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### Geometry

Chapter 11

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 ofShape

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

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

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.

Point

- Line
- Collinear
- Plane

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

- Parallel Lines

Line segment

- Endpoint
- Length

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

Degrees

- Protractor

Zero Angle: 0°

- Straight Angle: 180°
- Right Angle: 90°

Acute Angle: between 0° and 90°

- Obtuse Angle: between 90° and 180°

Complementary angles

- Adjacent complementary angles

Supplementary Angles

- Adjacent Supplementary Angles

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

Transversal

- Corresponding Angles

Transversal

- Corresponding Angles

Parallel lines Cut by a Transversal

- Corresponding Angles

Parallel lines Cut by a Transversal

- Corresponding Angles

Describe the relative position of angles 3 and 5.

- What appears to be true about their measures?

Describe the relative positions of angles 1 and 7.

- What appears to be true about their measures?

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

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

Curve

- Simple Curve

Curve

- Closed Curve

Curve

- Simple Curve
- Closed Curve
- Simple Closed Curve

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

Concave

- Convex

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

Classifying Polygons

- Polygons are classified according to the number of sides.

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 Polygons

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

Classifying Triangles

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

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 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

- Trapezoid – Quadrilateral with at least one pair of parallel sides.

Classifying Quadrilaterals

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

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 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 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.

Equilateral

- All sides are congruent
- Equiangular
- Interior angles are congruent

Figure 11.20, Page 689

- Regular Polygons are equilateral and equiangular.

Interior Angles

- Exterior Angles – The sum of the measures of the exterior angles of a polygon is 360°.

Interior Angles

- Exterior Angles
- Central Angles – The sum of the measure of the central angles in a regular polygon is 360°.

Interior Angles

- Exterior Angles
- Central Angles

Circle

- Compass

Circle

- Compass
- Center
- Radius
- Chord
- Diameter
- Circumference
- Tangent

How do you find the sum of the measure of the interior angles of a polygon?

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

Space

- Half Space
- A plane separates space into 3 disjoint sets, the plane and 2 half spaces.

Parallel Planes

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

Polyhedron

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

Prism

- Pyramid
- Apex
- Cylinder
- Cone
- Apex

Right Prisms, Pyramids, Cylinders and Cones

- Oblique Prisms, Pyramids, Cylinders and Cones

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 Polyhedron

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

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 _____.

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.

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.

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 __ __.

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 __ __.

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:

__ __ __

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.

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 __ __.

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. __

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.

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 __ __.

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. __

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? __

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)

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

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

Traversable

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

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

- 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

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 Equivalent

A doughnut and a coffee cup are topologically equivalent.

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

We will consider 3 attributes that any two topologically equivalent objects will share:

- Number of sides
- Number of edges
- Number of punctures or holes

Consider one strip of paper

- How many sides does it have?
- How many edges does it have?

Consider one strip of paper

- How many sides does it have? 2
- How many edges does it have? 1

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?

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 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 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.

How many sides does your result have?

- How many edges?

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?

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?

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.

Compare the number of sides and edges of the strip of paper, the loop, and the Mobius strip.

- Are any of those topologically equivalent?

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