Translations

Transformations in Mario!By: Yuval Timen, Kevin Wilson, Michaela Byrne, and Simon UnTable of Contents:

Translations Kevin Wilson

Dilations Yuval Timen

Rotations Simon Un

Reflections Michaela Byrne

Tessellations Michaela Byrne, Kevin Wilson

TranslationsTranslationsTranslations 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.

VocabularyPre-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.

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.

Theorem 7.5

If you reflect the object over line k and then reflect it again over line m it is a translation.kmReflectionMatricesMatrix 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.

Original points. X coordinates are on top, y on bottom.Translation Matrix. You add these numbers to the corresponding numbers of the original points to get the new points below.

ABCDABCDIf 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 . If you have a square for example, and you write , you would move every point 6 points right and 4 points up. If you do another translation, for example , you can add the vectors together to get a new one like .

Up and overComponent Form + VectorsComponent Form + Vectors

To each free vector (or translation), there corresponds a position vectorwhich is the image of theoriginunder 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.Coordinate NotationCoordinate notation is just another way of writing translations.Coordinate notation is written like this, (x+a, y+b) or (x-a, y-b)

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.

(x-3, y-2)Mario ApplicationThere are actually a lot of translation in Mario games such asMushrooms translating on the ground.

Moving blocks.

Boo enemies following you.

In Mario 3s 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.

FireballsMario 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.

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Translation ActivityA 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 form for the translation that occurred.

Translation Activity AnswersTranslation Matrix =

Matrix =

Coordinate Notation= (x+2,y+1)

Component Form =

DilationsDilationsA 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:

VocabularyScale 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.Vocabulary ContImage- 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.

Identifying DilationsThis dilation is an enlargement, because X (the pre-image) is the smaller triangle, and X ( the final image) is the larger triangle:

Identifying DilationsThis dilation is a reduction, because the final image is smaller than the pre-image.

Scale FactorThe 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 Scalar MultiplicationYou 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){}ApplicationIn 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.

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Mega MushrroomActivityTry 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-Leys shadow? Draw and solve the dilation on a piece of paper.Application AnswerThe shadow is 151.667 inches tall, and 28 inches wide.

Rotations

VocabularyRotation: Transformation in which a figure is turned about a fixed pointCenter of Rotation: The fixed point of a rotationAngle of Rotation: The angle formed when rays are drawn from the center of rotation to a point and its imageRotational symmetry: when an image can be mapped onto itself by a clockwise rotation of 180o or less

TheoremsTheorem 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.Rotation Using a Protractor

700 RotationMake 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 rotatedUse the distance to make a new point along the angleRepeat the steps for each point until the shape is remade

Coordinate Plane Rotation

Original (x, y)R900 (x, y) = (-y, x)counterclockwiseR1800 rotation (x, y) = (-x, -y)counterclockwiseR2700 = R-900 (x, y) = (y, -x)R2700 = counterclockwiseR-900 = clockwise

Preimage Reflection Over Two Lines (Theorem 7.3)

Acute:700 Angle(the angle of rotation is 2x0, where x0 is the measure of the acute angle formed by k and m)Angle of Rotation: 1400Rotational Symmetry

1800 rotationFigureAngle of RotationALL REGULAR POLYGONS360/number of sidesTriangle1200Square900Rotational symmetry: when an image can be mapped onto itself by a clockwise rotation of 180o or less

Mario Applications

Rotation is used in Mario several times:Fire bars (previous slide)Marios jump when he has a star powerupThe 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.

Activityhttp://www.mangahigh.com/en_us/maths_games/shape/transforming_shapes/rotationReflectionsImportant VocabularyA Reflection is a transformation which uses a line that acts as a mirror, with an image reflected across the lineThe line of reflection is a line that acts like a mirror in a reflectionThe line of symmetry is a line that acts as a mirror within a figureCoordinatesReflection 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)

Reflections

A reflection can happen over the y-axis,thex-axis,or even a line***Go to presentation mode to see the triangles move 38

Mario SymmetryThis mushroom has 1 line of symmetry

Mario has 1 line of symmetry

Mario Line of reflection

When Mario reaches the flagpole he reflects to the other side.***Go to Presentation mode to see Mario reflect over the flagplole40Reflections gameClick here to play a reflection game!

TessellationsA 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 hexagons by themselves.

You can tessellate with pentagons by themselves but the pentagons will not be regular.Types of TessellationsTranslation 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

Tessellations in Mario!In Mario, there are TWO main translation tessellations.

If you look closely you can see that the floor tessellates and so do the stairs leading to the flagpole.

WHOA! Even more!Tessellation Activityhttp://www.shodor.org/interactivate/activities/Tessellate/Bibliographyhttp://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/

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