FROM REFLECTIONS TO CONGRUENCE
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1 FROM REFLECTIONS TO CONGRUENCE A 10 th GRADE GEOMETRY UNIT UCSMP GEOMETRY, 2 ND EDITION, CHAPTER 4 By Robin A. Merrill Teacher, Buffalo Traditional School AOL.COM APPROX. TIME FRAME: 5-40 min. periods
2 SPECIAL TOOLS NEEDED: MIRA TI-83 WITH APPS GEOMASTER OVERALL OBJECTIVES: 1. Draw figures by applying the definition of reflection image. 2. Apply the definition of reflection to make and justify conclusions. 3. Apply properties of reflections to make conclusions, using one or more of the following justifications: A. Reflections preserve: angle measure betweenness collinearity distance B. Reflections switch: orientation C. Figure Reflection Theorem 4. Find Coordinates of reflection images of points over the coordinate axes. 5. Draw or identify images of figures under composite of two reflections. NCTM AND NYS STANDARDS: 1. Modeling / Multiple Representation 2. Measurement
3 MATERIALS AND EQUIPMENT NEEDED: 1. Mira 2. Geometric Constructions & Investigations with A Mira, by Ernest Woodward & Thomas Hamel; J. Weston Walch, Publisher, copyright UCSMP Geometry textbook, 2 nd edition. 4. TI-83 with Apps - Geomaster 5. TI-83 Plus GeoMaster p. ti.com 6. Pencil, protractor with ruler, graph paper
4 OVERVIEW OF UNIT: DAY 1: Introduce the Mira to the students. Handling and positioning of Mira. Simple reflection exercises. DAY 2: With Mira, 1. Reflect triangles over lines of reflection. 2. Measure sides and angles of pre-image and image. 3. Measure distances from A to A, B to B, etc. DAY 3: Introduce students to Geomaster and its capabilities. Practice drawing and storing a simple figure, like a triangle. DAY 4: In Geomaster, Have students take a stored triangle, and 1. Reflect it. 2. Translate it. 3. Rotate it. Note measurements of pre-images and images from 1-3 above. DAY 5: With Mira, Have students take a triangle, and 1. Reflect it. 2. Translate it. 3. Rotate it.
5 Day #1 Mira, mira, on the desk I. Objective: Students will draw figures by applying the definition of reflection image. II. Standard: Modeling/Multiple Representations III. Materials: Mira Pencil Copies of Lesson 1, p. 3 of Introduction to the Mira IV. Procedure: 1. Hand out miras and Lesson 1, p Explain how to use the mira: handle by sides, beveled edge at bottom towards belly, turn papers to get desired effect, not the mira; reach around to draw, still looking through the mira. 3. Let students practice with the mira on the boy: reflect legs onto head; reflect whole boy as if line of mira is a gymnastic bar. 4. Have students move paper around until they determine how to get the boy on the swing. 5. Trace image of boy onto swing. V. Reflections on Reflections: 1. What does moving the mira closer to and farther away from a pre-image do to the image? the 2. Where does the mira have to be located on the paper in order to get the boy onto swing? 3. How is the pre-image and the image different? How are they the same? VI. Homework Assignment: UCSMP Geometry textbook, P , #1-17 (all)
6 Day #2 Double Vision or Go Cross-eyed? I. Objective: Students will apply properties of reflections to make conclusions, using one or more of the following justifications: A. Properties preserved under a reflection B. Properties not preserved under a reflection C. Figure Reflection Theorem II. Standard: Measurement; Modeling/Multiple Representations III. Materials: Mira Pencil Copies of Lesson 2, p. 8, Lesson 3, p Protractor with ruler IV. Procedure: 1. Hand out miras and copies of lessons. (use p. 8, but white out directions) 2. Have students reflect triangle ABC over line of reflection, labeling it triangle A B C. *Note: is it easier to trace points, and then draw segments with ruler??? 3. Note terms and notation: pre-image, line of reflection, image, orientation 4. Measure angle B, and angle B. 5. Draw segment from point B to point B. Call point of intersection X. 6. Find BB, BX, and B X. 7. Measure the four angles at X. (are measuring all four necessary? Why or why not?) 8. What is the orientation of triangle ABC? Of triangle A B C? V. Reflections on Reflections: 1. Which is the easiest way to draw the image of a figure? 2. How are the measures of the angles B and B related? What can you conclude about angles A and A? C and C? 3. How are BB, BX, and B X related? What can you conclude about the point X?
7 4. What were the measures of the angles at X? What can you conclude about a line of reflection? 5. What does a reflection do to the orientation of a figure and its image? VI. Homework Assignment: UCSMP Geometry textbook, P , #2-20 (even)
8 Day #3 Welcome the Master of Geometry I. Objective: Students will be able to perform basic procedures in Geomaster on the TI-83, such as draw, store, and retrieve triangles, and move them around. II. Standard: Modeling/Multiple Representations III. Materials: TI-83 with Apps - Geomaster IV. Procedure: 1. In TI-83, go into Apps - Geomaster. Discuss and go through general descriptions of sub files of FILE, DRAW, MEAS, TRFM, and MISC. 2. Have students: A. Draw 5: Triangle - Discuss keystrokes to draw a triangle on screen, and 2: Line - horizontal? Vertical? Oblique? B. FILE: 3: Save file C. TRFM: 2: Reflection - Discuss keystrokes to select object to reflect, line around which the object will be reflected, and then watch the reflection occur. 3. Have students repeat process, perhaps this time the line of reflection goes through the figure. V. Reflections on Reflections: 1. What kind of triangle did you draw on the calculator? How could you tell? 2. What kind of line did you draw? How could you tell? 3. When you reflected your figure, where did the points go that were A. on the line of reflection? B. not on the line of reflection? VI. Homework Assignment: Repeat procedure done in class, but this time: 1. Add a second line of reflection parallel to the first. 2. Reflect your triangle again over the second line. 3. What did you notice about your original triangle, and the second reflected triangle?
9 Day 4 Flip, Slide, and Turn I. Objective: Students will reflect, translate, and rotate objects using Geomaster on the TI- 83. II. Standard: Measurement, and Modeling/Multiple Representations III. Materials: TI-83 with Apps - Geomaster IV. Procedure: 1. Discuss students homework from previous night - What did you notice about reflecting an object over 2 parallel lines? 2. Review order of keystrokes for creating, storing, and moving a figure. 3. Review TRFM: 2: Reflections 4. Have students do other transformations: A. 1: Translation (what they did for HW???) B. 3: Rotation (create angle of ) C. 4: Dilation (Size change) 5. Note notation on composite reflections. V. Reflections on Reflections: 1. When doing translations (slides), what do you notice about the distance between a point and its reflection, and the distance between the two parallel lines? 2. What do you notice about their orientation? 3. When doing rotations, what do you notice about the orientation of the figures? 4. What are the similarities and differences of the pre-image and the image under a dilation? VI. Homework Assignment: UCSPM Geometry textbook, p. 208 #7-13 (all) and p. 214 #9-16 (all)
10 Day 5 Back to the Drawing Board er, Mira I. Objective: Find coordinates of reflection images of points over the coordinate axes. II. Standard: Modeling/Multiple Representations III. Materials: Mira Graph paper Pencil Ruler IV. Procedure: 1. Have students draw a simple polygon on the coordinate system. Note the coordinates. 2. Using the mira, reflect the figure over the x-axis. Note the coordinates of the image. 3. Have students draw a second triangle on a new graph, and note coordinates. 4. Using the mira, have students reflect the figure over the y-axis. Note coordinates of the image. 5. Have students draw a third triangle on a new graph, trying to make one of the axes intersect the figure, and note the coordinates. 6. Have students reflect figure over the x-axis and then the y-axis (or vice versa) and note coordinates of image. V. Reflections on Reflections: 1. When reflecting over the x-axis, what changed with regards to each points coordinates? 2. When reflecting over the y-axis, what changed? 3. Can you make a general rule about reflections in the coordinate system? VI. Homework assignment: UCSMP Geometry textbook, p #1-39 (odd)
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