Math Background

Lesson: Transformations
Introducing the Concept

The use of transformations can be found in designs for linoleum, wallpaper, and wall and window murals. This section provides an opportunity to work with the art department to make some tessellations, which are based on translations, rotations, or reflections.

Materials: overhead transparencies with coordinate grids on them, overhead washable pen, and ruler; graph paper, ruler, and pencil for each student

Preparation: none

Prerequisite Skills and Background: Students should be familiar with various types of triangles and quadrilaterals. They should also have some familiarity with graphing points on a coordinate grid. Students should know the various types of angles and be able to use a ruler and protractor.

  • Say: Today we are going to investigate some geometric transformations. The transformations we will work with move figures in the plane. The first transformation is called a translation. A translation slides a figure along a line in one direction.

    Draw a triangle on a piece of grid paper on the overhead and label the vertices A, B, and C. Move all the points three units to the right and up two units. To do this, you need only move the three vertices and then connect them. Label the vertices of the new triangle A', B', and C'.

  • Say: Let's look at triangle ABC, which I've drawn on this graph paper. If I move each of the three vertices three units to the right and up two units, I form a new triangle, A´B´C.
  • Ask: How do the two triangles compare? (They are congruent.)
  • Say: On your graph paper, draw any quadrilateral and label it “ABCD”. Now move each of the four vertices four units to the left and four units up, with point A going to A´, B going to B´, and so on. Connect the points A´, B´, C´, and D´. What relationship do you see between the two quadrilaterals? (They are congruent.)
  • Say: Now we'll perform a transformation called a rotation. On your graph paper, draw a parallelogram so that one side is parallel to the bottom of the paper. Label the parallelogram “WXYZ.” Now count over four units to the left of the lower left-hand vertex of the parallelogram. (Do this on a piece of overhead graph paper, so that students can follow along. See the diagram below.) Label that point “P.”
  • Say: Now rotate that parallelogram 90 degrees in a counterclockwise direction like this.
    (Demonstrate the rotation.) Label the new figure “W´X´Y´Z´.”
    graph diagrams
  • Ask: How does the new figure compare with the old one? (They are congruent.)
  • Say: Now we're going to work with a transformation called a reflection. When you look in a mirror, what do you see? (Students should say a picture or an image of themselves.) Yes, and that is called a reflection. When you look in the mirror and move your right hand, what hand does your reflection seem to be moving? (your left hand)
  • Say: Let's look at this triangle, ABC. Draw this triangle on your sheet of graph paper. (Give students a minute to draw the triangle.) Now draw a line MN to the right of your triangle. (Show students, using the overhead transparency, the diagram below.) Now fold your paper along line MN and look through the back side of the paper to see if you can see points A, B and C. If you can, mark them on the paper. If you can't, make your lines darker on the original triangle and then try again. Now open up your paper and draw the new triangle on the other side of line MN.
    Have students who are able to draw the triangle help the students who are struggling.
    graph diagrams
  • Ask: How far is point B from the line MN? (1 unit) How far is B´ from the line MN? (1 unit) How far is A from the line MN? (7 units) How about A´? (7 units) How far is C from the line MN? (5 units) What about C´? (5 units)
  • Say: Another way to find the reflective image is to count how many units away from the line of reflection a given point is and count the same number of units on the other side. Let's try that method instead of paper folding to find the image of a rectangle about a line of reflection.
    graph diagram

Houghton Mifflin Math Grade 6