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Math 103 Lecture 10 notes page 1
Math 103 Lecture #10 notes
Geometry comes from two Greek words, ge meaning “earth” and metria meaning “measuring.”
The approach to Geometry developed by the Ancient Greeks has been used for over 2000 years as
the basis of Geometry.
The work of 2 Dutch educators, Dina van Hiele-Geldof and Pierre van Hiele, is influencing the
teaching of geometry. They were concerned about the difficulties their students were having with
geometry, so they conducted research aimed at understanding children’s levels of geometric
thinking to determine the kinds of instruction that can best help children. Each Van Hiele level
describes how children think about geometric concepts.
Level
Level
Level
Level
Level
0:
1:
2:
3:
4:
Levels of Mental Development in Geometry
P.H. Van Hiele, 1959
Recognition or Visualization
Analysis
Ordering or Informal Deductive
Deduction or Formal Deductive
Rigor
An effective teacher will use the Van Hiele levels to develop five skill areas for geometry.
Visual Skills
Verbal Skills
Drawing Skills
Logical Skills
Applied Skills
Level 0: Recognition or Visualization (visual skills)
Children at the visualization level think about shapes in terms of what they resemble. At this level,
children are able to sort shapes into groups that look alike to them in some way.
Ex:
“a mountain”
Level 1: Analysis (drawing and verbal skills)
Children at the analysis level think in terms of properties.
They can list all of the properties of a figure but don’t see any relationships between the
properties, and don’t realize that some properties imply others.
Math 103 Lecture 10 notes page 2
Level 2: Ordering or Informal Deductive (verbal skills)
Children at the informal deduction level not only think about properties but also are able to notice
relationships within and between figures. At this level, children are able to formulate meaningful
definitions. At this level, children are also able to make and follow informal deductive arguments.
Ex: “All squares are rectangles, but not all rectangles are squares.”
Level 3: Deduction or Formal Deductive (logical skills)
Children at the formal deductive level think about relationships between properties of shapes and
also understand relationships between axioms, definitions, theorems, corollaries, and postulates.
They understand how to do a formal proof and understand why it is needed. For successful
completion, the typical high school geometry course requires geometric understanding at the formal
deduction level.
Level 4: Rigor (applied skills)
Children at the rigor level can think in terms of abstract mathematical systems. College
mathematics majors and mathematicians are at this level.
In 1990, two researchers (Clements and Battista) hypothesized that a level exists that is below
visualization. They called it “pre-recognition.” Children operating at this level wouldn’t be able to
distinguish a three-sided figure from a four-sided figure.
Most elementary schoolers are at the visualization or analysis level; some middle-school children are
at the informal deduction level. The 6th NAEP noted that most students (4th, 8th and 12th grades)
appeared to be performing at the visualization level. It is desirable to have a child at the informal
deduction level or above by the time he or she finishes middle school.
Comments on Levels:
• The levels are not age dependent, but rather, are related more to the experiences students
have had.
• The levels are sequential; children must pass through the levels in order as their understanding
increases (except for gifted children).
• To move from one level to the next, children need to have many experiences in which they are
actively involved in exploring and communicating about their observations of shapes, properties,
and relationships.
• For learning to take place, language must match the child’s level of understanding. If the
language used is above the child’s level of thinking, the child may only be able to learn
procedures and memorize without understanding.
• It is difficult for two people who are at different levels to communicate effectively. A teacher
must realize that the meaning of many terms is different to the child than it is to the teacher
and adjust his or her communication accordingly.
Ex: Given a “square”
A person at the visualization level will think of a CD case, because that is what a square looks like.
A person at the informal deduction level thinks about he fact that a square has four congruent
sides and four congruent angles and will know the properties of a square such as opposite sides
parallel and the diagonals perpendicular bisectors.
Math 103 Lecture 10 notes page 3
Identify the Van Hiele level of development that a child is at when:
i) the child sees a square and can state “All squares are rectangles, but not all rectangles are
squares.”
ii) the child sees a square and says “it’s a floor tile.”
iii) the child sees a square and can write a proof that the diagonals are perpendicular.
iv) the child sees a square and can’t distinguish it from a triangle.
v) the child sees a square and can tell you that all angles are right angles, and opposite sides are
parallel, but may not realize that either of these properties imply the other.
vi) the child sees a square and can describe it as a vector space.
Self-check:
1. A child at the recognition level might say
2. A child at the analysis level might say
3. A child at the ordering level might say
4. Most elementary students are at what level?
5. It is desirable to have a child at what level by the time she/he finishes middle school?
Literature link:
The Shape of Things
The Greedy Triangle
Shapes, Shapes, Shapes,
Bear in a Square
Brown Rabbit’s Shape Book
Spaghetti and Meatballs for All!
According to NCTM Standards: The study of geometry in grades 3-7 requires thinking and doing.
As students sort, build, draw, model, trace, measure, and construct, their capacity to visualize
geometric relationships will develop. This exploration requires access to a variety of tools, such as
graph paper, rulers, compass and straightedge, pattern blocks, geoboards, and geometric solids, and
is greatly enhanced by electronic tools that support explorations, such as dynamic geometry
software.
Little is known of Euclid of Alexandria (300 B.C.) although legend has it that he studied geometry
for its beauty and logic. Euclid is best known for The Elements, a work so systematic and
encompassing that many earlier mathematical works were simply discarded and lost to all future
generations. The Elements, composed of 13 books included not only geometry, but arithmetic and
topics in algebra. Euclid set up a deductive system by starting with a set of statements that he
assumed to be true and showing that geometric discoveries followed logically from these
assumptions.
Euclidean geometry, or plane geometry, is a complete axiomatic system. Non-Euclidean geometry is
also a complete axiomatic system.
Math 103 Lecture 10 notes page 4
An axiomatic (or mathematical) system is a set of undefined terms, some definitions and assumed
truths (axioms), and theorems which are logically deduced from these definitions and axioms.
Plane (Euclidean) Geometry begins with 3 basic undefined terms,
Point = a point is a point.
Line = a line is a line.
Plane = a plane is a plane.
and adds defined terms, properties, axioms (statements assumed true without proof), and theorems
(statements which can be proved true or false through logical reasoning using definitions, axioms, or
other theorems) to create a complete axiomatic system.
To illustrate what "a complete axiomatic system means" notice, that the solutions to the following
problems can be deduced logically from established geometric theorems and axioms.
Triangles ABC and CDE are equilateral triangles with points A, C and E collinear. Find the measure
of ∠BCD.
D
B
C
A
E
Find the measures of angles 1, 2, and 3 given that TRAP is a trapezoid with TR || PA
T
2
1
P
30°
3
R
40°
A
The study of Euclidean Geometry begins with 3 fundamental definitions/terms: point, line, plane.
Linear Notions: (see page 464 Table 9-2)
Collinear points
A point between two points on a line
Line segment
Ray
Planar Notions:
Coplanar
Skew lines
Intersecting lines
Concurrent lines
Parallel lines
Question: Why is a tripod considered “level?”
Math 103 Lecture 10 notes page 5
Properties of Points, Lines, and Planes
1. There is exactly one line that contains any two distinct points.
2. If two points lie in a plane, then the line containing the points lies in the plane.
3. If two distinct planes intersect, then their intersection is a line.
4. There is exactly one plane that contains any three distinct noncollinear points.
5. A line and a point not on the line determine a plane.
6. Two parallel lines determine a plane.
7. Two intersecting lines determine a plane.
Distance Properties:
1.
The distance between any two points A and B is greater than or equal to 0, written AB ≥ 0.
The length of AB is denoted by AB.
2.
The distance between any two points A and B is the same as the distance between B and A,
written AB = BA
3.
For any three points A, B, and C, the distance between A and B plus the distance between B
and C is greater than or equal to the distance between A and C, written AB + BC ≥ AC.
Notice the difference between AB, AB, and AB.
Suppose we have 3 points, A, B, C, in a plane and can accurately measure the distance between any
two of them. How can we determine if the points are collinear?
1. find length of the line segments AB, BC, and AC
• If the points are not collinear, the sum of any two lengths will be greater than the third. If the
points are collinear, the sum of two of the line segments is equal to the third.
Example:
A
B
C
AB + BC = AC
What if AB + BC > AC ?
Triangle Inequality Property: The sum of the measures of any two sides of a triangle must be
greater than the measure of the third side.
Problem: If two sides of a triangle are 31 cm and 85 cm long and the measure of the third side must
be a whole number of centimeters,
• what is the longest the third side can be?
• What is the shortest the third side can be?
Pythagorean Theorem:
2
2
2
If a right triangle has legs of lengths a and b and hypotenuse of length c, then c = a + b .
There are hundreds of known proofs of the Pythagorean Theorem. Henry Perigal, a London
stockbroker, discovered what has been called the "paper and scissors" proof of the Theorem.
Math 103 Lecture 10 notes page 6
1
2
3
5
4
Use the Pythagorean Theorem to find x.
17
8
x
Special right triangles:
45° - 45° - 90° right triangle; sides have ratio 1 - 1 - √2
30° - 60° - 90° right triangle; sides have ratio 1 - √3 - 2
Problem: Construct a square with an area of exactly 2 units, given the length of 1 unit.
2
2
2
Pythagorean Triples are three natural numbers a, b, and c that satisfy the relationship c = a + b .
The least three numbers that are Pythagorean triples are 3-4-5. Another triple is 5-12-13 because
2
2
2
13 = 5 + 12 . Find two more Pythagorean triples.
Does doubling each number in a Pythagorean triple result in a new Pythagorean triple?
Does adding a constant to each number in a Pythagorean triple result in a new Pythagorean triple?
Determine whether the following is a right triangle.
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Math 103 Lecture 10 notes page 7
Angle Measurement
An angle is measured according to the amount of "opening" between its sides. The degree is
commonly used to measure angles (but radians and grads are also used).
A complete rotation about a point has a measure of 360°. A degree is subdivided into 60 parts,
called minutes, and each minute is further subdivided into 60 parts, called seconds. The
measurement 29 degrees, 47 minutes, 13 seconds is written 29°47'13".
Problem:
Find 47°45' – 29°58'
Express 47°45' as a decimal number of degrees
Express 32.6° in degrees and minutes, & seconds, without a decimal
A right angle is an angle whose measure is 90°.
An acute angle is an angle whose measure is less than 90°.
An obtuse angle is an angle whose measure is more than 90° but less than 180°.
When a transversal intersects two parallel lines, many types of angles are formed: exterior,
interior, vertical, corresponding, supplementary.
When a transversal intersects two parallel lines, the alternate interior angles are congruent.
When a transversal intersects two parallel lines, the corresponding angles are congruent.
A closed figure is one which can be traced, beginning and finishing at the same point.
Ex:
A simple figure is a curve which does not cross itself.
Ex:
A polygon is a simple closed figure consisting entirely of segments.
If it is possible to select two points inside a closed figure but the segment with these points as
endpoints is not entirely inside the figure, the figure is said to be concave. If the segment is
entirely inside the figure, the figure is said to be convex.
Ex:
Math 103 Lecture 10 notes page 8
A regular polygon has all sides congruent and all angles congruent.
See definitions, table 9-6 Billstein.
See hierarchy among polygons, figure 9-18 Billstein.
There are TWO definitions of a trapezoid:
a) A quadrilateral with exactly one pair of parallel sides.
b) A quadrilateral with at least one pair of parallel sides.
How does a trapezoid fit into a quadrilateral hierarchy using each definition?
See Quadrilaterals and Triangles True-False handout
Dissection Puzzle:
Divide a regular hexagon into
1. 3 identical rhombi
2. 6 identical kites
3. 4 identical trapezoids
4. 8 identical shapes (any shape)
5. 12 identical shapes (any shape)
How can you figure out the measure of the interior angle of any regular polygon? (see worksheet)
The Sum of the Measures of the Interior Angles of a Convex Polygon with n Sides is 180n – 360, or
(n–2)180°.
The measure of a single interior angle of a regular n-gon is
(180n – 360)/n or (n–2)180°/n
The Sum of the Measures of the Exterior Angles of any Convex Polygon is 360°.
Problems:
Find the measure of each angle of a regular decagon.
Find the number of sides of a regular polygon, each of whose angles has a measure of 175°.
The perimeter of a simple closed curve is the length of the curve, that is, the distance around the
figure.
The perimeter of a circle is its circumference. The ratio of a circumference C to diameter d is
symbolized as π. This relationship C/d = π is normally written as C = πd or C = 2πr. The ancient
Greeks discovered that if they divided the circumference of any circle by the length of its
diameter, they always obtained approximately the same number. This number is approximately 3.14.
For most practical purposes, π is approximately 22/7, 3 1/7, or 3.14, but π is an irrational number.
Note: www.cecm.sfu.ca/pi.pi.html
Math 103 Lecture 10 notes page 9
Area formulas:
Area of a rectangle:
Area of a parallelogram:
Area of a triangle:
Area of a trapezoid:
Area of a regular polygon:
Area of a circle:
Area is measured using square units and the area of a region is the number of square units that
cover the region without overlapping.
Area activities on a geoboard or dot paper teach the concept of area intuitively, and should precede
the development of formulas.
addition method: sum the areas of smaller pieces
rectangular method: construct a rectangle around the figure and subtract the areas of outside
regions.
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Draw a triangle on the geoboard grid with an area of 7.
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Hero or Heron’s Formula for area of any triangle:
5
4
7
Area =
s(s " a)(s " b)(s " c)
where s = semi-perimeter = 1/2 (a+b+c)
For the problem above,
!s = 1/2 (5 + 4 + 7) = 8
A=
!
8(8 " 5)(8 " 4)(8 " 7) # 9.8
Basic area relationships:
If each side of a square is doubled, area is increased by ?
Math 103 Lecture 10 notes page 10
If each side of a square is tripled, the area is increased by ?
If each side of a square is multiplied by n, the area is increased by ?
Does this property apply to other geometric figures?
Networks:
Network Theory is one of the most practical topics in topology, with applications to electrical
circuitry and economics.
Network skills:
• Count the number of vertices and edges
• Traversable graphs
• Euler circuits/graphs
• Euler path
• Isomorphic graphs
• Complete graphs
A network is traversable if there is a path through the network beginning at some vertex and
ending at the same or another vertex such that each arc is traversed exactly once.
The points in a network are called vertices, and the curves/lines are called arcs.
A network that is traversable in such a way that the starting point and the stopping point are the
same is an Euler circuit.
See figure 9-47 Billstein.
Koningsberg Bridge Problem handout.