Geometry

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Geometry

• Chapter 11

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Informal Study of Shape

• Until about 600 B.C. geometry was pursued in
  response to practical, artistic and religious
  needs. Considerable knowledge of geometry was
  accumulated, but mathematics was not yet an
  organized and independent discipline.
• Beginning in about 600 B.C. Pythagoras, Euclid,
  Thales, Zeno, Eudoxus and others began organizing
  the knowledge accumulated by experience and
  transformed geometry into a theoretical science.
• NOTE that the formality came only AFTER the
  informality of experience in practical, artistic
  and religious settings!
• In this class, we return to learning by trusting
  our intuition and experience. We will discover
  by exploring using picture representations and
  physical models.

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Informal Study of Shape

• Shape is an undefined term.
• New shapes are being discovered all the time.
• FRACTALS

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Informal Study of Shape

• Our goals are
• To recognize differences and similarities among
  shapes
• To analyze the properties of a shape or class of
  shapes
• To model, construct and draw shapes in a variety
  of ways.

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NCTM Standard Geometry in Grades Pre-K-2

• Children begin forming concepts of shape long
  before formal schooling. They recognize shape by
  its appearance through qualities such as
  pointiness. They may think that a shape is a
  rectangle because it looks like a door.
• Young children begin describing objects by
  talking about how they are the same or how they
  are different. Teachers will then help them to
  gradually incorporate conventional terminology.
  Children need many examples and nonexamples to
  develop and refine their understanding.
• The goal is to lay the foundation for more formal
  geometry in later grades.

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• 
  

7

• Point
• Line
• Collinear
• Plane

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• If two lines intersect, their intersection is a
  point, called the point of intersection.
• Parallel Lines

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• Concurrent

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• Skew Lines nonintersecting lines that are not
  parallel.

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• Line segment
• Endpoint
• Length

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• Congruent

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• Midpoint

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• Half Line
• A point separates a line into 3 disjoint sets
• The point, and 2 half lines.

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• Ray - the union of a half line and the point.

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• Angle the union of two rays with a common
  endpoint.

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• Vertex W Common endpoint of the two rays.
• Sides

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• The angle separates the plane into 3 disjoint
  sets The angle, the interior of the angle, and
  the exterior of the angle.

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• Degrees
• Protractor

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• Zero Angle 0
• Straight Angle 180
• Right Angle 90

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• Acute Angle between 0 and 90
• Obtuse Angle between 90 and 180

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• Reflex Angle

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• Perpendicular Lines

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Adjacent Angles
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Adjacent Angles
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Vertical Angles
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Vertical Angles
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• The sum of the measures of Complementary Angles
  is 90.

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• Complementary angles
• Adjacent complementary angles

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The sum of the measures of Supplementary Angles
is 180.
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• Supplementary Angles
• Adjacent Supplementary Angles

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• Lines cut by a Transversal these lines are not
  concurrent.

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• Transversal
• Corresponding Angles

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• Transversal
• Corresponding Angles

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• Parallel lines Cut by a Transversal

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• Parallel lines Cut by a Transversal
• Corresponding Angles

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• Parallel lines Cut by a Transversal
• Corresponding Angles

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• Describe the relative position of angles 3 and 5.
• What appears to be true about their measures?

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• Alternate Interior Angles

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• Describe the relative positions of angles 1 and
  7.
• What appears to be true about their measures?

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• Alternate Exterior Angles

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Triangle
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• The sum of the measure of the interior angles of
  any triangle is 180.

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Exterior Angle
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• The measure of the exterior angle of a triangle
  is equal to the sum of the measure of the two
  opposite interior angles.

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Note Homework Page 672 37
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DAY 2
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Homework QuestionsPage 667
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• Curve

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• Curve
• Simple Curve

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• Curve
• Closed Curve

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• Curve
• Simple Curve
• Closed Curve
• Simple Closed Curve

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• A simple closed curved divides the plane into 3
  disjoint sets The curve, the interior, and the
  exterior.

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Jordans Curve Theorem
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Jordans Curve Theorem
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Jordans Curve Theorem
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• Concave
• Convex

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• Polygonal Curve

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• Polygon Simple, closed curve made up of line
  segments. (A simple closed polygonal curve.)

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Classifying Polygons

• Polygons are classified according to the number
  of sides.

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Classifying Polygons

• TRIANGLE 3 sides
• QUADRILATERAL 4 sides
• PENTAGON 5 sides
• HEXAGON 6 sides
• HEPTAGON 7 sides
• OCTAGON 8 sides
• NONAGON 9 sides
• DECAGON 10 sides

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Classifying Polygons

• A polygon with n sides is called an n-gon
• So a polygon with 20 sides is called a 20-gon

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Classifying Triangles

• According to the measure of the angles.
• According to the length of the sides.

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Classifying Triangles

• According to the measure of the angles.
• Acute Triangle A triangle with 3 acute angles.
• Right Triangle A triangle with 1 right angle
  and 2 acute angles.
• Obtuse Triangle A triangle with 1 obtuse angle
  and 2 acute angles.

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Classifying Triangles

• According to the length of the sides.
• Equilateral All sides are congruent.
• Isosceles At least 2 sides are congruent.
• Scalene None of the sides are congruent.

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Classifying Quadrilaterals

• Trapezoid Quadrilateral with at least one pair
  of parallel sides.

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Classifying Quadrilaterals

• Trapezoid Quadrilateral with at least one pair
  of parallel sides.
• Parallelogram A Quadrilateral with 2 pairs of
  parallel sides.

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Classifying Quadrilaterals

• Trapezoid Quadrilateral with at least one pair
  of parallel sides.
• Parallelogram A Quadrilateral with 2 pairs of
  parallel sides.
• Rectangle A Quadrilateral with 2 pairs of
  parallel sides and 4 right angles.

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Classifying Quadrilaterals

• Trapezoid Quadrilateral with at least one pair
  of parallel sides.
• Parallelogram Quadrilateral with 2 pairs of
  parallel sides.
• Rectangle Quadrilateral with 2 pairs of
  parallel sides and 4 right angles.
• Rhombus Quadrilateral with 2 pairs of parallel
  sides and 4 congruent sides.

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Classifying Quadrilaterals

• Trapezoid Quadrilateral with at least one pair
  of parallel sides.
• Parallelogram Quadrilateral with 2 pairs of
  parallel sides.
• Rectangle Quadrilateral with 2 pairs of
  parallel sides and 4 right angles.
• Rhombus Quadrilateral with 2 pairs of parallel
  sides and 4 congruent sides.
• Square Quadrilateral with 2 pairs of parallel
  sides, 4 right angles, and 4 congruent sides.

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• Equilateral
• All sides are congruent
• Equiangular
• Interior angles are congruent

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Figure 11.20, Page 689

• Regular Polygons are equilateral and equiangular.

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• Interior Angles

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• Interior Angles
• Exterior Angles The sum of the measures of the
  exterior angles of a polygon is 360.

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• Interior Angles
• Exterior Angles
• Central Angles The sum of the measure of the
  central angles in a regular polygon is 360.

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• Interior Angles
• Exterior Angles
• Central Angles

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Classifying Angles Lab
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Day 3
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• Circle
• Compass

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• Center

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• Radius

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• Chord

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• Diameter

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• Circumference

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• Tangent

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• Circle
• Compass
• Center
• Radius
• Chord
• Diameter
• Circumference
• Tangent

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Find and Identify

• 1. E 2. K
• 3. I 4. A
• 5. C 6. M
• 7. B 8. J
• 9. D 10. F
• 11. G 12. H
• 13. L

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Classifying Angles Lab
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Whats Inside?
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How do you find the sum of the measure of the
interior angles of a polygon?
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Example 11.8Page 679
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Example 11.9Page 680
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Classifying Quadrilateralsand Geo-Lingo Lab
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Day 4
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Make a Square!

• Tangrams Ancient Chinese Puzzle
• Tangrams, 330 Puzzles, by Ronald C. Read

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Sir Cumference Books

• Sir Cumference and the First Round Table
• by Cindy Neuschwander
• Also
• Sir Cumference and the Great Knight of Angleland
• Sir Cumference and the Dragon of Pi
• Sir Cumference and the Sword Cone

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Angle Practice
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Must Cant May Answers
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Homework QuestionsPage 688
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• Space
• Half Space
• A plane separates space into 3 disjoint sets, the
  plane and 2 half spaces.

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• Parallel Planes
• Dihedral Angle
• Points of Intersection
• If two planes intersect, their intersection is a
  line.

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• Simple Closed Surface
• Figure 11.26, Page 698
• Solid
• Sphere
• Convex/Concave

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Polyhedron

• A POLYHEDRON (plural - polyhedra) is a simple
  closed surface formed from planar polygonal
  regions.
• Edges
• Vertices
• Faces
• Lateral Faces Page 699

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• Prism
• Pyramid
• Apex
• Cylinder
• Cone
• Apex

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• Right Prisms, Pyramids, Cylinders and Cones
• Oblique Prisms, Pyramids, Cylinders and Cones

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Regular Polyhedron

• A three-dimensional figure whose faces are
  polygonal regions is called a POLYHEDRON (plural
  - polyhedra).
• A REGULAR POLYHEDRON is one in which the faces
  are congruent regular polygonal regions, and the
  same number of edges meet at each vertex.

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Regular Polyhedron

• Polyhedron made up of congruent regular polygonal
  regions.
• There are only 5 possible regular polyhedra.

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Make Mine Platonic
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Make Mine Platonic
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• As the number of sides of a regular polygon
  increases, what happens to the measure of each
  interior angle? __
• Because they are formed from regular polygons,
  our search for regular polyhedra will begin with
  the simplest regular polygon, the equilateral
  triangle.
• Each angle in the equilateral triangle measures
  _____.

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• Use the net with 4 equilateral triangles to make
  a polyhedron.
• To make a three-dimensional object, we need to
  engage 3 planes. Therefore, we begin with three
  triangles at each vertex.

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• What is the sum of the measures of the angles at
  any given vertex? __
• This regular polyhedron is called a TETRAHEDRON.
  A tetrahedron has __ faces. Each face is an __
  __. We made this by joining __ __ at each
  vertex.

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• Form a polyhedron with the net that has 8
  equilateral triangles. You will join 4 triangles
  at each vertex.
• What is the sum of the measure of the angles at
  any given vertex? __
• This regular polyhedron is called an OCTAHEDRON.
  An octahedron has __ faces. Each face is an __
  __. At each vertex, there are __ __.

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• Use the net with 20 equilateral triangles to form
  a polyhedron. You will join 5 triangles at each
  vertex.
• What is the sum of the measure of the angles at
  any given vertex? __
• This regular polyhedron is called an ICOSAHEDRON.
  An icosahedron has __ faces. Each face is an
  __ __. At each vertex, there are __ __.

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• When we join 6 equilateral triangles at a vertex,
  what happens? Can you make a polyhedron with 6
  equilateral triangles at a vertex? __
• Is it possible to put more than 6 equilateral
  triangles at a vertex to form a polyhedron? __
• Name the only three regular polyhedra that can be
  made using congruent equilateral triangles
• __ __ __

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• A regular quadrilateral is most commonly known as
  a __.
• Each angle in the square measures __.
• Use the net with squares to make a polyhedron.
• To make a three-dimensional object, we need to
  engage 3 planes. Therefore, we begin with three
  squares at each vertex.

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• What is the sum of the measures of the angles at
  any given vertex? __
• This regular polyhedron is called a HEXAHEDRON.
  A hexahedron has __ faces. Each face is a __.
  At each vertex, there are __ __.

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• When we join 4 squares at a vertex, what happens?
  Can you make a polyhedron with 4 squares at a
  vertex? __
• Is it possible to put more than 4 squares at a
  vertex to form a polyhedron? __
• Name the only regular polyhedron that can be made
  using congruent squares. __

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• A five-sided regular polygon is called a __.
• Each interior angle measures __.
• Use net with regular pentagons to make a
  polyhedron. To make a three-dimensional object,
  we need to engage 3 planes. Therefore, we begin
  with three pentagons at each vertex.

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• What is the sum of the measures of the angles at
  any given vertex? __
• This regular polyhedron is called a DODECAHEDRON.
  A dodecahedron has __ faces. Each face is a __.
  At each vertex, there are __ __.

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• Is it possible to put 4 or more pentagons at a
  vertex and still have a three-dimensional object?
  __
• Name the only regular polyhedron that can be made
  using congruent pentagons. __

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• A six-sided regular polygon is called a __.
• Each interior angle measures __.
• Is it possible to put 3 or more hexagons at a
  vertex and still have a three-dimensional object?
  __

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• Is it possible to use any regular polygons with
  more than six sides together to form a regular
  polyhedron? __
• (Refer to the table on page one for numbers to
  verify)

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• Only five possible regular polyhedra exist. The
  union of a polyhedron and its interior is called
  a solid. These five solids are called PLATONIC
  SOLIDS.

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Day 5
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Homework QuestionsPage 709
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Konigsberg Bridge Problem
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Networks

• A network consists of vertices points in a
  plane, and edges curves that join some of the
  pairs of vertices.

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Traversable

• A network is traversable if you can trace over
  all the edges without lifting your pencil.

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Konigsberg Bridge Problem
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The network is traversable.
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Skit-So Phrenia!
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Seeing the Third Dimension
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Day 6
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Homework QuestionsPage 722
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Topology

• Topology is a study which concerns itself with
  discovering and analyzing similarities and
  differences between sets and figures.
• Topology has been referred to as rubber sheet
  geometry, or the mathematics of distortion.

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Euclidean Geometry

• In Euclidean Geometry we say that two figures are
  congruent if they are the exact same size and
  shape.
• Two figures are said to be similar if they are
  the same shape but not necessarily the same size.

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Topologically Equivalent

• Two figures are said to be topologically
  equivalent if one can be bended, stretched,
  shrunk, or distorted in such a way to obtain the
  other.

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Topologically Equivalent

• A doughnut and a coffee cup are topologically
  equivalent.

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• According to Swiss psychologist Jean Piaget,
  children first equate geometric objects
  topologically.

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Mobius Strip
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• We will consider 3 attributes that any two
  topologically equivalent objects will share
• Number of sides
• Number of edges
• Number of punctures or holes

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• Consider one strip of paper
• How many sides does it have?
• How many edges does it have?

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• Consider one strip of paper
• How many sides does it have? 2
• How many edges does it have? 1

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• Now make a loop with the strip of paper and tape
  the ends together.
• How many sides does it have?
• How many edges does it have?

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• Now make a loop with the strip of paper and tape
  the ends together.
• How many sides does it have? 2
• How many edges does it have? 2
• Now cut the loop in half down the center of the
  strip. Describe the result.

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Mobius Strip

• This time make a loop but before taping the ends
  together, make a half twist. This is called a
  Mobius Strip.
• How many sides does it have?
• How many edges does it have?

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Mobius Strip

• This time make a loop but before taping the ends
  together, make a half twist. This is called a
  Mobius Strip.
• How many sides does it have? 1
• How many edges does it have? 1
• Now cut the Mobius strip in half down the center
  of the strip. Describe the result.

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• How many sides does your result have?
• How many edges?

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• How many sides does your result have? 2
• How many edges? 2
• What do you think will happen if we cut the
  resulting strip in half down the center?
• Try it! What happened?

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• Make another Mobius strip
• Draw a line about 1/3 of the distance from the
  edge through the whole strip.
• What do you think will happen if we cut on this
  line?
• Try it! What happened?

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• Use your last two strips to make two untwisted
  loops, interlocking.
• Make sure they are taped completely
• Tape them together at a right angle. (They will
  look kind of like a 3 dimensional 8.)
• Cut both strips in half lengthwise.

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• Did you know that 2 circles make a square?

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• Compare the number of sides and edges of the
  strip of paper, the loop, and the Mobius strip.
• Are any of those topologically equivalent?