Lesson 4 - page 1 Title: Rotations and Symmetry I. Before Engagement Duration: 2 days Knowledge & Skills Understand transformations as operations that map a figure onto an image Understand characteristics of rotations Perform rotations on objects in the coordinate plane Conjecture about the effects of rotations Identify and describe rotational symmetry Vocabulary for Student Discourse Pre-image Image Rotate Symmetry Transformation Rigid transformation Congruent Mapping Point of symmetry Misconceptions Rotations are done about the center of a figure. Students need to attend to all three necessary pieces of information necessary to describe a rotation of the plane: direction, center, and amount of rotation. Generally, assumptions should not be made all three descriptors should be accounted for. A regular hexagon has rotational symmetry of 60. It is necessary to describe rotational symmetry by naming all of the rotations (greater than 0 and less than 360 ) that will map the figure onto itself. Students should observe that these rotations will be multiples of the least amount of rotation necessary to map a figure onto itself. Figures that have no rotational symmetry have an order of rotational symmetry of zero. Rotational symmetry can be defined as the number of times a figure maps onto itself in one complete rotation. Since one complete rotation will always result in a figure mapping directly onto itself, all figures have an order of rotational symmetry of at least one. (However, the figure itself would be described as having no rotational symmetry, since the figure does not map to itself following a rotation about its center of 180 degrees or less.) For example, the isosceles triangle shown at right has no rotational symmetry and an order of rotational symmetry of 1. III. Evidence of Individual Sense-Making Text: McDougal-Littell s Geometry Concepts and Skills Section 11.8 (Rotations) pages 636-639 10-13 22-26 29, 30 33-36
Lesson 4 - page 2 Additional Problems: 24. Which of the following words or phrases are most useful in writing the definition of rotation? center point line segment perpendicular line angle parallel line circle Write a definition using the terms you selected. 25. Each of the following illustrations shows a plane figure consisting of the letters F, R, E, and D evenly spaced and arranged in a row. In each illustration, an alteration is shown. In some of the illustrations, the image is obtained from the original figure by a geometric transformation consisting of a single rotation. In others, this is not the case. a. Which illustrations show a single rotation? b. Some of the illustrations are not rotations or even a sequence of rigid transformations. Pick one such illustration and use it to explain why it is not a sequence of rigid transformations. 26. What fraction of a turn does the wagon wheel need to turn in order to appear the very same as it does right now? How many degrees of rotation would that be?
Lesson 4 - page 3 27. What fraction of a turn does the propeller need to turn in order to appear the very same as it does right now? How many degrees of rotation would that be? 28. What fraction of a turn does the model of a Ferris wheel need to turn in order to appear the very same as it does right now? How many degree of rotation would that be? 29. For each of the regular polygons, describe the rotational symmetry. What patterns do you notice in terms of the angles of rotation when describing the rotational symmetry in a regular polygon? 30. Using point P as a center of rotation, rotate point Q 120 clockwise about point P and label the image Q
Lesson 4 - page 4 31. Using point C as the center of rotation, rotate point R 270 counterclockwise about point C and label the image R. 32. Review: Solve each system of equations. a. b.
Lesson 4 - page 5 GeoGebra Activity: Rotations 1. Draw with coordinates A(-4, 2), B(-3, -1) and C(-1, 3). 2. Rotate in clockwise direction about the origin and write the coordinates for A, B and C. A (, ) B (, ) C (, ) 3. Write a conjecture in function notation that explains how each coordinate changes after the reflection 4. Open a new Geogebra project and Draw with coordinates A(-4, 2), B(-3, -1) and C(-1, 3). 5. Reflect in clockwise direction about the origin and write the coordinates for A, B and C. A (, ) B (, ) C (, ) 6. Write a conjecture in function notation that explains how each coordinate changes after the reflection 7. Open a new Geogebra project and Draw with coordinates A(-4, 2), B(-3, -1) and C(-1, 3). 8. Reflect in clockwise direction about the origin and write the coordinates for A, B and C. A (, ) B (, ) C (, ) 9. Write a conjecture in function notation that explains how each coordinate changes after the reflection 10. Repeat steps 1-9 using counterclockwise rotations 11. Explain the relationship between clockwise and counterclockwise rotations about the origin.
Lesson 4 - page 6 Guided Practice: Rotations 1. Identify the line symmetry and rotational symmetry (if any) of each word. 2. In the diagram, lines & are parallel. is reflected in line, and is reflected in line. a. A translation maps onto which triangle? b. Which lines are perpendicular to? c. Name two segments parallel to. d. If the distance between and is 2.6 inches, what is the length of? e. Is the distance from B to the same as the distance from B to? Explain.