Unit 1 Introduction/Constructions This unit covers the course introduction and class expectations....
If you can't read please download the document
Unit 1 Introduction/Constructions This unit covers the course introduction and class expectations. It lays the basic groundwork for the entire year with
Unit 1 Introduction/Constructions This unit covers the course
introduction and class expectations. It lays the basic groundwork
for the entire year with definitions and commonly used terms and
symbols. This unit also covers how to manually construct various
geometric figures using a compass and a straight edge.
Slide 2
Standards SPIs taught in Unit 1: SPI 3108.1.1 Give precise
mathematical descriptions or definitions of geometric shapes in the
plane and space. SPI 3108.1.4 Use definitions, basic postulates,
and theorems about points, lines, angles, and planes to
write/complete proofs and/or to solve problems. SPI 3108.4.1
Differentiate between Euclidean and non-Euclidean geometries. CLE
(Course Level Expectations) found in Unit 1: CLE 3108.1.4 Move
flexibly between multiple representations (contextual, physical
written, verbal, iconic/pictorial, graphical, tabular, and
symbolic), to solve problems, to model mathematical ideas, and to
communicate solution strategies. CLE 3108.1.6 Employ reading and
writing to recognize the major themes of mathematical processes,
the historical development of mathematics, and the connections
between mathematics and the real world. CLE3108.2.3 Establish an
ability to estimate, select appropriate units, evaluate accuracy of
calculations and approximate error in measurement in geometric
settings. CLE 3108.4.4 Develop geometric intuition and
visualization through performing geometric constructions with
straightedge/compass and with technology. CFU (Checks for
Understanding) applied to Unit 1: 3108.1.3 Comprehend the concept
of length on the number line. 3108.1.4 Recognize that a definition
depends on undefined terms and on previous definitions. 3108.1.5
Use technology, hands-on activities, and manipulatives to develop
the language and the concepts of geometry, including specialized
vocabulary (e.g. graphing calculators, interactive geometry
software such as Geometers Sketchpad and Cabri, algebra tiles,
pattern blocks, tessellation tiles, MIRAs, mirrors, spinners,
geoboards, conic section models, volume demonstration kits,
Polydrons, measurement tools, compasses, PentaBlocks, pentominoes,
cubes, tangrams). 3108.1.12 Connect the study of geometry to the
historical development of geometry. 3108.1.14 Identify and explain
the necessity of postulates, theorems, and corollaries in a
mathematical system. 3108.2.6 Analyze precision, accuracy, and
approximate error in measurement situations. 3108.4.1 Recognize
that there are geometries, other than Euclidean geometry, in which
the parallel postulate is not true and discuss unique properties of
each. 3108.4.6 Describe the intersection of lines (in the plane and
in space), a line and a plane, or of two planes. 3108.4.7 Identify
perpendicular planes, parallel planes, a line parallel to a plane,
skew lines, and a line perpendicular to a plane. 3108.4.21 Use
properties of and theorems about parallel lines, perpendicular
lines, and angles to prove basic theorems in Euclidean geometry
(e.g., two lines parallel to a third line are parallel to each
other, the perpendicular bisectors of line segments are the set of
all points equidistant from the endpoints, and two lines are
parallel when the alternate interior angles they make with a
transversal are congruent). 3108.4.22 Perform basic geometric
constructions using a straight edge and a compass, paper folding,
graphing calculator programs, and computer software packages (i.e.,
bisect and trisect segments, congruent angles, congruent segments,
a line parallel to a given line through a point not on the line,
angle bisector, and perpendicular bisector).
Slide 3
Euclidean Geometry Euclidean geometry is a mathematical system
attributed to the Greek mathematician Euclid of Alexandria (300
BC). Euclid's text Elements is the earliest known systematic
discussion of geometry. It has been one of the most influential
books in history, as much for its method as for its mathematical
content. The method consists of assuming a small set of intuitively
appealing axioms, and then proving many other propositions
(theorems) from those axioms. Although many of Euclid's results had
been stated by earlier Greek mathematicians, Euclid was the first
to show how these propositions could be fit together into a
comprehensive deductive and logical system.
Slide 4
Beyond Euclidean Geometry Or A boys dream to figure out the
weirdness in the world
Slide 5
Explaining those other things Think of Nature, or other shapes
that arent normally found/defined in Euclidean Geometry In other
words, weird shapes which have unusual 3 dimensional properties or
even 2 dimensional properties- that arent easily calculated with
our standard rules
Slide 6
The Geometry of Graphs In the early 1700s the city of
Konigsberg Germany was connected by 7 bridges. People wondered if
it was possible to walk through the city and only cross each bridge
only once After trying several times, you might think no But this
really isnt a proof without trying each and every possible
combination
Slide 7
A simple look at the City
Slide 8
Leonhard Euler A famous Swiss mathematician at the time They
took the problem to him and asked him if there was a mathematical
model he might be able to devise to solve the problem of the
bridges He invented a whole new kind of geometry called Graph
Theory. Graph theory is now used to design city streets, analyze
traffic patterns, and determine the most efficient public
transportation routes i.e. buses
Slide 9
He Couldnt do it either, but Euler recognized that in order to
succeed, a traveler in the middle of the journey must enter a land
mass via one bridge and leave by another, thus that land mass must
have an even number of connecting bridges. Further, if the traveler
at the start of the journey leaves one land mass, then a single
bridge will suffice and upon completing the journey the traveler
may again only require a single bridge to reach the ending point of
the journey. The starting and ending points then, are allowed to
have an odd number of bridges. But if the starting and ending point
are to be the same land mass, then it and all other land masses
must have an even number of connecting bridges. Alas, all the land
masses of Konigsberg have an odd number of connecting bridges and
the journey that would take a traveler across all the bridges, one
and only one time during the journey, proves to be impossible!
Slide 10
New Terminology Vertex: This is a point Edge: This is a line
segment or curve that starts and ends at a vertex Graph: Formed by
vertexes and edges Odd Vertex: A vertex with an odd number of
attached edges Even Vertex: A vertex with an even number of
attached edges Traversable: A graph is traversable if it can be
traced without lifting the pencil from the paper and without
tracing an edge more than once
Slide 11
Rules of Traversability 1.A graph with all even vertices is
traversable. You can start at any vertex and end where you began
2.A graph with two odd vertices is traversable. You must start at
either of the odd vertices and finish at the other 3.A graph with
more than two odd vertices is not traversable
Slide 12
Example Lets try a simple one: Is this graph traversable? If it
is, describe the route Solution Determine the number of even / odd
vertices It has 2 odd, and one even vertice According to the 2 nd
rule, it is traversable
Slide 13
Can you go through this building, and only go through each door
only once?
Slide 14
Topology A branch of modern geometry which looks at shapes in a
new way In Euclidean Geometry, shapes are rigid and unchanging In
Topology, shapes can be twisted, stretched, bent and shrunk A
topologist does not know the difference between a coffee cup and a
doughnut
Slide 15
Topology Classification In Topology, objects are classified
according to the number of holes in them This is called their genus
Since Coffee Cups and Donuts both have one hole, they are
considered the same The genus gives the largest number of complete
cuts that can be made in the object without cutting the object into
two parts Objects with the same genus are topologically
equivalent
Slide 16
Hyperbolic Geometry Developed by Russian mathematician Nikolay
Lobachevsky (1792-1856) and Hungarian mathematician Janos Bolyai
(1802-1860) Based on the assumption that given a point not on a
line, there are an infinite number of lines that can be drawn
through the point parallel to the given line
Slide 17
Elliptic Geometry Proposed by German Mathematician Bernhard
Riemann (1826-1866) Assumes that there are no parallel lines Based
on a sphere Used by Albert Einstein when he created his theory of
the universe One aspect of this theory is that if you begin a
journey in space, and go in the same direction, eventually youll
come back to where you started This is where we get the idea that
space is curved
Slide 18
Fractal Geometry An attempt to replicate or describe nature A
close look at nature reveals patterns, repeated over and over, in
smaller and smaller detail Self Similarity: A pattern that repeats,
as well as adding new and unexpected patterns to the whole Fractal:
Comes from the Latin word Fractus, meaning broken up or
fragmented.
Slide 19
Fractal Geometry Iteration: The process of repeating a rule
(rules are used to create patterns) to create a self similar
pattern Computers can easily create fractals because you establish
rules, and they can repeat those rules thousands or millions of
times http://www.coolmath.com/fractals/gallery.ht m
http://www.coolmath.com/fractals/gallery.ht m
Slide 20
Fractal
Slide 21
Slide 22
Slide 23
Slide 24
Points Point this is a location. A point has NO SIZE. It is
represented by a small dot, and named by a capital letter. A
geometric figure consists of a set of points. Space this is defined
as the set of all points in existence.. A. B
Slide 25
Lines Line this can be thought of as a series of points that
extends in two opposite directions forever. You can name a line by
choosing any two points on the line, such as AB we read this as
Line AB. A. B Another way to name a line is with a single lowercase
letter, such as Line t Points that lie on the SAME line are
Collinear Points
Slide 26
Example Are points E, F, and C collinear? If so, what line do
the lie on? Are points E,F and D Collinear? Name line m in three
other ways. E P F C D n m l What do you think arrowheads are used
to show when drawing a line, or naming a line such as EF?
Slide 27
Planes Plane A plane is a flat surface that has NO thickness. A
plane extends forever in the directions of all of its lines. How
many lines do you think a plane may contain? How many points do you
think a plane may contain? You can name a plane by a single Capital
letter, or by at least three of its noncollinear points. Points and
lines that are within the same plane are called coplanar.
Slide 28
Example Plane P P Plane ABC D C B A
Slide 29
Another Example A B C D EF G H Name 3 planes
Slide 30
Postulates and Axioms A postulate or axiom is an accepted
statement of fact It is something we hold to be true, and we do not
need to prove it -it has either been proven already, or the proof
is self evident
Slide 31
Postulate 1-1,1-2,1-3, 1-4 1-1 Through any two points there is
EXACTLY ONE line 1-2 If two lines intersect, then they intersect in
EXACTLY ONE point 1-3 If two planes intersect, then they intersect
in EXACTLY ONE line 1-4 Through any three non-collinear points
there is EXACTLY ONE plane
Slide 32
Example Imagine you have a cube a dice for example. Just using
edges, sides, and corners, answer this: How many lines are there?
remember, what does it take to make a line? How many planes are
there? While there are infinite numbers of points, we also know
that the intersection of two lines creates a point. How many
intersections are there? i.e. how many points can you name?
Slide 33
Segment and Ray Segment: A part of a line. It consists of two
endpoints (which we have to label) and all of the points in
between. We would write this as AB or segment AB Ray: A ray is the
part of a line consisting of one endpoint and all of the points of
the line on one side of the endpoint. This is AB, or Ray AB A B AB
Endpoint A
Slide 34
More Rays Opposite Ray: These are two collinear rays with the
same endpoint. Opposite rays always form a line. Name the two
opposite rays presented here: Ray QR, and Ray QS. To be opposite,
we must imagine them going away from each other, so we must use the
center point as our starting point R Q S
Slide 35
Example Question: Ray LP and Ray PL form a line. Are they
opposite rays? Why or why not? Name the segments and rays formed by
this figure: A B C
Slide 36
A Closer Look at Lines Parallel Lines: These are coplanar, and
do not intersect. Are all lines that do not intersect Parallel? No.
What if they are not in the same plane, and do not intersect? Skew:
Lines that do not intersect, but are NOT coplanar.
Slide 37
Example Name a pair of parallel lines. Then name another pair.
Name one pair of skew lines, then name another pair. A B C D EF G
H
Slide 38
Unit 1 Quiz 1 1.Name a Point 2.Name a Line 3.Name a Plane
4.Name 2 Lines that are parallel 5.Name 2 Lines that are skew
6.Name 3 Points that are Coplanar 7.Name 2 Points that are
Collinear 8.Opposite Rays are ______ 9.(T/F) Opposite Rays have the
same endpoint 10.(T/F) Line Segments have one end point A B C D EF
G H
Unit 1 Quiz 2 1.How many points fit on the head of a pin? 2.An
intersection of 2 lines is a _______________? 3.An intersection of
2 planes is a ______________? 4.For a 2 lines to be parallel, they
must be _______________ and ______________. 5.For 2 lines to be
skew, they must _______________ and ______________. 6.Coplanar
means _______________________________ 7.Collinear means
_______________________________ 8.Imagine 3 nonlinear points. Can
you draw a plane through them? 9.Imagine 4 nonlinear points. Do
they have to be coplanar? 10.What is a postulate?
Slide 41
Perpendicular Lines Perpendicular Lines: 2 lines that intersect
to form Right Angles. The symbol means is perpendicular to. In the
diagram below, line AB line CD, and line CD line AB. A D C B This
symbol means Right Angle There are actually 4 symbols here, since
there are 4 right angles
Slide 42
The Ruler Postulate The points of a line can be thought of as
points on a ruler. We can therefore measure the distance between
two points. The distance between any two points is the absolute
value of the difference of the corresponding numbers. This is
common sense. If you have a nail at 4 inches on a board, and
another nail at 7 inches, how far apart are they?
Slide 43
Congruent Two items (in math) are considered congruent if they
have the same dimensions (size) and shape. This is a loose
definition, and we narrow it for various concepts. -in other math
disciplines this means almost equal to here, it means the same
length Congruent segments: Segments which have the same length.
They do not have to be on the same line, just have the same
measure.
Slide 44
Example We use hashmarks to indicate congruency in Geometry. We
will do this all year, to show segments are congruent, angles are
congruent, triangles are congruent and so on. A B C D When we look
at this picture, we can conclude that segment AB is congruent to
segment CD
Slide 45
Segment Addition Postulate If three points, A,B, and C are
collinear and B is between A and C, then AB +BC = AC If DT = 60,
find the value of x. Then find DS and ST. A B C D S T 2x-8
3x-12
Slide 46
Midpoint Midpoint: the point of a segment that divides the
segment into two congruent (equal length) segments. A midpoint, or
any line, ray, or other segment through a midpoint is said to
bisect the segment. If segment AB Segment BC, then. B is a
midpoint. A B C
Slide 47
Examples B is a midpoint, what is X? A x B 5 C A x B 5x-4
C
Slide 48
Assignment Page 24 8-25 (Guided Practice) Keep this assignment
we will add to it.
Slide 49
Angles An Angle ( ) is formed by two rays with the same
endpoint. The rays are the sides of the angle. The endpoint is the
vertex of the angle. The sides of the angle shown here are BT and
BQ. The vertex is B. You can name this angle 4 ways: B 1 B, TBQ,
QBT, or 1. Note that the vertex B is always the middle letter. T
Q
Slide 50
Examples Name 1 in two different ways Angle ABC Angle CBA Name
2 in two different ways Angle EBC Angle CBE 1 A B C D E 2
Slide 51
Measurement (m) We often measure angles in degrees. To indicate
the size or degree measure of an angle, we write a lower case m in
FRONT of the angle symbol. Here the degree measure is 80. We would
show this by writing m A = 80. 80 o A
Slide 52
Protractor Postulate You can add or subtract the measure of
angles. And all angles on one side of a line will add up to 180
degrees The measure of a straight line is 180 degrees. We also call
this a straight angle.
Slide 53
Example What is the m of 1? 2? 3? 4? ABC? 2 14 3 A B C
Slide 54
Congruent Angles Angles with the same measure are congruent
angles. In other words, if m 1 = m 2, then 1 2 We consider these
statements interchangeable: they mean the same thing. 1 2
Congruency symbol for angles
Slide 55
Classifying Angles There are 4 basic angles: Acute angle 0 <
x < 90 Right angle X = 90 Obtuse angle 90 < x < 180
Straight Angle X = 180 xoxo xoxo xoxo xoxo
Slide 56
BellRinger OK, time to evaluate the class and the teacher.
Write what you like, and dislike about the class (and me) so far. I
expect to see a paragraph. Not two sentences. At least three. 10
minutes tops
Simon Says Point Line Plane Parallel Lines Skew Lines
Intersecting Lines Intersecting Planes Intersection of a Line and a
Plane How many Lines are there between 2 points How many points are
there on the head of a pin? Perpendicular Lines Congruent Obtuse
Angle Acute Angle Right Angle When 2 lines intersect, how many
points do they intersect on?
Slide 59
Basic Constructions A construction requires a compass and a
straight edge to draw geometric figures. You can use a ruler, or
the side of your protractor for the straight edge. You will NOT get
credit for hand drawn figures. You will NOT get credit for figures
you measured with your ruler, instead of using the compass
properly.
Slide 60
Constructing Congruent Segments Given Segment AB Construct
Segment CD so that CD AB Draw a Ray Measure AB with your compass
Put the point of your compass on the endpoint of your ray, and make
a mark on the ray with the pencil end of the compass You now have a
new segment the same length as the first one A B CD
Slide 61
Constructing Congruent Angles Given angle A, construct angle B
so that angle b angle A Draw a ray Draw an arc on angle A, then
draw the SAME arc on ray B. Place the point of the compass on the
intersection of the arc and the side of the angle, and measure
across the angle to the other side and mark it lightly. Mark this
second arc on the first arc of the ray Draw a second line from the
endpoint of the ray Through the intersecting arcs, thus recreating
the arc A B Point of Compass Point of Compass
Slide 62
Perpendicular Bisector A perpendicular bisector of a segment is
a line, segment, or ray that is perpendicular to the segment at its
midpoint, thereby bisecting the segment into two congruent
segments. A M C C D
Slide 63
Constructing a Perpendicular Bisector Given segment AB
Construct segment XY so that XY AB at the midpoint M of AB (which
we will determine) Put the point of the compass at point A, and
draw a long arc make sure the arc is pas the half way point of the
line. Using the same compass setting, put the compass point on
point B and draw another long arc. Label the points of intersection
Draw a line between points X and Y A B X Y You now have a
perpendicular bisector you have made 4 right angles, and cut the
segment in half
Slide 64
Angle Bisector An angle bisector is a ray that divides an angle
into two congruent coplanar angles. Its endpoint is the angle
vertex. Within the ray, a segment with the same endpoint is also an
angle bisector in other words a segment or a ray can bisect an
angle but they both start with a ray. You can also say that a ray
or segment bisects the angle.
Slide 65
Construct the Angle Bisector Construct the bisector of an angle
Given angle A, construct ray AX, the bisector of angle A. Put the
compass point on vertex A, and draw an arc that intersects the
sides of the angle. Label the points of intersection B and C. Put
the compass point on Point B and draw an arc Without changing the
compass, put the point on Point C and draw another arc so that it
intersects the first arc Label the point of intersection Point X
Draw the ray XY AX is the bisector of Angle A. B C X A
Unit 1 Quiz 3 (2 Points each) Identify the Construction
1.Congruent Segments ____ 2.Congruent Angles ____ 3.Perpendicular
Lines ____ 4.Bisected Angles ____ 5.Solve this (no calculator):
Hint: write it out, and look for a pattern answer is a fraction B C
X A D) A B X Y A) AA C) A B A B B) 1/2 x 2/3 x 3/4 x 4/5 x 5/6 x
6/7 x 7/8
Slide 68
Distance Formula The Pythagorean Theorem States that: A 2 + B 2
= C 2 If we wanted to find C, we would square root both sides, to
get: (A 2 + B 2 ) = C On a graph in the coordinate plane, the value
of A is the distance between x values which we calculate by
subtracting the smaller x value from the larger x value or x 2 - x
1 Therefore A 2 is the same as (x 2 - x 1 ) 2 We do the same for Y
values, and get (y 2 - y 1 ) 2 Finally, we can calculate diagonal
distance on a graph by stating that: C = (x 2 - x 1 ) 2 + (y 2 - y
1 ) 2 Or we can use the calculator.
Slide 69
Unit 1 Quiz 3 In your own words define: 1.A Point 2.A Line 3.A
Plane 4.A Postulate/Axiom 5.Collinear 6.Coplanar 7.Skew 8.Parallel
Draw 9.The intersection of 2 planes and a line 10.The intersection
of 2 planes
Slide 70
Unit 1 Quiz 3 (5 points each) Write the Distance Formula State
where the Distance Formula comes from (What equation did I derive
it from in class?)
Slide 71
Unit 1 Quiz 2 Calculate the distance between these points
(leave answer in decimal form): 1.A (2,5) B (4,9) 2.C (1,4) D
(-3,5) 3.E (0,0) F (-3,-3) 4.G (5,5) H (-1,8) 5.J (2,11) K (2, 19)
6.Draw a right angle 7.Draw a straight angle 8.Draw an acute angle
9.Draw an obtuse angle 10.Angle ABC is 135 degrees. What kind of
angle is it?
Slide 72
Unit 1 Final Exam Extra Credit (5 points) On the answer sheet
provided, CONSTRUCT a 135 degree angle, using the techniques taught
in class. Show all work, and highlight the finished angle (I have
highlighters if you need to borrow one).
Slide 73
Unit 1 Final Exam Extra Credit (2 points per sentence) 1.Write
a sentence explaining the difference between coplanar and collinear
2.Write a sentence explaining three ways to tell if a point is a
midpoint on a line segment 3.Write a sentence explaining what
perpendicular means 4.Write a sentence explaining what parallel
means 5.Write a sentence explaining what a postulate is
