Handout 1: Viewing an Animation

Transcription

1 Handout 1: Viewing an Animation Answer the following questions about the animation your teacher shows in class. 1. Choose one character to focus on. Describe this character s range of motion and emotions, using descriptive words or sketches.. If you were to sketch this character, what geometric shapes would you draw? Draw one or more of those shapes below: 3. Throughout this unit, you will work with geometric transformations that allow you to create the illusion of motion through slides, flips, turns, and changes in the size of a -D object. Keep a record of your character s moves by picking one motion (such as a slide or a turn) and counting the number of times that your character moves in that particular way. 4. As you watch, determine the elements in the picture that do not move or seem to change as the animation progresses. What do you notice? 5. How do you think mathematical concepts or tools might be applied in animation? (Note: It s OK if you re not sure about this question you ll know how to answer it very soon!) 1

2 Handout : TI Calculator Animation Program #1 You can experiment with the programming capabilities of a TI graphing calculator to create short animations. Enter Program #1 below into your graphing calculator, run the program, and discuss what you see with your classmates. You can also experiment with other DRAW functions to create different images. You may want to consult the calculator manual to determine the parameters for each new function. Program #1: Moving Smile To create a new program: Press PRGM on the keypad of the graphing calculator. Move the cursor to NEW and press ENTER. Enter a name for the first program (such as MOVETRIANGLE ) and press ENTER. Note: The graphing calculator is automatically in ALPHA mode in order to allow you to type letters. The letters are written in blue or green above individual calculator keys. Wait for the screen to display a colon, then enter each command below. Press ENTER after each new command. Command Function and Location :0 Xmin Sets up the viewing window. The triangle in this program will only move within the first quadrant of the coordinate system. :5 Xmax The arrow assigns the number 0 to Xmin by using the store key, STO>. This :0 Ymin value will stay in Xmin until reassigned. :5 Ymax To find this command: Use the Window menu within the VARS key. Turns off the display of any equations entered in the Y= editor. :FnOff :PlotsOff To find this command: Press VARS, move the cursor to Y-VARS #4, and press ENTER. Turns off all plots. To find this command: Press nd [STAT PLOT] 4. 1

3 Command :AxesOn Function and Location Makes the coordinate axes visible during the animation. To find this command: Press nd [FORMAT]. :1 P Initial values are stored within variables: :0 Q :4 R :16 S :4 T :16 U :7 V :0 W :For(N, 1, 8) :ClrDraw :Line(P, Q, R, S) :Line(T, U, V, W) :For(A, 1, 50) :End :P+3 P :R+3 R :T+3 T :V+3 V :Q- Q :S- S :U- U :W- W :End P and Q represent the x- and y-coordinates of one endpoint of the segment. R and S represent the x- and y-coordinates of the second endpoint of the first line segment. T and U represent the x- and y-coordinates of one endpoint of the second segment. V and W represent the x- and y-coordinates of the second endpoint of the second segment. Use the Alpha keys to enter letters. Use the STO> key to assign values to each. This begins a loop with counter N that ranges from 1 to 8. The loop repeats eight times. To find For and End: Both are listed in alphabetical order in the calculator s CATALOG. To find ClrDraw and Line: Press nd [DRAW], #1, and #. The Line function sketches the two segments that form the object in this program; the input of this function includes the x- and y-coordinates of each segment s endpoints. ClrDraw at the beginning of the For loop clears the previous image from the screen. Each coordinate of the two endpoints of the line segments is changed: The segments are translated (shifted) to the right three units and down two units a total of eight times, completing the For loop. To run the program: Press PRGM. Move the cursor over the name of the program. Press ENTER. The program is executed.

4 Handout 3: Unit Project Description In this unit you will learn how to move objects on a -D plane by applying mathematical transformations, such as slides, flips, turns, and enlargements. For the unit project, you will design and create a flipbook that tells an animated story. You will create a character using geometric shapes and decide what movements the character will make in your story. You will represent the movements mathematically with transformations and then apply those transformations to animate your character. Project Requirements Your unit project includes three parts: a flipbook, a mathematical guide to your flipbook, and a short written reflection. Part 1: Flipbook with 0 35 Frames Your flipbook should tell a simple, animated story. You will apply geometric transformations to at least six frames in your flipbook, and then sketch the remaining frames in any way you wish in order to tell the story. Your tasks: Create a character or object composed of simple shapes. Animate the character by applying transformations. Include at least two of each type of transformation: Translation (slide or turn) Reflection (flip) Dilation (enlargement or shrinking) Align the flipbook frames so that you see an animation when you flip the pages. Part : Mathematical Guide to Your Flipbook Create a guide that documents and explains each transformation in your flipbook. For each transformation: Represent the shape(s) in your character with a matrix or matrices of coordinates Represent the transformation with mathematical notation Explain the action performed in the transformation Note: The matrices and transformations in your mathematical guide are similar to the programming that an animator would use to create a computer animation. 1

5 Part 3: Written Reflection Write a response to the framing questions below, using examples from your work throughout the unit: How can I use mathematical concepts and tools to create the illusion of motion in animation? How can I use mathematical transformations in a systematic way to apply changes to many different figures?

6 Assessment Checklist: Unit Project Use this checklist to help you plan and assess your project. Make sure that you include all the requirements. Your teacher will use this checklist to help evaluate your work. Requirements Percentage of Total Grade Comments Flipbook Student Comments Teacher Comments The flipbook has % frames and tells a simple story. Each frame is aligned with the next frame. 5% Two of each kind of transformation translation, reflection, and dilation are applied to objects in the flipbook 5% An animated sequence is perceived when pages are flipped. 5% Total 0% 1

7 Requirements Percentage of Total Grade Comments Mathematical Guide Student Comments Teacher Comments Each object in the flipbook is 5% represented with a matrix of coordinates. The discussion of each transformation is accurate and complete. 5% The explanation of the action that each transformation performs is clear and accurate. 10% Mathematical notation is used to represent each transformation. 5% Each transformation in the mathematical guide corresponds clearly to a flipbook frame. 5% Each translation in the mathematical guide includes: 10% A translation matrix Correct calculations performed on geometric objects, using matrix addition to complete the translation A translation vector with horizontal and vertical components A graphical representation that shows the action performed

8 Requirements Percentage of Total Grade Comments Mathematical Guide Student Comments Teacher Comments Each reflection includes: 10% A reflection matrix Correct calculations performed on geometric objects, using matrix multiplication to complete the reflection A line of reflection with its corresponding equation A graphical representation that shows the action performed Each dilation includes: 10% A dilation matrix Correct calculations performed on geometric objects, using scalar multiplication to complete the dilation A specified center of dilation and scale factor A graphical representation that shows the action performed Total 60% 3

9 Requirements Percentage of Total Grade Comments Reflection Student Comments Teacher Comments Responses to the framing 5% questions are supported by evidence from work throughout the unit. Discussion includes examples of how matrices facilitate the systematic use of geometric transformations for animation. 5% Reflection is well-organized and addresses the framing questions clearly and thoroughly. 10% Total 0% 4

10 Handout 4: What Are Transformations? Working with Notation Create a transformation or a function as described below. Use the examples from class to express each mapping with the appropriate notation. 1. A transformation T that moves any point on the plane five units to the left and three units down. A transformation R that reflects every point on the plane over the x-axis 3. A transformation D that enlarges every figure on the plane by a factor of your choice 4. A function f that maps every number to a number that is 10 more than its triple Using your mappings T, R, and f above, find the following: 1. The pre-image of 15 under the function f. The image of (, ) under T 3. The image of (4, 5) under R 4. The image of (, 0) under R 1

11 5. The pre-image of ( 5, 3) under T Representing Transformations with Graphs Given the polygon below, perform each transformation: SLIDE, MIRROR, and ENLARGE. Although the names used for each transformation are now whole words, the notation that follows is consistent with what you learned in class. Mathematicians often name functions and geometric transformations using letters or words that correspond to the output or action. SLIDE: (x, y) (x 3, y 3) MIRROR: (x, y) (y, x) ENLARGE: (x, y) (1.5x, 1.5y)

12 Note: One way to perform each transformation is to apply it to the coordinates of the figure s vertices. For example, to apply SLIDE to the roof of the house, first record the coordinates of the three vertices for the roof: A (0, 5) B (1, 4) E ( 1, 4) Then SLIDE each point: Subtract 3 from the x-coordinate and 3 from the y-coordinate to obtain the following image points: A ( 3, ) B (, 1) E ( 4, 1) Notation: Each image point is often labeled using the original letter followed by prime, or the symbol. This helps in keeping track of each point and its corresponding image. For subsequent transformations, you could re-use the letters but add two primes,, or three primes,, to distinguish image points under different transformations from pre-image points. Plot A, B, and E on the graph to obtain the image of the roof under SLIDE. Use the graph on page 41 and different-colored pencils to complete the SLIDE and apply MIRROR and ENLARGE to the polygon. Reflection Write short responses to the following prompts: Something I found interesting during this activity was: Something that surprised me was: Something that I want to know more about is: 3

13 Vocabulary Function: A rule that maps every element from one set to exactly one element in another set. Functions are frequently used in algebra to represent mappings of sets of numbers. Image: A figure that has undergone a transformation. One-to-one mapping: A function from set A to set B where every member of set B has exactly one preimage in set A. Pre-image: The original shape or figure. Transformation: A one-to-one mapping from the whole plane onto the whole plane. Translations, reflections, rotations, and dilations are all types of transformations. Transformations are frequently used in geometry to establish a one-to-one correspondence between sets of points. 4

14 Handout 5: Shifting Objects Translation Vectors Perform the specified translation for each problem. 1. Use the given translation vector to shift the polygon shown below. Assume that the squares on the grid paper are exactly the same size. Vector Polygon A B C E D

15 . Using your own polygon on separate graph paper, perform the following translations: (a) T: (x, y) ( 5 + x, y ) (b) L: (x, y) (x + 1, y + 1) Then describe the translations in your own words (for example, Move the polygon right units. ) 3. Write in mathematical notation how you can apply a translation of the plane that moves every point five units to the left and three units down. Sketch a translation vector that represents this shift and label its horizontal and vertical components. 4. Your friend Erica has been asked by her teacher to translate a triangle two units to the right and one unit down. She has come to you because she thinks that she will have to slide every point on the triangle, and she knows that s an impossible task! Can you help her? What would you tell Erica that might make her task do-able? Be sure to explain your reasoning, as Erica always wants to understand why something works the way it does.

16 Investigating Properties For Problems 5 7, use the trapezoid below. 5. Translate the trapezoid four units down and three units to the right. Write the translation in mathematical notation. Then label and record the coordinate vertices of the image, and draw a translation vector with its horizontal and vertical components. 6 4 A D 5 5 B 4 6 C

17 6. Optional: Using a separate sheet of paper, verify that the image is congruent to the pre-image under this translation. The distance formula is included below for reference: The distance, d, between two points (x 1, y 1 ) and (x, y ) is: 7. What other properties of the trapezoid remain unchanged by the translation? 4

18 Vocabulary Congruent polygons: Polygons that have the same size and shape, that is, their corresponding angles and sides are equal in measure. Congruent segments: Segments that have the same length. In mathematics, the symbol for congruence is. To write that segment AB is congruent to segment CD, you write. To denote that they have equal length, you can write AB = CD. Horizontal component: A vector representing only the horizontal movement of an object along a given vector. Isometry: A transformation whose image is congruent to its pre-image. An isometry maps every segment to a congruent segment. Translation: A transformation that glides every point of the plane the same distance in the same direction (also known as a slide). Vector: A quantity that has both direction and magnitude and is represented by a directed line segment. Vertical component: A vector representing only the vertical movement of an object along a given vector. 5

19 Handout 6: Translation Matrices Adding Matrices: Practice Practice adding matrices by completing each problem. 1. Determine which of the five given matrices can be added. Explain your choices. M = N = 7 3 O = P = Q = Add the matrices you chose in Problem 1. 1

20 3. The last page of this handout shows a graph of a figure representing a square person with a balloon. The coordinates of the figure are given. If you need to, use separate sheets of lined paper and graph paper to answer (a), (b), and (c). (a) Represent the person and the balloon, using a matrix for each. (b) Write a translation matrix for the person and balloon that moves the pre-image three units to the right. (c) Apply the translation to the pre-image, first using matrices and then sketching a new graph.

21 8 H: (1, 3) I: (1, 1) J: (0, 0) 6 K: (1, 0) L: (, 0) A1 M: (3, 0) N: (4, 0) 4 Z O: (4, 1) H P P: (4, 3) Q: (3, ) R Q R: (, ) S: (3, 1) I O T S T: (, 1) Y 5 U: (3, -) J K L M N 5 10 V: (3.5, -) W: (, -) X X: (1.5, -) W U V 4 6 Y: Z: A1: (5, 0) (5, 4) (5, 5.5) 8 Vocabulary Matrix: A rectangular array of mathematical elements with a fixed number of rows and columns. 3

22 Handout 7: Shifty Bird Use the diagram to perform translations on Shifty Bird, using the following steps: Step 1: Represent the translation vector that slides an object two units up and one unit left. Represent this vector in the corner of the graph paper with a directed line segment. Use the Pythagorean theorem to calculate the magnitude of this vector. Step : Use transformation notation to represent the translation in Step 1. Step 3: Copy the coordinates of the Shifty Bird on a x 9 matrix. Place the x-coordinates in the top row and the y-coordinates in the bottom row. Step 4: Write a mathematical expression with matrices that performs the translation described in Step 1. Step 5: Use the expression from Step 4 to perform the translation and obtain a new matrix. Name the new matrix Shifted Bird and then sketch Shifted Bird

23 Handout 8: Reflecting a Quadrilateral In this activity, you will reflect a quadrilateral over several lines. Complete the following instructions as you explore the effects of a reflection on a geometric object. 1. Label each vertex of the quadrilateral shown

24 . On a separate sheet of paper, record the coordinates of each vertex of the quadrilateral on the coordinate system. 3. Set up a matrix that corresponds to the coordinates of the quadrilateral. 4. Reflect each vertex over the x-axis. Imagine that the x-axis acts like a mirror and that you are interested in capturing the image. To represent the image, use the same letters as the original (pre-image) but with primes. For example, if a pre-image has vertex A, the equivalent vertex on the image is labeled A. For subsequent reflections, label the equivalent vertex A, and so on. 5. Record the coordinates of the image. How do the x- and y-coordinates of each point on the image differ from the x- and y-coordinates of each point on the pre-image? 6. Represent the reflection with transformation notation and write the coordinates of the reflected image in a new matrix.

25 7. Repeat this process as you reflect the quadrilateral over the following lines. Use different colors for each reflection. a. The y-axis b. The line y = x c. The line y = x 8. Generalize your findings by responding to the following prompts: a. In order to verify that one of these reflections is an isometry, I would need to show that... b. The patterns I noticed in the matrix representations of the reflections include For one of your reflections, draw a line from each vertex of the pre-image to its corresponding vertex in the image. Note the relationship between each of these segments and the original line of reflection. Do you think this relationship exists for each reflection you made? 3

26 Vocabulary Perpendicular bisector: A figure that divides a line segment into two congruent parts and forms right angles with it. Reflection in line m: A transformation, denoted R m, that maps each point in the pre-image of a geometric object with a corresponding point in the image, so that the line of reflection is the perpendicular bisector of the segment that connects the point in the pre-image to its image. Using mathematical notation, we can also write the definition as follows: A reflection in line m is a transformation that maps every point P to a point P, such that (1) if P is not on line m, then m is the perpendicular bisector of PP, and () if P is on line m, then P = P. 4

27 Handout 9: Properties of Reflections Do the following reflections to explore the properties of a geometric object that stay the same and the properties that change under a reflection. Your teacher will assign you a line of reflection. Do your work on both graph paper and chart paper. Be prepared to present your geometric object and your findings to the class. 1. On graph paper, plot a simple geometric figure and the line of reflection.. Reflect your object over the line. 3. Store the coordinates of the pre-image in one matrix and the coordinates of the image in another matrix. 4. On chart paper, set up a table with two columns as labeled below: Properties that change Properties that do not change (invariants) 5. Look carefully at your reflected image and its pre-image and fill in the columns on the chart paper. 6. (Optional) Verify one of the properties listed in the second column, Properties that do not change. Record your work on the chart paper. 1

28 Handout 10: Matrix Multiplication Matrix multiplication is used in pure mathematics, business applications, graphic design, computer programming, and medicine. By completing this handout, you will learn to multiply matrices, which will allow you to reflect and rotate geometric objects. Study the matrices below and answer the questions that follow. Use additional graph paper if you need more room for your work. 1. AB is the product of the two matrices A and B. What do you notice about the multiplication? Pay special attention to the subscripts and the positions they represent.. Use the matrix diagram above as a pattern to help you multiply matrices D and E (in other words, find the product DE). Specify the dimension of each matrix, including the resulting product matrix. 1

29 3. Can you multiply the matrices above in reverse order, given the multiplication diagram provided? If so, find ED. If not, explain your response. 4. The following are two additional examples of matrix multiplication. Calculate each entry and check it for accuracy. Pay close attention to the dimensions of the original matrices and of the product matrix. Example 1: Example : What can you deduce about matrix multiplication from these examples?

30 5. For the matrices below, find the product MN and the product NM. What can you conclude about matrix multiplication from your results? 3 4 0, M = N =

31 6. For the matrices below, find the product AI and the product IA. Find a real number x that has the same effect as the matrix I when multiplying it with any other real number, in any order , A = I =

32 7. Use subscript notation to write a diagram (similar to the one used above) that represents the multiplication of two square matrices. (A square matrix is one that has the same number of rows and columns for example, a x matrix or a 3 x 3 matrix.) 5

33 Handout 11: Matrix Multiplication Summary The findings below summarize properties of matrix multiplication. Check the accuracy of each point using the examples in Handout 10. In order to multiply two matrices, the number of columns of the first matrix must match the number of rows of the second. Order matters when multiplying matrices. Matrix multiplication is not commutative. With the matrices A and B given, we can make the following generalizations: Multiplying the elements in row 1 of matrix A by the corresponding elements in column 1 of matrix B and taking the sum of those products results in the element in row 1, column 1 of the product matrix AB. Multiplying the elements in row of matrix A by the corresponding elements in column 1 of matrix B and taking the sum of those products results in the element in row, column 1 of the product matrix AB. cut along dotted line Handout 11: Matrix Multiplication Summary The findings below summarize properties of matrix multiplication. Check the accuracy of each point using the examples in Handout 10. In order to multiply two matrices, the number of columns of the first matrix must match the number of rows of the second. Order matters when multiplying matrices. Matrix multiplication is not commutative. With the matrices A and B given, we can make the following generalizations: Multiplying the elements in row 1 of matrix A by the corresponding elements in column 1 of matrix B and taking the sum of those products results in the element in row 1, column 1 of the product matrix AB. Multiplying the elements in row of matrix A by the corresponding elements in column 1 of matrix B and taking the sum of those products results in the element in row, column 1 of the product matrix AB. 1

34 Handout 1: Applying Reflections When you worked with translations, you experienced the power of using matrices to transform objects. Computer animation takes advantage of this power. For example, to slide an object across a screen, you define a translation and create a loop that repeats the translation. The object moves repeatedly until it has crossed the screen. How do you do it? You represent the object and the translation in matrices and then add additional matrices. Now you will use matrices to transform objects in another way by reflecting objects over a line. You use matrix multiplication to create reflections. Part 1: Reflecting the Kite Figure The coordinates of the kite figure are stored in matrix K1. The new kite (A B C D ) is the image of kite ABCD reflected over the x-axis. Its coordinates are in matrix K. 1

35 8 6 B K1 4 A C 5 D = D' 5 4 A' C' 6 B' K 8 Vertices of K1: A(, 4), B(4, 6), C(6, 4), D(4, 0) Vertices of K (obtained by reflecting K1 over the x-axis): A (, 4), B (4, 6), C (6, 4), D (4, 0) K1 = K =

36 Task 1 Find a matrix R x that when right-multiplied by K1 produces K. (Right-multiplied means that K1 multiplies R x on its right side. Remember that order matters in matrix multiplication!) The algebraic expression for this operation is R x K1 = K. What should the dimension of R x be in order to perform this multiplication? What is R x? (Be sure to perform the matrix multiplication as you did on Handout 10: Matrix Multiplication. Use the space below for your work.) Explain why you think your result for R x makes sense. 3

37 Task Reflect kite ABCD over the y-axis. Find a matrix R y that when right-multiplied by K1 will produce the image of K1 reflected over the y-axis. Algebraically, the problem is stated like this: Find R y such that R y K1 = K3, where K3 = Does it matter whether you multiply K1 on the right or on the left, or can you pick either side? Why or why not? Task 3 Reflect kite ABCD over the line m whose equation is y = x. Find a matrix R m that when right-multiplied by K1 produces the image of K1 over line m. Algebraically, the problem is stated like this: Find R m such that R m K1 = K4, where K4 = Task 4 Reflect kite ABCD over line n whose equation is y = x. Find a matrix R n that when right-multiplied by K1 produces the image of K1 over line n. Algebraically, the problem is stated like this: Find R n such that R n K1 = K5, where K5 =

38 Part : Generalizing 1. List each reflection matrix found in the previous tasks and describe its function (what did it do to the kite?).. What patterns do you see in the matrices? 3. Recall that we have stored the x-coordinates of each point of the geometric object in the first row of the matrices and the y-coordinates of each point of the geometric object in the second row. Why do the patterns you found make sense? 4. Is there a relationship between the placement of specific numbers in particular positions of your reflection matrices and the multiplication process? If so, what is the relationship? 5

39 Part 3: Extension 1. Create a reflection matrix or a combination of matrices that will reflect any triangle ABC over the vertical line x = 1. Explain how this matrix differs from the reflection matrices you worked on in class.. Generalize your findings: Create a reflection matrix or a combination of matrices that will reflect any object over the line x = a. 3. How do you think the reflection matrix or the combination of matrices will differ if you are reflecting over a horizontal line whose general equation is y = b? 6

40 Handout 13: Shrinking and Enlarging How can you change the size of a geometric object without distorting its image? You can use a transformation called a dilation. Dilations allow you to create the perception of depth. With dilations, you can make images in your flipbook appear to change their original positions, moving closer to or farther away from you as they change in size. By dilating a given right triangle, you can explore the properties of the figure that change and the properties that remain the same after dilation. Part 1: Dilating a Right Triangle Task 1 Triangle ABC and point P are shown in the following graph. Follow the steps below to transform the triangle. 1. Use a ruler to draw a ray from point P through point A on triangle ABC. Repeat for the other two vertices of the triangle. 1

41 A C B P. Count the vertical change and the horizontal change from point P to point A, point P to point B, and point P to point C.

42 3. Count the same vertical and horizontal change that you found from P to A, starting from point A and moving in the opposite direction from P. Label the resulting point A. (Note that, by the Pythagorean theorem, the distance from P to A is equal to the distance from A to A.) Count the same vertical and horizontal change that you found from P to B, starting from point B and moving in the opposite direction from P. Label the resulting point B. (Note that, by the Pythagorean theorem, the distance from P to B is equal to the distance from B to B.) Count the same vertical and horizontal change that you found from P to C, starting from point C and moving in the opposite direction from P. Label the resulting point C. (Note that, by the Pythagorean theorem, the distance from P to C is equal to the distance from C to C.) 4. Sketch triangle A B C on the graph. 5. Comparing sizes: (a) Compare the lengths of the corresponding sides of your two triangles. In other words, compare the length of AB to the length of A B, the length of AC to the length of A C, and so on. What do you notice? (b) Compare the perimeters of the two triangles. 3

43 (c) Find the area of ABC and the area of A B C. What is the relationship between the two areas? (d) Use a protractor to measure and compare the corresponding angles of the two triangles. Record your observations. 6. Describe the relationship between ABC and A B C by noting the properties of the pre-image that changed and those that stayed the same after the dilation. 4

44 Task 1 1. Write a procedure for dilating ABC by a scale factor of, using P as the center. Apply this dilation to the sketch of ABC below and write the dilation using transformation notation, assuming P is the origin. Label the new triangle DEF, where D corresponds with A, E corresponds with B, and F corresponds with C. A C B P 5

45 . Predict and justify the differences in the corresponding side lengths and the perimeters of the two triangles. 3. How does the area of DEF compare with the area of ABC? Task 3 Use evidence from Tasks 1 and to explain why dilations are not isometries. Check your Glossary for the definition of an isometry, if necessary. 6

46 Part : Generalizing 1. Using transformation notation, write an expression for a dilation of the plane with scale factor 4.. Using transformation notation, write an expression for a dilation of the plane with scale factor k. 3. Suppose a dilation with scale factor k transforms the plane. By what factor will the perimeter of a geometric object on the plane change? 4. Suppose a dilation with scale factor k transforms the plane. By what factor will the area of a geometric object on the plane change? Vocabulary Dilation: A transformation with center C and scale factor k that maps every point A in the plane to a point A such that (a) if A is not point C, then CA is coincident with CA and CA = k(ca), where k 1, and (b) if A is point C, then A = A. Scale factor: A number that describes by how much a figure is enlarged or reduced. A figure that is dilated by a scale factor of n has the same shape as the original figure, but each length is n times the corresponding length in the original figure. Similar polygons: Polygons whose corresponding angles are congruent and whose corresponding sides are proportional. 7

47 Handout 14: Moving the Center What happens to a dilated image when the center is moved? Moving the center of dilation changes the location of the dilated image with respect to the original image, or pre-image. You have already dilated images using the origin as the center of dilation. Using other points as the center of dilation, you will be able to make images in your flipbook appear to move closer and farther away with respect to each other. By finding the centers of dilation for the triangles given below, you can explore dilating images that use a point other than the origin as a center of dilation. Part I Below are four dilations of triangle ABC, enlarged by a scale factor of. Each dilation uses a different center of dilation. For each diagram, find the center of dilation. Dilation #1 8 D 6 4 A F E C B O

48 Dilation # 1 D F 4 A E C B 5 5 4

49 Dilation # A = D C B O 5 10 F E 4 6 3

50 Dilation #4 D A C B O 5 10 F E 4 Part II Work with your group to come up with a procedure for finding the center of any dilation. Write your procedure on chart paper. Choose one person to present your procedure to the class. Be ready to answer questions. How do you know that your procedure works for all dilations? 4

51 Handout 15: Dilating with Matrices Perform each dilation using matrices. Show all of your matrix operations on a separate sheet of paper Dilate kite ABCD below by a scale factor of, using the origin as the center of dilation. Write a matrix equation for the specified dilation.. Dilate kite ABCD below by a scale factor of, using the origin as the center of dilation. (You may need to sketch the pre-image and its image on a separate sheet of graph paper.) Write a matrix equation for the specified dilation. 1

52 B 4 A C D 5 10

53 3. Dilate kite ABCD below by a factor of 1 equation for the specified dilation., using (4, ) as the center of dilation. Write a matrix 6 B 4 A C D

54 Handout 16: TI Calculator Animation Program # Enter the program into your graphing calculator, run the program, and discuss what you see with your classmates. You will use this program as a model to develop your own calculator animation. Program #: Zig Zag To create a new program: 1. Press PRGM on the keypad of the graphing calculator.. Move the cursor to NEW and press ENTER. 3. Enter a name for the program (such as ZIGZAG ) and press ENTER. Note: The graphing calculator is in ALPHA mode automatically in order to allow you to type letters. The letters are written in blue or green above individual calculator keys. 4. Wait for the screen to display a colon, and then enter one of the commands below. Press ENTER after each new command. Note: Any time a matrix is called within the code of the program, you must enter it as a matrix by using nd [MATRIX] and moving your cursor to the matrix name needed. The calculator will not recognize your entry as a matrix if you use brackets rather than the MATRIX menu. Command Function and Location Standardizes the part of the screen that will be displayed. The standard window is set as follows: Xmin= 10 :ZStandard Xmax= 10 Ymin= 10 Ymax= 10 To find this command: Press [Zoom] 6. Makes the x- and y-axes appear in the display. :AxesOn :FnOff To find: Press nd [FORMAT], move the cursor down to AxesOn, and press ENTER. Turns off the display of any equations entered in the Y= editor. To find: Press VARS, move the cursor to Y-VARS #4, and press ENTER. 1

55 Command Function and Location :PlotsOff Turns off all statistics plots so that data in the calculator will not interfere with the resulting program s graph. To find: Press nd [STAT PLOT] 4. :[[, 0][, 0][, 0]] [H] Stores values into matrix H, the translation matrix. H = To find: Enter the values of the matrix, using brackets, above the multiplication and subtraction keys. To assign values to H, use STO>. :[[ 8, 0][ 7, ][ 6, 0]] [F] Stores values into matrix F, the object matrix. Each row contains the x- and y-coordinates of a point. F = To find: Enter the values of the matrix, using brackets, and use STO> to assign values to F. :Line([F](1, 1), [F](1, ), [F](, 1), [F](, )) Creates a line segment between two endpoints whose positions in matrix F are specified. [F] (i, j) refers to the element in the i th row, j th column. To find: Press nd [DRAW]. :Line([F](, 1), (, ), (3, 1), (3, )) Creates a line segment between two endpoints whose positions in matrix F are specified. [F] (i, j) refers to the element in the i th row, j th column. To find: Press nd [DRAW]. Stores values in matrix G, the reflection matrix. :[[1, 0][0, 1]] [G] G = To find: Enter the values of the matrix, using brackets, and use STO> to assign values to G.

56 Command Function and Location :For (A, 1, 15) Begins a loop with counter A that ranges from 1 to 15. The loop repeats 15 times. To find: Press nd [CATALOG] and move cursor to For. :[F][G] [F] F and G are multiplied, and the resulting product is stored in F. :[F] + [H] [F] The translation matrix H is added to the new F from the previous line. :Line([F](1, 1), [F](1, ), [F](, 1), [F](, )) Creates a line segment between two endpoints whose positions in matrix F are specified. [F] (i, j) refers to the element in the i th row, j th column. To find: Press nd [DRAW]. :Line([F](, 1), [F](, ), [F] (3, 1), [F](3, )) Creates a line segment between two endpoints whose positions in matrix F are specified. [F] (i, j) refers to the element in the i th row, j th column. To find: Press nd [DRAW]. :For(D, 1, 30) Begins a delay loop. :End Ends the delay loop. To find: Press nd [CATALOG] and move the cursor to End. :End Ends the first For... loop. 5. To run the program: Press PRGM. Move the cursor over the name of the program. Press ENTER. The program is executed. 3

57 Appendix A: Figure 1: Pop 1 10 H: (1, 3) I: (1, 1) J: (0, 0) 8 K: (1, 0) L: (, 0) M: (3, 0) 6 D N: (4, 0) A1 C 1 E 1 1 I O: (4, 1) J 1 B F P: (4, 3) H 1 Z 1 H P Q: (3, ) G 1 R: (, ) R Q S: (3, 1) I O T S T: (, 1) Y 5 U: (3, -) J K L M N 5 10 V: (3.5, -) B 1: (8, 4.5) W: (, -) X V C 1: (8.5, 4.5) W U X: (1.5, -) D : (9, 5) 1 4 E 1: (10, 5) F 1: (10, 4) 6 G Y: (5, 0) 1: (10, 3.5) H : (9, 4) Z: (5, 4) 1 I : (9.5, 4.6) A : (5, 5.5) 1 1 J : (9, 4.5) 1 8 1