his backyard. Cody is excited about the project and can’t wait to start building. He wrote to his penpal Trevor, in New Zealand, and told him all about the clubhouse. Trevor asked Cody to email him some pictures, and Cody said that he would. In fact, Cody told him that he would email them right away.

The only problem is that they haven’t started yet. Cody’s Dad has insisted that he draw a complete plan of the clubhouse. Cody did that and thought that they would be able to start, but then Cody’s Dad said that he wanted Cody to draw a plan of the backyard and where the clubhouse was going to be built. This made Cody a bit frustrated, but he decided to do it anyway in hopes that they could start building the clubhouse on the weekend. Then he could take pictures and email them to Trevor.

Cody decided to use a coordinate grid to map out the backyard. He drew the following plan and went to show his Dad.

Cody has created a scale where each unit on the coordinate grid represents 5 feet. His clubhouse dimensions are \begin{align*}14.1 \ ft \times 14.1 \ ft\end{align*}. The point on the grid at (4, -3) represents the back door of Cody’s house. His family has a large backyard, so the clubhouse is the perfect size.

Cody brings his drawing to his Dad and shows him the plan for the clubhouse.

“That looks great, except the clubhouse needs to move to the right 10 feet. Your Mom wants to plant her garden right where the clubhouse is now. I think if you move it over 10 feet the drawing will work,” Dad explains.

**Cody gets back to work. How can he draw the clubhouse if he moves it over 10 feet to the right? This lesson is all about transformation. Using a transformation, Cody can redraw the clubhouse. We’ll come back to this problem at the end of this lesson to help Cody with his work.**

*What You Will Learn*

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

- Identify transformations in the coordinate plane as translations (slides), reflections (flips) and rotations (turns).
- Graph paired transformations of figures given coordinates of vertices.
- Describe transformations as \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}coordinate changes.
- Identify equivalent transformations with different coordinate changes.

*Teaching Time*

I. **Identify Transformations in the Coordinate Plane as Translations (slides), Reflections (flips) and Rotations (turns)**

**Figures can be transformed three different ways on the coordinate plane.** Remember that the coordinate plane is represented by the coordinate grid. So when you are transforming figures on the coordinate plane, you are moving them on the coordinate grid.

**What is a transformation?**

A ** transformation** is the movement of a figure on the coordinate grid.

Figures can be transformed in three different ways: a ** translation,** a

**or a**

*reflection***.**

*rotation*
**Translations**

A translation is a slide of a figure. When a figure stays in the same position, and it is simply slid from one part of the coordinate grid to another part of the coordinate grid, we call it a translation.

Here is an example of a translation.

**You can see here that the figure was not changed at all. It simply slid from one point to another point.**

**Reflection**

A reflection is a flip of a figure. We think of reflections when we think about a mirror. One half is like the other half, but they are reflected. When we reflect a figure on the coordinate grid, we flip it. Figures can be flipped over the \begin{align*}x\end{align*} axis or over the \begin{align*}y\end{align*} axis. Here is an example of a reflection.

Here you can see that the figure was reflected or flipped over the \begin{align*}y\end{align*} axis. Let’s look at an example where a figure was reflected over the \begin{align*}x\end{align*} axis.

**Rotations**

A rotation is a turn. When you turn a figure on the coordinate grid, you are rotating the figure. We can rotate figures in terms of degrees. Let’s look at an example.

**Notice that the triangle was turned in each quadrant. Each turn is a rotation of the figure.**

**Identify each example as a rotation, translation or reflection.**

*Take a few minutes to check your work with a neighbor.*

II. **Graph Paired Transformations of Figures Given Coordinates of Vertices**

In the last section we worked on identifying different transformations. We can also graph them using given vertices. Once you have graphed the figures, then you can identify whether you have a reflection, a rotation or a translation. Let’s begin.

Example

Graph Figure \begin{align*}ABC, A(-1, 5) \ B(-1, 1) \ C(-3, 3)\end{align*}

Then graph figure \begin{align*}DEF, D(1, 5) \ E(1, 1) \ F(3, 3)\end{align*}

**First, graph figure \begin{align*}ABC\end{align*}, then graph \begin{align*}DEF\end{align*} and compare the two figures. Here is a graph of the two figures.**

**In looking at these two figures, you can see that they represent a reflection.**

Let’s look at another one.

Example

Graph Figure \begin{align*}JGHI, J(1,1) \ G(3, -2), \ H(6,-1), \ I(6, 2)\end{align*} and figure \begin{align*}LMNK, L(-1,2), M(1, -1), N(4,0), K(4,3)\end{align*}

**Here is the graph of the two figures.**

**You can see that the two figures represent a translation or a slide.**

III. **Describe Transformations as \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*} Coordinate Changes**

We can describe translations by looking at how the \begin{align*}x\end{align*} and the \begin{align*}y\end{align*}coordinate have changed from one figure to the other corresponding figure. Let’s look at an example and then talk through this.

**Here we have two figures. We have triangle \begin{align*}ABC\end{align*} and we have triangle \begin{align*}A'B'C'\end{align*}. Triangle \begin{align*}ABC\end{align*} is the figure that we started with. We translated or slid the figure and created \begin{align*}A'B'C'\end{align*}.**

**We can describe this translation as the change in the \begin{align*}x\end{align*} value and as the change in the \begin{align*}y\end{align*} value.**

**We do that by writing the change in the coordinates of each vertex. This becomes an ordered pair of integers that expresses the translation.**

Looking at the two figures, you can see that each vertex was moved +3 on the 'x' axis and -2 on the 'y' axis.

**The ordered pair that expresses the change is (3, -2).**

**You can describe other transformations according to the change in \begin{align*}x\end{align*} and \begin{align*}y\end{align*} too.**

**Practice by describing the following transformation as the change in \begin{align*}x\end{align*} and the change in \begin{align*}y\end{align*}.**

(Notice that you will need to use more than one type of transformation on the figure below. Just apply them one at a time, and the answer will be correct.)

1.

*Take a few minutes to check your answer with a partner. Is your work accurate? Did you write your answer as an ordered pair?*

IV. **Identify Equivalent Transformations with Different Coordinate Changes**

In the last few sections, we identified ** equivalent** or equal transformations using graphing and using the change of the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} coordinates. We can also identify equivalent transformations without using graphing. We can look at the coordinate changes and determine whether or not two figures are equivalent.

**How can we do this?**

That is a good question. Let’s look at the coordinates of two figures and determine whether or not the two figures are equivalent by examining the coordinate changes.

Figure \begin{align*}XYZ\end{align*} has the following vertices.

\begin{align*}& X(4, 5)\\ & Y(2, 7)\\ & Z (3, 6)\end{align*}

Figure \begin{align*}ABC\end{align*} has the following vertices.

\begin{align*}& A(1, 7)\\ & B (-1, 9)\\ & C (0, 8)\end{align*}

**Are these two figures equivalent?**

To figure this out, we have to figure out the change in the \begin{align*}x\end{align*} coordinate from one vertex to the other and the change in the \begin{align*}y\end{align*} coordinate from one vertex to the other. If the change is the same for all three vertices, then the two figures are equivalent.

\begin{align*}X(4, 5)\end{align*} to \begin{align*}A(1, 7)\end{align*} From \begin{align*}x\end{align*} to \begin{align*}x\end{align*} is a change of -3, from \begin{align*}y\end{align*} to \begin{align*}y\end{align*} is change of +2. We can write this as ordered pair (-3, 2).

**If \begin{align*}Y\end{align*} to \begin{align*}B\end{align*} and \begin{align*}Z\end{align*} to \begin{align*}C\end{align*} also have a change of (-3, 2), then the two figures are equivalent. Each vertex must have the same change or the figures are not equivalent.**

\begin{align*}Y(2, 7)\end{align*} to \begin{align*}B (-1, 9)\end{align*} From \begin{align*}x\end{align*} to \begin{align*}x\end{align*} is a change of -3, from \begin{align*}y\end{align*} to \begin{align*}y\end{align*} is a change of +2. This vertex also has a change of (-3, 2).

\begin{align*}Z(3, 6)\end{align*} to \begin{align*}C (0, 8)\end{align*} from \begin{align*}x\end{align*} to \begin{align*}x\end{align*}, 3 to 0 is a change of -3, from \begin{align*}y\end{align*} to \begin{align*}y\end{align*}, 6 to 8 is a change of +2. This vertex also has a change of (-3, 2).

**The change in each vertex is the same from the first figure to the second figure. Therefore, the two figures are equivalent.**

**Try one of these on your own. Are these two figures equivalent? Why or why not?**

Figure \begin{align*}BCD\end{align*} has the following vertices. \begin{align*}B(1, 4) \ C(2, -1) \ D(4, 3)\end{align*}

Figure \begin{align*}LMN\end{align*} has the following vertices. \begin{align*}L(4, 2) \ M(5, -3) \ N(7, -1)\end{align*}

*Take a few minutes to check your answer with a partner. Be sure to explain why the figures are or are not equivalent.*

## Real Life Example Completed

*The Clubhouse*

**Let’s look at the beginning problem once again and help Cody with his transformation. Reread the problem and underline the important information.**

Cody and his Dad are working on building a clubhouse in his backyard. Cody is excited about the project and can’t wait to start building. He wrote to his penpal Trevor, in New Zealand, and told him all about the clubhouse. Trevor asked Cody to email him some pictures, and Cody said that he would. In fact, Cody told him that he would email them right away.

The only problem is that they haven’t started yet. Cody’s Dad has insisted that he draw a complete plan of the clubhouse. Cody did that and thought that they would be able to start, but then Cody’s Dad said that he wanted Cody to draw a plan of the backyard and where the clubhouse was going to be built. This made Cody a bit frustrated, but he decided to do it anyway in hopes that they could start building the clubhouse on the weekend. Then he could take pictures and email them to Trevor.

Cody decided to use a coordinate grid to map out the backyard. He drew the following plan and went to show his Dad.

Cody has created a scale where each unit on the coordinate grid represents 5 feet. His clubhouse dimensions are \begin{align*}14.1 \ ft \times 14.1 \ ft\end{align*}. The point on the grid at (4, -3) represents the back door of Cody’s house. His family has a large backyard, so the clubhouse is the perfect size.

Cody brings his drawing to his Dad and shows him the plan for the clubhouse.

“That looks great, except the clubhouse needs to move to the right 10 feet. Your Mom wants to plant her garden right where the clubhouse is now. I think if you move it over 10 feet the drawing will work,” Dad explains.

**First, let’s think about which type of transformation Cody needs to move the clubhouse. If the clubhouse is going to move 10 feet to the right, Cody needs to slide the clubhouse over. A slide is another name for a translation.**

**To complete the translation, Cody needs to move each of the vertices of the clubhouse two units to the right. He needs to move them each two units because each unit is worth 5 feet and Cody’s Dad has told him to move the clubhouse 10 feet to the right. Here is Cody’s redesign of the clubhouse location.**

**The arrows show where Cody moved each vertex.**

**Cody shows his Dad the drawing. Cody’s Dad is pleased with Cody’s perfect translation. Together, the two of them begin to work on building the clubhouse.**

## Vocabulary

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

- Transformation
- when a figure is moved in some way on the coordinate plane, the way that the figure is moved is called a transformation.

- Translation
- a slide, this when a figure slides on the coordinate plane from one place to the next

- Reflection
- the flip of a figure. Figures can be reflected over the \begin{align*}x\end{align*} axis or over the \begin{align*}y\end{align*} axis

- Rotation
- a turn. A figure can be turned in various directions on the coordinate plane.

- Equivalent
- another name for equal

## Time to Practice

Directions: Identify each image as a translation, rotation or reflection.

1.

2.

3.

4.

5.

Directions: For numbers 6 – 12 Draw your own figures, and demonstrate three different translations, three different rotations and three different reflections.

Directions: Complete the following reflections according to the directions.

13. Reflect this image across the \begin{align*}y\end{align*} axis.

14. Rotate the following image.

15 – 20 Go on a scavenger hunt and find examples of each of the different types of transformations in real world contexts.

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