8cmp06se KH1 qxd 6 8 06 7 16 AM Page 5 1 Three Types of Symmetry When part of a design is repeated to make a balanced pattern we say the design has symmetry Artists use symmetry to make designs that are pleasing to the eye Architects use symmetry to produce a sense of balance in their buildings Symmetry is also a feature of animals plants and mechanical objects The butterfly fan and ribbon below illustrate three kinds of symmetry Getting Ready for Problem 1 1 What part of each design is repeated to make a balanced pattern that allows us to say the three figures have symmetry How do the figures suggest different kinds of symmetry Investigation 1 Three Types of Symmetry 5 8cmp06se KH1 qxd 6 8 06 7 16 AM Page 6 1 1 Reflection Symmetry You have probably made simple heart shapes by folding and cutting paper as shown below The resulting heart shape has reflection symmetry which is sometimes called mirror symmetry or line symmetry The fold shows the line of symmetry A line of symmetry divides a figure into halves that are mirror images If you place a mirror on a line of symmetry you will see half of the figure reflected in the mirror The combination of the half figure and its reflection will have the same size and shape as the original figure You can use a mirror to check a design for symmetry and to locate the line of symmetry You can also use tracing paper to check for reflection symmetry Trace the figure and the possible line of symmetry Then reflect the tracing over the possible line of symmetry If the reflected tracing fits exactly on the original figure the figure has reflection symmetry What happens to the line of symmetry when you reflect the tracing and match it with the original figure Does its location change 6 Kaleidoscopes Hubcaps and Mirrors 8cmp06se KH1 qxd 6 8 06 7 16 AM Page 7 Problem 1 1 Reflection Symmetry Use a mirror tracing paper or other tools to find all lines of symmetry in each design or figure A B C D E F G H Homework starts on page 15 Investigation 1 Three Types of Symmetry 7 8cmp06se KH1 qxd 6 8 06 7 16 AM Page 8 Rotation Symmetry 1 2 The pinwheel design at the right does not have reflection symmetry However it can be turned less than a full turn around its center point in a counterclockwise direction to positions in which it looks the same as it does in its original position Figures with this property are said to have rotation symmetry The windmill snowflake and wagon wheel pictured below also have rotation symmetry Which two of the three objects pictured above also have reflection symmetry To describe the rotation symmetry in a figure you need to specify two things The center of rotation This is the fixed point about which you rotate the figure The angle of rotation This is the smallest angle through which you can turn the figure in a counterclockwise direction so that it looks the same as it does in its original position There are several rotation angles that move the pinwheel design above to a position where it looks like the original In this problem you will consider how these angles are related to the angle of rotation 8 Kaleidoscopes Hubcaps and Mirrors 8cmp06se KH1 qxd 6 8 06 7 16 AM Page 9 Problem 1 2 Rotation Symmetry A List all the turns of less than 360 that will rotate the pinwheel design to a position in which it looks the same as what is pictured What is the angle of rotation for the pinwheel design B In parts 1 3 list all the turns of less than 360 that will rotate the object to a position in which it looks the same as what is pictured Then give the angle of rotation 1 the windmill 2 the snowflake 3 the wagon wheel C Look at your answers for Questions A and B For each object or figure tell how the listed angles are related to the angle of rotation D The hubcaps below have rotation symmetry Complete parts 1 and 2 for each hubcap Hubcap 1 Hubcap 2 1 On a copy of the hubcap mark the center of rotation Then find all the turns of less than 360 that will rotate the hubcap to a position in which it looks the same as what is pictured 2 Tell whether the hubcap has reflection symmetry If it does draw all the lines of symmetry E Draw a hubcap design that has rotation symmetry with a 90 angle of rotation but no reflection symmetry F Draw a hubcap design that has rotation symmetry with a 60 angle of rotation and at least one line of symmetry For Hubcap Maker Visit PHSchool com Web Code apd 5102 G Investigate whether rectangles and parallelograms have rotation symmetry Make sketches For the shape s with rotation symmetry give the center and angle of rotation Homework starts on page 15 Investigation 1 Three Types of Symmetry 9 8cmp06se KH1 qxd 6 8 06 7 16 AM Page 10 1 3 Symmetry in Kaleidoscope Designs A kaleidoscope kuh ly duh skohp is a tube containing colored beads or pieces of glass and carefully placed mirrors When you hold a kaleidoscope up to your eye and turn the tube you see colorful symmetric patterns The kaleidoscope was patented in 1817 by the Scottish scientist Sir David Brewster Brewster was intrigued by the science of nature He developed kaleidoscopes to simulate the designs he saw in the world around him M A TI M E H Five of the designs below are called kaleidoscope designs because they are similar to designs you would see if you looked through a kaleidoscope M A TI H ME M A ME TI H 10 A B C D E F Kaleidoscopes Hubcaps and Mirrors 8cmp06se KH1 qxd 6 8 06 7 16 AM Page 11 Problem 1 3 Analyzing Symmetries Use what you know about reflection and rotation symmetry to analyze the six designs A Locate all the lines of symmetry in the designs B Give the angles of rotation for the designs with rotation symmetry C 1 Make a table showing the number of lines of symmetry and the angle of rotation for each design 2 What relationship if any do you see between the number of lines of symmetry and the angle of rotation 3 Analyze the kaleidoscope design below to see whether it confirms your relationship D Each of the designs can be made by repeating a small piece of the design We call this piece the basic design element For each design sketch or outline the basic design element E One of the designs is not a kaleidoscope design That is it is not similar to a design you would see if you looked through a kaleidoscope Which design do you think it is Why Homework starts on page 15 Investigation 1 Three Types of Symmetry 11 8cmp06se KH1 qxd 6 8 06 7 16 …