Essential Question: How do you draw the image of a figure under a rotation? Explore Exploring Rotations

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1 OMMON ORE Locker LESSON ommon ore Math Standards The student is epected to: OMMON ORE G-O.. Develop definitions of rotations... in terms of angles, circles,... and line segments. lso G-O.., G-O..5, G-O..6 Mathematical ractices OMMON ORE 17.3 Rotations M.5 Using Tools Language Objective Students work in small groups or pairs to identif and label the transformation shown on a coordinate plane and if a rotation, identif the point of rotation. Name lass Date 17.3 Rotations Essential Question: How do ou draw the image of a figure under a rotation? Eplore Eploring Rotations You can use geometr software or an online tool to eplore rotations. Draw a triangle and label the vertices,, and. Then draw a point. Mark as a center. This will allow ou to rotate figures around point. Resource Locker ENGGE Essential Question: How do ou draw the image of a figure under a rotation? ossible answer: To draw the image of a figure under a rotation of m around point, choose a verte of the figure, for eample, verte. Draw. Use a protractor to draw a ra that forms an angle of m with _. Use a ruler to mark point ' along the ra so that ' =. Repeat the process with the other vertices of the figure. onnect the images of the vertices (,, etc.) to draw the image of the figure. If the figure is on a coordinate plane, use an algebraic rule to find the image of each verte of the figure. Then connect the images of the vertices. REVIEW: LESSON ERFORMNE TSK View the online Engage. Discuss the motion of the minute hand of the clock with students. Then preview the Lesson erformance Task. D Select and rotate it 90 around point. Label the image of as. hange the shape, size, or location of and notice how changes. Draw,, and. Measure these angles. What do ou notice? Does this relationship remain true as ou move point? What happens if ou change the size and shape of? The measure of each angle is 90 ; this remains true regardless of the location of point or the size and shape of. Measure the distance from to and the distance from ' to. What do ou notice? Does this relationship remain true as ou move point? What happens if ou change the size and shape of? = '; this remains true regardless of the location of point or the size and shape of. Module Lesson 3 Name lass Date 17.3 Rotations Essential Question: How do ou draw the image of a figure under a rotation? G-O.. Develop definitions of rotations in terms of angles, circles, and line segments. lso G-O.., G-O..5, G-O..6 Eplore Eploring Rotations You can use geometr software or an online tool to eplore rotations. Draw a triangle and label the vertices,, and. Then draw a point. Mark as a center. This will allow ou to rotate figures around point. Select and rotate it 90 around point. Label the image of as. hange the shape, size, or location of and notice how changes. Resource Draw,, and. Measure these angles. What do ou notice? Does this relationship remain true as ou move point? What happens if ou change the size and shape of? The measure of each angle is 90 ; this remains true regardless of the location of point or the size and shape of. Measure the distance from to and the distance from ' to. What do ou notice? Does this relationship remain true as ou move point? What happens if ou change the size and shape of? = '; this remains true regardless of the location of point or the size and shape of. Module Lesson 3 HRDOVER GES Turn to these pages to find this lesson in the hardcover student edition. 857 Lesson 17.3

2 Reflect 1. What can ou conclude about the distance of a point and its image from the center of rotation? point and its image are both the same distance from the center of rotation.. What are the advantages of using geometr software or an online tool rather than tracing paper or a protractor and ruler to investigate rotations? Sample answer: Software or an online tool makes it eas to observe the effect of changing the shape or location of the preimage or changing the location of the center of rotation. Eplain 1 Rotating Figures Using a Ruler and rotractor rotation is a transformation around point, the center of rotation, such that the following is true. Ever point and its image are the same distance from. ll angles with verte formed b a point and its image have the same measure. This angle measure is the angle of rotation. In the figure, the center of rotation is point and the angle of rotation is 110. Eample 1 Draw the image of the triangle after the given rotation. ounterclockwise rotation of 150 around point 110 Step 1 Draw _. Then use a protractor to draw a ra that forms a 150 angle with _. EXLORE Eploring Rotations INTEGRTE TEHNOLOGY If time permits, students can use the software to eperiment with different angles of rotation. In particular, ask students to investigate a 360 angle of rotation. Students should discover that the image of a figure after a 360 rotation coincides eactl with the preimage. oint out that this means a 360 rotation is equivalent to a 0 rotation. Students ma also eplore angles of rotation greater than 360. In this case, students should conclude that an equivalent rotation can be found b subtracting 360 (or multiples of 360 ) from the angle of rotation. QUESTIONING STRTEGIES In what direction does the software rotate figures? How could ou use the software to produce a 90 clockwise rotation? ounterclockwise; enter 70 as the angle of rotation. EXLIN 1 Rotating Figures Using a Ruler and rotractor Module Lesson 3 ROFESSIONL DEVELOMENT Learning rogressions In this lesson, students etend the informal concept of a rotation as a turn to a more precise definition. Rotations are one of the three rigid motions that students stud in this module (translations and reflections are the other two). Rotations are somewhat more difficult to draw than the other rigid motions and predicting the effect of a rotation ma be more difficult for students than predicting the effect of a reflection or a translation. Geometr software is a useful tool for investigating rotations. Students will need a solid understanding of transformations, including rotations, when the combine transformations to solve real-world problems. INTEGRTE MTHEMTIL RTIES Focus on Math onnections M.1 Encourage students to use their knowledge of right angles to visualize rotations. Remind students that a 90 rotation is a quarter turn; a 5 rotation is half that. For eample, suggest students visualize what is approimatel a triangle after a rotation of 0 around. Rotations 858

3 QUESTIONING STRTEGIES How can ou use tracing paper to check our construction? Trace the figure; place a pencil s point on, and rotate the paper counterclockwise for the given angle of rotation. The traced version of the figure should lie on top of the rotated figure. Step Use a ruler to mark point along the ra so that =. Step 3 Repeat Steps 1 and for points and to locate points and. onnect points,, and to draw. lockwise rotation of 75 around point Q Step 1 Draw _ QD. Use a protractor to draw a ra forming a clockwise 75 angle with _ QD. Step Use a ruler to mark point D along the ra so that QD = QD. Step 3 Repeat Steps1 and for points E and F to locate points E and F. onnect points D, E, and F to draw D E F. E F D Q D E F Reflect 3. How could ou use tracing paper to draw the image of in art? ut a piece of tracing paper on the page and trace and point. ut the point of a pencil on point and use a protractor to rotate the tracing paper counterclockwise 150. Trace over in the new location, pressing firml to make an impression on the page below. Module Lesson 3 OLLORTIVE LERNING Small Group ctivit Have students work in small groups to write together a description of the similarities and differences the observe among the three transformations: translations, reflections, and rotations. Sample answer: ll three transformations preserve the size and shape of the original figure. Each transformation uses a different geometric object (vector, line, or point) to perform the transformation. Translations alwas preserve the orientation of the original figure, while reflections and rotations ma alter the orientation. 859 Lesson 17.3

4 Your Turn Draw the image of the triangle after the given rotation.. ounterclockwise rotation of 0 around point 5. lockwise rotation of 15 around point Q L Eplain K K L Drawing Rotations on a oordinate lane You can rotate a figure b more than 180. The diagram shows counterclockwise rotations of 10, 0, and 300. Note that a rotation of 360 brings a figure back to its starting location. When no direction is specified, ou can assume that a rotation is counterclockwise. lso, a counterclockwise rotation of is the same as a clockwise rotation of (360 - ). The table summarizes rules for rotations on a coordinate plane. J J T U S 300 Q S 0 10 T U EXLIN Drawing Rotations on a oordinate lane QUESTIONING STRTEGIES How can ou predict the quadrant in which the image of the quadrilateral will lie? Ever 90 of rotation moves the preimage around the origin b one quadrant, so a 70 rotation moves the preimage from Quadrant I to Quadrant IV. How can ou use the rule for rotation to show that the origin is fied under the rotation? The rule is (, ) (, -), so (0, 0) (0, 0), which shows that the origin is fied. Rules for Rotations round the Origin on a oordinate lane 90 rotation counterclockwise (, ) (-, ) 180 rotation (, ) (-, -) 70 rotation counterclockwise (, ) (, -) 360 rotation (, ) (, ) Eample Draw the image of the figure under the given rotation. Quadrilateral D; 70 The rotation image of (, ) is (, ). Find the coordinates of the vertices of the image. D (0, ) (, 0) (1, ) ' (, -1) (, ) '(, -) D (3, 1) D' (1, -3) Module Lesson 3 VOID OMMON ERRORS Some students ma confuse the direction of a rotation (clockwise or counterclockwise). Remind students that the direction is assumed to be counterclockwise unless otherwise stated. ssociate this default direction with the wa the quadrants are numbered. OMMUNITING MTH Students analze pictures of preimages and images, and discuss what kind of transformation is shown. The group must agree before labeling each picture. If a transformation is identified as a rotation, the group must determine the point of rotation. Each picture should be labeled, and this sentence completed: This shows a (translation/reflection/ rotation) because. rovide ke terms to help students complete the statement. Rotations 860

5 redict the quadrant in which the image will lie. Since quadrilateral D lies in Quadrant I and the quadrilateral is rotated counterclockwise b 70, the image will lie in Quadrant IV. lot,,, and D to graph the image. D - 0 D - KLM; 180 The rotation image of (, ) is (, ). Find the coordinates of the vertices of the image. K (, -1) K' (, 1 ) L (, -1) L' ( -, 1 ) M (1, -) M' ( -1, ) - - M K - 0 IV redict the quadrant in which the image will lie. Since KLM lies in Quadrant the triangle is rotated b 180, the image will lie in Quadrant II. L - M K L and lot K', L', and M' to graph the image. Reflect 6. Discussion Suppose ou rotate quadrilateral D in art b 810. In which quadrant will the image lie? Eplain. Quadrant II; the quadrilateral D is in Quadrant 1. Ever rotation of 360 brings the quadrilateral back to Quadrant I, and since 810 = , the 810 rotation is equivalent to a 90 rotation. This maps the quadrilateral to Quadrant II. Module Lesson 3 LNGUGE SUORT The words rotation and transformation (as well as function and notation) are all cognates with Spanish. The contain the same Latin root and have similar spellings and identical meanings. oint out that all these words in English end with tion, and in Spanish the all end with ción. This is a word pattern that ma be useful to students who speak English and Spanish. 861 Lesson 17.3

6 Your Turn Draw the image of the figure under the given rotation. 7. QR; Quadrilateral DEFG; 70 Eplain 3 Eample 3 Specifing Rotation ngles Find the angle of rotation and direction of rotation in the given figure. oint is the center of rotation. D Q R Q R G D D F E - G E F EXLIN 3 Specifing Rotation ngles QUESTIONING STRTEGIES When a drawing of a rotation shows two figures, how can ou tell which is the preimage and which is the image? The vertices of the image will have prime marks, for eample,. VOID OMMON ERRORS Some students ma have difficult identifing the direction of the rotation. Suggest that the visualize as the center of a clock, with the minute hand pointing to a verte on the preimage. s the minute hand moves to point at the corresponding verte, which wa is it moving (going the shortest wa)? Draw segments from the center of rotation to a verte and to the image of the verte. Measure the angle formed b the segments. The angle measure is 80. ompare the locations of the preimage and image to find the direction of the rotation. The rotation is 80 counterclockwise. D D D Module Lesson 3 Rotations 86

7 S R Q S Q R Draw segments from the center of rotation to a verte and to the image of the verte. Measure the angle formed b the segments. The angle measure is 135. The rotation is 135 (clockwise/counterclockwise). Reflect 9. Discussion Does it matter which points ou choose when ou draw segments from the center of rotation to points of the preimage and image? Eplain. No, as long as the points are corresponding points (i.e., a point and its image), the angle of rotation will be the same. 10. In art, is a different angle of rotation and direction possible? Eplain. Yes; a rotation of 80 counterclockwise is equivalent to a rotation of 80 clockwise. Your Turn Find the angle of rotation and direction of rotation in the given figure. oint is the center of rotation. 11. N K N K M L M L The transformation is a 70 clockwise rotation. Module Lesson Lesson 17.3

8 Elaborate 1. If ou are given a figure, a center of rotation, and an angle of rotation, what steps can ou use to draw the image of the figure under the rotation? Sample answer: Draw a segment from the center of rotation,, to one verte of the figure,. Use a protractor to draw a ra that forms an angle with _ that is equal to the angle of rotation. Use a ruler to mark a point along the ra so that ' =. Repeat the process with the other vertices of the figure. onnect the images of the vertices to draw the image of the figure. 13. Suppose ou are given DEF, D'E 'F', and point. What are two different was to prove that a rotation around point cannot be used to map DEF to D'E 'F'? (1) Show that D D', E E ', or F F '. () Show that m DD' m EE ', m EE m FF ', or m DD' m FF '. 1. Essential Question heck-in How do ou draw the image of a figure under a counterclockwise rotation of 90 around the origin? For each verte of the figure, use the rule (, ) (-, ) to find the coordinates of the image of the verte. lot the images of the vertices, then connect these points to draw the image of the figure. Evaluate: Homework and ractice ELORTE QUESTIONING STRTEGIES Given a line segment _ W, describe how ou would draw _ W under a rotation of 10 around. Draw a ra from that forms a 10 angle with the segment. Mark point W along this ra such that W = W. SUMMRIZE THE LESSON How do ou draw the image of a figure under a clockwise rotation of 90 around the origin? For each verte of the figure, use the rule (, ) (,-) to find the coordinates of the image of the verte. lot the images of the vertices; then connect these points to draw the image of the figure. 1. lberto uses geometr software to draw STU and point, as shown. He marks as a center and uses the software to rotate STU 115 around point. He labels the image of STU as S'T 'U '. Which three angles must have the same measure? What is the measure of these angles? Online Homework Hints and Help Etra ractice SS ', TT ', and UU '; all three angles measure 115 since this is the amount b which the triangle was rotated around the point. EVLUTE SSIGNMENT GUIDE oncepts and Skills Eplore Eploring Rotations ractice Eercise 1 Module Lesson 3 Eercise Depth of Knowledge (D.O.K.) 1 1 Recall of Information M.5 Using Tools OMMON ORE Mathematical ractices Eample 1 Rotating Figures Using a Ruler and rotractor Eample Drawing Rotations on a oordinate lane Eample 3 Specifing Rotation ngles Eercises Eercises 5 8 Eercises Recall of Information M. Modeling Recall of Information M.5 Using Tools Skills/oncepts M. Reasoning 1 Skills/oncepts M.3 Logic 15 Skills/oncepts M. Reasoning Rotations 86

9 INTEGRTE MTHEMTIL RTIES Focus on ommunication M.3 Remind students that in mapping a figure onto itself, the center of rotation is inside the figure. If students are confused, show a square rotating around a point at its center and a square rotating around a point outside the square. Help students verbalize the difference. Draw the image of the triangle after the given rotation.. ounterclockwise rotation of 30 around point 3. lockwise rotation of 55 around point J M L K L J ONNET VOULRY M K Have students complete a chart like the following vocabular chart, filling in the blank areas with pictures and words to describe the transformation.. ounterclockwise rotation of 90 around point F Transformations Translation Reflection Rotation D F Define or describe: Define or describe: Define or describe: E Draw an eample: Draw an eample: Draw an eample: D E Draw the image of the figure under the given rotation. 5. ; RST; R T S R S T Module Lesson 3 Eercise Depth of Knowledge (D.O.K.) OMMON ORE Mathematical ractices 16 Skills/oncepts M. Reasoning 17 0 Skills/oncepts M. Modeling 1 Skills/oncepts M. Reasoning 3 Strategic Thinking M.5 Using Tools 3 3 Strategic Thinking M.3 Logic 5 3 Strategic Thinking M.1 roblem Solving 865 Lesson 17.3

10 7. Quadrilateral EFGH; Quadrilateral QRS; 70 G F S H E Q - E 0 H - 0 R Q F - G R - S VOID OMMON ERRORS Some students ma rotate a figure around its center or around one of its vertices, not around point. Reread the instructions together. sk students to eplain the differences between rotating a figure around an eterior point, around a point on the figure, and around an interior point. Find the angle of rotation and direction of rotation in the given figure. oint is the center of rotation. 9. Z Y 10. U V X Z W X Y W U V The rotation is 50 counterclockwise. The rotation is 180. Write an algebraic rule for the rotation shown. Then describe the transformation in words G E F R S D S 0 D F E T T G - R - D (-, ) D' (, ), E (-1, 3) E '(1, -3), R (-, 3) R' (-3, -), S (-1, 3) F (-1, -3) F '(1, 3), G (-3, -) G '(3, ) S '(-3, -1), T (-3, -3) T '(3, -3), so so the image of (, ) is (-, -). The rule the image of (, ) is (-, ). The rule is (, ) (-, -) and the transformation is (, ) (-, ) and the transformation is a rotation of 180. is a counterclockwise rotation of 90. Module Lesson 3 Rotations 866

11 INTEGRTE MTHEMTIL RTIES Focus on Math onnections M.1 Discuss the fact that ever rotation can be epressed as a composition of reflections across intersecting lines. 13. Vanessa used geometr software to appl a transformation to, as shown. ccording to the software, m ' = m ' = m '. Vanessa said this means the transformation must be a rotation. Do ou agree? Eplain. No; according to the definition of a rotation, ever point and its image must be the same distance from, and that is not the case in the given figure. 1. Make a rediction In which quadrant will the image of FGH lie after a counterclockwise rotation of 1980? Eplain how ou made our prediction. Quadrant I; a rotation of 360 brings the figure back to its original location, so ou can subtract multiples of 360 from the angle of rotation = 1800 and - F = 180, so the rotation is equivalent to a rotation of 180, which maps FGH to Quadrant I. G H 15. ritical Thinking The figure shows the image of MN after a counterclockwise rotation of 70. Draw and label MN. The rule for the rotation is (, ) (, -). M M' has coordinates (, ), so the coordinates of M are (-, ). M N N N ' has coordinates (, 1), so the coordinates of N are ( 1, ). ' has coordinates (, 1), so the coordinates of are ( 1, ). 16. Multi-Step Write the equation of the image of line l after a clockwise rotation of 90. (Hint: To find the image of line l, choose two or more points on the line and find the images of the points.) Line l passes through (, 0) and (0, 1). The rule for a clockwise rotation of 90 is (, ) (, -), so (, 0) ' (0, ) - 0 l and (0, -1) ' (-1, 0). The line through ' and ' has slope 0 - () = and -intercept, so the equation of the line -1-0 is = -. - Module Lesson 3 DIFFERENTITE INSTRUTION Modeling To reinforce the meaning of rotation, show some eamples of rotations that might be familiar to students for eample, the turn of a steering wheel or Earth s orbit around the sun. sk students to give other eamples of turns or rotations in the real world, such as the motion of a DVD in a plaer, or of a doorknob. Discuss what all of the motions have in common. Help students see that all involve moving points around a fied point. 867 Lesson 17.3

12 17. Ferris wheel has 0 cars that are equall spaced around the circumference of the wheel. The wheel rotates so that the car at the bottom of the ride is replaced b the net car. how man degrees does the wheel rotate? 18 ; there are 360 in a complete rotation and there are 0 equall-spaced cars, so the amount of rotation is = The Sklon Tower, in Niagara Falls, anada, has a revolving restaurant 775 feet above the falls. The restaurant makes a complete revolution once ever hour. While a visitor was at the tower, the restaurant rotated through 135. How long was the visitor at the tower? Set up a proportion. 135 = =.5 The visitor was at the tower for.5 minutes. 19. mani plans to use drawing software to make the design shown here. She starts b drawing Triangle 1. Eplain how she can finish the design using rotations. ossible answer: Starting with triangle 1, rotate clockwise 60º around 1 the verte at the center of the heagon. Repeat the process using each 5 3 successive image as a preimage. 0. n animator is drawing a scene in which a ladbug moves around three mushrooms. The figure shows the starting position of the ladbug. The animator rotates the ladbug 180 around mushroom, then 180 around mushroom, and finall 180 around mushroom. What are the final coordinates of the ladbug? (, -); the 180 rotation around moves the ladbug from - (-, ) to (0, ); the 180 rotation around moves the ladbug from (0, ) to (, 0); the 180 rotation around moves the 0 - ladbug from (, 0) to (, -). 1. Determine whether each statement about the rotation (, ) (, -) is true or false. Select the correct answer for each lettered part. a. Ever point in Quadrant I is mapped to a point in Quadrant II. True b. oints on the -ais are mapped to points on the -ais. False True False c. The origin is a fied point True under the rotation. False Module 17 IN1_MNLESE38976_U7M17L3.indd d. The rotation has the same effect as a 90 clockwise rotation. True False e. The angle of rotation is 180. True False f. point on the line = is mapped to another point on the line =. True False Image redits: (t) Yuttasak Jannarong/Shutterstock; (b) Joe Sohm/Visions of merica/hotodisc/gett Images 6 Lesson 3 9/16/1 1:1 M Rotations 868

13 JOURNL Have students list some everda eamples of rotations and how the are used, and then describe how a rotation is like the original object and how it is different. H.O.T. Focus on Higher Order Thinking. ommunicate Mathematical Ideas Suppose ou are given a figure and a center of rotation. Describe two different was ou can use a ruler and protractor to draw the image of the figure after a 10 counterclockwise rotation around. ossible answer: (1) Use the ruler and protractor to draw a 150 clockwise rotation of the figure. () First draw a 180 rotation of the figure. Then draw a 30 counterclockwise rotation of the image. 3. Eplain the Error Kevin drew the image of after a rotation of 85 around point. Eplain how ou can tell from the figure that he made an error. Describe the error. ossible answer: '' ' should be rotated so that ' is at the top of the figure. fter correctl locating the image of point, Kevin translated the figure rather than rotating it. 85. ritique Reasoning Isabella said that all points turn around the center of rotation b the same angle, so all points move the same distance under a rotation. Do ou agree with Isabella s statement? Eplain. No; although all points rotate through the same angle, points closer to the center of rotation move a shorter distance than points father from the center of rotation. 5. Look for a attern Isaiah uses software to draw DEF as shown. Each time he presses the left arrow ke, the software rotates the figure on the screen 90 counterclockwise. Eplain how Isaiah can determine which quadrant the triangle will lie in if he presses the left arrow ke n times. Sample answer: Make a table that shows the quadrant the triangle will lie in for various values of n. F E D - 0 n Quadrant II III IV I II III IV I The remainder after dividing n b defines a pattern for the table. remainder of 0 QI remainder of QIII remainder of 1 QII remainder of 3 QIV Module Lesson Lesson 17.3

14 Lesson erformance Task tourist in London looks up at the clock in ig en tower and finds that it is eactl 8:00. When she looks up at the clock later, it is eactl 8:10. a. Through what angle of rotation did the minute hand turn? Through what angle of rotation did the hour hand turn? b. Make a table that shows different amounts of time, from 5 minutes to 60 minutes, in 5-minute increments. For each number of minutes, provide the angle of rotation for the minute hand of a clock and the angle of rotation for the hour hand of a clock. a. complete rotation around the face of the clock is 360. The face of the clock is divided into 1 equal parts, each representing 5-minute intervals. So the angle of rotation of the minute hand is = 30 for ever 5 minutes. During an interval of 10 minutes, the angle of rotation of the minute hand is 60. In one hour, the hour hand moves from one number on the face of the clock to the net, which is an angle of rotation of 30. Since an hour is 60 minutes, a 10-minute interval represents 1_ of this angle of rotation, or 5. 6 b. s above, the angle of rotation of the minute hand is 30 for ever 5 minutes. The angle of rotation of the hour hand is 5 for ever 10 minutes, or.5 for ever 5 minutes. Use these values to complete the table below. mount of Time (min) ngle of Rotation, Minute Hand ngle of Rotation, Hour Hand Image redits: ndrew Ward/Life File/hotodisc/Gett Images INTEGRTE MTHEMTIL RTIES Focus on Reasoning M. sk students to look at the table the made for the angles of rotation of the hour hand and minute hand of ig en. sk: How, if at all, would the table change if ou made a similar table for a small pocket watch? Eplain our reasoning. The table would not change. complete rotation around the face of an clock or an circle, no matter how big or how small, equals 360. The values in the table are based on that fact, not on the size of the circle. QUESTIONING STRTEGIES Have students refer to their angles of rotation tables. When the amount of time doubles, how are the angles of rotation affected? The double. When the amount of time increases b 15 minutes, how are the angles of rotation affected? The angle of rotation of the minute hand increases b 90. The angle of rotation of the hour hand increases b 7.5. If ou know the angle of rotation of the minute hand, how can ou find the angle of rotation of the hour hand? Divide the angle of rotation of the minute hand b 1. Module Lesson 3 EXTENSION TIVITY ig en isn t the biggest clock in the world. Have students conduct research to find five clocks bigger than ig en. sk them to present their findings as five congruent circles drawn to scale, with labels detailing the sizes of the clocks, and giving interesting facts about each. Students should include information about the angles formed on each clock as the time changes. Have students consider how often during a 1-hour period the hour and minute hands of each clock are at a 180 degree angle to each other. The can then calculate or estimate one or more times that this would occur, other than at 6 o clock. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Rotations 870