Transformations Lesson Summary: Students will explore rotations, translations, and reflections in a plane. Key Words: Transformation, translation, reflection, rotation Background knowledge: Students should be familiar with Cabri software. NCTM Standards addressed in this lesson: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Objectives: Students will be able to describe some of the properties of a plane figure and its reflection. Students will gain an understanding for what the reflection tool does in Cabri. Students will develop an appropriate definition for the word rotation. Students will analyze the properties of rotations. Students will recognize the difference between positive and negative rotations. Students will be able to describe a vector using ordered pair notation. Students will be able to translate objects by a given vector. Students will be able to derive the formula for the composition of two translations. Materials: Computer lab/calculators equipped with Cabri Paper Pencils Procedure: (suggestions) Pair students of varied ability levels. Allow students to work on this lab for one class period. Collect the lab sheets and discs from the students. Assessment: Grade questions based on clarity and demonstration of knowledge. Students who complete the lab extension could present their solutions in front of the class. Transformations
Activity One: Reflections Team member s names: File name: Goals: After completing this lab, you should: Be able to describe some of the properties of a plane figure and its reflection. Gain an understanding for what the reflection tool in Cabri does. Definitions: When a transformation is performed on a geometrical figure, the starting figure is called the preimage and the resulting figure is called the image. Procedure (Using Cabri): Part A: Reflecting a point over a line 1. Construct a line l. [Use the line and label tools] 2. Create a point not on l and label it A. [Use the point and label tools] 3. Reflect A over l and label the image A. [Use the reflection and label tools] 4. Measure the distance from A to l. [Use the distance and length tool] 5. Measure the distance from A to l. [Use the distance and length tool] 6. Compare these two distances. What do you notice? 7. Now draw a line from point A to point A. [Use the line tool] 8. Measure the four angles created by the [Use the angle tool] intersection of the two lines. 9. What do you notice? Repeat steps 1-8 with a new point. Do you get the same results? Why or why not? 10. Print and Save As Reflection4. Part B: Reflecting a line segment over a line 1. Go to File New.
2. Construct a line l. [Use line and label tools] 3. Draw a line segment and label it AB ' ' [Use the segment and label tools] 4. Reflect the segment AB over line l. [Use reflection tool] 5. Label the new segment AB. ' ' 6. Measure the length of segments AB and AB. ' ' [Use label tool] [Use dis tance and length tool] 7. Compare these two lengths. What do you notice? 8. Print and Save As Reflection5. Part C: Reflecting a pentagon over a line 1. Go to File New. 2. Construct a regular pentagon and label it ABCDE. [Use polygon and label tools] 3. Construct a line l that does not intersect the pentagon [Use line and label tools] ABCDE. 4. Reflect ABCDE over l. [Use reflection tool] 5. Label the new polygon A B C D E. [Use label tool] 6. What do you think the ratio of the area of ABCDE to the area of A B C D E will be? Why do you think this? 7. Test your hypothesis. Were you correct? [Use area tool] 8. Measure the sides in each pentagon. What do you notice about the corresponding sides? 9. Now measure the angles in each pentagon. What do you notice about the corresponding angles?
10. Repeat steps 1-8 with an irregular pentagon. Does your answer for #8 and #9 change? Why or why not? 11. In a paragraph of your own words, describe what the reflection tool in Cabri does. When an object is reflected, what changes? What remains the same? 12. In your own words, define reflection. 13.An isometry is a transformation in which the original figure and its image are congruent. Is reflection an isometry? Explain your reasoning. 14. Print and Save As Reflection6. Extension: Given a point A and its reflection A, How would one find the line over which point A was reflected? (It may be helpful to draw a picture.) Transformations
Activity Two: Translations Team member s names: File name: Goals: After completing this lab, you should be able to: Describe a vector in ordered pair notation. Translate objects by a given vector. Derive the formula for the composition of two translations. A vector like the one shown has an initial point (in this case A) and a terminal (or end) point (B). We write vectors as <x,y> where x is the horizontal change from the initial point to the terminal point, and y is the vertical change from the initial point to the terminal point. In the above example we would write the vector as <3,2>. When we translate an object by a vector <x,y> we move each point of the object x units horizontally and y units vertically. A negative x value will cause the image point to be moved to the left and a negative y value will cause the image point to be moved down. This is similar to the Cartesian coordinate system with which you are already familiar. Procedure (Using Cabri): Part A: Labeling and Constructing Vectors 1. Turn on the grid and axes. [Use the show axes and define grid tools] 2. Plot two vectors whose initial points [Use vector tool] and terminal points are points on your grid. 3. Using Cartesian coordinates, label the [Use label tool] initial points and terminal points of the vectors. 4. Label each vector using the ordered pair [Use label tool] notation: <x,y>.
5. Print and Save As Translation1 Part B: Translating Objects 1. Go to File New. 2. Draw a vector <x,y>on the screen. [Use vector tool] 4. Draw a point not on the vector and label it A. [Use point and label tools] 5. Translate the point by that vector. [Use translation tool] 6. Label the image A. [Use label tool] 7. Describe what you see. Do you notice any relationships between the objects? 8. Print and Save As Translation2 9. Go to File New. 10. Draw a line segment on the screen and label [Use the segment and label tools] it BC. 11. Draw a vector on the screen. [Use the vector tool] 12. Translate the segment by the vector. [Use the translations tool] 13. Label the image BC. ' ' [Use the label tool] 14. What do you notice? Is there a relationship betweenbc, BC, ' ' and the vector? 15. On the same screen, draw a quadrilateral. [Use the polygon tool] 16. Translate the quadrilateral by the vector. [Use the translation tool] 17. Describe what you see. Here are some things to consider. Are the images still arranged in the same order? Are the images the same size as their preimages? Are they oriented in the same way?
Part C: Compositions of Translations For any two translations, when the second translation is performed on the image of the first translation the resulting translation is called composition of translations. 1. Go to File New. 2. Turn on the grid and axes. [Use the show axes and define grid tools] 3. Construct a polygon wherever you choose [Use the polygon and vector tools] on the grid and two vectors whose terminal points and initial points are grid points. We will refer to this polygon as P. 4. Label the vectors using ordered pair notation. [Use the label tool] 5. Translate polygon P by one of the vectors. [Use the translate tool] We will refer to this image as P. 6. Translate P by the other vector. We will [Use the translate tool] refer to this image as P. 7. There exists a vector <x,y> that when P is translated by <x,y> the result is P. What is this vector? Use the grid to help you find it. 8. Repeat part C with a few more examples. Do you notice any patterns? Given two vectors <a,b> and <c,d>, what is the vector that would perform the same translation as the composition of these two translations? 9. An isometry is a transformation in which the original figure and its image are congruent. Is translation an isometry? Explain your reasoning. 10. Print and Save As Translation3 Extension: Draw a polygon and two parallel lines. Reflect the polygon over one line. Then reflect it over the other. Is there a translation that would produce the same image? If no, why not? If yes, show an example.
Transformations Activity Three: Rotations Team member s names: File name: Team member s names: Goals: After completing this lab you should have discovered the definition of a rotation and what its properties are. Procedure: (Using Cabri) Part A: Rotation of a polygon by 62 degrees 1. Draw a five-sided polygon and label each vertex point. We will refer to this polygon as P. [Use the polygon tool and label tool] 2. Use the numerical edit tool to insert the number 62 on the screen. [Use the numerical edit tool] 3. Rotate P about the vertex of your choice by 62 degrees. Label this polygon P. [Use the rotation tool] 4. Pick one of the vertices and measure the distances between it and its corresponding vertex in the image. Record the values on a blank section of the page [Use the distance and length tool] 4. Print and Save As Rotation1 Part B: Rotation of a polygon by negative 77 degrees 1. Go to File New. 2. Construct another polygon and label each vertex point. We will refer to this polygon as Q. [Use the polygon tool and the label tool] 3. Rotate Q about the vertex of your choice by 77 degrees. Don t forget to use the numerical edit tool before you do the rotation! Label this polygon Q.
[Use the rotation and numerical edit tools ] 4. Rotate Q about the vertex that corresponds to the one chosen in Part B #2 by negative 77 degrees. Don t forget to use the numerical edit tool before you do the rotation! [Use the rotation and numerical edit tools ] 5. Print and Save As Rotation2 Summarize your findings: 1. Consider P and P. What happened when you rotated by 62 degrees? 2. Consider Q and Q. What happened when you rotated by negative 77 degrees? 3. What were the similarities and differences between the two polygons when they were rotated? 4. An isometry is a transformation in which the original figure and its image are congruent. Is rotation an isometry? Explain your reasoning. 5. What did you notice about the distances between the original vertex point you picked and the corresponding vertex of the rotated polygon? Why do you think this is?
6. What do you see happen in part B? Extensions: 7. Using what you have learned, make up your own definition of rotation. Why do you think your definition is complete? 1. In the next class period you will be in groups of four. In your groups you will be asked to compare and contrast your definition of rotation. Try to formulate a definition within your groups that incorporates everybody s thoughts. Then the class will collectively decide on an appropriate definition. 2. Consider some art that you have seen. What are some famous paintings that use rotations? How can you tell?