Humans & Transformations

Humans & Transformations Emily DiMaulo-Milk Emma Halecky Troy Karanfilian AJ Perlowin Matt Monaghan

Table of Contents Rotation Translation Reflection Tessellations Dilations

Emily DiMaulo-Milk Rotations

Rotations A transformation in which a figure is turned about a fixed point

Center of rotation The fixed point around which the preimagerotates. Can be inside, outside, or on the figure.

Angle of rotation 180º Rays drawn from the center of rotation to a point and its image form the angle at which the figure rotated Rotations are expressed in degrees, and the amount of degrees a figure rotates is the angle of rotation.

Rotational Symmetry 180º When a figure can be mapped onto itself by a clockwise rotation of 180º or less

Rotational Symmetry All 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 angle • Irregular 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.

Let’s Try It! Yes Yes No Does the shape have rotational symmetry?

Rotate a Shape Step 1. Draw a line connecting the original point to the center of rotation

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

Rotate a Shape Step 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

Let’s Try It! Using a coordinate plane, draw triangle ABC. A(1,9) B(5,8) C(8,8) Rotate the figure 50º clockwise.

Regular Rotations on a Coordinate Plane Just use these equations- -Counterclockwise Clockwise R90- (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)

Let’s Try It! Without drawing the shape, rotate figure ABCDE 180º counterclockwise. A(1,0) B(3,8) C(4,9) D(-7,9) E(-8,0)

Answers A’(-1,0) B’(-3,-8) C’(-4,-9) D’(7,-9) E’(8,0)

Rotations In the Human Body When you turn your head from left to right, that’s approximately 120º of rotation You can roll your eyes 360º You can rotate your entire body by spinning in a circle

Let’s play a game! Simon Spins This game is a lot like “Simon Says” The 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 Fact Simon Says… A 2008 study found that Simon Says actually has a psychological benefit and helps young children learn to suppress impulsiveness.

Troy Karanfilian Translations

Vocabulary Translation- 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 collinear As 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 parallel Also, 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 <-4,3>)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-coordinates Ex (Fig 4): Entry- A slot in a matrix Ex (Fig. 5):

Describing a Translation Let’s 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- <a,b>

Describing a Translation- Examples Let’s 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- <-4, 3> Let’s say that triangle DEF is moved to the right 5 units and down 2 units Fig. 7: Coordinate notation- ex: (x,y)-->(x+5, y-2) Component Form- <5, -2>

Graph an Image on a Coordinate Plane Given a Preimageand 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 Rule Using 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: <a,b>

Find the Coordinates of the Preimage given the Coordinates of the Image and the Rule If 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- Example Triangle KJL istranslated (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 Image Let’s say that the rule <4,-3> is applied to triangle MNO M(1,2) N(-3,2) O(-5,-1) You would put the points into a matrix Ex. (fig. 10): Then you would put the rule into a matrix Ex. (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 them M’(5,-1) N’(1,-1) O’(-1,-4)

Translations in the Human Body One translation in the human body is a dislocated shoulder: The humerustranslates down past the scapulacausing 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!

AJ Perlowin reflections

Reflection An 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.

Reflecting shapes A B C D If we reflect the quadrilateral ABCD in a mirror line we label the image quadrilateral A’B’C’D’. A’ B’ Pre-image Image C’ D’ Axis of reflection This transformation is isometric.

Reflecting shapes A A’ B B’ Pre-image Image C C’ D D’ Axis of reflection If we draw a line from any point on the object to its image the line forms a perpendicular bisector to the mirror line.

Reflection on a coordinate grid A’(–2, 6) B’(–7, 3) C’(–4, –1) y The 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. A(2, 6) 7 6 5 B(7, 3) 4 3 2 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x –1 –2 C(4, –1) Notice that this reflection follows the rule (x,y)  (-x,y) This is true for all reflections over the y-axis –3 –4 –5 –6 –7

Reflection on a coordinate grid D’(–5, –3) B’(4, –5) A’(–4, –6) The 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. y A(–4, 6) 7 B(4, 5) 6 5 4 3 D(–5, 3) 2 C(2, –2) 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x –1 C’(2, –2) –2 Notice that this reflection follows the rule (x,y)  (x,-y) This is true for all reflections over the x-axis –3 –4 –5 –6 –7

Reflection on a coordinate grid B’(–1, 7) C’(–6, 2) y x = y The 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. 7 6 A’(4, 4) 5 4 A(4, 4) 3 2 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x B(7, –1) –1 –2 Notice that this reflection follows the rule (x,y)  (-x,-y) This is true for all reflections over the line y = x. –3 –4 –5 –6 C(2, –6) –7

Finding the axis of reflection A A’ B B’ Pre-image Image C C’ D D’ Axis of reflection To 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.

Lines of Symmetry A 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.

Lines of Symmetry In Triangles How 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.

Lines of Symmetry How 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.

Lines of Symmetry How 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.

Minimum Distance Minimum 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. A B Line C

Minimum Distance To find the minimum distance you have to first reflect point A over the line to get A’. Then you have to draw a strait line between A’ and B. Lastly you draw in point C where the line intersects with your connecting line segment. A B Reflection  Line C    Connecting Line A’

Reflections In Real Life

Word Scramble immuinmdtaiencsis 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 rmoirrnielor ixsafofelocitern to produce an image of the object. This transformation is tmiisceor. A ilenfoyerymtmsis the line that a shape reflects on to its self. There can be many in a shape or none all depending on the shape.

Humans & Transformations