Making Sense of Transformations with Communicators and the Graphing Calculator

1. Making Sense of Transformations with Communicators and the Graphing Calculator Jim Rahn LL Teach, Inc. www.jamesrahn.com www.llteach.com James.rahn@verizon.net

2. Communicators • Dry Erase Markers • Eraser Cloths • Handouts • Rulers Supplies at the table

3. available at http://jamesrahn.com/workshops/acnctm.htm Workshop Handout and Powerpoint

4. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. Grade 6

5. Understand congruence and similarity using physical models, transparencies, or geometry software. 1. Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. 2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. Grade 8

6. 3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two dimensional figures, describe a sequence that exhibits the similarity between them. Grade 8

7. Exploring Reflections of Points

8. II I IV III Place the Transformation Template in your communicator. We usually name the quadrants I, II, III, and IV. Label your quadrants with large Roman Numerals and then be ready to hold your communicators up so I can see your responses.

9. Place three points in Quadrant I. Name them. Switch communicators with a partner and check the naming of the points.

10. Place three points in Quadrant II. Name them. Switch communicators with a partner and check the naming of the points.

11. Place three points in Quadrant III and IV. Name them. Switch communicators with a partner and check the naming of the points.

12. Repeat the steps several times. Confirm that your conjecture is correct. Locate a point in Quadrant I. Name the point. Reflect the point over the x-axis. Name the point. Make an observation and a conjecture. Describe your observation.

13. Repeat the steps several times. Confirm that your conjecture is correct. Locate a point in Quadrant I. Name the point. Reflect the point over the y-axis. Name the point. Make an observation and a conjecture. Describe your observation.

14. Locate points in other quadrants and create their reflections over the x- and y-axes. Confirm your conjecture about what happens to the coordinates when your reflect a point over the x-axis and y-axis.

15. Suppose a point (-4,5) is reflected over the x-axis, what point does it become? Write your response on the communicator in the right quadrant. Checking for Understanding

16. Suppose a point (4,-3) is reflected over the y-axis, what point does it become? Write your response on the communicator in the right quadrant. Checking for Understanding

17. What happens when we reflect a polygon?

18. Remove the template from the communicator. Draw a triangle in Quadrant I on the template that has no vertical or horizontal sides. Place your communicator on top of the template. Draw the axes on the communicator and your triangle. We will use the communicators this way for the rest of the workshop.

19. Write the coordinates for the three vertices. Create a reflection of your triangle over the x-axis by flipping your communicator vertically. Trace the reflection. Flip your communicator again and name the vertices of the new figure. What do your observe?

20. Erase your communicator. Trace the axes and your original triangle on the communicator. Create a reflection of your triangle over the y-axis by flipping your communicator horizontally. Trace the reflection. Flip your communicator again and name the vertices of the new figure. What do your observe?

21. Draw a new triangle on the template in Quadrant III that has no vertical or horizontal sides. Use your conjecture to name the new coordinates if that triangle is reflected over the x-axis. Confirm your answer by performing the reflection.

22. Erase your communicator. Trace the triangle in Quadrant III and the axes on your communicator. Use your conjecture to name the new coordinates if that triangle is reflected over the y-axis. Confirm your answer by performing the reflection.

23. Erase your communicator. Reflect your triangle in Quadrant I over the y-axis. Trace the reflection. Make a list of the coordinates on the reflected triangle. What do your observe?

24. What other properties can we observe?

25. Erase your communicator. Trace the axes and your original triangle on the communicator. Create a reflection of your triangle over the y-axis by flipping your communicator horizontally. How are the figures related? Pick up your communicators and compare the two triangles.

26. Draw the line segments between to corresponding points on the triangles. Make some observations about these line segments. What do your observe?

27. Creating a Rotation

28. Place your communicator on top of the template. Draw the axes on the communicator and your triangle in Quadrant I. Rotate your triangle 90o counterclockwise. Record the coordinates of both figures. Make an observation about the coordinates.

29. Place your communicator on top of the template. Draw the axes on the communicator and your triangle in Quadrant I. Rotate your triangle 90o counterclockwise. Record the coordinates of both figures. Make an observation about the coordinates.

30. Place your communicator on top of the template. Draw the axes on the communicator and your triangle in Quadrant I. Rotate your triangle 180o counterclockwise. Record the coordinates of both figures. Make an observation about the coordinates.

31. Integrating Technology

32. Turn the Calculator ON and Press Graph Press Zoom 4. Decimal Press 2nd Zoom and Turn the Grid on Setting Up the Calculator

33. Press Graph Your graph nearly resembles the graph we have been working with on the communicators. Press Window and change Ymin=-5 and Ymax=5. Let’s build the same triangle you had on your template. Press LIST to see L1, L2, and L3.

34. In L1 place the x-coordinates for your triangle Repeat the first coordinate as the fourth coordinate. In L2 place the corresponding y-coordinate for your triangle. Repeat the first coordinate as the fourth coordinate. Press 2nd plot (Y=). Notice all the plots are currently off. Press 1 to turn on plot 1. Set up Plot 1 as illustrated.

35. Press Graph. You should see your original triangle graphed in quadrant I. To create a second triangle press LIST and enter two new lists. What numbers should we enter?

36. If we want to reflect the triangle over the x-axis what should the new ordered pairs be? Which lists should we select for x and which for y? Press 2nd Plot (y=) and select plot 2. Enter this list that you believe should produce a triangle that is the reflection over the x-axis.

37. Now work to change plot 2 so the triangle is reflected over the y-axis.

38. Now work to change plot 2 so the triangle is rotated 90o counterclockwise.

39. Now work to change plot 2 so the triangle is rotated 90o clockwise.

40. Now work to change plot 2 so the triangle is rotated 180o .

41. how communicators can help build students understanding of change coordinates change when an object is reflected over the x- or y-axis or rotated. • How communicators can help building understanding for some of the properties of reflections. • How we can apply our new knowledge to create reflections and rotations on a graphing calculator We experienced: