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Transformati ons in Carnivals By Hannah Lin, Joseph Choi, Kaitlyn Choi and Kevin Walker Table of Contents – Welcome! • Rotations by Hannah Lin……………… 3 to 22 • Reflections by Joe Choi……….23 to 34 • Translations by Kaitlyn Choi……..35 to 53 • Tessellations by Kaitlyn Choi and Hannah Lin……..54 to 65 • Dilations by Kevin Walker………….66 to 77 • Bibliography………………………………78 to 82 tion s By Hannah Lin What is a rotation? • A rotation is a transformation in which a figure is turned about a fixed point. Key Vocabulary Terms • The fixed point of a rotation is called the center of rotation. • Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a clockwise rotation of 180 ° or less. Some polygons that have rotational symmetry: square, equilateral triangle, regular pentagon and rectangle. The equilateral triangle can be rotated 120 degrees and 180 degrees and map onto itself. Examples of shapes that have rotational symmetry: Practice Problems Determine whether each polygon has rotational symmetry. If yes, describe the rotations that map the polygon onto itself. 1) 2) 3) 4) Equilateral triangle Kite Regular hexagon Square Answer Key 1) 2) 3) 4) Yes, 120° and 180° No. Yes, 60°, 120° and 180° Yes, 90° and 180° Rotation Theorems • Theorem 7.2 Rotation Theorem: A rotation is an isometry. (original shape and after shape are congruent) • Theorem 7.3: If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is a rotation about point P. The angle of rotation is 2x°, where x° is the measure of the acute or right angle formed by k and m. Theorem 7.3 If the angle between line l and m is 32 degrees x, then the angle of rotation about O would be 2x, 64 degrees. The rotation is a “double reflection” of shape XYZ: one reflection across line l, and one reflection across line m, which is equal to one rotation around point O. *not drawn to scale To find the new coordinates of an image after being rotated a certain degree clockwise or counterclockwise, you would use these equations: R90°(x, y)= (-y, x) {counterclockwise} R180°(x, y)= (-x, -y) R270°(x, y)= (y, -x) R-90°(x, y)= (y, -x) {clockwise} Do it Yourself! Find the new coordinates of the vertices of the image after a clockwise rotation of 90 degrees around the origin, using the equations from the previous slide: Answer 13) A’(1,3) B’(1,5) C’(4,6) D’(4,2) Rotating and Drawing the New Figure Say we want to rotate this carnival ticket 60 degrees counter-clockwise around point P. When you rotate a figure, you need to do it vertex by vertex. First, draw a light line from P to your designated vertex. In this example, we will start with A. The line is about 2.6 cm long. Remember that. Next, line up the protractor onto the newly drawn line as you see below. Make sure your angle is going in the counter-clockwise direction. Mark 60 degrees and take away the protractor like so. Then, using a ruler, line up P with the marked angle line. Remember the distance between A and P? It’s 2.6 cm. Keeping the ruler lined up with P and the marked angle line, mark A` at 2.6 cm so the distance from P to A` is also 2.6 cm. Do the same process for points B, C and D and you will get your new shape, rotated 60 degrees counterclockwise around point P! *not to scale on computer, make sure you do it on paper Do it Yourself! Rotate figure WXYZ 120 degrees counter-clockwise around Point P. Challenge: Rotate figure WXYZ 50 degrees counter-clockwise around point Z. Answers 120 Degrees 50 Degrees (Challenge) Real-Life Application After a long day at the carnival, you decide to end the day by riding a peaceful Ferris wheel. One rotation of the wheel takes 20 minutes. If you want to find the measure of the angle between each seat, you would divide 360 by the number of pieces the wheel is divided into. In this case, it would be 8. So the measure of the angle between each seat is 45 degrees. Real-life application continued If you want to find out the degree of rotation to get to any seat, multiply 45° by the number of “leaps” you take to get to that seat. For example, what is the degree of rotation to get from seat 1 to 5(clockwise)? You are going to take 4 leaps, so 45 * 4= 180 degrees. Challenge: How many minutes would it take to get to the top of the Ferris wheel, seat 4, from seat 1? (clockwise) {refer back to the description} Answer Write a proportion. (45 * 3= 135) The answer is 7.5 minutes. Reflections in Carnivals By Joseph Choi What is a Reflection? • A reflection is a transformation which uses a line that acts like a mirror, with an image reflected in the line • A shape is reflected over the line of reflection, which acts like a mirror for the shape • Theorem 7.1 Reflection Theorem – A reflection is an isometry Key Vocabulary • Line of Reflection – The line that a shape is reflected over in a transformation • Line of Symmetry – The line that goes through a shape and allows the shape to be mapped out onto itself Equations Reflecting Over Lines: • rx-axis (x , y) = (x , -y) • ry-axis (x , y) = (-x , y) • ry=x (x , y) = (y , x) • ry=-x (x , y) = (-y, -x) Point Reflection • R180 (x , y) = (-x , -y) Reflections are Everywhere Whenever there is a symmetric figure in real life, such as a Ferris wheel, that figure can be created by reflecting half of the figure on the line of symmetry Reflection: True or False True or False True or False True or False Minimum Distance The tent is on fire and the clown must go to the well to put it out! The shortest distance to go to the well and then the tent would be the minimum distance. Minimum Distance In order to find this we would first draw a line at the well (point C) parallel to the ground. Then, we would reflect the clown (point A) over the line that we drew earlier. Then in order to find the distance, we would connect point A’ to the tent (point B). Minimum Distance So to recap… Step 1: Draw a parallel line to the ground from point C. Step 2: Reflect point A over the line you drew. Step 3: Connect point A’ to point B Step 4: Measure the line from point A’ to point B Try it Yourself Point C Solutions Minimum Distance: √65 Cotton Candy Reflections: 1) False 2) False 3) False Fun in Carnivals Pin the Tail on the correctly reflected point Translati ons By Kaitlyn Choi What is a translation? • A translation is a transformation that maps two points, A and B, in the plane to points A’ and B’, so that the following properties are true: 1. AA’ is congruent to BB’ 2. AA’ is parallel to BB’, or AA’ and BB’ are collinear. Examples: Key Vocabulary Words Vector – a quantity that has both direction and magnitude, or size. Initial point – the starting point Terminal point – ending point terminal point Initial point Isometry – a transformation that has congruent pre-images and images Key Vocabulary Words cont. Component form – combines the horizontal and vertical components of a vector To find the component form of a vector, you can use this equation: < x2 – x1, y2 – y1 > OR you can count the number of units it takes to reach your terminal point, both horizontally and vertically. Ex. 1 < x2 – x1, y2 – y1 > = < 4 – 1, 5 – 1 > < 3 , 4 > You can also see that it was moved three units to the right, and four units up. So the component form of the vector would be < 3 , 4 >. Translation Theorems Theorem 7.4: A translation is an isometry. Theorem 7.5: If lines x and y are parallel, then a reflection in line x followed by a reflection in line y is a translation. If B” is the image of B, then the following is true: 1. BB” is perpendicular to x and y. 2. BB” = 2d, where d is the distance between x and y. Theorem 7.5 Keep in mind that: - Lines x and y are parallel. - The images shown from each axis over is a reflection. x y A A’ B B’ A” As you can see here, there was a translation taken place from AB to A”B” after two reflections. Also, BB” is perpendicular to lines x and y. B” Remember that the distance between lines x and y is d, and the distance between B and B” is 2d. So for example, if d was four, the length of BB” would have to equal eight. Equation To translate a figure, use this equation: Ta,b(x, y) = (x + a, y + b) A and b are constants. A moves horizontally while b moves vertically. The number of units you move to the right or left equals a, and the number of units you move up or down equals b. Next, x and y are the coordinate points of the pre-image. This is all explained in coordinate notation. The equation for this is: (x + a, y + b) Plug in to solve! Translate each line one by one. Example Let’s start off with an easy example. Let’s go piece by piece and start off with just one line–just one segment of a triangle. Refer to the graph to the left. What is the image after the translation (x + 5, y – 2)? To translate this line, move five units to the right and two units down. Matrices A matrix is another way to find the image of a translated figure, given coordinate notation or a vector in component form. It is a rectangular arrangement of numbers in rows and columns. Matrix addition is what we use to find the coordinates of an image. So first, set up the translation rule in the matrix. If the coordinate notation is (x + 2, y + 4), or if the vector/component form is < 2 , 4 >, insert it like this: [2 2 2 2] [4 4 4 4] [ -3 -4 1 0 ] [ 0 -2 -1 -3 ] The amount of times you must insert this is the amount of vertices you have. So if you have four vertices, like in this case, you’d have to write it four times. Next, take the coordinates of your pre-image: (-2, 0) (-2, -4) (1, -1) (0, -3) And write it like this. Then add them together. Add the corresponding numbers. So for example, add the first number of the first matrix to the first number of the second matrix, and so on. [2 2 2 2] [4 4 4 4] + [ -3 -4 1 0 ] [ 0 -2 -1 -3 ] = [ -1 -2 3 2 ] [ 4 2 3 1] So the new coordinates of the image Would be: (-1, 4) (-2, 2) (3, 3) (2, 1) Do It Yourself! Describe each translation with coordinate notation. 1. 8 units to the right and 3 units down 2. 5 units to the left and 4 units up Find the coordinates of the image with the given picture and information. Refer back to the picture again and find the coordinates of the image using matrices. 1. (x + 1, y – 4) 1. (x – 6, y + 7) 2. (x + 5, y – 2) 2. (x – 1, y – 3) Do It Yourself! <ANSWER KEY> Describe each translation with coordinate notation. 1. 8 units to the right and 3 units down (x + 8, y – 3) 2. 5 units to the left and 4 units up (x – 5, y + 4) Find the coordinates of the image with the given picture and information. 1. (x + 1, y – 4) P: (0, -3) Q: (3, 0) R: (7, -1) S: (3, -5) 2. (x + 5, y – 2) P: (4, -1) Q: (7, 2) R: (11, 1) S: (7, -3) Refer back to the picture again and find the coordinates of the image using matrices. 1. (x – 6, y + 7) P: (-7, 8) Q: (-4, 11) R: (0, 10) S: (-4, 6) 2. (x – 1, y – 3) P: (-2, -2) Q: (1, 1) R: (5, 0) S: (1, -4) Translate the Ball At carnivals, there are lots of fun games that require throwing or translating a ball. Describe the translation [in component form AND coordinate notation] we can use to get the ball to hit the target. (2, 4) (-5, 2) Translate the Ball <ANSWER KEY> At carnivals, there are lots of fun games that require throwing or translating a ball. Describe the translation [in component form AND coordinate notation] we can use to get the ball to hit the target. (2, 4) (-5, 2) Component form: <7, 2> Coordinate notation: (x + 7, y + 2) Word Search Complete the word search. Componentform Coordinatenotation Image Initialpoint Isometry Preimage Terminalpoint Transformation Translation Vector Word Search <ANSWER KEY> Real Life Application D (6,8) . Emily is at a carnival. She is getting on an hot air balloon. The trip is supposed to go from Point A to Point B, taking a stop, then stop at point D. However, because of the wind, it goes off course to point C. . . B (5,3) A (0,0) . C (7, 5) Real Life Application Cont. 1. Write the component form of the vectors shown in the image. D (6,8) . 2. Write the component form of the vector that describes the path the balloon can take to arrive at its terminal point. 3. Write the component form of the vector that describes the path the hot air balloon could take to go straight from point A to point D. . . B (5,3) A (0,0) . C (7, 5) Real Life Application Cont. <ANSWER KEY> 1. Write the component form of the vectors shown in the image. <5, 3> ; <2, 2> D (6,8) . 2. Write the component form of the vector that describes the path the balloon can take to arrive at its terminal point. <-1, 3> . 3. Write the component form of the vector that describes the path the hot air balloon could take to go straight from point A to point D. <6, 8> . B (5,3) A (0,0) . C (7, 5) TESSELLATIO NS By Kaitlyn Choi What is a tessellation? • A tessellation is a transformation that forms a pattern with a repeated shape with no gaps or overlaps. KEY vocabulary words • Frieze pattern or border pattern – A pattern that extends to the left and right in such a way that the pattern can be mapped onto itself by a horizontal translation. Some frieze patterns can be mapped onto themselves by other transformations as well. They are classified with different letters. - Translation: T - 180 degrees rotation: R - Reflection in a horizontal line: H - Reflection in a vertical line: V - Horizontal Glide Reflection: G Key vocabulary words cont. • Glide reflection – a transformation in which every point is mapped onto a point by the following steps: 1. A translation maps A onto A’. 2. A reflection in a line parallel to the direction of the translation maps A’ onto A”. Here’s a little somethi ng to help you out: Example Classify the frieze pattern below. So in order to see what this pattern can be classified as, Evaluate each transformation. Was it translated? YES. Was it rotated 180 degrees? YES. Was it reflected horizontally? YES. Was it reflected vertically? YES. Was there a horizontal glide reflection? YES. So this is a TRHVG. How do you know if A SHAPE tessellates or not? Use the equation: (n-2)*180 / 360 If it comes out to be a whole number, the shape will tessellate. For example: Let’s try a quadrilateral, which has four sides. (4-2)*180 / 360 = 360 / 360 1 It can be tessellated. Now, let’s try a pentagon, which has five sides. (5-2)*180 / 360 = 540 / 360 1.5 It cannot be tessellated. Do It Yourself! Classify the following frieze patterns. 1. 3. 2. 4. Do It Yourself! <ANSWER KEY> Classify the following frieze patterns. 1. 3. Translation, vertical line reflection 2. Translation, 180 degrees rotation 4. Translation, rotation, vertical line reflection, horizontal glide reflection Translation, 180 degrees rotation, horizontal line reflection, vertical line reflection, horizontal glide reflection Unscramble the words! _______________ _______________ _______________ _______________ _______________ _______________ _______________ _______________ _______________ Unscramble the words! <ANSWERPATTERN KEY> TESSELLATION TRANSFORMATION VERTICAL FRIEZE HORIZONTAL CLASSIFICATION BORDER GLIDEREFLECTION Real Life Application Have you ever seen eye illusions? In carnivals, there are fun houses that purposely confuse your senses—they are really exciting! This tessellated optical illusion has a trick. Stare at the picture and you will see gray dots appear. This eye illusion is formed by black tessellating squares on a white background which make seeing the gray dots possible. Dilations In a carnival By: Kevin Walker Dilation defined • Dilation- A transformation in which a polygon is enlarged or reduced by a given factor around a given center point. Key Vocab • Scale factor- the factor in which the measurements of a shape are dilated by • Reduction- If the Scale Factor is less than 1 but greater than 0 • Enlargement- If the Scale Factor is greater than 1 • Preimage- shape being dilated • Image- Shape that is a result of a dilation that is similar to the preimage. • Center of dilation- A fixed point on a coordinate plane in which all points are dilated around Need to know • All circles dilate because all circles are similar. ( Look below for example balloons) • If the scale factor is 1 then the Image and the preimage are identical The Equation • Dk(x,y)=(kx,ky) • This means that in order to find the new, dilated coordinates, you simply multiply the original coordinates by the scale factor. Matrices • In a matrix you organize the points of a vertices in rows and columns, rows going from left to right and columns going up and down. • In a 2 dimensional shape the top row is the x coordinates and the y is the bottom. • The rows represent different points, so in any given quadrilateral there will be 2 rows and 4 columns. Look below for an example Scalar Multiplication • Now lets try dilating the vertices using a scale factor, we will work with the coordinates used on the previous slide. • With a scale factor of two, you simply multiply all the coordinates in the matrix to find your dilated shape. • This is called scalar multiplication How to dilate • There are 2 ways to dilate, either you can measure the distance from the center of dilation to the vertices and then multiply the distance using the scale factor to find the new point; or if you are dilating around the origin you can simply multiply the vertices using the scale factor like shown on the previous slide. Now you try! • Jumbo the Elephant is, well… jumbo! And his little brother, Tiny the Elephant is, well… tiny! Find out just how much tinier he is! Try Again! • Cousin Ed is even bigger than Jumbo! Find out the height and width of Ed using the scale factor Explanation • Tiny is exactly 1 half the size of jumbo because if you take jumbos coordinates and multiply them by ½ you get the coordinates of tiny since they were dilated around the origin • Ed is 24 feet tall and 16 feet wide, using the scale factor of 4 you multiply Jumbo’s characteristics to find Ed’s Real life applications • In the real world we see dilations everywhere. We can find dilations in a carnival through different drink sizes; small, medium and large. They are all similar to one another; however, they are different sizes, the size differences between them could be described using scale factors. THE END! Thanks for coming to the Transformation Carnival! Bibliography-translations • Key word definitions and several practice problems: McDougal Littell Inc. Geometry Textbook Images: • http://studentweb.wilkes.edu/jennifer.werner/FinalProjectWerner/translation%20triangle.jpg • http://hotmath.com/hotmath_help/topics/transformations/translation.gif • http://etc.usf.edu/clipart/49300/49310/49310_graph_blank_lg.gif • http://cache4.assetcache.net/xc/450468697.jpg?v=1&c=IWSAsset&k=2&d=B53F616F4B95E553AA045CD58424C7D1 0B075991A00C9A70E7F79BAA020FB585 • http://img3.wikia.nocookie.net/__cb20101014143929/clubpenguin/images/thumb/5/56/Circus_ Ball.PNG/120px-Circus_Ball.PNG • http://www.clipartbest.com/cliparts/yTk/egr/yTkegrREc.png Bibliography-reflections • http://misstaraleexo.deviantart.com/art/Cotton-Candy-174819102 • http://www.clipartguide.com/_pages/1386-0901-2323-2707.html • http://funny-pictures.picphotos.net/clown-clip-art-vector-onlineroyalty-free-public-domain/doblelol.com*thumbs*baby-girl-crawlingclip-art-vector-online-royalty-free-funny_4708006791349289.jpg/ • https://www.youtube.com/watch?v=w492-EVCHQo • http://www.freepik.com/free-vector/wishing-well-clipart_384788.htm • http://www.dreamstime.com/stock-images-grey-donkeyimage1681104 Bibliography-rotations • http://www.binghamton.edu/campus-activities/docs/ferriswheelcarousel.jpg • Key concepts and some problems: McDougal Littell Inc. Geometry Textbook • Pictures: http://www.illustrationsof.com/royalty-free-ferris-wheel-clipart-illustration1075355.jpg, http://www.pamsclipart.com/clipart_images/circus_clown_face_0515-1004-17040147_SMU.jpg, http://www.theory.cs.uvic.ca/~cos/venn/gifs/VictoriaColor.gif, http://dev.physicslab.org/img/062e65af-eca6-4ab5-be56-5e9c2d963a3c.gif, http://www.doolins.com/images/products/detail/yellowclownrollticket.jpg, http://stitchontime.com/osc/images/carnival2.gif • http://www.emathematics.net/transformations.php?def=rot ational Bibliography-tessellations Key word definitions and several practice problems: McDougal Littell Inc. Geometry Textbook Images • http://www.clker.com/cliparts/L/s/x/o/0/k/carnival-a2jaycees3-hi.png • http://gwydir.demon.co.uk/jo/tess/optical1.gif Bibliography-dilations • Cover- http://www.google.com/imgres?imgurl=http%3A%2F%2Fthumbs.dreamstime.com%2Fz%2Fcircus-tent17174190.jpg&imgrefurl=http%3A%2F%2Fwww.dreamstime.com%2Fstock-photo-circus-tentimage17174190&h=800&w=800&tbnid=ci6saSKJbBuR1M%3A&zoom=1&q=carnival%20tent&docid=KvpffsR9ObWJTM&ei=tQVpU9bWLYSVy AS-64DoAw&tbm=isch&ved=0CFsQMygHMAc&iact=rc&uact=3&dur=679&page=1&start=0&ndsp=12 • Dilation defined- http://www.mathopenref.com/dilate.html • Dilation definedhttp://www.google.com/imgres?imgurl=http%3A%2F%2Fblog.volunteerspot.com%2F.a%2F6a010535724f84970b010536f8e8fb970bpi&imgrefurl=http%3A%2F%2Fblog.volunteerspot.com%2Fvolunteer_guru%2F2009%2F01%2Fplan-a-school-carnival-to-raise-money-andbuild-community-spirit-its-easy.html&h=424&w=283&tbnid=m4cWV5pDTe0aM%3A&zoom=1&docid=X_MOssCgNe6lDM&ei=lyloU8X7KaHayAGrmIC4DA&tbm=isch&ved=0CKIBEDMoOTA5&iact=rc&ua ct=3&dur=201&page=4&start=40&ndsp=21 • Now you tryhttp://www.google.com/imgres?imgurl=http%3A%2F%2Fwww.clipartbest.com%2Fcliparts%2FRcG%2F6k4%2FRcG6k4ecL.png&imgrefurl=ht tp%3A%2F%2Fwww.clipartbest.com%2Fcartoon-elephantclipart&h=726&w=800&tbnid=w3b9zd0mF4f1GM%3A&zoom=1&docid=GNDZCuzFBYMpnM&ei=IjdoU46fB8K6yAHhjIHQDg&tbm=isch&ved =0CG0QMygGMAY&iact=rc&uact=3&dur=180&page=1&start=0&ndsp=11 • Real life applicationshttp://www.google.com/imgres?imgurl=http%3A%2F%2Fwww.i2clipart.com%2Fcliparts%2Fa%2F2%2Ff%2F8%2Fclipart-fast-food-drinkssoda-fountain-256x256-a2f8.png&imgrefurl=http%3A%2F%2Fwww.i2clipart.com%2Fclipart-fast-food-drinks-soda-fountaina2f8&h=550&w=256&tbnid=FBRGJxdXsKT0wM%3A&zoom=1&q=clip%20art%20soda&docid=jq5ImZIeEnRHXM&ei=ewtpU9aUB4W0yASnio LADQ&tbm=isch&ved=0CFMQMygBMAE&iact=rc&uact=3&dur=445&page=1&start=0&ndsp=24