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.
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). 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. Beyond Euclidean Geometry Or A boys dream to figure out the weirdness in the world 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 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 A simple look at the City 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
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!
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 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 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 2nd rule, it is traversable Can you go through this building, and only go through each door only once? 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 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 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
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 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. 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 Fractal Fractal Fractal Fractal 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 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 Example Are points E, F, and C collinear? If so, what line do the lie on? C n F E P D
l Are points E,F and D Collinear? Name line m in three other ways. m What do you think arrowheads are used to show when drawing a line, or naming a line such as EF? 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. Example C
B A D Plane ABC P Plane P Another Example Name 3 planes H G
E F C D A B 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 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 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? 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.
A B 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. Endpoint A A This is AB, or Ray AB B More Rays Opposite Ray: These are two collinear rays
with the same endpoint. Opposite rays always form a line. R Q S 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 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: C B A 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.
Example Name a pair of parallel lines. Then name another pair. Name one pair of skew lines, then name another pair. H G E F C D
A B Unit 1 Quiz 1 Name a Point Name a Line Name a Plane Name 2 Lines that are parallel 5. Name 2 Lines that are skew 6. Name 3 Points that are E 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 1. 2. 3. 4. H
G F C D A B Assignment Page 16/17 8-22, 27-32, 40-45 (guided practice) Worksheet 1-2 and 1-3 (independent practice)
Unit 1 Quiz 2 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. How many points fit on the head of a pin? An intersection of 2 lines is a _______________?
An intersection of 2 planes is a ______________? For a 2 lines to be parallel, they must be _______________ and ______________. For 2 lines to be skew, they must _______________ and ______________. Coplanar means _______________________________ Collinear means _______________________________ Imagine 3 nonlinear points. Can you draw a plane through them? Imagine 4 nonlinear points. Do they have to be coplanar? What is a postulate? 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
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? 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. 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 Segment Addition Postulate If three points, A,B, and C are collinear and B is between A and C, then AB +BC = AC A B C If DT = 60, find the value of x. Then find DS
and ST. 2x-8 D 3x-12 S T 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. A B C If segment AB Segment BC, then . B is a midpoint. Examples B is a midpoint, what is X? A
x B 5 C B is a midpoint, what is X? A x B
5x-4 C Assignment Page 24 8-25 (Guided Practice) Keep this assignment we will add to it. 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 T B, TBQ, QBT, or 1. Note that the vertex B is always the middle letter. Q Examples Name 1 in two different ways
Angle ABC Angle CBA Name A 1 2 in two different ways B C
2 Angle EBC Angle CBE E D 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.
80o A 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. Example What is the m of 2? 3? 4?
1? 3 2 1 A ABC?
4 B C 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. 2 Congruency symbol
for angles 1 Classifying Angles There are 4 basic angles: Acute angle x o 0 < x < 90 Right angle X = 90
Obtuse angle 90 < x < 180 Straight Angle X = 180 xo xo xo 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 Assignment Page 31 6-14,18-21 (Guided Practice) (add to previous assignment) Worksheet 1-4 (independent Practice) Simon Says
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?
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. Constructing Congruent Segments Given Segment AB
Construct Segment CD so that CD AB A B C D 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
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 A Draw a second line from the endpoint of the ray Through the intersecting arcs, thus recreating the arc B Point of Compass Point of Compass 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. C A M D C 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. You now have a
X Label the points of intersection perpendicular bisector you Draw a line between points have made 4 right angles, X and Y and cut the segment in half A B Y 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. 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 B and draw another arc so that it intersects the first arc Label the point of intersection Point X A Draw the ray XY
AX is the bisector of Angle A. X C Assignment Worksheet 1-5 (Guided/Independent Practice) Unit 1 Quiz 3 (2 Points each) Identify the Construction A 1. Congruent Segments ____ B) 2. Congruent Angles ____ A 3. Perpendicular Lines ____ C)
4. Bisected Angles ____ A 5. Solve this (no calculator): Hint: write it out, and look for a pattern answer is a fraction 1/2 x 2/3 x 3/4 x 4/5 x 5/6 x 6/7 x 7/8 A) X Y B B A
A D) B A X C Distance Formula The Pythagorean Theorem States that: A2 + B 2 = C2 If we wanted to find C, we would square root both sides, to
get: (A2 + B2) = 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 x2 - x 1 Therefore A2 is the same as (x2 - x1)2 We do the same for Y values, and get (y2 - y1)2 Finally, we can calculate diagonal distance on a graph by stating that: C = (x2 - x1)2 + (y2 - y1)2 Or we can use the calculator. Unit 1 Quiz 3 In your own words define:
Draw 1. 2. 3. 4. 5. 6. 7. 8. 9. The intersection of 2 planes and a line 10. The intersection of 2 planes
A Point A Line A Plane A Postulate/Axiom Collinear Coplanar Skew Parallel 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?)
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?
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). 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
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