Rigid Motion vs. Non-rigid Motion Transformations What are some things you think of when we say going to a theme park. Have you ever been to a theme park? If so, when and where was it? What was your best memory about the theme park? If not, where would be the first theme park you would like to go to and why? Rigid Motion Transformations: A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions preserve distances and angle measures. Referring to the virtual Theme Park: Describe the object, identify the transformation, describe the transformation and state if it is a rigid motion transformation or not. Object 1: Object 2: Object 3: Object 4: Rigid vs. Non-Rigid Transformations 1
1. Use the rule at the top of each column to find the coordinates of each new image. Plot each new triangle on the graph. (x,y) (2x,2y) (3x,3y) (4x,4y) (5x,5y) A(1,2) A (, ) A (, ) A (, ) A (, ) B(3,1) B (, ) B (, ) B (, ) B (, ) C(3, 3) C (, ) C (, ) C (, ) C (, ) 2. Explain what happened to Triangle ABC. 3. How much larger is the length of A B than the length of AB? 4. How much larger is the length of A B than the length of AB? 5. How much larger is the length of A B than the length of AB? 6. Verify your answers to questions 3 5 using a compass. Rigid vs. Non-Rigid Transformations 2
7. Find the slopes of each of the following segments. a) AB b) A B c) A B d) A B e) A B 8. What conclusion can you draw from your answers to question 7? 9. Draw a line through all the A s, then another line through all the B s and then another line through all the C s. Make sure your line extends across the entire coordinate plane. What do you notice? The point where all the lines intersect is called the center of dilation. The center of dilation is a fixed point in the plane about which all points are expanded or contracted. Label the center of dilation X. 10. How much larger is the length of A X than the length of AX? 11. How much larger is the length of A X than the length of AX? 12. How much larger is the length of A X than the length of AX? 13. Verify your answers to questions 10 12 using a compass. Definition of Dilation: Dilation is a transformation that moves each point along a ray through the point originating from a fixed center and multiplies distances from the center by a common scale factor; where a figure is made proportionally larger or smaller. 14. Are dilations rigid motions transformations? Explain. Rigid vs. Non-Rigid Transformations 3
Constructing Dilations (when center of dilation is outside a figure) 1. Point C is the center of dilation. Follow the following steps to dilate the triangle. Step 1: Draw rays CX, CY, and CZ. Step 2: With your compass, measure the CX. Transfer this distance along CX so that you find point X such that CX = 2(CX). Step 3: Repeat Step 2 with Points Y and Z. Rigid vs. Non-Rigid Transformations 4
Constructing Dilations (when center of dilation is on a figure) 2. Point C is the center of dilation. Follow the following steps to dilate the triangle. Step 1: Draw rays CX, CY, and CZ. Step 2: With your compass, measure the CX. Transfer this distance along CX so that you find point X such that CX = 3(CX). Step 3: Repeat Step 2 with Points Y and Z. Rigid vs. Non-Rigid Transformations 5
Constructing Dilations (when center of dilation is inside a figure) 3. Point C is the center of dilation. Follow the following steps to dilate the triangle. Step 1: Draw rays CX, CY, and CZ. Step 2: With your compass, measure the CX. Transfer this distance along CX so that you find point X such that CX = 2(CX). Step 3: Repeat Step 2 with Points Y and Z. Rigid vs. Non-Rigid Transformations 6
Materials: Teacher Directions Handouts 1 per student Straightedge Compass Computer/Overhead projector and (Geogebra Installed or internet access) Objective: Students will determine the difference between rigid and non-rigid transformations. Directions: Begin class by having students complete the introduction regarding theme parks. Show the geogebra applet after a few students have shared their responses about theme parks. If you have geogebra installed on your computer open file: Rigid vs Non Rigid Intro.ggb If you have internet access go to: http://www.geogebratube.org/student/m145556 Have a student read the definition of rigid motion transformation then describe object 1, object 2, object 3, object 4. Share answers and let students know that we will focus on non-rigid motion transformations today. Problems 1 5. Students should be able to complete Problems 1 5 on their own. (10 mins) Call on students to share answers to questions 3,4, and 5. Before continuing on in the lesson, demonstrate to students how they can use a compass to measure and compare lengths. Demonstrate how to copy a segment on the board. Then have students complete problem 6, and verify their answers to 5. Problems 6 13 Students should be able to complete 6 13 on their own. (15 minutes) Call on students to share their response to problem 8. Have a discussion around the relationship between corresponding segments of dilated figures. Corresponding segments are // to each other and there exists a scale factor that relates the lengths of the corresponding segments. Rigid vs. Non-Rigid Transformations 7
Call on students to share their response to problem 9. Have discussion around the center of dilation. Ask students, What would happen we moved the triangle so the center of dilation was inside of the triangle and did the problem again? What will our picture look like then? Think-Pair-Share Have a student read the definition of dilation out loud. Compare definition to our picture that students created. Ask what do we call the fixed center? What were the scale factors that we used in these problems? What scale factors would reduce or shrink our figure? Have students complete problem 15 and share answers aloud. Tell students that this unit will focus only on rigid motion transformations so we will not cover dilations again till later on. Our focus will be on transformations that preserve congruence: translations, reflections, and rotations. Have students complete the constructions on their own. Give students a few minutes to struggle with the directions for the constructions before modeling how to do the problem. Extension: You can have students write relationships between pre-image and image segments in regards to length and direction. Rigid vs. Non-Rigid Transformations 8