1. 2012-2013 Arrington 1 UNIT ONE OVERVIEWOFTRANSFORMATIONS

2. 2012-2013 Arrington 2 Goals for Activity Understand that a transformation is a function that maps the plane onto itself. Identify some basic properties of isometries and dilations. Link the concept of congruency to isometry. Link the concept of similarity to dilation.

3. 2012-2013 Arrington 3 GO (Warm-up) Materials needed: Notebook Paper, #2 Pencil, Color Pencil 1.With your pencil, draw the parallelogram shown below on a sheet of notebook paper. 2.Turn your paper over to the back. Using a regular colored pencil, draw over the parallelogram on the back of the paper. 3.Put your notebook paper on top of the original parallelogram. Slide the notebook paper about 2 inches to the “northeast” (i.e. slide right and then up). 4. Trace over the notebook paper parallelogram. Pencil markings from the back of the notebook paper should transfer to this paper. Use light markings to draw the resulting figure. This drawing technique is called the “two-sided transfer technique”.

4. 2012-2013 Arrington 4 Which pictures accurately depicts the results of sliding the patty paper?

5. 2012-2013 Arrington 5 ABOUT TRANSFORMATIONS A transformation of the plane is a one-to-one mapping (function) of the plane onto itself. Refer to the warm-up. You were given a parallelogram. Then, using notebook paper you performed a slide to locate a new parallelogram. This action can be thought of as a transformation. The resulting figure is called the image of the original figure.

6. 2012-2013 Arrington 6 1. Color the original figure blue. Label it parallelogram ABCD. 2. Color the image red. Label it parallelogram A’B’C’D’. Be sure to mark corresponding points with the same letter. A B C D A’ B’D’ C’

7. 2012-2013 Arrington 7 DISCUSSION QUESTIONS “Is distance preserved under this transformation?” What does this mean? YES, For this translation it does not matter what two points are chosen because the distance between any two points corresponding points on the original figure will be the same as the distance between corresponding points on the image.

8. 2012-2013 Arrington 8 DISCUSSION QUESTIONS Is parallelism preserved under this transformation? What does that mean? YES, because the lines are parallel in the original figure and mapped to images that are parallel lines as well.

9. 2012-2013 Arrington 9 DISCUSSION QUESTIONS In question two, Is collinearity preserved under this transformation? YES, In mathematical terms, the importance of this statement is that “lines are mapped to lines.” Example. Chose three points that lie on the line in the original figure. Then identify the corresponding points on the image.

10. 2012-2013 Arrington 10 DISCUSSION QUESTIONS Is angle measure preserved under this transformation? Angle measure is preserved. In other words, angles are taken to angles of the same measure. Betweeness is preserved. In other words, if B is between A and C on a line, then B’ is between A’ and C’ on the images. Is betweeness preserved under this transformation?

11. 2012-2013 Arrington 11 3. Using the chart below, write TRUE or FALSE for each statement. Under this transformation: TRUE OR FALSE 3. Angle measure is preserved. In other words, angles are taken to angles of the same measure. 4. Collinearity is preserved. In other words, if three points lie on the same line, then their images lie on the same line. 5. Betweeness is preserved. In other words, if B is between A and C on a line, then B’ is between A’ and C’ on the images. 1. Distance is preserved. In other words, lines are taken to lines, and line segments to line segments of the same length. 2. Parallelism is preserved. In other words, parallel lines are taken to parallel lines. TRUE

12. 2012-2013 Arrington 12 Introduction to Isometries We say two figures are congruent if there is a sequence of isometries of the plane onto itself that map one figure onto the other. In other words, congruent shapes have the same shape and size. Which of the pairs of figures above appear to be congruent? Write how could you justify the pairs you pick. Isometry is a transformation that preserves shape and size.

13. 2012-2013 Arrington 13 1.Yes, Why? 2.No, Why? 3.Yes, Why? 4.No, Why? 5.Yes, Why? 6.No, Why? Are images 1,2,3,4, or 5 an isometry?

14. 2012-2013 Arrington 14 Let’s Continue

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17. 2012-2013 Arrington 17 REVIEW GOALS Understand that a transformation is a function that maps the plane onto itself. Identify some basic properties of isometries and dilations. Link the concept of congruency to isometry. Link the concept of similarity to dilation.

18. 2012-2013 Arrington 18 Discussion Question A dilation is a function. Similarity is a property of two figures. These are related because a dilation creates similar figures. How is dilation different from similarity?