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Humans & TransformationsEmily DiMaulo-MilkEmma HaleckyTroy KaranfilianAJ PerlowinMatt MonaghanTable of ContentsRotationTranslationReflectionTessellationsDilationsTessellationsMatt MonaghanRotationsA transformation in which a figure is turned about a fixed point

Center of rotationThe fixed point around which the preimage rotates. Can be inside, outside, or on the figure.

Angle of rotationRays drawn from the center of rotation to a point and its image form the angle at which the figure rotatedRotations are expressed in degrees, and the amount of degrees a figure rotates is theangle of rotation.

180Rotational SymmetryWhen a figure can be mapped onto itself by a clockwise rotation of 180 or less

180Rotational SymmetryAll regular figures have rotational symmetry, but a figure can still have rotational symmetry even if it is not regular, just look at lines of symmetry. To find the angle of rotation needed to achieve rotational symmetry Regular figures- Find the measure of one interior angleIrregular figures- Look at the lines of symmetry, and divide the numbers of sides a figure has by the number of lines of symmetry. This works as a general rule and should be used with general logic.Lets Try It!Does the shape have rotational symmetry?YesYesNoRotate a ShapeStep 1.Draw a line connecting the original point to the center of rotation

Rotate a ShapeStep 2.Measure how many degrees you want to rotate the figure with a protractor

Rotate a ShapeStep 3. Measure the distance between the original point and the center of rotation, then draw the new point the same distance from the center of rotation at the proper distance. Repeat with each point

Lets Try It!Using a coordinate plane, draw triangle ABC.A(1,9)B(5,8)C(8,8)Rotate the figure 50 clockwise.Regular Rotations on a Coordinate PlaneJust use these equations--CounterclockwiseClockwiseR90- (x,y) = (-y, x)R90- (x,y) = (y, -x)R180- (x,y) = (-x, -y)R180- (x,y) = (-x, -y)R270- (x,y) = (y, -x)R270- (x,y) = (-y, x)

Lets Try It!Without drawing the shape, rotate figure ABCDE 180 counterclockwise.A(1,0)B(3,8)C(4,9)D(-7,9)E(-8,0)AnswersA(-1,0)B(-3,-8)C(-4,-9)D(7,-9)E(8,0)Rotations In the Human BodyWhen you turn your head from left to right, thats approximately 120 of rotation

You can roll your eyes 360You can rotate your entire body by spinning in a circle

Lets play a game!Simon SpinsThis game is a lot like Simon SaysThe leader calls out a motion or body part, and you only do the motion or touch the body part if it is a rotation in your body or has the potential to rotate. If it is not, stay still! Any motion will make you get out.

Interesting FactA 2008 study found that Simon Says actually has a psychological benefit and helps young children learn to suppress impulsiveness.Simon SaysTranslationsTroy KaranfilianVocabularyTranslation- A transformation that maps every 2 points P and Q in the plane to points P and Q so that the following properties are true:1) PP = QQ As you can see in Figure 1, the distances between the original points and the new points are the same.

2) PP is parallel to QQ OR PP and QQ are collinearAs you can see in Figure 2, the old points and new points create line segments. The slopes of these lines are the same, meaning that they are parallelAlso, in Figure 3, the line segments still have the same slope, but they are collinear. However, it is still a translation since the segments can be either parallel OR collinear. Fig 1:Fig 2: Fig 3:

Vocabulary cont.Vector- A quantity that has both direction and magnitude (NOTE: When a vector is written as vector PQ, P is the initial point and Q is the end point)Component Form (or Vector Form)- combines the horizontal and vertical components (If a point is translated 4 units to the left and 3 units up, it would be written as )Matrix- an array of numbers in brackets that represent points; each column represents a point, the first row represents the x-coordinates, and the second row represents the y-coordinatesEx (Fig 4): Entry- A slot in a matrix Ex (Fig. 5):

Describing a TranslationLets say that a represents the units a point is translated on the x-axis and b represents the units a point is translated on the y-axis.Coordinate Notation Format- (x,y) ---> (x+a,y+b)Component Form-

Describing a Translation- ExamplesLets say that triangle ABC is moved to the left 4 units and up 3 units Fig. 6:Coordinate notation- ex:(x,y)-->(x-4, y+3)Component Form-

Lets say that triangle DEF is moved to the right 5 units and down 2 unitsFig. 7:Coordinate notation- ex: (x,y)-->(x+5, y-2)Component Form-

Graph an Image on a Coordinate Plane Given a Preimage and a Rule

Take each point and apply the rule to them. Then connect the dots.Ex. Triangle GHI is translated (x,y)-->(x+2, y-3)G(4,2) H(3,1) Fig. 8:I(-1,-2)

G(4+2, 2-3)H(3+2, 1-3)I(-1+2, -2-3)

G(6,-1)H(5,-2)I(5,-2)

Write a RuleUsing Coordinate Notation:You would count how many units are added to the point across the x-axis (a) and how many units are added to the point across the y-axis (b). Then you would plug it into this format:(x,y)-->(x+a, y+b)

Using Component Form:You would count how many units are added to the point across the x-axis (a) and how many units are added to the point across the y-axis (b). Then you would plug it into this format:

Find the Coordinates of the Preimage given the Coordinates of the Image and the RuleIf you have the image and you are looking for the preimage, you would take a (amount of units that are added to the point across the x-axis) and b (amount of units that are added to the point across the y-axis) and multiply each of the by -1. You would take the solutions and make a new rule. Then, you would apply the new rule to the points of the image. Then, you get the preimage.Finding the Preimage- ExampleTriangle KJL is translated (x,y)-->(x+2, y-3)2(-1) = -2-3(-1) = 3(x,y)-->(x-2, y+3)Fig. 9:

K(4,2)J(3,1)L(-1,-2)

K(4-2, 2+3)J(3-2, 1+3)L(-1-2, -2+3)

K(2,5)J(1,4)L(-3,1)

Use Matrices to Find the Coordinates of a Translation ImageLets say that the rule is applied to triangle MNOM(1,2)N(-3,2)O(-5,-1)You would put the points into a matrixEx. (fig. 10):

Then you would put the rule into a matrixEx. (fig. 11):

Once you know the matrices, add them.Ex:+Continued on Next Slide

Matrix Example cont.To actually add the two matrices, you would add the corresponding entries (so the top left entries would be added together)Ex. (fig. 12):

------------------------Fig. 13:

Then, you would convert that matrix back into coordinates and label themM(5,-1)N(1,-1)O(-1,-4)

Translations in the Human BodyOne translation in the human body is a dislocated shoulder:

The humerus translates down past the scapula causing pain in the arm.

Translations in the Human Body cont.Another translation in the human body is chewed up food traveling through the esophagus:

The Bolus (or chewed food) is translating down through the esophagus

Games!Matrix 2048! - http://scratch.mit.edu/projects/21659646/(Login: Period3; Password: mathp3)Translation Rule Memory Game!- http://scratch.mit.edu/projects/21697690/(Login: Period3; Password: mathp3)

Translation Crossword Puzzle!

reflectionsAJ PerlowinReflectionAn object can be reflected in a mirror line or axis of reflection to produce an image of the object.For example,Each point in the image must be the same distance from the axis of reflection as the corresponding point of the original object.

36Reflecting shapesIf we reflect the quadrilateral ABCD in a mirror line we label the image quadrilateral ABCD.ABCDABCDPre-imageImageAxis of reflectionThis transformation is isometric.37ABCDABCDPre-imageImageAxis of reflectionReflecting shapesIf we draw a line from any point on the object to its image the line forms a perpendicular bisector to the mirror line.38Reflection on a coordinate gridThe vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, 1). Reflect the triangle in the y-axis and label each point on the image.

01234567123456712345672463571A(2, 6)B(7, 3)C(4, 1)A(2, 6)B(7, 3)C(4, 1)Notice that this reflection follows the rule(x,y) (-x,y)This is true for all reflections over the y-axisxy39Reflection on a coordinate gridThe vertices of a quadrilateral lie on the points A(4, 6), B(4, 5), C(2, 2) and D(5, 3). Reflect the quadrilateral in the x-axis and label each point on the image.01234567123456712345672463571A(4, 6)B(4, 5)C(2, 2)A(4, 6)B(4, 5)D(5, 3)Notice that this reflection follows the rule(x,y) (x,-y)This is true for all reflections over the x-axisD(5, 3)C(2, 2)xy40Reflection on a coordinate gridThe vertices of a triangle lie on the points A(4, 4), B(7, 1) and C(2, 6). Reflect the triangle in the line y = x and label each point on the image.01234567123456712345672463571A(4, 4)C(2, 6)A(4, 4)B(1, 7)C(6, 2)x = yNotice that this reflection follows the rule(x,y) (-x,-y)This is true for all reflections over the line y = x.xyB(7, 1)41ABCDABCDPre-imageImageAxis of reflectionFinding the axis of reflectionTo find the axis of reflection all you have to do is use the pre-image and the image and find the point of perpendicular bisection.42Lines of SymmetryA line of symmetry is the line that a shape reflects on to its self. There can be many lines of symmetry in a shape or none all depending on the shape.43Lines of Symmetry In TrianglesHow many lines of symmetry does each triangle have?Notice that the isosceles triangle has one line of symmetry and the equilateral triangle has two while the scalene triangle has none. This is true for all triangles.44Lines of SymmetryHow many lines of symmetry does each quadrilateral have?Notice that the square has four lines of symmetry, the rhombus and rectangle both have two while the kite and isosceles trapezoid both have one. This is true for all of the corresponding quadrilaterals.45Lines of SymmetryHow many lines of symmetry does each regular polygon have?Notice that all the regular polygons have the same number of lines of symmetry as they have sides. This is true for all regular polygons.46Minimum DistanceMinimum Distance is the point on a line that is the shortest distance between two points that the path intersects on a line. This is normally represented as point C.ACLineB47Minimum DistanceALineBATo find the minimum distance you have to first reflect point A over the line to get A. Reflection Connecting Line Then you have to draw a strait line between A and B.C Lastly you draw in point C where the lineintersects with your connecting line segment.48Reflections In Real Life

Word Scrambleimmuinm dtaiencs is the point on a line that is the shortest distance between two points that the path intersects on a line.An object can be reflected in a rmoirr niel or ixsa fo felocitern to produce an image of the object.This transformation is tmiisceor.A ilen fo yerymtms is the line that a shape reflects on to its self. There can be many in a shape or none all depending on the shape.

VocabularyTessellation- A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes with no gaps or overlaps. Tessellations are like a puzzle.Tessellations in the Human Body

All the cells fit together like a puzzle and tessellate.Frieze PatternsFrieze Pattern- A pattern that extends itself to the left and right in such a way that the pattern can be mapped onto itself by a horizontal translation.Classification of Frieze PatternsT - TranslationTR - Translation and 180TG - Translation and horizontal glide reflectionTV - Translation and vertical line reflectionTHG - Translation, horizontal line reflection and horizontal glide reflectionTRVG - Translation, 180 rotation, horizontal line reflection, vertical line reflection, and horizontal glide reflectionTRHVG - Translation, 180 rotation, horizontal line reflection, vertical line reflection, and horizontal glide reflection

TessellationsCan you tell what each of these are?

DilationsDilations- a type of transformation with center C and scale factor k, that maps every point P on a plane to a point P so that the following two properties are true:If P is not the center point C, then the image point P lies on line CP. The scale factor k is a positive number such that k=CP/CP and k is not equal to 1If P is the center point C, then P=PEquation: Dk(x,y)= (kx,ky)

Scalar Multipication

Dilations in the Human BodyDilations happen very often in your eyes. Your pupils dilate due to changes in the light and even changes in emotion. Your pupils would dilate if you just came out of a dark room into bright sunlight. They would also dilate if you get scared, excited, or see someone you like!

Practice Problems1) John goes to the movies on a sunny day. Outside the theater his pupils are are 0.25 cm wide. When he goes inside his eyes dilate by a scale factor of 2. How wide are his pupils inside the theater?

2) Kelly goes to the eye doctor. Her pupils are originally 6mm. When the doctor shines the light in her eyes they dilate to 2mm. What is the scale factor of the dilation?

Answers:1) 0.5cm2) 1/3Final GameFrankenstein's Art ProjectThis is like a color by number, except its color by transformation!Color all Rotations- greenTranslations- purpleReflection- yellowTessellation- blue (only if there is no other transformation)Dilation- red2 or more transformations- orange

Game Board for Frankenstines art Project

Save the photo to word and print if youd like to play.Interesting FactFrankenstein was actually the scientist, not the monster.

Bibliographyhttp://www.carnagill-school.ik.org/img/man-turn-head.gifhttp://www.msnpro.com/emoticons/best-emoticons/roll-eyes.gifhttp://upload.wikimedia.org/wikipedia/commons/a/a7/Frankenstein's_monster_(Boris_Karloff).jpghttp://en.wikipedia.org/wiki/Peristalsishttps://www.mathway.com/http://coolmath.com/algebra/24-matrices/01-whats-a-matrix-01.htmhttp://coolmath.com/algebra/24-matrices/02-adding-subtracting--01.htm

Bibliography (cont.)http://forum.woodenboat.com/showthread.php?112363-Voronoi-Diagrams-in-Naturehttp://langfordmath.com/ECEMath/Geometry/FriezePatternPractice.html http://www.fun-stuff-to-do.com/geometric-shapes-worksheets.html http://education-portal.com/academy/lesson/dilation-in-math-definition-meaning-quiz.html#lessonhttp://image.tutorvista.com/cms/images/38/dilation-graph.JPG

Bibliography (Cont.)http://www.mathwarehouse.com/transformations/dilations/images/picture-of-dilation-in-math2.pnghttp://www.mathwarehouse.com/algebra/matrix/images/matrix-multiplication/scalar-multiplication.pnghttp://drjoannabuckley.files.wordpress.com/2011/05/single-eye.jpghttp://www.nature.com/eye/journal/v24/n6/images/eye2009275f3.jpg

Bibliography (cont.)http://www.mathopenref.com/axis.html http://www.healthylifestylesliving.com/enlighten-the-soul/law-of-attraction/autosuggestion-and-the-person-in-the-mirror/http://therightathome.com/the-two-houses-with-identical-planninghttp://outlandishobservations.blogspot.com/2013/09/friday-fun-facts-9272013.html