Learning Task: Exploring Reflections and Rotations
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1 Learning Task: Exploring Reflections and Rotations Name Date Mathematical Goals Develop and demonstrate an understanding of reflections and rotations of figures in general and on a coordinate plane. Essential Questions How are reflections and rotations similar and different? Georgia Standards of Excellence MGSE9-12.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MGSE9 12.G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle measure to those that do not (e.g., translation versus horizontal stretch). MGSE9 12.G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. MGSE9 12.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. MGSE9 12.G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Standards for Mathematical Practice 5. Use appropriate tools strategically. 7. Look for and make use of structure. July 2016 Page 30 of 87
2 Learning Task: Exploring Reflections and Rotations Name Date Reflections: 1. Select the vertices of the square and go to Display Label Points. Start with C so that your square s vertices are labeled C, D, E, & F. Find the coordinates of each point: C (, ); D (, ); E (, ); F (, ) 2. Double click the line x = 2. You should see an animation on the line. This makes x = 2 the line of reflection (also called the line of symmetry). 3. Select the interior & the vertices of your square. Go to Transform Reflect. 4. Select the vertices of the new square and go to Display Show Labels. 5. How have the new points changed? 6. Double click on the y axis to change it to the line of reflection. Select the interior & vertices of your square. Go to Transform Reflect. 7. What has happened? Why is this reflection further away than the last one? What effect did changing the reflection line have? July 2016 Page 31 of 87
3 8. Write out the coordinates of each square. Original Square Reflection over x = 2 Reflection over y axis C (, ) C (, ) C (, ) D (, ) D (, ) D (, ) E (, ) E (, ) E (, ) F (, ) F (, ) F (, ) How far apart are the original square and the first reflection? The original square and the second reflection? How far is the original square from x = 2 and how far is the first reflection from x = 2? How far is the original square from the y axis and the second reflection from y axis? 9. Delete the two reflections so that only your original square remains. 10. Double click the line y = 0.5x 5 to make it the new line of reflection. If you were to reflect the square over y = 0.5x 5, predict where would the new vertices be? Original Square Prediction of Reflection over y = 0.5x 5 C (, ) C (, ) D (, ) D (, ) E (, ) E (, ) F (, ) F (, ) July 2016 Page 32 of 87
4 11. After you have made your prediction, select the vertices and interior of the square. Go to Transform Reflect. Select the vertices of the new square and go to Display Show Labels. How does your prediction compare with the actual reflection? 12. Make a general conclusion about what happens to coordinates of a point when they are reflected over a line. Rotations: 13. Delete the reflection so only the original square remains. 14. Using the Selection Arrow Tool, double click the point B. A brief animation should show. Point B is now your center of rotation. 15. Select the interior and the vertices of your square. Go to Transform Rotate. A box labeled Rotate should come up. (Move the box over to the side so you can still see your sketch.) You should see a lighter colored square that has appeared on your sketch. Change the number of degrees BUT DON T CLICK ANY BUTTONS. The image should change where it goes. Try different angles (between 0º and 360º). Explain what is happening to the square and the points. 16. Try negative angles (once again without clicking any buttons). What do you notice about 90º and 270º? 180º and 180º? 45º and 315º? Why do you think this is? July 2016 Page 33 of 87
5 Are there any other pairs of measures that have the same phenomenon? How could we predict additional pairs? 17. Type in 270º (clockwise) for the angle and click the button rotate. Write down the new coordinates and compare them to the old coordinates. Original Square Rotation around B C (, ) C (, ) D (, ) D (, ) E (, ) E (, ) F (, ) F (, ) What relationship is there between the vertices of the two figures? 18. What conclusions can you make about what happens to coordinates when rotated? EXTENSION 19. Is there ever a time when a rotation is the same as a reflection? Explain. 20. Create an animation in a new Geometer s Sketchpad window to support your answer to number 19. your animation to your teacher. July 2016 Page 34 of 87
Learning Task: Exploring Reflections and Rotations
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