Transformations in Baseball

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

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

3. Dilations By Nicholas Miller Table of Contents Main Menu

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

5. 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

6. 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

7. 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

8. 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 Contents Main Menu

9. 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

10. 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 Contents Main Menu

11. What are the new coordinates if the scale factor is 3? A--- (1,1) 1 x 3 = 3 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 Contents Main Menu

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

13. 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

14. Rotations By Jeen Kim Table of Contents Main Menu

15. Rotation Table of Contents • Rotation • Vocabulary/Key Concepts • Things to Know • Examples • Real Life Applications • Activity Table of Contents Main Menu

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

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

18. 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

19. Examples • Rotational Symmetry Before: After: 180° Table of Contents Main Menu

20. 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 Contents Main Menu

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

22. Rotation Activity Table of Contents Main Menu

23. Find the angle of rotation Draw the shape from the origin if the angle of rotation is 120° Table of Contents Main 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. Reflections By Mary Ham Table of Contents Main Menu

26. 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

27. Key Words Line of 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. Rectangle has 2 lines of Symmetry Table of Contents Main Menu

28. 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. Main Menu Table of Contents

29. 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 the line of reflection is y- axis, then the y value stay the same and x value become opposite. Main Menu Table of Contents

30. How to reflect pre-images • Use the following equation: • Rx-axis (x,y)=(x,-y) • Ry-axis (x,y)=(-x,y) • Ry=x (x,y)= (y,x) • Ry=-x (x,y)= (-y,-x) Or 1. Draw the perpendicular line of the line of reflection from a point. Table of Contents Main Menu

31. 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 Contents Main Menu

32. Determine Lines of Symmetry • 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. # of sides: 4 # of lines of symmetry:4 Square Table of Contents Main Menu

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

34. Line of Symmetry in Baseball A base ball and a baseball field have one line of symmetry Table of Contents Main Menu

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

36. 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

37. Translations By Amar Thakkar Table of Contents Main Menu

38. Table of Contents • Summary • Vocabulary • Example • Solution • Real World Examples • GSP Activity Main Menu

39. 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

40. 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: PP’ = QQ’ and 2)’ ll ’, or and’ are collinear. Table of Contents Main Menu

41. Vocabulary • 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” Main Menu Table of Contents

42. Vocabulary • The component form of a vector combines the horizontal and vertical components Component Form Ta,b(x,y)= (x+a) (y+b) Table of Contents Main Menu

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

44. Solution • Coordinate Notation (x-5, y+3) • Component Form <-5, 3> Table of Contents Main Menu

45. 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

46. 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

47. Tessellation By Group Main Menu Table of Contents

48. Table of Contents • Tessellation by Nick Miller • Vocabulary/Key Terms by Mary Ham • Real Life Examples by Amar Thakkar • Activity by Jeen Kim Main Menu

49. 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 Contents Main Menu

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