Chapter 7: Symmetry
Types of Symmetry We will be concerned with two types of symmetry.
Types of Symmetry We will be concerned with two types of symmetry. Reflective symmetry
Types of Symmetry We will be concerned with two types of symmetry. Reflective symmetry Rotational symmetry
Why Teach Symmetry? Children have an innate sense of symmetry, in that they look for balance and order in the real world naturally.
Why Teach Symmetry? Children have an innate sense of symmetry, in that they look for balance and order in the real world naturally. Learning about symmetry aids students in learning how to classify objects according to the arrangement of their constituent parts.
Why Teach Symmetry? Children have an innate sense of symmetry, in that they look for balance and order in the real world naturally. Learning about symmetry aids students in learning how to classify objects according to the arrangement of their constituent parts. Students learn concepts about geometric shapes at a very early age. They learn, first, about a shape as a whole, but, with the help of symmetry, they learn how to focus on the characteristics and parts of an object.
Where We See Symmetry
Where We See Symmetry
Where We See Symmetry
Where We See Symmetry
Where We See Symmetry
Where We See Symmetry
Where We See Symmetry
Where We See Symmetry
Where We See Symmetry
Reflective Symmetry Definition A figure has reflective symmetry if there is a line that the figure can be folded over so that one half of the figure matches the other half exactly.
Reflective Symmetry Definition A figure has reflective symmetry if there is a line that the figure can be folded over so that one half of the figure matches the other half exactly. We call the line the axis of symmetry. This line can have any orientation, horizontal, vertical, etc.
The Alphabet Place the capital letters of the alphabet into the correct location. Horizontal Symmetry Vertical Symmetry
The Alphabet Place the capital letters of the alphabet into the correct location. Z Q RS P N L J G F Horizontal Symmetry B C D E K H I O X A M T U V W Y Vertical Symmetry
Rotational Symmetry Definition A figure is said to have rotational symmetry if there is a center of rotation around which the figure can be rotated less than a full turn such that it matches the original figure exactly.
Rotational Symmetry Definition A figure is said to have rotational symmetry if there is a center of rotation around which the figure can be rotated less than a full turn such that it matches the original figure exactly. The rotational order is the number of times we rotate the figure before returning to the original orientation. The degree of rotation is the number of degrees the figure must be turned to look the same without turning a full 360.
Rotational Symmetry Definition A figure is said to have rotational symmetry if there is a center of rotation around which the figure can be rotated less than a full turn such that it matches the original figure exactly. The rotational order is the number of times we rotate the figure before returning to the original orientation. The degree of rotation is the number of degrees the figure must be turned to look the same without turning a full 360. If we have to turn 360, we say the figure has no rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Examples Example Determine how many lines of symmetry the figure has and the order of rotational symmetry.
Symmetry in 3-Dimensions Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron
Groups When we look at properties and characteristics of sets, there are some common ones we can look at: Closure
Groups When we look at properties and characteristics of sets, there are some common ones we can look at: Closure Associativity
Groups When we look at properties and characteristics of sets, there are some common ones we can look at: Closure Associativity Existence of unique identity element
Groups When we look at properties and characteristics of sets, there are some common ones we can look at: Closure Associativity Existence of unique identity element Existence of unique inverses
Groups When we look at properties and characteristics of sets, there are some common ones we can look at: Closure Associativity Existence of unique identity element Existence of unique inverses If all of the properties are satisfied, we have a group under a given operation.
Groups When we look at properties and characteristics of sets, there are some common ones we can look at: Closure Associativity Existence of unique identity element Existence of unique inverses If all of the properties are satisfied, we have a group under a given operation. Question is, do we have a group when we consider the set of rotations and reflections on a square under the order of composition?