Lesson 3Grid Moves

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.

3.1 Notice and Wonder: The Isometric Grid

What do you notice? What do you wonder?

A blank isometric grid.

3.2 Transformation Information

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.

  1. Translate triangle ABC 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.
  2. Translate triangle ABC so that C goes to C’ .

  3. Rotate triangle ABC 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.
  4. Reflect triangle ABC using line \ell .
    1. Select the Reflect about Line tool.
    2. Click on the figure to reflect, and then click on the line of reflection.
  1. Rotate quadrilateral ABCD 60^\circ counterclockwise using center B .
  2. Rotate quadrilateral ABCD 60^\circ clockwise using center C .
  3. Reflect quadrilateral ABCD using line \ell .
  4. Translate quadrilateral ABCD so that A goes to C .

Are you ready for more?

Try your own translations, reflections, and rotations.

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

Lesson 3 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 ABCD 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.

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

Lesson 3 Practice Problems

  1. Apply each transformation described to Figure A. If you get stuck, try using tracing paper.

    1. A translation which takes P to P’
    2. A counterclockwise rotation of A, using center P , of 60 degrees
    3. A reflection of A across line \ell
  2. Here is triangle ABC drawn on a grid.

    On the grid, draw a rotation of triangle ABC , a translation of triangle ABC , and a reflection of triangle ABC . Describe clearly how each was done.

    1. Draw the translated image of ABCDE so that vertex C moves to C’ . Tracing paper may be useful.
    2. Draw the reflected image of Pentagon ABCDE with line of reflection \ell . Tracing paper may be useful.
    3. Draw the rotation of Pentagon  ABCDE around C clockwise by an angle of 150 degrees. Tracing paper and a protractor may be useful.