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 (Alternative) Name Date Reflections: 1. On a piece of graph paper, graph the following points to create square CDEF C (3, 0); D (4, 1); E (5, 0); F (4, 1) 2. Draw the line: x = Using either Mira, patty paper or a transparency reflect the square over the x = 2 line. 4. How have the new points changed? 5. Using the original square, now reflect it over the y axis. 6. What has happened? Why is this reflection further away than the last one? What effect did changing the reflection line have? 7. 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 (, ) July 2016 Page 35 of 87
3 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 and the y axis? 8. Draw the line y = 0.5x 5. If you were to reflect the original square over y = 0.5x 5, predict where the new vertices would be. Original Square C (, ) Prediction of Reflection over y = 0.5x 5 C (, ) D (, ) D (, ) E (, ) E (, ) F (, ) F (, ) 9. After you have made your prediction, use the Mira, patty paper, or transparency to reflect the original square over the y = 0.5x 5 line. How does your prediction compare with the actual reflection? 10. Make a general conclusion about what happens to coordinates of a point when it is reflected over a line. July 2016 Page 36 of 87
4 Rotations: 11. Return to the original square. 12. Plot the point B(2,0). Point B is now your center of rotation. 13. Experiment with rotating the square about point B using either patty paper or a transparency. Try different numbers of degrees (less than 360) counter clockwise. Explain what is happening to the square and the points. 14. Try rotating negative degrees (clockwise). What do you notice about 90º and 270º? 180º and 180º? 45º and 315º? Why do you think this is? Are there any other pairs of measures that have the same phenomenon? How could we predict additional pairs? 15. Rotate the square about point B, 270º clockwise. 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 (, ) July 2016 Page 37 of 87
5 What relationship is there between the points of the two figures? 16. What conclusions can you make about what happens to coordinates when rotated? EXTENSION 17. Is there ever a time when a rotation is the same as a reflection? Explain. 18. Create a sequence of transformations to support your answer to number 17. July 2016 Page 38 of 87
Learning Task: Exploring Reflections and Rotations
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
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