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Geometry. Chapter 11. 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.

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geometry

Geometry

Chapter 11

informal study of shape
InformalStudy 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.
informal study of shape3
Informal Study ofShape
  • Shape is an undefined term.
  • New shapes are being discovered all the time.
  • FRACTALS
informal study of shape4
InformalStudyof 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.
nctm standard geometry in grades pre k 2
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.

slide7
Point
  • Line
  • Collinear
  • Plane
slide8
If two lines intersect, their intersection is a point, called the point of intersection.
  • Parallel Lines
slide11
Line segment
  • Endpoint
  • Length
slide14
Half Line
  • A point separates a line into 3 disjoint sets:

The point, and 2 half lines.

slide18
The angle separates the plane into 3 disjoint sets: The angle, the interior of the angle, and the exterior of the angle.
slide19
Degrees
  • Protractor
slide20
Zero Angle: 0°
  • Straight Angle: 180°
  • Right Angle: 90°
slide21
Acute Angle: between 0° and 90°
  • Obtuse Angle: between 90° and 180°
slide29
Complementary angles
  • Adjacent complementary angles
slide31
Supplementary Angles
  • Adjacent Supplementary Angles
slide33
Transversal
  • Corresponding Angles
slide34
Transversal
  • Corresponding Angles
slide38
Describe the relative position of angles 3 and 5.
  • What appears to be true about their measures?
slide40
Describe the relative positions of angles 1 and 7.
  • What appears to be true about their measures?
slide46
The measure of the exterior angle of a triangle is equal to the sum of the measure of the two opposite interior angles.
slide54
Curve
  • Simple Curve
slide55
Curve
  • Closed Curve
slide56
Curve
  • Simple Curve
  • Closed Curve
  • Simple Closed Curve
slide57
A simple closed curved divides the plane into 3 disjoint sets: The curve, the interior, and the exterior.
slide62
Concave
  • Convex
classifying polygons
Classifying Polygons
  • Polygons are classified according to the number of sides.
classifying polygons66
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
classifying polygons67
Classifying Polygons
  • A polygon with n sides is called an “n-gon”
  • So a polygon with 20 sides is called a “20-gon”
classifying triangles
Classifying Triangles
  • According to the measure of the angles.
  • According to the length of the sides.
classifying triangles69
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.
classifying triangles70
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.
classifying quadrilaterals
Classifying Quadrilaterals
  • Trapezoid – Quadrilateral with at least one pair of parallel sides.
classifying quadrilaterals75
Classifying Quadrilaterals
  • Trapezoid – Quadrilateral with at least one pair of parallel sides.
  • Parallelogram – A Quadrilateral with 2 pairs of parallel sides.
classifying quadrilaterals77
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.
classifying quadrilaterals79
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.
classifying quadrilaterals81
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.
slide83
Equilateral
    • All sides are congruent
  • Equiangular
    • Interior angles are congruent
figure 11 20 page 689
Figure 11.20, Page 689
  • Regular Polygons are equilateral and equiangular.
slide86
Interior Angles
  • Exterior Angles – The sum of the measures of the exterior angles of a polygon is 360°.
slide87
Interior Angles
  • Exterior Angles
  • Central Angles – The sum of the measure of the central angles in a regular polygon is 360°.
slide88
Interior Angles
  • Exterior Angles
  • Central Angles
slide91
Circle
  • Compass
slide98
Circle
  • Compass
  • Center
  • Radius
  • Chord
  • Diameter
  • Circumference
  • Tangent
find and identify
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

make a square
Make a Square!

Tangrams – Ancient Chinese Puzzle

Tangrams, 330 Puzzles, by Ronald C. Read

sir cumference books
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
slide121
Space
  • Half Space
  • A plane separates space into 3 disjoint sets, the plane and 2 half spaces.
slide122
Parallel Planes
  • Dihedral Angle
  • Points of Intersection
  • If two planes intersect, their intersection is a line.
slide123
Simple Closed Surface

Figure 11.26, Page 698

  • Solid
  • Sphere
  • Convex/Concave
polyhedron
Polyhedron
  • A POLYHEDRON (plural - polyhedra) is a simple closed surface formed from planar polygonal regions.
  • Edges
  • Vertices
  • Faces
  • Lateral Faces – Page 699
slide125
Prism
  • Pyramid
  • Apex
  • Cylinder
  • Cone
  • Apex
slide126
Right Prisms, Pyramids, Cylinders and Cones
  • Oblique Prisms, Pyramids, Cylinders and Cones
regular polyhedron
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.
regular polyhedron128
Regular Polyhedron
  • Polyhedron made up of congruent regular polygonal regions.
  • There are only 5 possible regular polyhedra.
slide131
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 _____.
slide132
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.
slide133
What is the sum of the measures of the angles at any given vertex? __
  • This regular polyhedron is called aTETRAHEDRON. A tetrahedron has __ faces. Each face is an __ __. We made this by joining __ __ at each vertex.
slide134
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 __ __.
slide135
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 __ __.
slide136
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:

__ __ __

slide137
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.
slide138
What is the sum of the measures of the angles at any given vertex? __
  • This regular polyhedron is called aHEXAHEDRON. A hexahedron has __ faces. Each face is a __. At each vertex, there are __ __.
slide139
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. __
slide140
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.
slide141
What is the sum of the measures of the angles at any given vertex? __
  • This regular polyhedron is called aDODECAHEDRON. A dodecahedron has __ faces. Each face is a __. At each vertex, there are __ __.
slide142
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. __
slide143
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? __
slide144
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)

slide145
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.
networks
Networks

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

traversable
Traversable

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

topology
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.”
euclidean geometry
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.
topologically equivalent
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.

topologically equivalent174
Topologically Equivalent

A doughnut and a coffee cup are topologically equivalent.

slide175
According to Swiss psychologist Jean Piaget, children first equate geometric objects topologically.
slide177
We will consider 3 attributes that any two topologically equivalent objects will share:
  • Number of sides
  • Number of edges
  • Number of punctures or holes
slide178
Consider one strip of paper
  • How many sides does it have?
  • How many edges does it have?
slide179
Consider one strip of paper
  • How many sides does it have? 2
  • How many edges does it have? 1
slide180
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?
slide181
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.

mobius strip182
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?
mobius strip183
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.

slide185
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?
slide186
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?
slide187
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.
slide189
Compare the number of sides and edges of the strip of paper, the loop, and the Mobius strip.
  • Are any of those topologically equivalent?