Lesson 3Grid Moves

Learning Goal

Let’s transform some figures on grids.

Learning Targets

I can decide which type of transformations will work to move one figure to another.
I can use grids to carry out transformations of figures.

Lesson Terms

counterclockwise
image
reflection
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 $ℓ$

Lesson Summary

When a figure is on a grid, we can use the grid to describe a transformation. For example, here is a figure and an image of the figure after a move.

Two identical quadrilaterals on a grid labeled B D C A and B prime D prime C prime A prime. In quadrilateral B D C A, point B is 3 units right and 3 units down from the edge of the grid. Point D is 1 unit left and 1 unit up from point B. Point C is 2 units up from point B. Point A is 2 units right from point B. In quadritaleral B prime D prime C prime A prime, point B prime is 3 units down and 4 units right from point B. Point D prime is 3 units down and 4 units right from point D. Point C prime is 3 units down and 4 units right from point C. Point A prime is 3 units down and 4 units right from point A.

Quadrilateral $A B C D$ is translated 4 units to the right and 3 units down to the position of quadrilateral $A^{'} B^{'} C^{'} D^{'}$ .

A second type of grid is called an isometric grid. The isometric grid is made up of equilateral triangles. The angles in the triangles all 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 '

Here is quadrilateral $K L M N$ and its image $K^{'} L^{'} M^{'} N^{'}$ after a 60-degree counterclockwise rotation around a point $P$ .