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Transformations 8th Grade Math 2D Geometry: ... 2D Geometry: Transformations 2013-12-09 Slide 3 / 168 Table of Contents · Reflections · Dilations · Translations Click on a topic

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    This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.

    Click to go to website: www.njctl.org

    New Jersey Center for Teaching and Learning

    Progressive Mathematics Initiative

    Slide 2 / 168

    8th Grade Math

    2D Geometry: Transformations

    www.njctl.org

    2013-12-09

    Slide 3 / 168

    Table of Contents

    · Reflections · Dilations

    · Translations

    Click on a topic to go to that section

    · Rotations

    · Transformations

    · Congruence & Similarity

    Common Core Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5

    · Special Pairs of Angles

    · Symmetry

    Slide 4 / 168

    Transformations

    Return to Table of Contents

    Slide 5 / 168

    Any time you move, shrink, or enlarge a figure you make a transformation. If the figure you are moving (pre-image) is labeled with letters A, B, and C, you can label the points on the transformed image (image) with the same letters and the prime sign.

    A B

    C

    A' B'

    C'

    pre-image image

    P ull

    P ull

    for transformation shown

    Slide 6 / 168

    The image can also be labeled with new letters as shown below.

    Triangle ABC is the pre-image to the reflected image triangle XYZ

    A B

    C

    X Y

    Z

    pre-image image

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    There are four types of transformations in this unit:

    · Translations · Rotations · Reflections · Dilations

    The first three transformations preserve the size and shape of the figure. They will be congruent. Congruent figures are same size and same shape.

    In other words: If your pre-image is a trapezoid, your image is a congruent trapezoid.

    If your pre-image is an angle, your image is an angle with the same measure.

    If your pre-image contains parallel lines, your image contains parallel lines.

    Slide 8 / 168

    There are four types of transformations in this unit:

    · Translations · Rotations · Reflections · Dilations

    The first three transformations preserve the size and shape of the figure.

    In other words: If your pre-image is a trapezoid, your image is a congruent trapezoid.

    If your pre-image is an angle, your image is an angle with the same measure.

    If your pre-image contains parallel lines, your image contains parallel lines.

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    Translations

    Return to Table of Contents

    Slide 10 / 168

    Slide 11 / 168

    A translation is a slide that moves a figure to a different position (left, right, up or down) without changing its size or shape and without flipping or turning it.

    You can use a slide arrow to show the direction and distance of the movement.

    Slide 12 / 168

    This shows a translation of pre-image ABC to image A'B'C'. Each point in the pre-image was moved right 7 and up 4.

    page1svg

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    Click for web page

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    Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

    A B

    CD

    A' B'

    C'D'

    To complete a translation, move each point of the pre-image and label the new point.

    Example: Move the figure left 2 units and up 5 units. What are the coordinates of the pre-image and image? PU

    LL

    Slide 15 / 168

    Translate pre-image ABC 2 left and 6 down. What are the coordinates of the image and pre-image?

    A

    B

    C

    Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

    PU LL

    Slide 16 / 168 Translate pre-image ABCD 4 right and 1 down. What are the coordinates of the image and pre-image?

    A

    B

    C

    D

    Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

    PU LL

    Slide 17 / 168

    A B

    C

    D

    Translate pre-image ABCD 5 left and 3 up.

    What are the coordinates of the image and pre-image?

    Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

    PU LL

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    A rule can be written to describe translations on the coordinate plane. Look at the following rules and coordinates to see if you can find a pattern.

    2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1)

    2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3)

    4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

    5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1)

    http://www.mathwarehouse.com/transformations/translations-interactive-activity.php http://www.mathwarehouse.com/transformations/translations-interactive-activity.php

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    Translating left/right changes the x-coordinate.

    Translating up/down changes the y-coordinate.

    2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1)

    2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3)

    4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

    5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1)

    Slide 20 / 168

    Translating left/right changes the x-coordinate. · Left subtracts from the x-coordinate

    · Right adds to the x-coordinate

    Translating up/down changes the y-coordinate. · Down subtracts from the y-coordinate

    · Up adds to the y-coordinate

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    2 units Left … x-coordinate - 2 y-coordinate stays rule = (x - 2, y)

    5 units Right & 3 units Down… x-coordinate + 5 y-coordinate - 3 rule = (x + 5, y - 3)

    click to reveal

    A rule can be written to describe translations on the coordinate plane.

    click to reveal

    Slide 22 / 168

    Write a rule for each translation.

    2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1)

    2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3)

    4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

    5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1)

    (x, y) (x-2, y+5) (x, y) (x-2, y-6)

    (x, y) (x-5, y+3) (x, y) (x+4, y-1)

    click to reveal click to reveal

    click to reveal click to reveal

    Slide 23 / 168

    1 What rule describes the translation shown?

    A (x,y) (x - 4, y - 6)

    B (x,y) (x - 6, y - 4)

    C (x,y) (x + 6, y + 4)

    D (x,y) (x + 4, y + 6)

    D E

    F

    G

    D' E'

    F'

    G' Pull Pull

    Slide 24 / 168

    2 What rule describes the translation shown?

    A (x,y) (x, y - 9)

    B (x,y) (x, y - 3) C (x,y) (x - 9, y) D (x,y) (x - 3, y)

    D E

    F

    G

    D' E'

    F'

    G'

    PullPull

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    3 What rule describes the translation shown?

    A (x,y) (x + 8, y - 5)

    B (x,y) (x - 5, y - 1)

    C (x,y) (x + 5, y - 8)

    D (x,y) (x - 8, y + 5) D

    E

    F

    G

    D' E'

    F'

    G'

    PullPull

    Slide 26 / 168

    4 What rule describes the translation shown?

    A (x,y) (x - 3, y + 2)

    B (x,y) (x + 3, y - 2)

    C (x,y) (x + 2, y - 3)

    D (x,y) (x - 2, y + 3) D

    E F

    G

    D' E' F'

    G'

    PullPull

    Slide 27 / 168

    5 What rule describes the translation shown?

    A (x,y) (x - 3, y + 2) B (x,y) (x + 3, y - 2)

    C (x,y) (x + 2, y - 3) D (x,y) (x - 2, y + 3) D

    E F

    G

    D' E'

    F'

    G'

    PullPull

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    Rotations

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    Slide 29 / 168 Slide 30 / 168

    A rotation (turn) moves a figure around a point. This point can be on the figure or it can be some other point. This point is called the point of rotation.

    P

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    Rotation

    The person's finger is the point of rotation for each figure.

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    When you rotate a figure, you can describe the rotation by giving the direction (clockwise or counterclockwise) and the angle that the figur