Lesson 3Making the Moves

Learning Goal

Let’s draw and describe translations, rotations, and reflections.

Learning Targets

I can use grids to carry out transformations of figures.
I can use the terms translation, rotation, and reflection to precisely describe transformations.

Lesson Terms

counterclockwise
image
reflection
sequence of transformations
transformation
vertex

Warm Up: Notice and Wonder: The Isometric Grid

What do you notice? What do you wonder?

Activity 1: Transformation Information

Your teacher will give you tracing paper to carry out the moves specified. Use $A^{'}$ , $B^{'}$ , $C^{'}$ , and $D^{'}$ to indicate vertices in the new figure that correspond to the points $A$ , $B$ , $C$ , and $D$ in the original figure.

Problem 1

Follow the directions below each statement to tell GeoGebra how you want the figure to move. It is important to notice that GeoGebra uses vectors to show translations. A vector is a quantity that has magnitude (size) and direction. It is usually represented by an arrow.

These applets are sensitive to clicks. Be sure to make one quick click, otherwise it may count a double-click.

After each example, click the reset button, and then move the slider over for the next question.

Translate triangle $A B C$ so that $A$ goes to $A^{'}$ .
1. Select the Vector tool.
2. Click on the original point $A$ and then the new point $A^{'}$ . You should see a vector.
3. Select the Translate by Vector tool.
4. Click on the figure to translate, and then click on the vector.
Translate triangle $A B C$ so that $C$ goes to $C^{'}$ .
Rotate triangle $A B C$ $90^{\circ}$ counterclockwise using center $O$ .
1. Select the Rotate around Point tool.
2. Click on the figure to rotate, and then click on the center point.
3. A dialog box will open; type the angle by which to rotate and select the direction of rotation.
4. Click on ok.
Reflect triangle $A B C$ using line $ℓ$ .
1. Select the Reflect about Line tool.
2. Click on the figure to reflect, and then click on the line of reflection.

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In Figure 1, translate triangle $A B C$ so that $A$ goes to $A^{'}$ .
In Figure 2, translate triangle $A B C$ so that $C$ goes to $C^{'}$ .
In Figure 3, rotate triangle $A B C$ $90^{\circ}$ counterclockwise using center $O$ .
In Figure 4, reflect triangle $A B C$ using line $ℓ$ .

A grid broken up into 4 sections with a copy of triangle ABC in them. Figures 1-3 have a point off the triangle. Figure 4 has a vertical line l to the right of the triangle.

Problem 2

Rotate quadrilateral $A B C D$ $60^{\circ}$ counterclockwise using center $B$ .
Rotate quadrilateral $A B C D$ $60^{\circ}$ clockwise using center $C$ .
Reflect quadrilateral $A B C D$ using line $ℓ$ .
Translate quadrilateral $A B C D$ so that $A$ goes to $C$ .

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In Figure 5, rotate quadrilateral $A B C D$ $60^{\circ}$ counterclockwise using center $B$ .
In Figure 6, rotate quadrilateral $A B C D$ $60^{\circ}$ clockwise using center $C$ .
In Figure 7, reflect quadrilateral $A B C D$ using line $ℓ$ .
In Figure 8, translate quadrilateral $A B C D$ so that $A$ goes to $C$ .

A grid broken up into 4 sections with a copy of quadrilateral ABCD in them. Figure 7 has a diagonal line l below the figure.

Are you ready for more?

Try your own translations, reflections, and rotations.

Make your own polygon to transform, and choose a transformation.
Predict what will happen when you transform the image. Try it - were you right?
Challenge your partner! Right click on any vectors or lines and uncheck Show Object. Can they guess what transformation you used?

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Make your own polygon to transform, and choose a transformation.
Predict what will happen when you transform the image. Try it - were you right?
Challenge your partner! Right click on any vectors or lines and uncheck Show Object. Can they guess what transformation you used?
Try to challenge your partner again.

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The effects of each move can be “undone” by using another move. For example, to undo the effect of translating 3 units to the right, we could translate 3 units to the left. What move undoes each of the following moves?

Translate 3 units up
Translate 1 unit up and 1 unit to the left
Rotate 30 degrees clockwise around a point $P$
Reflect across a line $ℓ$

Activity 2: A to B to C

Here are some figures on an isometric grid. Explore the transformation tools in the tool bar. (Directions are below the applet if you need them.)

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Translate

Select the Vector tool.
Click on the original point and then the new point. You should see a vector.
Select the Translate by Vector tool.
Click on the figure to translate, and then click on the vector.

Rotate

Select the Rotate around Point tool.
Click on the figure to rotate, and then click on the center point.
A dialog box will open. Type the angle by which to rotate and select the direction of rotation.

Reflect

Select the Reflect about Line tool.
Click on the figure to reflect, and then click on the line of reflection.

Name a transformation that takes Figure $A$ to Figure $B$ . Name a transformation that takes Figure $B$ to Figure $C$ .
What is one sequence of transformations that takes Figure $A$ to Figure $C$ ? Explain how you know.

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Here are some figures on an isometric grid.

Three trapezoid, blue A, green B and green C, on an isometric grid.

Name a transformation that takes Figure $A$ to Figure $B$ . Name a transformation that takes Figure $B$ to Figure $C$ .
What is one sequence of transformations that takes Figure $A$ to Figure $C$ ? Explain how you know.

Are you ready for more?

Experiment with some other ways to take Figure $A$ to Figure $C$ . For example, can you do it with …

No rotations?
No reflections?
No translations?

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Experiment with some other ways to take Figure $A$ to Figure $C$ . For example, can you do it with …

No rotations?
No reflections?
No translations?

Lesson Summary

A move or combination of moves is called a transformation. When we do 1 or more moves in a row, we often call that a sequence of transformations. When a figure is on a grid, we can use the grid to describe a transformation. We use the word image to describe the figure after a transformation. To distinguish the original figure from its image, points in the image are sometimes labeled with the same letters as the original figure, but with the symbol $^{'}$ attached, as in $A^{'}$ (pronounced “A prime”) is the image of $A$ after a transformation.

A translation can be described by two points. If a translation moves point $A$ to point $A^{'}$ , it moves the entire figure the same distance and direction as the distance and direction from $A$ to $A^{'}$ . The distance and direction of a translation can be shown by an arrow.

For example, here is a translation of quadrilateral $A B C D$ that moves $A$ to $A^{'}$ .

A quadrilateral A, B, C, D, and its translation to A prime, B prime, C prime, D prime.

A rotation can be described by an angle and a center. The direction of the angle can be clockwise or counterclockwise.

For example, quadrilateral $K L M N$ is rotated 60 degrees counterclockwise using center $P$ . This type of grid is called an isometric grid. The isometric grid is made up of equilateral triangles. The angles in the triangles each measure 60 degrees, making the isometric grid convenient for showing rotations of 60 degrees.

A grid with quadrilateral KLMN in blue and green K'L'M'N '

A reflection can be described by a line of reflection (the “mirror”). Each point is reflected directly across the line so that it is just as far from the mirror line, but is on the opposite side.

For example, pentagon $A B C D$ is reflected across line $m$ .

A pentagon A, B, C, D, E, and its reflection in a line m, to pentagon A prime, B prime, C prime, D prime, E prime.