Transformations in Mario!

1. Transformations in Mario! Table of Contents: Translations –Kevin Wilson Dilations –Yuval Timen Rotations –Simon Un Reflections –Michaela Byrne Tessellations– Michaela Byrne, Kevin Wilson By: Yuval Timen, Kevin Wilson, Michaela Byrne, and Simon Un

2. Translations

3. Translations Translations are a transformation that involves “sliding” a point or set of points (object/shape) Translations preserve length and size. The object is NOT flipped or rotated. The shape looks exactly the same except it is in a different place.

4. Vocabulary • Pre-image: The beginning point or set of points under a transformation, in this case a translation • Image: The resulting point or set of points under a transformation. • Isometry: An isometry is a transformation of the plane that preserves length.  • Matrix: A rectangular grid of numbers. • Vector: A quantity that has both direction and magnitude.

5. Vocabulary (cont.) Invariant: A figure or property that remains unchanged under a transformation of the plane is referred to as invariant.  No variations have occurred. Translation Theorem: A translation is an isometry. Theorem 7.5:If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is a translation.

6. Theorem 7.5 k m Reflection If you reflect the object over line k and then reflect it again over line m it is a translation.

7. Matrices Original points. X coordinates are on top, y on bottom. • Matrix addition and subtraction can be represented geometrically as a translation of a shape on the plane. • If you wanted to move the square below 2 units right and 1 unit up, you would apply the translation matrix which you would add to the original points to get the new points. B C D A Translation Matrix. You add these numbers to the corresponding numbers of the original points to get the new points below. D’ A’ B’ C’

8. Component Form + Vectors • If you have a square for example, and you write <6,4>, you would move every point 6 points right and 4 points up. • If you do another translation, for example <3,5>, you can add the vectors together to get a new one like <9,9>. • If you have a picture like the one below, you can write the translation (where the picture moves to) in component form. • You do this by seeing how far over and up the new point is. • Component form is written like this <x,y>. Up and over

9. Component Form + Vectors • To each free vector (or translation), there corresponds a position vector which is the image of the origin under that translation.  • Unlike a free vector, a position vector is "tied" or "fixed" to the origin.  A position vector describes the spatial position of a point relative to the origin. Any two vectors of the same length and parallel to each other are considered identical.  They need not have the same initial and terminal points.  This is called a free vector which is used here.

10. Coordinate Notation If you have originally have points (3,4) and (5,8) and you want to translate the points 3 points left and 2 points down coordinate notation would be written like this. • Coordinate notation is just another way of writing translations. • Coordinate notation is written like this, (x+a, y+b) or (x-a, y-b) (x-3, y-2)

11. Mario Application • There are actually a lot of translation in Mario games such as… • Mushrooms translating on the ground. • Moving blocks. • Boo enemies following you. • In Mario 3’s over world, Mario or Luigi translate to a new stage when you select it. • Mario or friends going down the pipes . • Coins coming out of blocks. • Fireballs

12. Mario Application (cont.) • The translation in Mario we are focusing on is the Bullet Bill enemy. • The Bullet Bill translates left or right across the screen, not up or down, without changing shape, size, and without flipping, or rotating. x’ x

13. Translation Activity • A flying question block is at points A(3,6) B(7,6) C(3,2) and D (7,2) • It flies to points A’(5,7) B’(9,7) C’(9,3) and D’ (5,3) • Write the translation matrix , the matrix form, coordinate notation, and component formfor the translation that occurred.

14. Translation Activity Answers • TranslationMatrix = • Matrix = • Coordinate Notation= (x+2,y+1) • Component Form = <2,1>

15. Dilations

16. Dilations • A dilation is when an original object is transformed in a way that allows it to get bigger or smaller proportionally. • This is the only translation that does not preserve the length, therefore it is not isometric. • Dilations use scale factor and scalar multiplication to change the pre-image into the final image. For example:

17. Vocabulary • Scale Factor- The number that shows the relation between the pre-image and the image. Is commonly written as a fraction –if it is a reduction, then the fraction is less than 1 (I.e. 1/3) and if the dilation is an enlargement, then it is written as a fraction greater than 1 (I.e. 3/1) • Pre-image- The original shape, before the dilation takes place.

18. Vocabulary Con’t • Image- The final image after the dilation has been applied. This can be an enlargement or a reduction of the pre-image. • Center- The originating point (usually (0,0) on a graph) that the dilation is based around. • Scalar Multiplication-The process of applying the scale factor to the vertices of the pre-image, giving you the coordinated of the final image.

19. Identifying Dilations This dilation is an enlargement, because X (the pre-image) is the smaller triangle, and X’ ( the final image) is the larger triangle:

20. Identifying Dilations This dilation is a reduction, because the final image is smaller than the pre-image.

21. Scale Factor • The scale factor tells you how much you must dilate the pre-image by. It gives you the factor for Scalar Multiplication. • The scale factor is written as a fraction; if the denominator is bigger, it is a reduction; if the numerator is bigger, it is an enlargement. • The scale factor can be found by the ratio of ANY linear measurement from the preimage to the image. The scale factor is written in one of two ways (y being the larger number; x being the smaller number): y/x = this is an enlargement, because the bigger number is on top x/y = this is a reduction, because the smaller number is on top

22. Scalar Multiplication You are given the pre-image on a graph. To find the final image, you take the coordinates of the vertices of the pre-image. Then, you multiply all of them by the scale factor, thus giving you the coordinates of the vertices of the final image: { } Scale Factor: 2/3 Dilation: Reduction A=( 6,0) x2/3 A’ =(4,0) B=(5,3) x2/3 B’ =(3.33, 2) C=(4,4) x2/3 C’ =(2.66, 2.66)

23. Application Mega Mushrroom In Mario, the main dilation is when he gets a . He gets bigger proportionally, and after the effects wear off, he returns to his normal size. M’ M

24. Activity Try your own: Dai-Ley Ting is standing alone on the playground, all by himself. The time is 6:30 PM, and his shadow is stretched across the blacktop. The scale factor from Dai-Ley to the shadow is 7/3. If Dai-Ley is 5’ 5” tall, and 1’ wide, how many inches tall and wide is Dai-Ley’s shadow? Draw and solve the dilation on a piece of paper.

25. Application Answer The shadow is 151.667 inches tall, and 28 inches wide.

26. Rotations

27. Vocabulary • Rotation: Transformation in which a figure is turned about a fixed point • Center of Rotation: The fixed point of a rotation • Angle of Rotation: The angle formed when rays are drawn from the center of rotation to a point and its image • Rotational symmetry: when an image can be mapped onto itself by a clockwise rotation of 180o or less

28. Theorems • Theorem 7.2: Rotation is an isometry (the figure does not change in size) • Theorem 7.3: If two 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 2x0, where x0 is the measure of the acute angle formed by k and m.

29. Rotation Using a Protractor • Make an angle (700) of the point you are rotating (A) to the point of rotation (the origin) • Measure the distance from the origin to the point that is being rotated • Use the distance to make a new point along the angle • Repeat the steps for each point until the shape is remade 700 Rotation

30. Coordinate Plane Rotation Original (x, y) R900 (x, y) = (-y, x) counterclockwise R2700= R-900 (x, y) = (y, -x) R2700 = counterclockwise R-900 = clockwise R1800 rotation (x, y) = (-x, -y) counterclockwise

31. Preimage Reflection Over Two Lines (Theorem 7.3) Acute: 700 Angle Angle of Rotation: 1400 (the angle of rotation is 2x0, where x0 is the measure of the acute angle formed by k and m)

32. Rotational Symmetry 1800 rotation Rotational symmetry: when an image can be mapped onto itself by a clockwise rotation of 180o or less

33. Mario Applications • Rotation is used in Mario several times: • Fire bars (previous slide) • Mario’s jump when he has a star powerup • The background rotates when Mario hits the “P” button. • The bar in Super Mario Bros. Wii allows the player to rotate the bar according to the rotation of the bar.

34. Activity http://www.mangahigh.com/en_us/maths_games/shape/transforming_shapes/rotation

35. Reflections

36. Important Vocabulary • A Reflection is a transformation which uses a line that acts as a mirror, with an image reflected across the line • The line of reflection is a line that acts like a mirror in a reflection • The line of symmetry is a line that acts as a mirror within a figure

37. Coordinates • Reflection on x-axis: (x,y) = (x,-y) Reflection on y-axis: (x,y) = (-x,y) Reflection on line y=x: (x,y) = (y,x) Reflection on line y=-x: (x,y) = (-y,-x)

38. Reflections A reflection can happen over the y-axis, the x-axis, or even a line

39. Mario Symmetry This mushroom has 1 line of symmetry Mario has 1 line of symmetry

40. Mario Line of reflection When Mario reaches the flagpole he reflects to the other side.

41. Reflections game Click here to play a reflection game!

42. Tessellations • A Tessellation is a pattern made from shapes that fit together without any gaps and do not over lap. • You can only tessellate with triangles, squares, and hexagonsby themselves. You can tessellate with pentagons by themselves but the pentagons will not be regular.

43. Types of Tessellations • Translation Tessellation- a tessellation where the shape repeats itself by moving or sliding • Rotation Tessellation- a tessellation where the shape repeats by rotating or turning • Reflection Tessellation- a tessellation where the shape repeats by reflecting or flipping Definitions are from https://sites.google.com/site/tessellationunit/tessellations/kinds-of-tessellations

44. Tessellations in Mario! WHOA! Even more! • In Mario, there are TWOmain translation tessellations. • If you look closely you can see that the floor tessellates and so do the stairs leading to the flagpole.

45. Tessellation Activity • http://www.shodor.org/interactivate/activities/Tessellate/

46. Bibliography • http://www.videogameobsession.com/videogame/ani-question-200-vgo.gif • http://www.themoderndaypirates.com/pirates/wp-content/uploads/20 • 10/10/mushroom.jpg • http://www.creativeuncut.com/gallery-04/art/nsmb-fire-flower.jpg • mario.wikia.com • en.wikipedia.org • http://www.mangahigh.com/en_us/games/transtar • www.regentsprep.org • www.geom.uiuc.edu • www.mathisfun.com • https://sites.google.com/site/tessellationunit/tessellations/kinds-of-tessellations • http://www.shodor.org/interactivate/activities/Tessellate/