Transformations 5-Day Unit Plan 10 TH Grade. by Carrie J. Magee

Transcription

1 Transformations 5-Day Unit Plan 10 TH Grade by Carrie J. Magee

2 Table of Contents Unit Objectives..3 NCTM Standards..3 NYS Standards..3 Resources..3 Materials 3 Day 1: Rigid Motion in a Plane 4 Day 2: Reflections 10 Day 3: Rotations 14 Day 4: Translations and Vectors 18 Day 5: Glide Reflections and Compositions..24 Magee Page 2

3 Unit Objectives- Students will be able to: identify types of rigid transformations. use properties of reflections. relate reflections and line symmetry. relate rotations and rotational symmetry. use properties of translations and glide reflections. NCTM Standards for Grades 9-12 Geometry Standard Communication Standard Connections Standard Representation Standard NYS Standards for Math A Key Idea 3-Operations Key Idea 4-Modeling/Multiple Representation Resources Textbook, Geometry by McDougall-Littell, 2001 Materials Compass, protractor & straight edge Mira Geometer s Sketchpad Magee Page 3

4 Day 1: Rigid Motion in a Plane Objective: Students will be able to: identify the three basic rigid transformations. identify isometries. use transformations in real-life. Materials Miras, overhead projector, transparencies Opening Activity- Have students copy definitions of image, preimage, transformation and isometry. Activity #1- Discuss the above definitions. Discuss the three basic rigid transformationsreflections, rotations and transformations. Activity #2- Had out worksheet #1. Using the mira, have the students reflect ABC over the y-axis and then answer the questions. Discuss the answers to the questions. Activity #3- Discuss rigid transformations. Have the students decide which ones of the following are isometries. A.) B.) Magee Page 4

5 C.) A and C are isometries while B is not. Explain why this is true. This is because the angle measure and lengths are preserved. Activity #4- Identifying and using transformations in real-life. Discuss examples 4 and 5 in the book on page 398. Have the students think of at least one other transformation. Closing Activity- Investigate the second example on worksheet #1. Homework- pp ; 12-17, 21-25, even, 34, 35 Magee Page 5

6 Rigid Motion in a Plane Image- the new figure that results from the transformation of a figure in a plane. Preimage- the original figure in the transformation of a figure in a plane. Transformation- the operation that maps, or moves, a preimage onto an image. Isometry- a transformation that preserves lengths. Three Basic Transformations 1. Reflections- a type of transformation that uses a line that acts like a mirror, called a line of reflection, with an image reflected in the line. 2. Rotations- a figure is turned about a fixed point, called a center of rotation. 3. Translations- maps every two points P and Q in the plane to P` and Q` so that the following two properties are true: PP` = QQ`. PP` QQ`, or line segments PP` and QQ` are collinear. Magee Page 6

7 Worksheet #1 D D' C E E' C' 1. Name and describe the transformation. 2. Name the coordinates of the vertices of the image. 3. Is DABC congruent to its image? Why or why not? Magee Page 7

8 A m B D C H G E F n 4. Describe the motion that moves ABCD onto EFGH. 5. Reflect EFGH over line m and name the corresponding vertices of the new figure JKLM. Is EFGH congruent to JKLM? Magee Page 8

9 A m B D C H G E F L n K M J 6. Describe the motion that maps ABCD onto JKLM. Is ABCD congruent to JKLM? 7. Can one flip be used to move ABCD onto JKLM? Explain why or why not? Magee Page 9

10 Day 2: Reflections Objective- Students will be able to: identify and use reflections in a plane. identify relationships between reflections and line symmetry. Materials- Miras, graph paper, overhead projector, transparencies. Opening Activity- Have the students copy definitions of reflection and line of reflection. Discuss the properties of a reflection. Activity #1- Have the students graph H(2,2) and G(5,4). They should then reflect H in the x-axis and G in the line y=4. Then discuss the properties of reflections in the coordinate axes. Activity #2- Explain Theorem 7.1 and then prove it. The proof is in the book on page 405 example 2. Closing Activity- Define line of symmetry. Have the students use the miras to find the lines of symmetry in the figures on worksheet #2. Homework- pp ; 15-17, 22-29, 31-33, 48, 49 Magee Page 10

11 Reflections Reflection- a type of transformation that uses a line which acts like a mirror, with an image reflected in the line. Line of Reflection- the mirror line. A reflection maps every point P in the plane to a point P`, so that the following properties are true: If P is not on m, then m is the perpendicular bisector of PP`. If P is on m, then P = P`. P m m P P` P` Magee Page 11

12 6 4 G G` y=4 2 H H` -4-6 Reflections in the coordinate axes have the following properties: If (x,y) is reflected in the x-axis, its image is the point (x,- y). If (x,y) is reflected in the y-axis, its image is the point (-x,y). Theorem 7.1 Reflection Theorem A reflection is an isometry. Line of Symmetry- a line that a figure in the plane has if the figure can be mapped onto itself by a reflection in the line. Magee Page 12

13 Using the Mira, find all line of symmetry of the following figures. 1.) 2.) Worksheet #2 3.) 4.) 5.) Magee Page 13

14 Day 3: Rotations Objectives- Students will be able to: identify rotations in a plane. identify rotational symmetry. use Geometer s Sketchpad to reflect and rotate images. use a compass and protractor to rotate a figure. Materials- compasses, protractors, computers equipped with Geometer s Sketchpad, overhead projector, transparency. Opening Activity- Using Geometer s Sketchpad, have the students construct a scalene triangle, then label the vertices A, B, and C. Next draw two lines that intersect but do not intersect the triangle. Label the lines k and m. Label the point of intersection P. Reflect DABC in line k to obtain DA`B`C`. Reflect DA`B`C` in line m to obtain DA``B``C``. How is DABC related to DA``B``C``? Activity #1- Have students copy definitions of rotation, center of rotation, and angle of rotation. Discuss the above definitions. Activity #2- Rotate a figure using a compass and protractor. First draw a triangle ABC and a point P. Then draw a segment connecting vertex A and the center of rotation point P. Next use a protractor to measure a 60 degree angle counterclockwise and draw a ray. Then place the point of the compass at P and draw an arc from A to locate A`. Repeat these steps for each vertex. Connect the vertices to form the image. Label the image A`B`C`. Activity #3- Rotations in a coordinate plane. Using Geometer s Sketchpad, draw a figure using the points A(2,-2), B(4,1), C(5,1), and D(5,-1). Rotate the figure 90 degrees counter clockwise about the origin. What are the coordinates of the new vertices? A`(2,2), B`(-1,4), C`(-1,5) and D`(1,5). Closing Activity- Discuss rotational symmetry. Have students identify rotational symmetry. Homework- pp ; 1-5, 13-19, 20, Magee Page 14

15 Rotations B' B C' A' A C m A'' p C'' B'' k Rotation- a transformation in which a figure is turned about a fixed point. Center of Rotation- the fixed point. Angle of Rotation- the angle formed when rays are drawn from the center of rotation to a point and its image. A rotation about a point P through x degrees is a transformation that maps every point Q in to plane to a point Q`, so that the following properties are true: If Q is not point P, then QP = Q`P and m QPQ`=x. If Q is point P, then Q = Q`. This transformation can be described as (x,y)æ(-y,x). Magee Page 15

16 B' C` 6 4 D` 2 A' B C A D -4-6 Rotational Symmetry- the figure can be mapped onto itself by a rotation of 180 or less. Magee Page 16

17 Which of the following have rotational symmetry and why? A.) B.) C.) Worksheet #3 A can be mapped onto itself by a clockwise or counterclockwise rotation of 45, 90, 135, or 180 about its center so therefore it has rotational symmetry. B can be mapped onto itself by a clockwise or counterclockwise rotation of 180 about its center so therefore it also has rotational symmetry. Magee Page 17

18 Objectives- Students will be able to: identify and use translations in the plane. identify vector components. find vectors. Day 4: Translations and Vectors Materials- overhead projector, transparencies, Geometer s Sketchpad. Opening Activity- Define translation. Discuss Theorem 7.5. Activity #1- Using Geometer s Sketchpad, sketch a triangle with vertices A(-1,-3), B(1,- 1), and C(-1,0). Then sketch the image of the triangle after the translation (x,y)æ(x-3,y+4). Shift each point 3 units to the left and 4 units up. The translated vertices should be A`(-4,1), B`(-2,3), and C`(-4,4). Activity #2- Define vector, initial point, terminal point and component form. Have students identify vector components. Closing Activity- Perform a translation using vectors. Homework- pp ; Magee Page 18

19 Translations and Vectors Translation- a transformation that maps every two points P and Q in a plane to points P` and Q`, so that the following properties are true: PP` = QQ` PP` QQ` or PP` and QQ` are collinear. Theorem 7.5- If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is a translation. If P`` is the image of P, then the following are true: PP`` is perpendicular to k and m. PP`` =2d, where d is the distance between k and m. Magee Page 19

20 Activity #1 6 C` 4 B` 2 A` C B -2 A -4-6 Magee Page 20

21 Vector- quantity that has both direction and magnitude. Initial point- starting point. Terminal point- ending point. Component form- a vector combines the horizontal and vertical components. Identify the vector components. A.) 6 4 K 2 J Magee Page 21

22 B.) 6 4 N x M C.) 6 4 T 2 S Magee Page 22

23 The component form of GH is <4,2>. Use GH to translate the triangle whose vertices are A(3,-1), B(1,1) and C(3,5). First graph ABC. GH is <4,2> so the vertices should be shifted to the right 4 units and up 2 units. Label the vertices A`, B`, and C`. 6 C` 4 C 2 B` B A` A -4-6 Magee Page 23

24 Day 5: Glide Reflections and Compositions Objectives- Students will be able to: identify glide reflections in a plane. represent transformations as compositions of simpler transformations. Materials- Geometer s Sketchpad, overhead projector. Opening Activity- Define glide reflection. Use the following information to find the image of a glide reflection. A(-1,-3), B(-4,-1), and C(-6,-4) Translation: (x,y)æ(x+10,y) Reflection: in the y-axis Activity #1- Define compostion. Find the image of a composition using the following: P(2,-2), Q(3,-4) Rotation: 90 degrees counterclockwise about the origin. Reflection: in the y-axis. Closing Activity- Repeat Activity #1 but reverse the order. Perform the reflection first and then the rotation. What do you notice? Homework- pp ; Magee Page 24

25 Glide Reflections and Compositions Glide Reflection- a transformation in which every point P is mapped onto a P`` by the following steps: a translation maps P onto P`. a reflection in a line k parallel to the direction of the translation maps P` onto P``. 6 4 C'' A'' 2 B'' B B' -2 A A' C -4 C' -6 Composition- when two or more transformations are combined to produce a single transformation. Magee Page 25

26 Activity #1 6 4 Q`` P`` 2 P` Q` P -4 Q -6 Magee Page 26

27 Closing Activity P` -2 P`` P Q` -4 Q Q`` -6 The order which the transformations are performed affects the final image. Magee Page 27