NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Why do you think the numbers 360 and 360 are used in reference to rotation. Rotating means that we are moving in a circular pattern and circles have 360. The rotation of degrees with center is defined by using transparencies On a piece of paper fix a point as the. center of rotation let be a point in the plane and let the ray. be drawn Let be a number between 360 and, Instructions for performing a rotation If there is a rotation of degrees with center the image is the. point described as follows On a piece of transparency trace and. in red Now use a pointed object e g the, leg with spike of a compass to pin the transparency at the point First suppose 0 Then holding the paper in. place rotate the transparency counterclockwise so that if we denote the final position of the rotated red point that was. by then the is degrees For example if 30 we have the following picture. As before the red rectangle represents the border of the rotated transparency Then by definition is the. MP 6 point, If however 0 then holding the paper in place we would now rotate the transparency clockwise so that if we. denote the position of the red point that was by then the angle is degrees For example if 30 we. have the following picture, Again we define to be in this case Notice that the rotation moves the center of rotation to itself. Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Exercises 1 4 4 minutes,Students complete Exercises 1 4 independently. 1 Let there be a rotation of degrees around center Let be a point other than Select so that Find. i e the rotation of point using a transparency, Verify that students have rotated around center in the counterclockwise direction. 2 Let there be a rotation of degrees around center Let be a point other than Select so that Find. i e the rotation of point using a transparency, Verify that students have rotated around center in the clockwise direction. 3 Which direction did the point rotate when, It rotated counterclockwise or to the left of the original point. 4 Which direction did the point rotate when, It rotated clockwise or to the right of the original point. Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Discussion 5 minutes, Observe that with as the center of rotation the points and lie on a circle whose center is and whose. Assume we rotate the plane degrees around center Let be a point other than Where do you think. will be located, The points and will be equidistant from that is is on the circumference of the circle with. center and radius The point would be clockwise from if the degree of rotation is negative. The point would be counterclockwise from if the degree of rotation is positive. If we rotated degrees around center several times where would all of the images of be located. All images of will be on the circumference of the circle with radius. Why do you think this happens, Because like translations and reflections rotations preserve lengths of segments The segments of. importance here are the segments that join the center to the images of Each segment is the. radius of the circle We discuss this more later in the lesson. Consider a rotation of point around center 180 degrees and 180 degrees Where do you think the. images of will be located, Both rotations although they are in opposite directions will move the point to the same location. Further the points and will always be collinear i e they will lie on one line for any point. This concept is discussed in more detail in Lesson 6. Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Concept Development 3 minutes, Now that we know how a point gets moved under a rotation let us look at how a geometric figure gets moved. under a rotation Let be the figure consisting of a vertical segment not a line and two points Let the. center of rotation be the lower endpoint of the segment as shown. Then the rotation of 30 degrees with center moves the point represented by the left black dot to the lower. red dot the point represented by the right black dot to the upper red dot and the vertical black segment to. the red segment to the left at an angle of 30 degrees as shown. Video Presentation 2 minutes, The following two videos show how a rotation of 35 degrees and 35 degrees with center respectively rotates a. geometric figure consisting of three points and two line segments. http www harpercollege edu skoswatt RigidMotions rotateccw html. http www harpercollege edu skoswatt RigidMotions rotatecw html. The videos were developed by Sunil Koswatta, Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Discussion 2 minutes, Revisit the question posed at the beginning of the lesson and ask students. What is the simplest transformation that would map one of the following figures to the other. We now know that the answer is a rotation, Show students how a rotation of approximately 90 degrees around a point chosen on the perpendicular bisector. bisector of the segment joining the centers of the two circles in the figures would map the figure on the left to the. figure on the right Similarly a rotation of 90 degrees would map the figure on the right to the figure on the left. Note to Teacher,Continue to remind students,that a positive degree of. rotation moves the figure,counterclockwise and a,negative degree of rotation. moves the figure clockwise, Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Exercises 5 6 4 minutes,Students complete Exercises 5 and 6 independently. 5 Let be a line be a segment and be an angle as shown Let there be a rotation of. degrees around point Find the images of all figures when. Verify that students have rotated around center in the counterclockwise direction. Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. 6 be a segment of length units and be an angle of size Let there be a rotation by degrees. where about Find the images of the given figures Answer the questions that follow. Verify that students have rotated around center in the clockwise direction. a What is the length of the rotated segment,The length of the rotated segment is units. b What is the degree of the rotated angle,The degree of the rotated angle is. Concept Development 4 minutes, Based on the work completed during the lesson and especially in Exercises 5 and 6 we can now state that rotations. have properties similar to translations with respect to Translation 1 Translation 3 of Lesson 2 and reflections with. respect to Reflection 1 Reflection 3 of Lesson 4, Rotation 1 A rotation maps a line to a line a ray to a ray a segment to a segment and an angle to an angle. Rotation 2 A rotation preserves lengths of segments. Rotation 3 A rotation preserves measures of angles. Also as with translations and reflections if 1 and 2 are parallel lines and if there is a rotation then the lines. 1 and 2 are also parallel However if there is a rotation of degree 180 and is a line. and are not parallel Note to teacher Exercises 7 and 8 illustrate these two points. Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Exercises 7 8 5 minutes,Students complete Exercises 7 and 8 independently. 7 Let and be parallel lines Let there be a rotation by degrees where about. Verify that students have rotated around center in either direction Students should respond that. 8 Let be a line and be the center of rotation Let there be a rotation by degrees where about Are. the lines and parallel, Verify that students have rotated around center in either direction any degree other than Students should. respond that and are not parallel, Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Closing 3 minutes, Summarize or have students summarize what we know of rigid motions to this point. We now have definitions for all three rigid motions translations reflections and rotations. Rotations move lines to lines rays to rays segments to segments angles to angles and parallel lines to parallel. lines similar to translations and reflections, Rotations preserve lengths of segments and degrees of measures of angles similar to translations and. reflections, Rotations require information about the center and degree of rotation whereas translations require only a. vector and reflections require only a line of reflection. Lesson Summary, Rotations require information about the center of rotation and the degree in which to rotate Positive degrees of. rotation move the figure in a counterclockwise direction Negative degrees of rotation move the figure in a. clockwise direction,Basic Properties of Rotations, Rotation 1 A rotation maps a line to a line a ray to a ray a segment to a segment and an angle to an. Rotation 2 A rotation preserves lengths of segments. Rotation 3 A rotation preserves measures of angles. When parallel lines are rotated their images are also parallel A line is only parallel to itself when rotated exactly. Terminology, ROTATION description For a number between and the rotation of degrees around center. is the transformation of the plane that maps the point to itself and maps each remaining point. so that and are the same, of the plane to its image in the counterclockwise half plane of ray. distance away from and the measurement of is degrees. The counterclockwise half plane is the half plane that lies to the left of. while moving along,the direction from to,Exit Ticket 5 minutes. Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Lesson 5 Definition of Rotation and Basic Properties. Exit Ticket, 1 Given the figure let there be a rotation by degrees where 0 about Let be. 2 Using the drawing above let 1 be the rotation degrees with 0 about Let 1 be. Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. Exit Ticket Sample Solutions, 1 Given the figure let there be a rotation by degrees where about Let be. Sample rotation shown above Verify that the figure has been rotated counterclockwise with center. 2 Using the drawing above let be the rotation degrees with about Let be. Sample rotation shown above Verify that the figure has been rotated clockwise with center. Problem Set Sample Solutions,1 Let there be a rotation by around the center. Rotated figures are shown in red, Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 2. 2 Explain why a rotation of degrees around any point never maps a line to a line parallel to itself. A degree rotation around point will move a given line to Parallel lines never intersect so it is obvious. that a degree rotation in either direction does not make lines and parallel Additionally we know that there. exists just one line parallel to the given line that goes through a point not on If we let be a point not on the. line must go through it in order to be parallel to does not go through point therefore and are not. parallel lines Assume we rotate line first and then place a point on line to get the desired effect a line. through This contradicts our definition of parallel i e parallel lines never intersect so again we know that. line is not parallel to, 3 A segment of length has been rotated degrees around a center What is the length of the rotated. segment How do you know, The rotated segment will be in length Rotation 2 states that rotations preserve lengths of segments so the. length of the rotated segment will remain the same as the original. 4 An angle of size has been rotated degrees around a center What is the size of the rotated angle How do. The rotated angle will be Rotation 3 states that rotations preserve the degrees of angles so the rotated. angle will be the same size as the original, Lesson 5 Definition of Rotation and Basic Properties. This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eureka math org This work is licensed under a. This file derived from G8 M2 TE 1 3 0 08 2015 Creative Commons Attribution NonCommercial ShareAlike 3 0 Unported License.

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