The Amazing Guide to Transformations (Using video games)

1. The Amazing Guide to Transformations (Using video games) Click to Continue…

2. What is a Transformation? The definition of transformation is the replacement of the variables in an algebraic expression by their values in terms of another set of variables, or a mapping of one space onto another or onto itself.

3. Table of Contents • Translations- Liam Hanrahan- Donkey Kong • Tessellations- Gina Kim - Tetris • Rotations- Tyler Yuen- Pipe Mania • Reflections- Christian Skroce- Pac-man • Dilations- Erin Hughes- Super Mario

4. Translations

5. Translation Key Terms • Translation- Moving a shape, without rotating or flipping it. "Sliding". • Initial Point- The starting point of the vector • Terminal Point- The end point of a vector • Vector- a quantity that has both direction and magnitude • Component form- is made up by the horizontal and vertical components of a vector. • Image- The new figure after you apply the transformation • Pre-image- The original figure before the transformation • Coordinate notion- it tells you how many spaces on the x and y axis you should move each point. For example, (x,y) >>>>>>> (x+3,y-2)

6. Donkey Kong Mario is trying to escape death and is originally at (-4, -6), he is then translated (7, 2) to point (3, -4). Image Pre-Image

7. Translation • If the pre image of a point is at (3, 2) and you want it to translate (-5, 6) than what are the coordinates of the point after the translation? (-2, 8)

8. Translation • If the final image of a triangle is at (3, 1), (-4, 7) and, (-2, 3) and it was just translated (4, 2) then what were the coordinates before the translation? (-1, -1), (-8, 5) and, (-6, 1)

9. Translation • If the pre-image of a rectangle has coordinates of (1, 4), (7, -2), (7, 4)and (1, -2). After a translation, the coordinates are (4, 2), (10, -4), (10, 2) and, (4, -4). What was the rule off the translation that occurred? (3, -2)

10. Donkey Kong Vectors Since the pre-Image is moved 7 spots to the right and 2 spots up, you can make a vector that describes the objects motion. The vectors component form would then be (7, 2). The Initial point is (-4, -6) and the terminal point is (3, -4) Image Pre-Image

11. Donkey Kong Matrices You can also describe Mario's motion by using matrices in the matrices format, you put the original coordinate for the x in the 1st column then next in the 2nd column you put the x rule, then in the third column you put the final coordinate in the x. You do the same thing below it but for the y coordinates. , the matrices form is: -4, 7, -3 -6, 2, -4 Image Pre-Image

12. Matrices Practice • A triangle with coordinates of (3, 2), (4, 7) and, (9, 0) and experienced a translation of (5, 6). Use the matrices format to find the new images coordinates. (3, 4, 9) (5, 5, 5) (8, 9, 14) (2, 7, 0) (6, 6, 6) (8, 13, 6) Coordinates= (8, 8), (9, 13) and, (14, 6)

13. Tessellations Tetris the game itself is not a tessellation, however its pieces are tetrominoes which can tessellate

14. Look at how Tetris pieces tessellate! These Tetris pieces are also known as a quadromino or tetromino which is a polyomino made from four squares. As you can see tetrominoes can tessellate. A polyomino is a polygon made from squares of the same size, connected only along complete edges.

15. Types of Tessellations Types of Tessellations • Regular • Tessellation a pattern made by repeating a regular polygon made of two or more regular polygons where the pattern at each vertex are the same; there are a total of eight kinds • Semi-Regular Tessellation • Demi-Regular Tessellation made of regular polygons in which there are either two or three different polygon arrangements, or consists of curved shapes A tessellation has a repeating pattern of figures that completely covers a plane without any gaps or overlaps. Click on the examples

16. Would it make a regular tessellation or not? Because the figures in a tessellation do not overlap or leave gaps, the sum of the measures of the angles around any vertex must be 360°. Use this formula for the measure of an angle of a regular polygon. Substitute the number of sides of the regular polygon you are testing for n. 180 (n – 2) n a = If the product of the expression is a factor of 360, then the regular polygon will tessellate.

17. As you have tried using the formula for multiple regular polygons, the truth came out to be that only triangles, squares, and hexagons tessellate. See for yourself. The interior angle of a pentagon is 108 degrees. . . 108 + 108 + 108 = 324 degrees . . . Not a factor of 360 120 + 120 + 120 = 360 degrees That’s why hexagons work.

18. Create your own tessellation! On a large piece of paper trace your tessellation stencil over and over, lining up each new tracing to the one you just completed. You can work top to bottom or left to right. Fill the page. Or click on the link below if you want to do it fast online http://www.pbs.org/parents/education/math/games/first-second-grade/tessellation/

19. How to name a tessellation To name a tessellation, simply work your way around one vertex counting the number of sides of the polygons that form that vertex. The trick is to go around the vertex in order so that the smallest numbers possible appear first. That's why we wouldn't call our 3, 3, 3, 3, 6 tessellation a 3, 3, 6, 3, 3!

20. Symmetries found in tessellations Translation Reflection Rotation Glide Reflection Match them!

21. Real life applications Gina has recently bought a house and need to retile some rectangular and square floors. She visits the local Home Depot and discovers that there are many shapes and sizes of tiles. She is particularly attracted to the tiles labeled “tetrominoes”. What is the minimum number of each tetromino she needs to completely tile a rectangle? A square? She cannot cut the tetrominoes.

22. Rotations

23. Key Terms • Vocabulary- • Center of rotation: the fixed point of a rotation • Angle of rotation: rays drawn from the center of rotation to a point and its image form an angle • Rotational Symmetry: a figure that can be mapped onto itself by a clockwise rotation of 180 degrees or less • Isometry: a transformation that preserves length • Theorems: • Theorem 7.2: A rotation is an isometry • 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 degrees, where x degrees is the measure of the acute or right angle formed by k and m

24. Rotational Symmetry • What is rotational symmetry and how do I know if a figure has it? • A figure has rotational symmetry if it can be mapped onto itself by a clockwise rotation of 180 degrees or less • Does this shape have rotational symmetry? Does this shape have rotational symmetry? NO! YES!

25. How to Rotate a Figure on a Coordinate Plane • In order to rotate a figure on a coordinate plane you will need this table: • Equations • R90 (x,y) = (-y,x) • R180 (x,y) = (-x,-y) • R­270 (x,y) = (y,-x) • R-90 (x,y) = (y,-x) So lets say we want to rotate this rectangle 90° A B D C

26. How to Rotate a Figure on a Coordinate Plane (cont.) • In ordertorotate a figure 90°youhavetochangeeachpointaccordingtothetable. • So forpoint A, its original coordinates are (1,3) • In ordertorotateit 90° we need to change the original x coordinate to the original y and make it negative, this is your new x coordinate • For the new y coordinate all you have to do is take the original x coordinate • The new point for A is now (-3,1) • The new point for B is now (-3,5) • The new point for C is now (-1,5) • The new point for D is now (-1,1) • Equations • R90 (x,y) = (-y,x) • R180 (x,y) = (-x,-y) • R­270 (x,y) = (y,-x) • R-90 (x,y) = (y,-x)

27. How to Rotate a Figure on a Coordinate Plane (cont.) • Quick! This water pipe needs to be rotated 270° in order to beat the level! What are the new coordinates in order to win? HINT • Equations • R90 (x,y) = (-y,x) • R180 (x,y) = (-x,-y) • R­270 (x,y) = (y,-x) • R-90 (x,y) = (y,-x) A B D C

28. Drawing a Rotation Image Using a Protractor • To draw a rotation image with a protractor on a point, you need to start with a point on the shape, then draw a line to the point you are going to rotate it around. Then create another line according to the angle you want to rotate it. You have to do this for every point of the shape you are rotating. When you finish creating the new lines, connect the end points and you should have your new shape.

29. Drawing a Rotation Image Using a Protractor (cont.) • This picture has fallen out of its place, rotate it 90° from point P in order to place it upright! B C P D A

30. Finding the Angle of Rotation When a Pre-image is Reflected Over Two Lines • In order to do this you need to find the middle angle of the three images, then double it to find the angle of rotation. D’ A’ C’ A D B’ B D’’ C C’’ 123° B’’ A’’

31. Answer Key • New coordinates: A=(-4,4) B=(-4,6) C=(-7,-6) D=(-7,-4) • Angle of Rotation = 114°

32. Reflections

33. What is a Reflection? • A reflection is a transformation where a line on a grid acts like a mirror. An image is reflected over this line. Key Terms Line of Reflection- the line that acts like a mirror in a reflection Line of Symmetry- a line in a figure that allows the figure to be mapped onto itself by a reflection in the line Isometry- a type of transformation where the figure retains its original shape

34. How to Find Lines of Symmetry • Different shapes have different lines of symmetry. • You can determine the number of lines of symmetry in a regular polygon by the number of sides it has. -Squares -4 sides -4 lines of symmetry -Regular Hexagon -6 sides - 6 lines of symmetry -Isosceles Triangle - 1 line of symmetry

35. How to reflect of pre-image over a line • In this example, the blue image is reflected over the y- axis. When reflecting an image over a line, it is important to make sure that opposite sides and point are equidistant from the line of reflection.

36. How to Find the Equation for the line of Reflection • To find the equation for the line of reflection of two images, you must find two midpoints between points in the original image and the reflected image. So the equation for this line of reflection is y=3.

37. Finding Minimum Distance • In this problem we were trying to find point C so that AC+BC is a minimum distance. You must reflect point A over a line of reflection (Ex. y= 2, x- axis). Then you connect the reflected A to B. You then find the equation for this line and set it equal to what y is in the line of reflection (Ex. 2 if it’s y=2, 0 if it is the x-axis. Whatever x equals is the x coordinate for C, which is located on the line you reflected A over.

38. Lines of Symmetry Activity • How many line of symmetry do the pac-man map and pac-man have?

39. Answers • Lines of Symmetry Problem- Pac-man map- 1 line of symmetry Pac-man- 1 line of symmetry

40. Reflecting a Pre-image Over a Line • GSP Activity- Step-by-Step • Step 1- Click the graph tab and click show grid. • Step 2- Construct the ghost figure as scene above. • Step 3- Click the transform tab and select REFLECT. This will reflect the figure over the x-axis. Make sure the corresponding points in each figure are equidistant from each other.

41. Answer • Ghost GSP Activity-

42. Writing Equations for the Line of Reflection • Find the equation for the line of reflection for these two triangles.

43. Answer • Writing equation for the line of reflection- • Equation- y=1/2x+1/2

44. Finding Minimum Distance(Real World Application) • In an arcade, there are two popular pac-man games. One at point A(-1,4) and the other at point B(6,3). You want to put a change machine somewhere in the first row of games (this is also the x-axis). Find where point C or the coin machine needs to be so that AC + BC is a minimum distance. Hint: Reflect point A over the x-axis.

45. Answer • Point C- (3,0)

46. Dilations

47. Key Words and Vocab • Dilation: a transformation that produces an image that is the same shape as the original but a different size. • Scale Factor: the amount by which the image grows or shrinks • Reduction: a dilation where the image shrinks and the scale factor is less than one • Enlargement: a dilation where the image grows and the scale factor is greater than one

48. Center or dilation: the point which the polygon is dilated off of • = K = scale factor

49. What is a dilation? • A dilation is taking a figure and either making it smaller or bigger using the scale factor. • When an object is dilated the new image and the previous image are similar but not congruent.

50. Properties Preserved in Dilations • 1. angle measures remain the same • 2. Parallel lines remain parallel • 3. Collinear points stay on the same line • 4. Midpoints remain the same in each figure • 5. Orientation (letter order) remains the same.