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# Unit 1 foundations of geometry

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• Ask: What do you suppose this picture is made of? (“color?” “dots?” “we call those dots, points”)
• Here are some more examples of art that use a collection of “points”. You can look up more works of art in these similar styles.
• Ask: What is this picture made of? (“Lines?” “Close, but what exactly is a line? Part of a line is called a line segment..”)
• Ask: “Can you find the ray?” then “what time(s) form opposite rays?”Reiterate that DIRECTION MATTERS on rays.
• Ask: “What are some planes? …”
• Start: Discovery Based Learning (ask “from our definitions, what is true? What can happen? How can this happen?” Think Pair Share activity, answer is student words)
• This should be a 5-10 minute group activity. If time permits.
• Ask: What is this picture made of, besides line segments? “Angles”
• Inquiry Based Learning: “What do these words mean?”Review: How do we label the points? Segments? Rays? Angles?
• Students are encouraged to paraphrase AFTER writing these exact steps. Extra examples on board with my GIANT yellow protractor.
• Inquiry Based Learning: “What do these words mean?”Review: How do we label the points? Segments? Rays? Angles?
• ### Unit 1 foundations of geometry

1. 1. Euclidean vsNon-Euclidian Geometry<br />Ms. Hayde Rivas<br />
2. 2. Bell Ringer<br />Take out Homework and put on top of desk<br />Parent form<br />article<br />Pick up a School Map <br />Correct your quiz (Answers are posted on walls)<br />
3. 3. Taxi-Cab<br />http://www.learner.org/teacherslab/math/geometry/shape/taxicab/index.html<br />
4. 4. Parallel<br />Computer designs run programs simultaneously <br />http://www.cse.psu.edu/~teranish/ri_02.html<br />Extra Europa – Parallel<br />By: DietmarTollerian<br />
5. 5. Spherical<br />
6. 6. Hyperbolic<br />
7. 7. Summarize the Article<br />“What is Non-Euclidean Geometry?”<br />By Joel Castellanos<br />
8. 8. 1.1 Euclidean Geometry<br />Euclid is a Greek mathematician.<br />Euclid lived in 300 B.C.<br />Euclid wrote a book “The Elements”<br />In high school, we study “The Elements” which is Euclid’s 2000 year old book.<br />Greeks used Euclidean Geometry to design buildings, predict locations and survey land.<br />
9. 9. 1.2 Non-Euclidean Geometry <br />Any geometry different from Euclidean geometry.<br />Each system of geometry has different definitions, postulates and proofs.<br />Spherical geometry and hyperbolic geometry are the most common Non-Euclidean Geometry.<br />The essential difference between Euclidean geometry and non-Euclidean is the nature of parallel lines.<br />
10. 10. 1.3 Spherical Geometry<br />Spherical Geometry is geometry on the surface of a sphere.<br />Lines are the shortest distance between two points. <br />All longitude lines are great circles. <br />Spherical Geometry is used by pilots and ship captains.<br />
11. 11. 1.4 Hyperbolic Geometry<br />Hyperbolic geometry is the geometry of a curved space.<br />Same proofs and theorems as Euclidean geometry but from a different perspective.<br />
12. 12. Let’s Summarize…..again<br />Euclidean Geometry<br />Non-Euclidean Geometry<br />Euclid was a Greek mathematician.<br />“The Elements” was written by Euclid sometime 300 BC<br />The concepts studied in Geometry today.<br />Non-Euclidean geometry is any geometry different from Euclidean geometry.<br />Three types of Non-Euclidean geometry are Taxi-Cab, Spherical geometry and Hyperbolic geometry.<br />This applications are used for maps, global traveling or space traveling.<br />
13. 13. Point, Line, Plane line segment or Ray<br />State Whether It is a…..<br />
14. 14. Unit 1: Points, Lines and Planes pg. 5<br />What are the undefined terms of Euclidean Geometry?<br />Points<br />
15. 15. George Seurat (1859-1891) – Paris, France Sunday Afternoon on Isle de La Grande Jatte<br />
16. 16. Points an object or location in space that has no size (no length, no width)<br />Art styles: neo-impressionism, pointillism, divisionism<br />Charles Angrand (1854 -1926) , NormandyCouple in the Street<br />Henri-Edmond Cross(1909)The Church of Santa Maria degli Angeli near Assisi<br />
17. 17. Lines a straight path (a collection<br /> of points) that has no thick- ness and extends forever. <br />Written: AB<br />Endpoint a point at one end of a segment or the starting point <br /> of a ray.<br />Line a straight path (a collection<br />Segment of points) that has no thick- nessand two endpoints.<br /> Written: AB<br />Pablo Picasso (1881-1973) –Malaga, Spain<br />The Guitar Player<br />
18. 18. Ray part of a line that has one<br /> endpoint and extends forever<br /> in one direction.<br /> Written: AB<br />Opposite two rays that have a common <br />Ray endpoint and form a line.<br />Collinear points that are on the same<br /> line. A<br /> B<br /> C<br />Non- not collinear<br />Collinear <br /> A,B, C as a group<br />
19. 19. Salvador Dali (1904-1989) – Figueres, Spain<br />Skull of Zarbaran<br />Plane a flat surface that has no<br /> thickness and extends forever.<br />Coplanar points that are on the same<br /> plane<br />Non- not coplanar<br />Coplanar <br />Art styles: cubism<br />
20. 20. Unit 1: Points, Lines and Planes pg. 6<br />Summary The three undefined terms of Euclidean geometry are___________, ________________ and ____________.<br />
21. 21. A TAUT PIECE OF THREAD<br />Line Segment<br />
22. 22. A KNOT ON A PIECE OF THREAD<br />Point<br />
23. 23. A PIECE OF CLOTH<br />Plane<br />
24. 24. THE WALLS IN YOUR CLASSROOM<br />Plane<br />
25. 25. A CORNER OF A ROOM<br />Point<br />Corner<br />
26. 26. THE BLUE RULES ON YOUR NOTEBOOK PAPER<br />Line Segments<br />
27. 27. YOUR DESKTOP<br />Plane<br />
28. 28. EACH COLOR DOT, OR PIXEL, ON A VIDEO GAME SCREEN<br />Point<br />
29. 29. A TELECOMMUNICATIONS BEAM TO A SATELITE IN SPACE<br />Ray<br />
30. 30. A CREASE IN A FOLDED SHEET OF WRAPPING PAPER<br />Line Segment<br />
31. 31. A SHOOTING STAR<br />Ray<br />
32. 32. THE STARS IN THE SKY<br />Point<br />
33. 33. Y=MX+B<br />Line<br />
34. 34. A CHOCALATE CHIP PANCAKE<br />Plane<br />
35. 35. THE CHOCOLATE CHIPS IN THE PANCAKE<br />Point<br />
36. 36. Remember<br />A point is an exact location without a defined shape or size<br /> A Line goes on forever<br />A Plane is a flat surface<br />A Ray has ONE endpoint<br />A Segment has TWO endpoints<br />
37. 37. Unit 1: Postulates and Theorems pg. 7<br />What are the defined terms of Euclidean Geometry?<br />Theorem A statement that requires proofs and previous postulates. This technique utilizes deductive reasoning. <br />Postulate A statement is accepted as truth without proof. Also called an axiom.<br />
38. 38. Unit 1: Postulates and Theorems<br />(At the bottom of pg 7) <br />Summary Answer Essential Question in Complete Sentences. What are the defined terms of Euclidean Geometry? <br />The defined terms of Euclidean Geometry ____________ and ____________ . The first term is defined as ____________ . The second term is defined as ____________ .<br />
39. 39. Glue the POSTULATE sheet so that it is able to flap open. Cut along the dotted lines. PG 8<br />Postulate Through any two points…<br />1-1-1 <br />There is exactly one line.<br />Postulate Through any three non-collinear<br />1-1-2 points… <br />There is exactly one plane containing them.<br />Postulate If two points lie in a plane, then <br />1-1-3 the line containing those <br /> points..<br />Lies in that plane.<br />Postulate If two lines intersect, then they <br />1-1-4 intersect…<br />In exactly one point.<br />Postulate If two planes intersect, then they <br />1-1-5 intersect…<br />In exactly one line.<br />
40. 40. Activity 1<br />Create a picture using only points<br />Create a picture using line segments (label endpoints)<br />Create a picture for each postulate.<br />
41. 41. Unit 1: Distance and Length pg. 9What does the Ruler Postulate mean and how does it define distance?<br />Parallel Lines Coplanar lines that do not intersect.<br />Perpendicular Lines that intersect to form a right<br />Lines angle<br />
42. 42. Unit 1: Distance and Length pg. 9What is the Ruler Postulate mean and how does it define distance?<br />Ruler Postulate Points on a line can be paired with real numbers and distance between the two points can be found by finding the absolute value of the difference between the numbers.<br />REMEMBER: All distance must be Positive (In GEOMETRY)!!! <br />LENGTH To measure the LENGTH of a <br />Distance (on a number line) segment, you can use a number line to find the DISTANCE between the two endpoints, or you can use the formula.<br />
43. 43. Unit 1: Postulates and Theorems<br />(At the bottom of pg 9) <br />Summary What does the Ruler Postulate and how does it define distance? <br />The Ruler Postulate states ______________. It defines distance as _____________.<br />
44. 44. Ruler Postulate Examples pg 10<br />
45. 45. Segment Addition Postulatepg. 11<br />
46. 46. Segment Addition Postulate pg 12<br />
47. 47. Unit 1: All About ANGLESpg. 1<br />How can you name and classify an angle?<br />Angle<br />
48. 48. Angle A figure formed by two rays <br /> with a common endpoint, <br /> called a vertex<br /> Written: A OR BAC <br /> Side<br />Vertex<br />Side<br /> Ray part of a line that has one<br /> (Sides) endpoint and extends forever<br /> in one direction.<br /> Written: AB<br /> Vertex the common endpoint of the <br />(End- sides of an angle<br /> point)<br />Pablo Picasso (1881-1973) –Malaga, Spain<br />The Guitar Player<br />
49. 49. Interior of<br />an Angle<br />Exterior of an Angle<br />Measure<br />Of an Angle<br />Congruent Angles<br />Degree <br />The set of all points between the sides of an angle<br />A<br />The set of all points outside an angle<br />B<br />Angles are measured in degrees. <br />C<br />Angles with equal measures.<br />of a complete circle<br />
50. 50. Construction A method of creating a mathematically precise figure <br /> using a compass and straight <br /> edge, software, or paper <br /> folding <br />How do I use 1.) Line up the center hole of A protractor? the protractor with the point <br /> or vertex (corner)<br /> 2.) Line up a side (line) with <br /> the straight edge of the <br /> protractor<br /> 3.) Read the number that is <br /> written on the protractor at <br /> the point of intersection <br /> (start from zero and count <br /> up). This is the measure of <br /> the angle in degrees. <br />
51. 51. Unit 1: All About ANGLESpg. 14<br />How can you name and classify an angle?<br /> Protractor Postulate<br /> When it’s a straight line the angles sum up to be <br />
52. 52. Protractor Postulate<br />Measure the Angles<br />How to Use a Protractor<br />1.) Line up the center hole of A protractor the protractor with the point or vertex (corner)<br />2.) Line up a side (line) with the straight edge of the protractor<br />3.) Read the number that is written on the protractor at the point of intersection (start from zero and count up). This is the measure of the angle in degrees. <br />Name a right angle and an acute angle:<br /> <br />Right = ________ Acute = __________ <br /> <br />What is the measure of the only obtuse angle shown? <br /> <br />Obtuse measure = ________°<br />
53. 53. An angle that measures greater than 0° AND less than 90°<br />An angle that measures EXACTLY 90°<br />An angle that measures greater than 90° AND less than 180°<br />An angle that measures EXACTLY 180°<br />Acute<br />Right<br />Obtuse<br />Straight<br />A<br />B<br />C<br />
54. 54. ANGLE ADDITION POSTULATE<br />The measure of angle DEG = 115º, and the measure of angle = 48º. Find the measure of angle FEG.<br />F<br />D<br />E<br />G<br />
55. 55. BISECTOR<br />Ray KM bisects angle JKL, measure of angle JKM = (4x + 6)º, and the measure of angle MKL = (7x – 12)º. Find the measure of angle JKL.<br />J<br />M<br />K<br />L<br />
56. 56.
57. 57. Theorems<br />Congruent<br />Congruent<br />Congruent<br />
58. 58. Vertical Angles<br />
59. 59. Complementary Angles<br />
60. 60. Sources<br />Geometry, Holt<br />Sarah Gorena<br />C-Scope<br />