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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-B;CA-B LESSON2 . 4 Name Class Date Investigating 2 . 4 Investigating Symmetry Symmetry Essential Question: How do you determine whether a figure has line symmetry or rotational symmetry? Resource Common Core Math Standards Locker 1 Explore 1 Identifying Line Symmetry The student is expected to: COMMON A figure has symmetry if a rigid motion exists that maps the figure onto itself. A figure Line of CORE G-CO.A.3 has line symmetry (or reflectional symmetry) if a reflection maps the figure onto itself. symmetry Given a rectangle, parallelogram, trapezoid, or regular polygon, describe Each of these lines of reflection is called a line of symmetry. the rotations and reflections that carry it onto itself. You can use paper folding to determine whether a figure has line symmetry. Mathematical Practices Trace the figure on a piece of tracing paper. COMMON CORE MP.7 Using Structure Language Objective Have students work with a partner to give clues about a figure, and identify whether figures have line symmetry, rotational symmetry, or both and draw the line(s) of symmetry. If the figure can be folded along a straight line so that one half of the figure exactly matches ENGAGE the other half, the figure has line symmetry. The crease is the line of symmetry. Place your shape against the original figure to check that each crease is a line of symmetry. Essential Question: How do you determine whether a figure has line symmetry or rotational symmetry? Possible answer: To identify line symmetry, look for a line of reflection, which is a line that divides the figure into mirror-image halves. To identify rotational symmetry, think of the figure rotating Sketch any lines of symmetry on the figure. one around its center. The figure has rotational The figure has line of symmetry. symmetry if a rotation of at most 180° produces the original figure. Mifflin Houghton © Company Harcourt Publishing PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the photo of the Module 2 through “File info” 101 Lesson 4 DO NOT EDIT--Changes must be made CorrectionKey=NL-B;CA-B Date Class flower with students. Consider whether you could Name Symmetry 2 . 4 Investigating whether a figure has line symmetry Resource Locker Essential Question: orHow rotational do you determinesymmetry? ezoid, or regular polygon, describe the rotations HARDCOVER PAGES 8996 COMMON CORE G-CO.A.3 Given a rectangle, parallelogram, trap and reflections that carry it onto itself. Line of turn the flower and have it still appear the same. to itself. A figure symmetry Identifying Line Symmetry 1 Explore 1 if a rigid motion exists) if that a reflection maps the maps figure the on figure onto itself. A figure has symmetry of symmetry. e symmetry. has line symmetry (or reflectional symmetry GE_MNLESE385795_U1M02L4 101 Each of these lines of reflection is calledetermine a line whether a figure has lin 5/14/14 7:32 PM You can use paper folding to d Then preview the Lesson Performance Task. tracing paper. Trace the figure on a piece of Turn to these pages to find this lesson in the f of the figure exactly matches f symmetry. Place your long a straight line so that one hal symmetry. The crease is the linee o of symmetry. If the figure can be folded a the other half, the figure has lineure to check that each crease is a lin hardcover student shape against the original fig edition. try on the figure. Sketch any lines of symme one line of symmetry. The figure has © Houghton Mifflin Harcourt Publishing Company Publishing Harcourt Mifflin Houghton © Lesson 4 101 5/14/14 7:33 PM Module 2 02L4 101 GE_MNLESE385795_U1M 101 Lesson 2 . 4 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A CorrectionKey=NL-A;CA-A Draw the lines of symmetry, if any, on each figure and tell the total number of lines of symmetry each figure has. EXPLORE 1 Figure Identifying Line Symmetry How many lines two zero one of symmetry? INTEGRATE TECHNOLOGY Reflect Have students use geometry software or cut out 1. What do you have to know about any segments and angles in a figure to figures to examine the symmetry of regular polygons. decide whether the figure has line symmetry? Pairs of segments in the figure must have the same length and pairs of angles must have Then have them use inductive reasoning to make the same measure, so that one half of the figure will coincide with the other half when the conjectures about the number of lines of symmetry a figure is folded across a line of symmetry. regular n-gon has. n lines of symmetry A circle 2. What figure has an infinite number of lines of symmetry? 3. Discussion A figure undergoes a rigid motion, such as a rotation. If the figure has line QUESTIONING STRATEGIES symmetry, does the image of the figure have line symmetry as well? Give an example. Yes. The line of symmetry also undergoes the rigid motion. For example, if the L-shape in What are the three rigid motions explained Step D is rotated into a V-shape, the line of symmetry is rotated the same way. in this module? What does a rigid motion transformation preserve? translation, reflection, 1 Explore 2 Identifying Rotational Symmetry rotation; shape and size A figure has rotational symmetry if a rotation maps the figure onto itself. The angle of rotational symmetry, which is greater than 0° but Angle of rotational less than or equal to 180°, is the smallest angle of rotation that maps a symmetry: 72° figure onto itself. EXPLORE 2 An angle of rotational symmetry is a fractional part of 360°. Notice that 360 ____ ° Mifflin Houghton © Company Harcourt Publishing every time the 5-pointed star rotates 5 = 72°, the star coincides with itself. The angles of rotation for the star are 72°, 144°, 216°, and 288°. If Identifying Rotational Symmetry a copy of the figure rotates to exactly match the original, the figure has rotational symmetry. Trace the figure onto tracing paper. Hold the center of the traced figure against the original INTEGRATE TECHNOLOGY figure with your pencil. Rotate the traced figure counterclockwise until it coincides again with the original figure beneath. Ask students to discuss the pros and cons of using geometry software to investigate properties of rotations and symmetry. Be sure students recognize that such software has the advantage of making it easy to change parameters (such as the angle of rotation) so that they can observe the effects of the 120 By how many degrees did you rotate the figure? ° changes. 120 , 240 What are all the angles of rotation? ° ° Module 2 102 Lesson 4 QUESTIONING STRATEGIES PROFESSIONAL DEVELOPMENT When you are testing a figure to see if it has GE_MNLESE385795_U1M02L4.indd 102 3/20/14 12:35 PM rotational symmetry, where is P, the center of Math Background rotation? at the center of the figure A wallpaper pattern is a planar repeating pattern. Mathematicians classify wallpaper patterns based on the symmetries they exhibit, for example, only translation symmetry; translation and reflection symmetry; or all these plus rotational symmetry. Every wallpaper pattern can be classified by identifying its symmetries. Surprisingly, there are precisely 17 different classifications. That is, any repeating pattern that covers a plane can be reduced to one of 17 basic types. This unusual mathematical fact has had far-reaching applications in a number of fields, including chemistry and crystallography. Investigating Symmetry 102 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Determine whether each figure has rotational symmetry. If so, identify all the angles of EXPLAIN 1 rotation less than 360°. Describing Symmetries Figure Angles of rotation 90°, 180°, 270° none 180° INTEGRATE TECHNOLOGY less than 360° Human faces appear to have symmetry, but most Reflect people’s faces aren’t perfectly symmetric. Photocopy a A circle 4. What figure is mapped onto itself by a rotation of any angle? picture of a face onto two transparencies and cut each 5. Discussion A figure is formed by line l and line m, which intersect at an angle of 60°. Does the figure one down the center of the face. Flip the pieces of one have an angle of rotational symmetry of 60°? If not, what is the angle of rotational symmetry? No, the angle of rotational symmetry for the figure is 180 . A rotation of 60 about the transparency over and put the two left sides together ° ° intersection will only map one of the lines onto the other line. and the two right sides together to create two different faces with perfect symmetry. Discuss with students how to tell if a figure has symmetry. Explain 1 Describing Symmetries A figure may have line symmetry, rotational symmetry, both types of symmetry, or no symmetry. Example 1 Describe the symmetry of each figure. Draw the lines of symmetry, name QUESTIONING STRATEGIES the angles of rotation, or both if the figure has both. How can you find the center point of a regular polygon? The center is the point that is equidistant from each vertex or corner. AVOID COMMON ERRORS Step 1 Begin by finding the line symmetry of the figure.
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