Making Mathematical Connections Activity 1 6/3/2002 Activity 1 Isometries of the Plane We will investigate transformations of the plane that preserve distance. These transformations will take the entire plane onto a "copy" of itself. For this activity you will need to draw a scalene triangle on a two transparencies so that you have two "identical" triangles. Here's a definition. Definition: A one-to-one, onto function f from the plane to the plane is called an isometry of the plane if, for any two points P and Q, the segment PQ is congruent to the segment fPfQ()(). 1. Give an explanation of the meaning of congruent as used above. Try to be precise. 2. Using your transparencies, experiment to find ways to move a triangle from one location to another. How many types of isometries of the plane can you identify? Is there one that is a combination of two others? What properties of your triangles are preserved by the isometries? 3. Consider an isometry of the plane as a function from the plane to itself. Craft a careful definition for each type of isometry of the plane. 4. Is there a type of transformation of the plane that is not an isometry but preserves some of the same properties preserved by isometries? Making Mathematical Connections Activity 1 6/3/2002 Teacher's Notes for Activity 1 • Allot a 50-minute class period for this activity. • Students should work in groups of three or four. • The idea of congruent triangles on transparency sheets is to give the students the image of a transformation taking the entire plane to itself. • Item 2 is meant to allow students to "discover" the four basic isometries, rotation, reflection, translation, and glide. They might need some prompting on the last. • Items 1 and 3 are to reinforce the place of definition in mathematics. Sharing definitions and the ensuing discourse is likely to bring out the importance of careful wording. There are other issues to be discussed regarding mathematical definitions. These include minimal conditions and elegance. These issues might begin to surface here. • Item 4 tries to get students to identify similarity transformations. These will be addressed again in Activity 5. • A minor goal of this activity is to help students begin to make connections between functions and geometric transformation. Making Mathematical Connections Activity 2 6/3/2002 Activity 2 Rotations, Reflections, Translations, and Glides These are the four main types of isometries of the plane. They are called isometries ("same measure") because distance is preserved. Here are some exercises investigating isometries. 1. For a reflection F and a point P, if F(P)=P', what is F(P')? 2. Let F be a reflection in line m. What is F(m)? 3. For any transformation T, a line left invariant, k, is a line such that T(k)=k. a) Find a line other than m that is invariant under the reflection F in line m. b) What are the invariant lines for a reflection in line m? How many are there? c) What are the invariant lines for a rotation of angle α radians with center O? d) What are the invariant lines for a translation? e) What are the invariant lines for a glide? 4. Given a geometric figure and it's image under a reflection, is there a way to find the line of reflection? Explain. 5. Given a geometric figure and it's image under a rotation, is there a way to find the angle and center of the rotation? Explain. Does the answer depend on the angle? 6. Given a geometric figure and it's image under a translation, is there a way to find the translation vector? Explain. 7. Given a geometric figure and it's image under a glide, is there a way to find the line of reflection and the translation vector? Explain. Making Mathematical Connections Activity 2 6/3/2002 Teacher's Notes for Activity 2 • Allot a 50-minute class period for this activity. • Students should work in groups of three or four. • Provide students with rulers, protractors, compasses, graph paper, and MIRAs (or some such device). • The basic goal of this activity is to acquaint students with some basic properties of various isometries. • Before Item 6, introduce the vocabulary "translation vector." • Students might need guidance on Item 3a and Item 7, both pertaining to glides. Making Mathematical Connections Activity 3 6/3/2002 Activity 3.1 Compositions Now we will experiment with composing isometries. 1. What kind of transformation is the composition of two translations? 2. What kind of transformation is the composition of two reflections? 3. What kind of transformation is the composition of two rotations? 4. Can you find a transformation of the plane that takes triangle ABC to triangle A'B'C'? Can you do it with a single transformation? What is the minimum number of translations, reflections, and/or rotations needed? Can you do it with only reflections? A B C C' A' B' Activity 3.2 • Complete the Connected Geometry Investigation 2.11, A Reflection Puzzle. • Analyze the investigation from a high school teacher's perspective (Activity A, item 2). • Try to prove your conjecture about reflections. You may need to use a lemma that states "the image of three non-collinear points determines an isometry." Making Mathematical Connections Activity 3 6/3/2002 Teacher's Notes for Activity 3 • Allot a 50-minute class period for Part 3.1. Students may need to finish as homework. • Students should work in groups of three or four. • Part 3.1, Item 3 might be hard. You may want to give a hint about using cases (same- center or different-center rotations). • The last question in Item 4 foreshadows the conjecture we'd like students to make in Part 3.2. That is, that any isometry can be expressed as the composition of three reflections. • Part 3.2 should be assigned for homework with class discussion of the results. Making Mathematical Connections Activity 4 6/3/2002 Activity 4 Proof with Isometries Here are some theorems to prove using isometries. 1. An isometry preserves angle measure. 2. If ∆ABC and ∆DEF are two triangles such that AB = DE, AC = DF and ∠BAC = ∠EDF, then BC = EF, ∠ABC = ∠DEF, and ∠ACB = ∠DFE. 3. If ∆ABC and ∆DEF are two triangles such that AC = DF, ∠BAC = ∠EDF, and ∠ACB = ∠DFE, then BC = EF, AB = DE, and ∠ABC = ∠DEF. 4. Suppose that ∆ABC is a triangle in which ∠ABC = ∠ACB. Prove that AB = AC. 5. Suppose that A and B are two distinct points on a circle and that the tangents to the circle at A and B meet at point P. Prove that AP = BP. (Hint: Consider the line that passes through P and the center of the circle.) 6. Any isometry is a composition of reflections, translations, and rotations. 7. Any isometry is a composition of reflections. Making Mathematical Connections Activity 4 6/3/2002 Teacher's Notes for Activity 4 • This activity will take more than a 50-minute class period. For some parts students will probably need some hints or steering. Try to work with their ideas rather than give them too big a push. Make sure they work with isometries, not measure. • Students should work in groups of three or four. You might want to ask students to read this activity for homework before working on it in class. Ask them to bring in written ideas or sketches. • Finishing parts two and three requires the use of Euclid's postulates. For a hint, ask them which ones. • Part five has a hint. Part six will lay the foundation for Activity 6. Making Mathematical Connections Activity 5 6/3/2002 Activity 5 The Human Vertices We will investigate transformations of a square of side one in the coordinate plane. Each of you will have a starting coordinate. You will connect the vertices with taught rope for straight lines. Person Starting Coordinates (x,y) A (1,0) B (0,0) C (1,1) D (0,1) The square will be transformed under the functions described below. Function (ti) ti (x,y) t1 ()xy, − t2 ()2xyy+ , t3 ()xy−+21, t4 2 −+2 2 (),()xy2 xy t5 ()−xy, t 1 1 6 ()2 yx, 2 t −1 +−+ 7 ()2 xyx, 2 y t8 ()yx, t9 1 + 3 − 3 + 1 2 xyxy2 , 2 2 t −+ −1 + 10 ()xy2 , 2 xy 1 Making Mathematical Connections Activity 5 6/3/2002 1) Describe the image and location of the square under each transformation. What happens in each transformation to characteristics such as distance and angle? What happens to pairs of parallel lines? Are there other characteristics that change or stay unchanged? 2) Classify the transformations into categories according to characteristics of the square that change or remain unchanged. How many different categories do you have? What are they? 3) Which transformations are isometries? 4) Write the transformations that are isometries as matrix equations. That is, think of a x vertex (x,y) as a vector, . Write an equation with a 2 by 2 matrix A and a vector y i e x x e so that t = A + . f ii y y f 5) What are the matrix equations for the non-isometric transformations? 6) Calculate the determinant of each matrix. What relationship do you notice between the determinant of the matrix and the type of transformation? 2 Making Mathematical Connections Activity 5 6/3/2002 Teacher's Notes for Activity 5 • The physical part of this activity should be completed in a 50-minute class period.