Grades 5 - 8

What is Symmetry?What is Symmetry?• Symmetrical = Same on both sidesSymmetrical = Same on both sides

• Symmetry: balanced proportions, or Symmetry: balanced proportions, or capable of division by a capable of division by a

longitudinal longitudinal plane into similar plane into similar halveshalves

• Symmetries preserve distances, Symmetries preserve distances, angles, angles, sizes, and shapessizes, and shapes

Basic Types of Basic Types of SymmetrySymmetry

Vertical Line

Horizontal Line

Line Symmetry: when one half of an image is the mirror image of the other half.

Where can we find Where can we find symmetry?symmetry?• Symmetry can be found in art, nature, sports, math, and even in the mirror.

How many lines of symmetry can you find in these pictures?

• Types of symmetry that transform Types of symmetry that transform the the plane the object is onplane the object is on

• Four main types of Euclidean PlanesFour main types of Euclidean Planes•TranslationTranslation•RotationRotation•ReflectionReflection•Glide ReflectionGlide Reflection

• Similar concepts can be found in 3-D Similar concepts can be found in 3-D or or spatial symmetries also. spatial symmetries also.

The Euclidean PlaneThe Euclidean Plane

Translation:Translation:• To translate an object means to move it To translate an object means to move it

without rotating or reflecting it. without rotating or reflecting it.

• Every translation has a direction and a Every translation has a direction and a distance.distance.

• Example of Translation:Example of Translation:

– If a family of ducks all lined up in a row is If a family of ducks all lined up in a row is moved forward or back by one, two, or more moved forward or back by one, two, or more ducks they have translated on the plane. This ducks they have translated on the plane. This translation does not change the appearance of translation does not change the appearance of the procession, it just moves it on the plane. the procession, it just moves it on the plane.

Rotation:Rotation:• To rotate an object means to turn it To rotate an object means to turn it

around.around.

• Every rotation has a center and an angle. Every rotation has a center and an angle.

Before Rotation

After Rotation

Center

[Angle 90˚]

• Rotational Symmetry:Rotational Symmetry:– The kaleidoscope image of a flower below as well as The kaleidoscope image of a flower below as well as

the wheel are both examples of rotational symmetry. the wheel are both examples of rotational symmetry. They can rotate around the center point up to 360 They can rotate around the center point up to 360 degrees and still look exactly the same. degrees and still look exactly the same.

– What other items can you think of that have rotational What other items can you think of that have rotational symmetry? You might be surprised how many there symmetry? You might be surprised how many there are. are.

The object does not have to rotate 360 degrees to be rotational symmetry, as long as it has a center that it rotates around and an angle or amount of rotation, it is considered rotational.

ReflectionReflection::• To reflect an object means to produce its mirror To reflect an object means to produce its mirror

image. Every reflection has a mirror line. image. Every reflection has a mirror line.

• A reflection of a “B" is a backwards “B". A reflection of a “B" is a backwards “B".

Mirror Line

• Reflection:Reflection:– Most common form of symmetryMost common form of symmetry– Reflection symmetry is formed when any Reflection symmetry is formed when any

image is reflected or a mirror of itself on image is reflected or a mirror of itself on either side of the mirror line. either side of the mirror line.

The Taj Mahal has both horizontal and vertical reflection symmetry.

The Vitruvian Man by Leonardo DaVinci has vertical reflection symmetry.

Glide Reflection:Glide Reflection:• A glide reflection combines a A glide reflection combines a reflectionreflection with a with a

translationtranslation along the direction of the mirror line. along the direction of the mirror line. • Glide reflections are the only type of symmetry Glide reflections are the only type of symmetry

that that involve more than one step. involve more than one step.

[Before Glide Reflection]

[After Glide Reflection]

Mirror Line

Symmetry in 2D Shapes:Symmetry in 2D Shapes:

• In shapes such as the In shapes such as the triangle, square, triangle, square, pentagon, etc. there are pentagon, etc. there are approximately the same approximately the same number lines of number lines of symmetry as there are symmetry as there are sides of the shape. To sides of the shape. To test this cut out shapes test this cut out shapes and fold them to find and fold them to find how many lines of how many lines of symmetry there really symmetry there really are. are.

Exercises to Extend Exercises to Extend Learning:Learning:• Take a shape and translate it forward or backward on a plane, Take a shape and translate it forward or backward on a plane,

example move it forward 2 inches, or back 2 inches. Repeat this example move it forward 2 inches, or back 2 inches. Repeat this process, only use glide reflection instead of translation. See if you process, only use glide reflection instead of translation. See if you are able to create a symmetrical pattern by repeating this are able to create a symmetrical pattern by repeating this technique several times. technique several times.

• Take paper and fold it in half repeatedly until the desired size. Take paper and fold it in half repeatedly until the desired size. Then cut shapes and designs into it (make a snowflake). Unfold Then cut shapes and designs into it (make a snowflake). Unfold and observe the rotational symmetry, then discuss the rotational and observe the rotational symmetry, then discuss the rotational symmetry created.symmetry created.

• Using the letters of the alphabet, see how many fall under each Using the letters of the alphabet, see how many fall under each category of the Euclidean Planes. category of the Euclidean Planes.

• Complete the words shown in the image. How were you able to Complete the words shown in the image. How were you able to know what letters were appearing there?know what letters were appearing there?

References:References:

• Clip art and Animated Graphics. Clip art and Animated Graphics. http://www.adrianbruce.com/Symmetry/

• Clip Art. Clip Art. Microsoft Office Clip Art GalleryMicrosoft Office Clip Art Gallery• "Euclidean plane isometry." "Euclidean plane isometry." Wikipedia, The Free EncyclopediaWikipedia, The Free Encyclopedia. 17 Mar 2008, . 17 Mar 2008,

16:40 UTC. Wikimedia Foundation, Inc. 12 May 2008 <16:40 UTC. Wikimedia Foundation, Inc. 12 May 2008 <http://en.wikipedia.org/w/index.php?title=Euclidean_plane_isometry&oldid=198885950>.>.