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Transformations in Baseball Nick Miller Jeen Kim Mary Ham Amar Thakkar Inning 6.

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Presentation on theme: "Transformations in Baseball Nick Miller Jeen Kim Mary Ham Amar Thakkar Inning 6."— Presentation transcript:

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2 Transformations in Baseball Nick Miller Jeen Kim Mary Ham Amar Thakkar Inning 6

3 Main Menu Dilations by Nick Miller Rotations by Jeen Kim Reflections by Mary Ham Translations by Amar Thakkar Tessellation by Group

4 Dilations By Nicholas Miller Table of Contents Main Menu

5 Table of Contents Dilation Scale Factor What is the Scale Factor? Matrices New Image and Preimage Theorem Problems Dilations for Baseball Main Menu

6 Dilation Dilation is a similarity transformation in which a figure is enlarged or reduced using a scale factor ≠ 0, without altering the center Table of Contents Main Menu

7 Scale Factor The amount by which an object enlarges or reduces is known as the scale factor If the scale factor is a number of a fraction larger than 1, the new figure is an enlargement of the pre image If the scale factor is a number or a fraction less than one, than the new figure is a reduction of the pre image Main Menu Table of Contents

8 What is the Scale Factor? To find the scale factor, put the B’ coordinates over the pre image B coordinates: -2/-1 for x 6/3 for y They should both come out to the same thing In this case, the scale factor is 2 If you are given the scale factor in a problem, and you need to find out the coordinates either the pre image or the new figure, either multiply or divide the coordinates of the figure you already have, and you get the new coordinates Table of Contents Main Menu

9 Matrices What you see in the top left corner of the picture is called a matrices, which gives people a better organization of finding the new coordinates In the photo, 2 is the scale factor, and you have the coordinates of the preimage Table of ContentsMain Menu

10 New Image and Preimage In a dilation, the new image created from the preimage is similar to the preimage. The figures would be congruent only if the scale factor is 1 Table of Contents Main Menu

11 Theorem If B is not the center point O, then the image point P’ lies on line CB The scale factor k is a positive number such that k= OB’/OB and K doesn’t = 1 If B is the center point O, then B=B Table of ContentsMain Menu

12 What are the new coordinates if the scale factor is 3? A--- (1,1) 1 x 3 = 3 B--- (2,3) 2 x 3 = 6 3 x 3 = 9 C--- (4,1) 4 x 3 = 12 1 x 3 = 3 A= (3,3) B= (6,9) C= (12,3) Table of ContentsMain Menu

13 What are the coordinates of the baseball field if the scale factor is 1/3 with the center point (-2,0) A= (-5,1) B= (-2,4) C= (-2,3) D= (1,1) Table of ContentsMain Menu

14 Dilations for Baseball In baseball, the field is a diamond. An example using the baseball diamond is if a little league field wanted to be renovated, and be made bigger, the dilation would be an enlargement from the little league size field to the middle school size field. Table of Contents Main Menu

15 Rotations By Jeen Kim Table of Contents Main Menu

16 Rotation Table of Contents Rotation Vocabulary/Key Concepts Things to Know Examples Real Life Applications Activity Table of ContentsMain Menu

17 Rotation Rotation: o A transformation where a figure is turned around a center of rotation. o A rotation is an isometry (a transformation where the figure stays congruent) Table of Contents Main Menu

18 Vocabulary/Key Concepts Center of Rotation: o A fixed point anywhere that the figure rotates about Angle of Rotation: o The angle created by rays drawn from the center of rotation to a point and its image. Theorem: o Line K and line M intersect at point P. Then a reflection in line K and the line M is a rotation about point P. o The angle of rotation is double the angle formed by K and M. Rotational Symmetry: o A figure in the plane has rotational symmetry if the figure can be mapped onto itself by clockwise rotation of 180 degrees or less. Table of Contents Main Menu

19 Things to Know Angle of Rotation: – R90° (x,y) = (-y, x) – R180° (x,y) = (-x,-y) – R270° (x,y) = (y,-x) – R-90° (x,y) = (y,-x) Table of Contents Main Menu

20 Examples Rotational Symmetry Before: After: 180° Table of Contents Main Menu

21 Real Life Applications Rotations can be found in Baseball! From the pitching mound to the bases, the base runner is the object that rotates and the pitcher is the center of rotation. The batter ran from the home plate to second base, which is a rotation of 180°. Table of ContentsMain Menu

22 Rotation Activity Rotate the shape by 60° CLOCKWISE. TIP: Use a protractor and a ruler. Table of Contents Main Menu

23 Rotation Activity Table of ContentsMain Menu

24 Find the angle of rotation Draw the shape from the origin if the angle of rotation is 120° Table of Contents Main Menu

25 Find the angle of rotation Draw the shape from the origin if the angle of rotation is 120° Table of ContentsMain Menu

26 Reflections By Mary Ham Table of Contents Main Menu

27 Table of Contents Key Words What is a Reflection? Reflection How to Reflect Pre-images Determine Lines of Symmetry How to Find Line of Symmetry Line of Symmetry in Baseball Reflect Your Own Baseball! Main Menu

28 Key Words Line of reflection- A line that acts like a mirror in a reflection. Line of reflection- A line that acts like a mirror in a reflection. Line of symmetry- An imaginary line that you could fold the image and both halves match exactly. Line of symmetry- An imaginary line that you could fold the image and both halves match exactly. Line of Reflection Rectangle has 2 lines of Symmetry Table of ContentsMain Menu

29 What is reflection?? Reflection is a transformation which uses a line that acts like a mirror, with an image reflected in the line. Reflection is an isometry. – Isometry is a transformation that preserves lengths. Table of Contents Main Menu

30 Reflection If a figure is reflected over the y-axis, then the y value stays the same but x value becomes opposite and vice versa. If a figure is reflected over the y-axis, then the y value stays the same but x value becomes opposite and vice versa. If the line of reflection is y- axis, then the y value stay the same and x value become opposite. Table of Contents Main Menu

31 How to reflect pre-images Use the following equation: Use the following equation: R x-axis (x,y)=(x,-y) R x-axis (x,y)=(x,-y) R y-axis (x,y)=(-x,y) R y-axis (x,y)=(-x,y) R y=x (x,y)= (y,x) R y=x (x,y)= (y,x) R y=-x (x,y)= (-y,-x) R y=-x (x,y)= (-y,-x)Or 1. Draw the perpendicular line of the line of reflection from a point. Main MenuTable of Contents

32 How to reflect pre-images (Cont) 2. Draw the point in the different side of line of reflection; the distance from the image and line of reflection has to be the same distance from the pre-image from the line reflection. Table of ContentsMain Menu

33 Determine Lines of Symmetry There’s not really a way to find lines of symmetry by equation. There’s not really a way to find lines of symmetry by equation. However, if a figure is a regular polygon, then the number of sides is equal to the number of lines of symmetry. However, if a figure is a regular polygon, then the number of sides is equal to the number of lines of symmetry. Square # of sides: 4 # of lines of symmetry:4 Table of ContentsMain Menu

34 How to find line of reflection 1. Find pairs of reflecting (corresponding) points. 1. Find pairs of reflecting (corresponding) points. 2. Find the midpoint of the pair of reflecting points. 2. Find the midpoint of the pair of reflecting points. 3. Connect the midpoints; the line has to be a straight line. 3. Connect the midpoints; the line has to be a straight line. Table of ContentsMain Menu

35 Line of Symmetry in Baseball A base ball and a baseball field have one line of symmetry Table of ContentsMain Menu

36 Reflect your own baseball Stuff you need: Graph paper or coordinate grid, pencil, eraser, and Computer if using GSP 1.Draw a pre-image of a baseball that you would like to reflect on the coordinate grid. 2.Draw the line of reflection where you want to reflect the baseball. Table of Contents Main Menu

37 Make your own reflection (Cont.) 3. Reflect the pre-image over the line of reflection, the shape has to be exact. 4. Then you got yourself a reflection Table of Contents Main Menu

38 Translations By Amar Thakkar Table of Contents Main Menu

39 Table of Contents Summary Vocabulary Example Solution Real World Examples GSP Activity Main Menu

40 Summary Translations are basically the sliding of a shape from one section to another section. On a grid you can find a translated figure if you have a coordinate notation or component form or you can find these by calculating the distance traveled x and y or y and x to find the coordinate notation or component form. Table of Contents Main Menu

41 Vocabulary A translation is a transformation that maps every two points P and A in the plane to points P’ and Q’, so that the following properties are true: A translation is a transformation that maps every two points P and A in the plane to points P’ and Q’, so that the following properties are true: PP’ = QQ’ and 2)’ ll ’, or and’ are collinear. PP’ = QQ’ and 2)’ ll ’, or and’ are collinear. Table of ContentsMain Menu

42 Vocabulary A vector is a quantity that has both direction and magnitude, or size. A vector is a quantity that has both direction and magnitude, or size. When a vector is drawn, the initial point, or starting point, of the vector is drawn point P and the terminal point, or ending point, of the vector is point Q. is read “vector PQ” When a vector is drawn, the initial point, or starting point, of the vector is drawn point P and the terminal point, or ending point, of the vector is point Q. is read “vector PQ” Main Menu Table of Contents

43 Vocabulary The component form of a vector combines the horizontal and vertical components Component Form T a,b (x,y)= (x+a) (y+b) Table of ContentsMain Menu

44 EXAMPLE Find the coordinate notation and component form of this translation. Table of Contents Main Menu

45 Solution Coordinate Notation Coordinate Notation (x-5, y+3) Component Form Component Form Table of Contents Main Menu

46 Real world Examples When a runner runs around the bases each time he or she reaches a new base, a translation of 60 units has happened. Table of Contents Main Menu

47 GSP Activity Click on Graph show grid Draw three points on the grid and connect them with line segments Select all three sides with the cursor and copy the triangle by pressing control c Go to another section of the grid and press control v to paste Find the coordinate notation and component form. Table of Contents Main Menu

48 Tessellation By Group Main Menu Table of Contents

49 Tessellation by Nick Miller Tessellation Vocabulary/Key Terms by Mary Ham Vocabulary/Key Terms Real Life Examples by Amar Thakkar Real Life Examples Activity by Jeen Kim Activity Main Menu

50 What is a Tessellation? A tessellation is a repeating pattern of figures that completely covers a plane without any gaps or overlaps To tessellate is to cover a plane surface by repeated use of a single shape, without any gaps or overlapping Table of ContentsMain Menu

51 Key Terms Tessellation Tessellation- A repeating pattern of figures that completely covers a plane without any gaps or overlaps. Edge Edge- Intersection between two bordering tiles. Vertex Vertex- Intersection of three or more bordering tiles. Main Menu Table of Contents

52 Key Terms (cont) Regular Tessellation Regular Tessellation- When a tessellation uses only one type of regular polygon to fill up a plane. Semi-regular Tessellation Semi-regular Tessellation- When a tessellation uses more than one type of regular polygon to fill up a plane. Regular tessellationSemi-regular tessellation Main MenuTable of Contents

53 Real-Life Example 4 random bases joined together form a tessellation. Table of ContentsMain Menu

54 Tessellation Activity Use 4 regular hexagons and 6 regular triangles to create a tessellation. Table of Contents Main Menu

55 Tessellation Activity Example Answer: Table of ContentsMain Menu

56 LAST INNING Design and Editing by: Jeen Kim Dilations by: Nick Miller Rotations by: Jeen Kim Reflections by: Mary Ham Translations by: Amar Thakkar Reflections by: the Group

57 Rotation Bibliography  eometry/GT4/ROTATEPIC3.gif eometry/GT4/ROTATEPIC3.gif  Clip Art on Microsoft PowerPoint  Blank Coordinate Plane from Mrs. Haemmerle  McDougal Littell Inc. Geometry Chapter 7 Resource Book

58 Reflection Bibliography Images  Baseball 1:http://www.sullivanil.us/SYB.htmlhttp://www.sullivanil.us/SYB.html  Coordinate: BRA/MultipleChoiceReview/Shapes.html BRA/MultipleChoiceReview/Shapes.html  Baseball 2:http://mypixelpress.com/photo- baseball-red-thread-63.htmlhttp://mypixelpress.com/photo- baseball-red-thread-63.html  Baseball field: eball/baseball-diamond-field.gif.php eball/baseball-diamond-field.gif.php

59 Tessellation Bibliography  Clip Art on Microsoft Powerpoint

60 GIFs/JPEGs  /baseball_player__runningA.gif /baseball_player__runningA.gif  content/uploads/2013/04/Rutledge.gif.opt_.gif content/uploads/2013/04/Rutledge.gif.opt_.gif  content/uploads/2013/02/Berry2B.gif.opt_.gif content/uploads/2013/02/Berry2B.gif.opt_.gif  SzlHhVj3i0o/UQ_E52pAMaI/AAAAAAAADZs/v- 3FVFHcNCY/s1600/image1.jpg SzlHhVj3i0o/UQ_E52pAMaI/AAAAAAAADZs/v- 3FVFHcNCY/s1600/image1.jpg


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