Presentation of Unit Lesson Plans on Geometric Transformations

Presentation of Unit Lesson Plans on Geometric Transformations EDUC 366 June 27, 2007 Unit on Geometric Transformations Lesson 1 – Reflections Lesson 2 – Translations Lesson 3 – Rotations Reflections Who is this man? Who is this woman? What are the words in the red box? How are they all connected? Click here to find out! S.W.B.A.T. •Identify properties of reflection transformations •Identify and locate reflection images of figures •Use a coordinate system to formulate general mappings for reflections. •Construct reflections on the Cartesian coordinate system Reflections are the first examples of transformations that we will be studying in this unit. A transformation is a change in position, shape, or size of a figure. Did you take a look in the mirror this morning? You were actually the pre-image, or the original figure. The view you caught in the mirror was your image, or the figure after a transformation. Leonardo da Vinci‟s writings were the preimage. In order to read them, you must hold them up to a mirror, then read the image. ???????????? What are some types of reflections? Well, I‟m glad you asked……. There are reflections along a vertical line, Reflections along a horizontal line, And reflections along a diagonal line! Have some practice with the characteristics of reflections! Click here to experiment with an activity. Now that you‟ve had a chance to work with some reflections, let‟s discuss what you saw. A transformation maps the pre-image onto its image. Take a look at the example below. F Mirror line F‟ U N N‟ U‟ FUN  F 'U ' N ' The above statement would be read “Triangle FUN maps to Triangle F prime, U prime, N prime.” Notice that prime notation is used to identify points on the image. F Mirror line F‟ U N N‟ U‟ FUN  F 'U ' N ' Is FUN congruent to F ' U' N' ? For a review of congruency, click here. F Mirror line F‟ U N N‟ U‟ FUN  F 'U ' N ' If the original figure and its image are congruent, then the transformation is an isometry. Are reflections isometries? Have you ever noticed how ambulances have reverse printing on their fronts? Why do you think that is? A reflection reverses the orientation of a figure. The image will always appear reversed from the pre-image. Remember….. Reflections are flips!!! Based on what we‟ve learned so far, can anyone predict what the two main properties of reflections are? Have you given it a try? Check your predictions here. Just to make sure it‟s crystal clear……… Is the figure on the right side of the mirror line a reflection of the figure on the left side of the mirror line? Yes or no I love Geometry! I love Geometry! Is the figure on the right side of the mirror line a reflection of the figure on the left side of the mirror line? Yes or no Activity Now that you‟ve learned more about transformations and reflections, show what you know! Complete this activity on reflections of polygons, including the six questions on the activity page. Let‟s relate our knowledge of reflections to what we know about the coordinate system. Suppose we have coordinate points A(3, 1), B(4, 2), and C(3, 5). If we reflected these points over the xaxis, what would be the coordinates of their images, A‟, B‟, and C‟? Can you find a pattern and make a generalization for the mapping of any general point (x, y) to its image? If you need help, click here for the solution. Let‟s try another one. Use the same points as above with A(3, 1), B(4, 2), and C(3, 5). If we reflected these points over the y-axis, what would be the coordinates of their images, A‟, B‟, and C‟? Can you find a pattern and make a generalization for the mapping of any general point (x, y) to its image? Give it your best shot, then check here for the solution. Activity Test your skills on mirror lines and coordinates of reflections in the following online worksheet. Click on the shaded boxes at the top of the page to advance to the next problem. The problem you‟re working on currently will be displayed by an „x‟ in the shaded box. Good luck! Let‟s wrap it up……… Who can give me an example of a reflection with a pre-image and image? Why are you certain it is a reflection? What properties does it have? Given a figure, how can you map a reflection on the Cartesian coordinate system? Something to think about……. According to the Wall Street Journal, most drugs are made up of two versions of the same molecule. One version is called an Risomer, the other version is called an S-isomer, and they are mirror images of each other with different healing properties. In an effort to produce drugs with fewer side effects, researchers have learned to produce pure batches of these isomers. They run tests to see which version has the least amount of side effects, then produce the drug from this batch. For example, the R-isomer of Albuterol treats asthma, but its mirror version (the S-isomer) has been shown to increase the risk of future attacks. Pretty neat, huh? Geometry is EVERYWHERE! Lesson 2 – Translations Back to Main Page Translations Click on the image below to watch the video of OK Go‟s “Here It Goes Again” on YouTube. How would you describe the motion of the band members that makes this video unique? S.W.B.A.T. •Locate translation images of figures •Represent translations using vectors •Represent translations using matrix addition •Model translation images on the Cartesian coordinate system •Formulate a translation based on coordinates of pre-images and images Activity Let‟s start with an exercise. Click here to complete ten questions that will help us determine the characteristics of our next type of transformation. Please do the five green questions and the five purple questions by clicking on the colored squares at the top of the page. F F ’ U N U’ N’ Take a look at triangle FUN and triangle F‟U‟N‟ above. Does the transformation FUN  F ' U' N' seem to be an isometry? Think about it, then click here for the answer. F F ’ U N U’ N’ Does the transformation FUN  F ' U' N' change the orientation? Give it some thought and click here. F F ’ U N U’ N’ The picture above is an example of a transformation called a translation. A translation is a transformation that moves points the same distance and in the same direction. Every point on the blue triangle was moved ten units to the right to produce the image of the red triangle. To put it another way……. Translations are SLIDES! Activity Try some translations out for yourself! Take a few minutes to complete these three activities (use the arrows above the questions to move from one activity to the next), then we‟ll move on to describing the distance and direction of translations. F U U’ F’ N N’ The orange lines in the diagram above represent vectors. A vector expresses the distance and direction of a translation. The vectors in the triangle diagram are FF' , UU' , and NN' . F U U’ F’ N N’ Be careful -vectors are not the same as rays! Although the notation may look identical, vectors have a fixed length. They do not go on forever like rays. F U U’ F’ N N’ Like rays, vectors have initial points. In this case the initial points are F, U, and N. But unlike rays, vectors also have terminal points. In this case the terminal points are F‟, U‟, and N‟. F U U’ F’ N N’ Vectors can be expressed in ordered pair notation . The x-value is the horizontal change between the initial and terminal points, and the y-value is the vertical change between the initial and terminal points. In the example above, the blue triangle is translated ten units to the right and five units down. F U U’ F’ N N’ The value for vector FF' is <10,-5>. What is the value for vector UU' ? What is the value for vector NN' ? Think it over and explain your answer, then check it here. F U U’ F’ N N’ You‟re probably asking “How do I represent a translation?” That‟s an excellent question! Mathematically, this translation can be expressed in three different ways: The first way F U U’ F’ N N’  (x,y)  (x+10, y-5) On coordinate axes, x will become x+10, and y will become y-5. The second way F U U’ F’ N N’ T(10,5 ) The “10” means add 10 to all x-coordinates, and the “-5” means subtract 5 from all ycoordinates. And the third way… Sometimes a translation is just represented as a vector of a certain length and direction. This vector moves all x-coordinates 10 units to the right and all y-coordinates 5 units down and would have the notation <10, -5>. F U U’ F’ N N’ The coordinates of FUN are F(-3, 7), U(-5.5, 4.5), and N(-3, 1). If, under the translation, x becomes x+10 (we add 10 to all x-coordinates), and y becomes y-5 (we subtract 5 from all y-coordinates), then what are the coordinates of F‟, U‟, and N‟? Click to check your answer step by step. Activity Let‟s practice vectors and translations using the information we just learned. Complete the problems on the interactive worksheet to test your knowledge. Addition and subtraction on individual coordinates is only one way to translate figures in the coordinate plane. We can also use addition and subtraction through matrices. Consider the following example: 4 S M 2 W I 2 -4 -2 -2 -4 4 We have rectangle SWIM. On our coordinate system, where would the image of SWIM be under the translation <-11, 4>? S 2 -2 -2 W M 2 4 I First step: Set up a matrix with the x-coordinates of the vertices in the first row and the ycoordinates of the vertices in the second row. Vertices of Pre-Image in Matrix Form x-coordinate y-coordinate 2 8 8 2 4 4 1 1   SW I M S 2 -2 -2 W M 2 4 I Next step: Set up a translation matrix for the translation <-11, 4>. Translation Matrix  11  11  11  11  4  4 4 4   S 2 -2 -2 W M 2 4 I Last step: Add these two matrices together using matrix addition. For an algebra review of matrix addition, click here. 2 8 8 2 4 4 1 1 +    11  11  11  11  4 4 4 4      9  3  3  9 =  8 8 5 5   S’ M’ W’ I’ 4 2 S W -4 -2 -2 M 2 4 I The matrix sum is the image, where the first row contains the x-coordinates of the image vertices and the second row contains the y-coordinates. Vertices of Image in Matrix Form S‟ W‟ I‟ M‟ - 9 - 3 - 3 - 9 8 8 5 5   Exercise Now you try one! Figure YOU has vertices (-7, -3), (-9, -7), and (-5, -7). Find the image of this figure under the translation <14, 2> by using matrices. After you have sketched out the figure and its image, check your answer here. Show what you know and exercise your brain! Try this interactive activity on vectors and translations. Let‟s Wrap It Up…… What are the different ways can we locate an image under a translation if we are given the pre-image? What is a vector and how is it used to represent a translation? Using matrices, what are the steps we take to find an image if we are given a translation? If you are given the coordinates of a point, how do you find the coordinates of its image under a translation? Worksheet Please print out this worksheet, complete the questions, and turn it in to me when completed. If accommodations are necessary, please choose the appropriate version. Worksheet Worksheet – Visual Impairment Worksheet – Autism Did you know….. Translations are a critical part of animation. Your favorite animated movies wouldn‟t look the same without it. But animation isn‟t only used for entertainment. It‟s used in control systems and flight simulators for pilot training, and also in scientific research. It can help surgeons practice without putting a real life in danger. Automobile companies use it for making 3D models of cars. It‟s used by architects to make model houses so their clients can take virtual tours. And it‟s critical for your video games. Pretty neat, huh? Geometry is EVERYWHERE! Moving on! Lesson 3 – Rotations Lesson 1 – Reflections Back to Main Page Rotations Can anyone tell me the name of the game pictured above? What is the object of the game? How is it played? What happens if you let the pieces fall without changing them? What are some strategies? For anyone who is not familiar with the game, let‟s take a look at an example. S. W. B. A. T. • • • • Identify properties of rotation transformations Identify and locate rotation images of figures Construct a rotation of a figure and formulate a general mapping on the Cartesian coordinate system Determine whether rotations have symmetry Rotations are the final examples of transformations that we will be studying in this unit. To rotate an object means to turn it around or to spin it. Every rotation has a rotocenter, or center of rotation, and an angle. In the example below, the pre-image (red R) was rotated around a rotocenter (which also happens to be the center of the circle) for 90 degrees in order to produce the image (blue R). In each of these pair examples, the image on the right is a rotation of the pre-image on the left. Activity Have some practice with the characteristics of rotations! Click here to experiment with an activity. Please complete all three sub-activities listed in the right-hand frame of the activity page and make sure your answer all of the questions! The direction of a rotation can be either clockwise or counterclockwise. Think about your locker combination. You need to turn it clockwise to a certain number, then counterclockwise to another number, then clockwise again to the final number before the lock will open. You use rotations every day! Positive angles are typically measured counterclockwise. If a rotation is clockwise, it is usually a negative angle. To work with rotations, you need to be able to recognize angles of certain sizes and understand the basic workings of a unit circle. Remember that a unit circle moves in a counterclockwise direction! Activity Try this activity. Click on the „rotation‟ button at the top of the page when it loads. Then click on the buttons to experiment with triangles under ¼ turn rotations clockwise and counterclockwise, and also ½ turn rotations. Notice the x and y coordinates of the images under each transformation. The rotocenter may be located in three different areas in a rotation. The center of rotation could be inside the pre-image, as in the following picture. Arrow A was rotated 120 degrees about Point C to produce Arrow A‟ A’ C A The center of rotation could also be outside of the pre-image, as shown in the picture below. Arrow A was rotated 120 degrees about Point C to produce Arrow A‟. A’ C A As another option, the center of rotation could be on the pre-image, as shown in this picture. Arrow A was rotated 120 degrees about Point C to produce Arrow A‟. A’ A C Under a rotation, does each point move the same distance? Why or why not? A’ A C Is Arrow A congruent to Arrow A‟? For a review of congruency, click here. If the original figure and its image are congruent, then the transformation is an isometry. Are rotations isometries? Does a rotation change orientation? So, what are the two properties of a rotation? So, what are the two properties of a rotation? 1) A rotation is an isometry. 2) A rotation does not change orientation. Let‟s work through some problems together. P A X T N E This is regular pentagon PENTA and it has been divided into five congruent triangles. P A X T N E What is the image of E under a 72 degree rotation about X? Have you thought about it? Click for an explanation. P A X T N E What is the rotation that maps E to N? Check your answer here. P A X N E T What is the image of any point under a 360 degree rotation? Are you ready for the answer? P A X N E T What do you notice about regular pentagon PENTA each time it is rotated 72 degrees? Does it look different? What do YOU think? Here‟s what I‟ve noticed. See if it‟s different from your ideas. Activity Let‟s practice what you‟ve learned about rotation symmetry. Please link to this activity. Complete question B1, parts A and B, then check your answers. Let‟s relate our new knowledge about rotations to what we know about the coordinate system. Suppose we have points A(3, 1), B(4, 2), C(3, 5). If we rotated these points 90 degrees counterclockwise, what would be the coordinates of their images, A‟, B‟, and C‟? Can you find a pattern and make a generalization for the mapping of any general point (x, y) to its image? If you need help, click here for the solution. Let‟s try another one. Use the same points as above with A(3, 1), B(4, 2), and C(3, 5). If we rotated these points 180 degrees, what would be the coordinates of their images, A‟, B‟, and C‟? Can you find a pattern and make a generalization for the mapping of any general point (x, y) to its image? Give it your best shot, then check here for the solution. Activity Now try this matching game. You will be presented with 16 squares. Eight squares will have a description, such as „Rotate 180 degrees‟. The other eight squares will have a mapping, such as (x, y)  (-x, -y). You need to click on the two squares that match each description to its correct mapping. Activity Now that you are a master of rotations, test your skills in the following online worksheet. Click on the shaded boxes at the top of the page to advance to the next problem. The problem you‟re working on currently will be displayed by an „x‟ in the shaded box. Please complete the five brown questions and the five red questions. Good luck! Let‟s wrap it up…… What are the properties of a rotation? What is an example of a rotation where the center is inside the pre-image? What is an example of a rotation where the center is outside of the pre-image? What is rotation symmetry? Extra Activity Try a fun game called Pentominoes. It‟s said to be part of the inspiration for the game Tetris. Choose a 10 x 6 grid, and rotate and move the pieces to fill the entire grid. Good luck! Something to think about……. Why do clocks run “clockwise”? Before clocks were invented, sundials were used to tell time. In the northern hemisphere, the sun's shadow rotated in the direction we now call clockwise. The clock hands were built to mimic the natural movement of the sun. However, if clocks had been invented in the southern hemisphere, clockwise would have been in the opposite direction. Pretty neat, huh? Geometry is EVERYWHERE! Lesson 1 – Reflections Lesson 2 – Translations Back to Main Page This man is Leonardo da Vinci. This woman is Mona Lisa. This is the sentence “I love geometry.” written from right to left. How are they all connected? Leonardo da Vinci created both of them! In his notebooks, da Vinci wrote using "mirror writing," writing that went backwards from right to left, instead of from left to right. In order to read his writing normally, you must place a mirror beside the writing and read the reversed image in the mirror. No one knows why he wrote this way, but some possibilities have been suggested: •He was trying to make it harder for people to read his notes and steal his ideas. •He was hiding his scientific ideas from the powerful Roman Catholic Church, whose teachings sometimes disagreed with what Leonardo observed. •Leonardo wrote with his left hand. Writing left handed from left to right was messy because the ink just put down would smear as his hand moved across it. Leonardo chose to write in reverse because it prevented smudging. Try it out for yourself! Back to lesson ABSOLUTELY! The pre-images and images are exactly the same size and shape, only reversed. Back to lesson The two main properties of reflections are: •A reflection is an isometry (The figure and its image are congruent). •A reflection reverses orientation (The image appears “backwards”). Did you get it? Nice job! Back to lesson Nice try, but no. The image on the right does not have a reversed orientation. Try again! back to lesson Nice job! The figure on the right is not a reflection because the orientation of the preimage is not reversed. Back to lesson Nice try, but no. The transformation is not an isometry. The figure on the right is not congruent to the figure on the left. Its dimensions are larger. Try again! Back to lesson Very nice! The transformation is not an isometry. The figure on the right of the mirror line has reverse orientation, but it is not congruent to the figure on the left. Its dimensions are much larger. Way to go! Back to lesson The first thing we need to do is graph the points we are given. C B 2 A 2 We need to reflect the points over the x-axis (remember, it‟s the horizontal one). This means Point A would keep the same x-coordinate. Its ycoordinate would be the same distance away from the x-axis, but below the axis. So the coordinates for Point A‟ would be (3, -1). If we follow the same logic, the coordinates for Point B‟ would be (4, -2) and the coordinates for Point C‟ would be (3, -5). C B A A’ B’ C’ Now let‟s take a look at the pre-image and image coordinates: A (3, 1)  A‟ (3, -1) B (4, 2)  B‟ (4, -2) C (3, 5)  C‟ (3, -5) What generalization can we make about coordinates that are reflected over the x-axis? Well, it looks like the x-coordinate stays the same, but the y-coordinate becomes negative. So, the mapping becomes……….. (x, y)  (x, -y) Back to lesson The first thing we need to do is graph the points we are given. C B 2 A 2 We need to reflect the points over the y-axis (remember, it‟s the vertical one). This means Point A would keep the same y-coordinate. Its x-coordinate would be the same distance away from the y-axis, but left of the axis. So the coordinates for Point A‟ would be (-3, 1). If we follow the same logic, the coordinates for Point B‟ would be (-4, 2) and the coordinates for Point C‟ would be (-3, 5). C’ C B’ A’ 2 B A 2 Now let‟s take a look at the pre-image and image coordinates: A (3, 1)  A‟ (-3, 1) B (4, 2)  B‟ (-4, 2) C (3, 5)  C‟ (-3, 5) What generalization can we make about coordinates that are reflected over the y-axis? Well, it looks like the y-coordinate stays the same, but the x-coordinate becomes negative. So, the mapping becomes……….. (x, y)  (-x, y) Back to lesson Sure it is! Remember from Lesson 1, an isometry is a transformation in which the original figure and its image are congruent. These two triangles have sides of the same length and the same angles. F F’ U N U’ N’ Back to lesson No, the transformation does not change the orientation. The order of the vertices in the pre-image is counterclockwise (FUN), and so is the order of the vertices in the image (F‟U‟N‟). F F’ U N U’ N’ Back to lesson Since translation is an isometry, the original figure and its image are congruent. Therefore, each point of the image will be an equal distance from the original figure. So the value of vectors FF',UU' , and NN' will all be <10, -5>. Back to lesson First we‟ll calculate the coordinates of F‟. The x-coordinate of F is -3. We want to add 10 to all x-coordinates, so the xcoordinate of F‟ is (-3 + 10), or 7. The y-coordinate of F is 7. We want to subtract 5 from all y-coordinates, so the y-coordinate of F‟ is (7 – 5), or 2. Thus the coordinates of F’ are (7,2). Now we‟ll calculate the coordinates of U‟. The x-coordinate of U is -5.5. We want to add 10 to all x-coordinates, so the xcoordinate of U‟ is (-5.5 + 10), or 4.5. The y-coordinate of U is 4.5. We want to subtract 5 from all y-coordinates, so the ycoordinate of U‟ is (4.5 – 5), or -0.5. Thus the coordinates of U’ are (4.5,-0.5). Finally we‟ll calculate the coordinates of N‟. The x-coordinate of N is -3. We want to add 10 to all x-coordinates, so the x-coordinate of N‟ is (-3 + 10), or 7. The y-coordinate of N is 1. We want to subtract 5 from all y-coordinates, so the y-coordinate of N‟ is (1 – 5), or -4. Thus the coordinates of N’ are (7,-4). Back to lesson 2 -4 -2 -2 -4 2 4 Y’ Y O’ O U U’ Let‟s take a look at the matrices we needed for our transformation. 5 9  7  9  5 14 14 14  7  3  7  7   2 2 2    1  5  5       Back to lesson The distance each point moves depends on how close it is to the center of rotation. A’ A C The front point of Arrow A‟ traveled a farther distance than a point on the arrow very close to Point C. So the points farthest from the center of rotation move the farthest. Return to lesson ABSOLUTELY! A’ A’ C A C A The pre-images and images are exactly the same size and shape, only in different positions. Rotating them doesn‟t change size or orientation. Back to lesson No, rotation does not change orientation. A’ C A Rotation just changes the angle the figure is viewed. It does not change the way the figure would be described. Back to lesson Well, let‟s think about it step by step. Without a protractor, how would we know what a 72 degree angle would be? 360 degrees P divided by 5 equal angles = 72 degrees per angle E A X T N We want to rotate vertex E through one segment of the pentagon. Doing this would move vertex E to vertex P. Therefore, P is the image of E under a 72 degree rotation about X. Back to lesson Let‟s look at the info we know…… P A X 360 degrees divided by 5 equal angles = 72 degrees per angle E T N We‟re rotating through 4 of the angles around X. Remember, we‟re going counterclockwise!!! If each angle is 72 degrees, then the angle of rotation from E to N would be 72 X 4 = 288 degrees. Back to lesson It‟s the same point! Choose any point to start, say Point P. P 360 degrees divided by 5 equal angles = 72 degrees per angle E A X T N We know there are 72 degrees in each central angle about X. We want to rotate 360 degrees. How many 72 degree rotations would that be? 360  72 = 5 P A X E 360 degrees divided by 5 equal angles = 72 degrees per angle T N So, we need to make five 72 degree rotations. First rotation - Point A Second rotation - Point T Third rotation - Point N Fourth rotation - Point E The fifth and final rotation puts the image on Point P. Back to lesson Here we have regular pentagon PENTA. P A X T N 360 degrees divided by 5 equal angles = 72 degrees per angle E Suppose we rotate PENTA 72 degrees clockwise about Point X. Here‟s the image: E P X N A T It looks the same! What if we rotate it another 72 degrees? N E X P A T It still looks the same! This is called rotation symmetry. Here‟s the test for rotation symmetry: If you can rotate a figure less than 360 degrees around a center point, and its image is identical to its pre-image, then the figure has rotation symmetry. The center point is called the center of rotation, and angle used to turn the figure is the angle of rotation. In our PENTA example, the center of rotation was Point X (also the center of the pentagon), and the angle of rotation was 72 degrees. P A X T N 360 degrees divided by 5 equal angles = 72 degrees E per angle Rotational symmetry plays a big part in the study of crystals and also in design arts. Back to lesson The first thing we need to do is graph the points we are given. C B 2 A 2 We need to rotate the points counterclockwise 90 degrees around the origin. This would move all of the points from quadrant 1 into quadrant 2. Imagine that we took the entire coordinate system and turned it ¼ turn. So the coordinates for Point A‟ would be (1, 3). If we follow the same logic, the coordinates for Point B‟ would be (-2, 4) and the coordinates for Point C‟ would be (-5, 3). B’ C’ A’ B 2 A 2 C Now let‟s take a look at the pre-image and image coordinates: A (3, 1)  A‟ (-1, 3) B (4, 2)  B‟ (-2, 4) C (3, 5)  C‟ (-5, 3) What generalization can we make about coordinates that are rotated 90 degrees around the origin? Well, it looks like the y-coordinate of the pre-image takes the opposite sign, then the x- and y-coordinates switch places. So, the mapping becomes……….. (x, y)  (-y, x) Back to lesson The first thing we need to do is graph the points we are given. C B 2 A 2 We need to rotate the points counterclockwise 180 degrees around the origin. This would move all of the points from quadrant 1 into quadrant 3. Imagine that we took the entire coordinate system and turned it ½ turn. So the coordinates for Point A‟ would be (-3, -1). If we follow the same logic, the coordinates for Point B‟ would be (-4, -2) and the coordinates for Point C‟ would be (-3, -5). C B A 2 2 A’ B’ C’ Now let‟s take a look at the pre-image and image coordinates: A (3, 1)  A‟ (-3, -1) B (4, 2)  B‟ (-4, -2) C (3, 5)  C‟ (-3, -5) What generalization can we make about coordinates that are rotated 180 degrees around the origin? Well, it looks like the x- and y-coordinates just take the opposite sign. So, the mapping becomes……….. (x, y)  (-x, -y) Back to lesson