Graphing and Describing “Rotations” Be sure to read notes section on each slide for additional instructions and answers. Transformations Day 5

we spent a day focusing on “Translations” 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.

We spent a day on “Reflections” A reflection (flip) creates a mirror image of a figure.

Today… We will focus on “Rotations” 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

Congruency 1) Translations 2) Reflections 3) Rotations 4) Dilations The first 3 transformations preserve the size and shape of the figure. That means… If your pre-image (the before) is a polygon, your image (the after) is a congruent polygon. If your pre-image contains parallel lines, your image contains congruent parallel lines If your pre-image is an angle, your image is an angle with the same measure. Could this represent a translation? Point out in the top image that you can determine congruency by counting the number of units that make up each corresponding line segment. Then click mouse and see if students can determine if the right triangles could represent a translation. Answer: No because translations preserve the size and shape. Figure A is bigger than Figure B.

Let’s Get Started Section 1: Describing Rotations Section 2: Rules for Rotating 180º

Describing a Rotation When you rotate a figure, you can describe the rotation by… Giving the direction: clockwise or counter-clockwise Giving the degrees that the figure is rotated around the point of rotation: 90º, 180º, 270º

NOTE: 180º rotations turn the figure upside down! Describing a Rotation pre-image What are 2 ways to describe the rotation from A to B ? What are 2 ways to describe the rotation from A to D ? What are 2 ways to describe the rotation from A to C ? 90º rotations move the figure over 1 quadrant (CW or CCW) 180º rotations move the figure over 2 quadrants 270º rotations move the figure over 3 quadrants Explain the different rotations: 90, 180 and 270 (both clockwise and counter-clockwise) Make sure students understand that a 90 degree CW turn is same as 270 degree CCW. Make sure students understand that a 90 degree CCW turn is the same as a 270 degree CW. Click mouse and go through questions. Answer B is a 90 degree clockwise or 270 degree CCW C is a 180 degree turn clockwise or CCW D is a 90 degree CCW or 270 degree clockwise NOTE: 180º rotations turn the figure upside down!

Turn and Talk Which can be described as a 180º rotation around the origin? Be prepared to share your reasoning with the class. A B A B Figure #2 is the 180 degree rotation. You can tell because the figure has been turned upside down. Figure #1 Figure #2

Turn and Talk Which describes the rotation of the cell phone? Select ALL that apply. 90° clockwise 180º clockwise 270° clockwise 90° counter clockwise 180º counter clockwise 270° counter clockwise HANDS ON… As students Turn and Talk, give them the opportunity to pull out their own smart phones and do a similar rotation. Answer: A, F If you look at the word “Samsung” you can tell that the phone has been turned 90 degrees clockwise (or 270 CCW).

You Try #1 Describe the given rotation. Give both the CW and CCW description. Figure Q is the pre-image. It was rotated 90 degrees clockwise or 270 degrees counter-clockwise.

You Try #2 Analyze the given transformation. Which of the following could map the Blue square onto the Green? Reflection across the x-axis. 180º rotation around the origin. A translation 6 left and 6 down. Both B and C. Answer: C You can tell that this is not a 180 degree rotation by looking at the letters. The Green box has not been turned upside down. Show students how the translation 6 left and 6 down maps the Blue box on top of the Green box.

You Try #3 Describe the rotation from Figure A to Figure B. Give both CW and CCW description. 90 degrees counter-clockwise or 270 degrees clockwise

You Try #4 Which describes the rotation of the cell phone? Select ALL that apply. 90° clockwise 180º clockwise 270° clockwise 90° counter-clockwise 180º counter-clockwise 270° counter-clockwise Answer: C, D

Moving on… Section 1: Describing Rotations Section 2: Rules for Rotating 180º

Rotate the figure below 180° clockwise about the origin. Graphing rotations Rotate the figure below 180° clockwise about the origin. Graphing a rotation can be very challenging. Most of us have difficulties envisioning what a figure will look like after a 90º, 180º or 270º turn. Students will not get tracing paper on state test, so we will be using RULES to teach students how to rotate.

Graphing rotations Students will not get tracing paper on state test, so we will be using RULES to teach students how to rotate.

Rotate the figure below 180° clockwise about the origin. Determine a beginning line axis and color it. *Think about an axis you would walk to get to most/all of your points. Draw a second color on the axis 180° clockwise. Imagine you are a person walking your original line starting at the origin. How far do you walk the line to reach point V? Do you turn left tor right off your line to reach point V? How many spaces? Now repeat the same steps on your new line, 180° clockwise. Continue the steps above for each point to create your new image. Graphing rotations Rotate the figure below 180° clockwise about the origin. Students will not get tracing paper on state test, so we will be using RULES to teach students how to rotate.

Guided Practice Rotate ABC 180º clockwise. RULE: Keep the coordinates; Change both signs to the opposite. First find coordinates using the RULE, then graph the rotation. Preimage Image A(7, -2) A’(-7, 2) B(11, -5) B’(-11, 5) C(3, -5) C’(-3, 5) You can point out afterwards: The signs that we determined from our RULE do indeed make sense. The image ended up in Quadrant II. In Quadrant II, x is always negative and y is always positive. If you look at the signs that we came up with from our rule, it coincides with this fact. Rotate ABC 180º clockwise.

You Try Rotate ∆ABC 180º clockwise. RULE: Keep the coordinates; Change both signs to the opposite. Preimage Image A(-4, 1) A’(4, -1) B(1, 3) B’(-1, -3) C(-2, 5) C’(2, -5) D(-5, 3) D’(5, -3) Notice, part of the image ends up in Quadrant III and Part ends up in Quadrant IV. The rule still came up with correct signs for the quadrant in which the point fell! Rotate ∆ABC 180º clockwise.

Rotating 180º clockwise or counter-clockwise RULE: Keep the same coordinates; Change both signs to the opposite. (x, y) → (-x, -y) EXAMPLE: Preimage Image X(1, 2) X’(-1, -2) Y(3, 5) Y’(-3, -5) Z(-3,4) Z’(3, -4) Notice how image is upside down with a 180º rotation! These means “take the opposite”, not “make it negative” ! Explain the RULE for rotating 180 degrees clockwise (same as 180 degrees CCW). Make sure students understand that we are not turning everything to negatives. The negatives is this case mean that we are changing the signs to be the opposite of what they are. If the sign is positive, change it to negative. If the sign is negative, change it to a positive. Review example with students. If able… Play YouTube video for students. Watch as teacher performs 2 different 180 degree rotations. You will probably need to click on the “Activate Adobe Flash” and then “Allow and Remember” for it to run. You will also need speakers. I would also make sure that there is a link on your website to this video. https://www.youtube.com/watch?v=8ZeeDYIlNFk

Guided Practice #1 Rotate ∆EFG 180º clockwise. Remember… turning 180 degrees means change both signs. Pre-image Image E(-3, 7) E’(3, -7) F(-7, 3) F’(7, -3) G(-9, 6) G’(9, -6) Point out that image falls Q4 where x is always positive and y is always negative. RULE gives the correct signs!

You Try #1 Rotate ∆EFG 180º clockwise. Remember… turning 180 degrees means change the sign of both coordinates. Pre-image Image Q(2, -2) Q’(-2, 2) R(9, -2) R’(-9, 2) S(9, -6) S’(-9, 6) Point out that image falls Q2 where x is always negative and y is always positive. RULE gives the correct signs!

End of Presentation