Aspects of Geometry. Finite models of the projective plane and coordinates

Size: px
Start display at page:

Download "Aspects of Geometry. Finite models of the projective plane and coordinates"

Transcription

1 Review Sheet There will be an exam on Thursday, February 14. The exam will cover topics up through material from projective geometry through Day 3 of the DIY Hyperbolic geometry packet. Below are some practice problems. Students are expected to study outside of class, but some class time will be set aside on Tuesday and Wednesday for students to ask questions about topics related to the exam. We will do some practice problems in class on several days leading up to the exam. Formulas: Students are expected to know the formulas and definitions relevant to the class. A brief summary of topics covered: Finite models of the projective plane and coordinates Infinite models of the projective plane including: the three dimensional model, the Euclidean plane with a line at infinity, the unit sphere Homogeneous coordinates, graphing lines and planes Converting from Euclidean plane with a line at infinity (sitting at z = 1) to homogeneous coordinates Projections from lines to lines geometrically and algebraically. Fractional linear transformations. The cross ratio Models of hyperbolic space and their geodesics: the homogeneous model approximated by triangles fitting seven to a point, the Poincare disc, the upper half plane Mobius transformations: in particular, the analysis of the map f(z) = (z i)/(z + i) taking the upper half plane to the Poincare disc The hyperbolic distance formula for the upper half plane Areas of spherical and hyperbolic triangles

2 Projective Plane 1. Do all points on the projective plane look like all other points? Why or why not? 2. In the model of the Euclidean plane with a line at infinity, we say that parallel lines meet at infinity. When we embed this model into the three dimensional model, with the Euclidean plane sitting at z = 1, at what point do parallel lines meet? 3. Why do we draw the Euclidean plane at z = 1 usually? Where else could the Euclidean plane sit inside the three dimensional model of projective space? Homogeneous Coordinates 4. Write four different names for the point [3 : 4 : 1]. 5. What is the Euclidean coordinates for the point corresponding to the projective point [5 : 1 : 3]? 6. What is the equation for the projective line whose image on the Euclidean plane is given by y = 4x + 2? 7. What is the intersection in homogeneous coordinates of the lines 3x + 2y z = 0 and 6x 4y 5z = 0? Is this a point at infinity or on the Euclidean plane? 8. What is the intersection in homogeneous coordinates of the lines x + 2y 3z = 0 and 3x 5z 2y = 1? Is this a point at infinity or on the Euclidean plane? 9. Suppose we took the plane at z = 1 and looked at the projective line y + 2x = 1. What is its point at infinity? What is the equation for the 3D plane defining this line? Now, we want to look at where this projective line intersects the 3D plane z 4x = 2 instead. Where does the point (1, 1) go? What about the point ( 2, 5)? What projective points are fixed under projection from z = 1 to z 4x = 2? 10. Draw the projective line 3z 2y = 0 in the three dimensional model. What is its image in the Euclidean plane at z = 1? 11. What points at infinity should the projective curve y = x 3 have? 12. In the spherical model of the projective plane, draw the curve x 2 + y 2 4z 2 = Try redoing any of the problems from the worksheet on homogeneous coordinates (the typed ones with problems (a) through (g).)

3 Projections 14. Construct two projections which give the function f(x) = 2x + 5. For each step, tell whether the projection is between parallel or perpendicular lines, and whether it is from a point at infinity or a finite point. 15. Prove that the composition of any number of projections is a fractional linear transformation. 16. Give a picture for the projection f(x) = 1/x. Where do the points x = 0 and x = go under this projection? 17. Is every fractional linear transformation a projection? What is the relationship between fractional linear transformations and projections? 18. Is it possible for a projection to fix just one point? That is, is it possible to have a fractional linear transformation f(x) with only one x satisfying f(x) = x? What about exactly two? What about exactly zero, three, and four? The Cross Ratio 19. Show that the cross ratio is preserved under projections. 20. Consider four points of the form x, x + kx, x + k 2 x, x + k 3 x where x and k are real numbers. For what values of k can this be the projection of four equally spaced points? 21. What is the relationship between [p, q, ; r, s] and [p, r; s, q]? 22. Try reproving one or two of the theorems from the cross ratio reading without looking at the proof. 23. Suppose f is a function which preserves the cross ratio, and has values f(0) = 1, f(1) = 2, and f(3) = 1. What is the value of f(5)? Write an equation for f. Hyperbolic Geometry Models 24. Do all points in the hyperbolic plane look like all other points? Why or why not? 25. In the Poincare disc, what are the geodesics? Practice finding a geodesic between two points. 26. In the Poincare disc, draw a line l and a point not on that line P. Draw several straight lines through P which do not intersect l.

4 27. Show that the combinatorial circumference of a disc of radius r in the hyperbolic plane is bounded above by 5 r (7/5) and below by 2 r (7/2). 28. In the upper half plane model, what are the geodesics? Practice finding a geodesic between two points. 29. Find two parallel lines which diverge as you go towards infinity in both directions. 30. Find two parallel lines which converge on one side as you go to infinity. Mobius Transformations 31. Show that Mobius transformations preserve cross ratio. 32. Draw the images of the vertical lines from real points 2, 1, 0, 1, and 2 under the map f(z) = (z i)/(z + i) taking the half plane to the disc. 33. Show that the map g(z) = (z 1/2)/(1 (1/2)z) takes the disk to the disk. Describe where the boundary goes under g and where 0 goes. 34. Consider the map h(z) = 1/z. Describe what this does to the unit circle, z C such that z < 1. What does it do to the boundary of the unit circle z C such that z = 1? Hyperbolic Distance Formula 35. What is the ratio between the hyperbolic distance between 2i 0.1 and 2i + 0.1? What about between 3i 0.1 and 3i + 0.1? 36. What is the hyperbolic distance between the points 2 + i and 4 + i in the upper half plane? Do you expect this to be bigger or smaller than the distance between 2 + 2i and 4 + 2i? What about compared to the distance between 1 + i and 3 + i? 37. Show that d(z, w) = d(w, z) for any two points z, w in the upper half plane. 38. Draw the intersection of circle on the upper half plane model centered at 2i with radius 1 and the imaginary axis. Areas of Spherical and Hyperbolic Triangles 39. Prove that you cannot make a rectangle in the projective plane.

5 40. Suppose I cut a sphere into six slices (six lunes), like a clementine, using lines of longitude that are equally spaced. Then I slice it at the equator. This makes twelve equally sized triangles. What are their angle measurements and area? 41. Draw a triangle with one, two, or three angles equal to 0 degrees on the Poincare disc. 42. What is the area of a pentagon with five right angles? 43. Why does the concept of similarity not mean much in the hyperbolic plane?

WUSTL Math Circle Sept 27, 2015

WUSTL Math Circle Sept 27, 2015 WUSTL Math Circle Sept 7, 015 The K-1 geometry, as we know it, is based on the postulates in Euclid s Elements, which we take for granted in everyday life. Here are a few examples: 1. The distance between

More information

MATERIAL FOR A MASTERCLASS ON HYPERBOLIC GEOMETRY. Timeline. 10 minutes Exercise session: Introducing curved spaces

MATERIAL FOR A MASTERCLASS ON HYPERBOLIC GEOMETRY. Timeline. 10 minutes Exercise session: Introducing curved spaces MATERIAL FOR A MASTERCLASS ON HYPERBOLIC GEOMETRY Timeline 10 minutes Introduction and History 10 minutes Exercise session: Introducing curved spaces 5 minutes Talk: spherical lines and polygons 15 minutes

More information

INTRODUCTION TO 3-MANIFOLDS

INTRODUCTION TO 3-MANIFOLDS INTRODUCTION TO 3-MANIFOLDS NIK AKSAMIT As we know, a topological n-manifold X is a Hausdorff space such that every point contained in it has a neighborhood (is contained in an open set) homeomorphic to

More information

274 Curves on Surfaces, Lecture 5

274 Curves on Surfaces, Lecture 5 274 Curves on Surfaces, Lecture 5 Dylan Thurston Notes by Qiaochu Yuan Fall 2012 5 Ideal polygons Previously we discussed three models of the hyperbolic plane: the Poincaré disk, the upper half-plane,

More information

12 m. 30 m. The Volume of a sphere is 36 cubic units. Find the length of the radius.

12 m. 30 m. The Volume of a sphere is 36 cubic units. Find the length of the radius. NAME DATE PER. REVIEW #18: SPHERES, COMPOSITE FIGURES, & CHANGING DIMENSIONS PART 1: SURFACE AREA & VOLUME OF SPHERES Find the measure(s) indicated. Answers to even numbered problems should be rounded

More information

Module 1 Session 1 HS. Critical Areas for Traditional Geometry Page 1 of 6

Module 1 Session 1 HS. Critical Areas for Traditional Geometry Page 1 of 6 Critical Areas for Traditional Geometry Page 1 of 6 There are six critical areas (units) for Traditional Geometry: Critical Area 1: Congruence, Proof, and Constructions In previous grades, students were

More information

1 Appendix to notes 2, on Hyperbolic geometry:

1 Appendix to notes 2, on Hyperbolic geometry: 1230, notes 3 1 Appendix to notes 2, on Hyperbolic geometry: The axioms of hyperbolic geometry are axioms 1-4 of Euclid, plus an alternative to axiom 5: Axiom 5-h: Given a line l and a point p not on l,

More information

Optimal Möbius Transformation for Information Visualization and Meshing

Optimal Möbius Transformation for Information Visualization and Meshing Optimal Möbius Transformation for Information Visualization and Meshing Marshall Bern Xerox Palo Alto Research Ctr. David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science

More information

Geometry Vocabulary. acute angle-an angle measuring less than 90 degrees

Geometry Vocabulary. acute angle-an angle measuring less than 90 degrees Geometry Vocabulary acute angle-an angle measuring less than 90 degrees angle-the turn or bend between two intersecting lines, line segments, rays, or planes angle bisector-an angle bisector is a ray that

More information

Copyright. Anna Marie Bouboulis

Copyright. Anna Marie Bouboulis Copyright by Anna Marie Bouboulis 2013 The Report committee for Anna Marie Bouboulis Certifies that this is the approved version of the following report: Poincaré Disc Models in Hyperbolic Geometry APPROVED

More information

Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND

Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 1 AND Chapter 9 Geometry Copyright 2009 Pearson Education, Inc. Chapter 9 Section 7 - Slide 2 WHAT YOU WILL LEARN Transformational geometry,

More information

Möbius Transformations in Scientific Computing. David Eppstein

Möbius Transformations in Scientific Computing. David Eppstein Möbius Transformations in Scientific Computing David Eppstein Univ. of California, Irvine School of Information and Computer Science (including joint work with Marshall Bern from WADS 01 and SODA 03) Outline

More information

7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

More information

Lesson Polygons

Lesson Polygons Lesson 4.1 - Polygons Obj.: classify polygons by their sides. classify quadrilaterals by their attributes. find the sum of the angle measures in a polygon. Decagon - A polygon with ten sides. Dodecagon

More information

Geometry. (F) analyze mathematical relationships to connect and communicate mathematical ideas; and

Geometry. (F) analyze mathematical relationships to connect and communicate mathematical ideas; and (1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is (A) apply mathematics to problems arising in everyday life,

More information

Course Number: Course Title: Geometry

Course Number: Course Title: Geometry Course Number: 1206310 Course Title: Geometry RELATED GLOSSARY TERM DEFINITIONS (89) Altitude The perpendicular distance from the top of a geometric figure to its opposite side. Angle Two rays or two line

More information

Straight and Angle on Non-Planar Surfaces: Non-Euclidean Geometry Introduction A module in the Algebra Project high school curriculum

Straight and Angle on Non-Planar Surfaces: Non-Euclidean Geometry Introduction A module in the Algebra Project high school curriculum Straight and Angle on Non-Planar Surfaces: Non-Euclidean Geometry Introduction A module in the Algebra Project high school curriculum David W. Henderson, lead writer Notes to teachers: pg 2 NE1. Straight

More information

Geometry Practice. 1. Angles located next to one another sharing a common side are called angles.

Geometry Practice. 1. Angles located next to one another sharing a common side are called angles. Geometry Practice Name 1. Angles located next to one another sharing a common side are called angles. 2. Planes that meet to form right angles are called planes. 3. Lines that cross are called lines. 4.

More information

3D Hyperbolic Tiling and Horosphere Cross Section

3D Hyperbolic Tiling and Horosphere Cross Section 3D Hyperbolic Tiling and Horosphere Cross Section Vladimir Bulatov, Shapeways Joint AMS/MAA meeting San Diego, January 10, 2018 Inversive Geometry Convenient container to work with 3 dimensional hyperbolic

More information

A VERTICAL LOOK AT KEY CONCEPTS AND PROCEDURES GEOMETRY

A VERTICAL LOOK AT KEY CONCEPTS AND PROCEDURES GEOMETRY A VERTICAL LOOK AT KEY CONCEPTS AND PROCEDURES GEOMETRY Revised TEKS (2012): Building to Geometry Coordinate and Transformational Geometry A Vertical Look at Key Concepts and Procedures Derive and use

More information

STEINER TREE CONSTRUCTIONS IN HYPERBOLIC SPACE

STEINER TREE CONSTRUCTIONS IN HYPERBOLIC SPACE STEINER TREE CONSTRUCTIONS IN HYPERBOLIC SPACE DENISE HALVERSON AND DON MARCH Abstract. Methods for the construction of Steiner minimal trees for n fixed points in the hyperbolic plane are developed. A

More information

Answer Key: Three-Dimensional Cross Sections

Answer Key: Three-Dimensional Cross Sections Geometry A Unit Answer Key: Three-Dimensional Cross Sections Name Date Objectives In this lesson, you will: visualize three-dimensional objects from different perspectives be able to create a projection

More information

Optics II. Reflection and Mirrors

Optics II. Reflection and Mirrors Optics II Reflection and Mirrors Geometric Optics Using a Ray Approximation Light travels in a straight-line path in a homogeneous medium until it encounters a boundary between two different media The

More information

Formal Geometry Unit 9 Quadrilaterals

Formal Geometry Unit 9 Quadrilaterals Name: Period: Formal Geometry Unit 9 Quadrilaterals Date Section Topic Objectives 2/17 9.5 Symmetry I can identify line and rotational symmetries in twodimensional figures. I can identify line and rotational

More information

Study Guide and Review

Study Guide and Review State whether each sentence is or false. If false, replace the underlined term to make a sentence. 1. Euclidean geometry deals with a system of points, great circles (lines), and spheres (planes). false,

More information

Sec 4.1 Coordinates and Scatter Plots. Coordinate Plane: Formed by two real number lines that intersect at a right angle.

Sec 4.1 Coordinates and Scatter Plots. Coordinate Plane: Formed by two real number lines that intersect at a right angle. Algebra I Chapter 4 Notes Name Sec 4.1 Coordinates and Scatter Plots Coordinate Plane: Formed by two real number lines that intersect at a right angle. X-axis: The horizontal axis Y-axis: The vertical

More information

104, 107, 108, 109, 114, 119, , 129, 139, 141, , , , , 180, , , 128 Ch Ch1-36

104, 107, 108, 109, 114, 119, , 129, 139, 141, , , , , 180, , , 128 Ch Ch1-36 111.41. Geometry, Adopted 2012 (One Credit). (c) Knowledge and skills. Student Text Practice Book Teacher Resource: Activities and Projects (1) Mathematical process standards. The student uses mathematical

More information

Geometry Assessment. Eligible Texas Essential Knowledge and Skills

Geometry Assessment. Eligible Texas Essential Knowledge and Skills Geometry Assessment Eligible Texas Essential Knowledge and Skills STAAR Geometry Assessment Reporting Category 1: Geometric Structure The student will demonstrate an understanding of geometric structure.

More information

Carnegie Learning Math Series Course 2, A Florida Standards Program

Carnegie Learning Math Series Course 2, A Florida Standards Program to the students previous understanding of equivalent ratios Introduction to. Ratios and Rates Ratios, Rates,. and Mixture Problems.3.4.5.6 Rates and Tables to Solve Problems to Solve Problems Unit Rates

More information

CURRICULUM CATALOG. GSE Geometry ( ) GA

CURRICULUM CATALOG. GSE Geometry ( ) GA 2018-19 CURRICULUM CATALOG Table of Contents COURSE OVERVIEW... 1 UNIT 1: TRANSFORMATIONS IN THE COORDINATE PLANE... 2 UNIT 2: SIMILARITY, CONGRUENCE, AND PROOFS PART 1... 2 UNIT 3: SIMILARITY, CONGRUENCE,

More information

Grade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe

Grade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles October 16 & 17 2018 Non-Euclidean Geometry and the Globe (Euclidean) Geometry Review:

More information

Number/Computation. addend Any number being added. digit Any one of the ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9

Number/Computation. addend Any number being added. digit Any one of the ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 14 Number/Computation addend Any number being added algorithm A step-by-step method for computing array A picture that shows a number of items arranged in rows and columns to form a rectangle associative

More information

Inversive Plane Geometry

Inversive Plane Geometry Inversive Plane Geometry An inversive plane is a geometry with three undefined notions: points, circles, and an incidence relation between points and circles, satisfying the following three axioms: (I.1)

More information

Topology of Surfaces

Topology of Surfaces EM225 Topology of Surfaces Geometry and Topology In Euclidean geometry, the allowed transformations are the so-called rigid motions which allow no distortion of the plane (or 3-space in 3 dimensional geometry).

More information

Topics in geometry Exam 1 Solutions 7/8/4

Topics in geometry Exam 1 Solutions 7/8/4 Topics in geometry Exam 1 Solutions 7/8/4 Question 1 Consider the following axioms for a geometry: There are exactly five points. There are exactly five lines. Each point lies on exactly three lines. Each

More information

MAT175 Overview and Sample Problems

MAT175 Overview and Sample Problems MAT175 Overview and Sample Problems The course begins with a quick review/overview of one-variable integration including the Fundamental Theorem of Calculus, u-substitutions, integration by parts, and

More information

Refraction at a single curved spherical surface

Refraction at a single curved spherical surface Refraction at a single curved spherical surface This is the beginning of a sequence of classes which will introduce simple and complex lens systems We will start with some terminology which will become

More information

Slope, Distance, Midpoint

Slope, Distance, Midpoint Line segments in a coordinate plane can be analyzed by finding various characteristics of the line including slope, length, and midpoint. These values are useful in applications and coordinate proofs.

More information

Grade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe

Grade 6 Math Circles October 16 & Non-Euclidean Geometry and the Globe Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles October 16 & 17 2018 Non-Euclidean Geometry and the Globe (Euclidean) Geometry Review:

More information

Provide a drawing. Mark any line with three points in blue color.

Provide a drawing. Mark any line with three points in blue color. Math 3181 Name: Dr. Franz Rothe August 18, 2014 All3181\3181_fall14h1.tex Homework has to be turned in this handout. For extra space, use the back pages, or blank pages between. The homework can be done

More information

Reflection & Mirrors

Reflection & Mirrors Reflection & Mirrors Geometric Optics Using a Ray Approximation Light travels in a straight-line path in a homogeneous medium until it encounters a boundary between two different media A ray of light is

More information

Mathematics As A Liberal Art

Mathematics As A Liberal Art Math 105 Fall 2015 BY: 2015 Ron Buckmire Mathematics As A Liberal Art Class 26: Friday November 13 Fowler 302 MWF 10:40am- 11:35am http://sites.oxy.edu/ron/math/105/15/ Euclid, Geometry and the Platonic

More information

Name: Period 2/3/2012 2/16/2012 PreAP

Name: Period 2/3/2012 2/16/2012 PreAP Name: Period 2/3/2012 2/16/2012 PreP UNIT 11: TRNSFORMTIONS I can define, identify and illustrate the following terms: Symmetry Line of Symmetry Rotational Symmetry Translation Symmetry Isometry Pre-Image

More information

Amphitheater School District End Of Year Algebra II Performance Assessment Review

Amphitheater School District End Of Year Algebra II Performance Assessment Review Amphitheater School District End Of Year Algebra II Performance Assessment Review This packet is intended to support student preparation and review for the Algebra II course concepts for the district common

More information

Describe Plane Shapes

Describe Plane Shapes Lesson 12.1 Describe Plane Shapes You can use math words to describe plane shapes. point an exact position or location line endpoints line segment ray a straight path that goes in two directions without

More information

EXPERIENCING GEOMETRY

EXPERIENCING GEOMETRY EXPERIENCING GEOMETRY EUCLIDEAN AND NON-EUCLIDEAN WITH HISTORY THIRD EDITION David W. Henderson Daina Taimina Cornell University, Ithaca, New York PEARSON Prentice Hall Upper Saddle River, New Jersey 07458

More information

Mathematics Scope & Sequence Geometry

Mathematics Scope & Sequence Geometry Mathematics Scope & Sequence Geometry Readiness Standard(s) First Six Weeks (29 ) Coordinate Geometry G.7.B use slopes and equations of lines to investigate geometric relationships, including parallel

More information

Mathematics Scope & Sequence Geometry

Mathematics Scope & Sequence Geometry Mathematics Scope & Sequence 2016-17 Geometry Revised: June 21, 2016 First Grading Period (24 ) Readiness Standard(s) G.5A investigate patterns to make conjectures about geometric relationships, including

More information

Simplicial Hyperbolic Surfaces

Simplicial Hyperbolic Surfaces Simplicial Hyperbolic Surfaces Talk by Ken Bromberg August 21, 2007 1-Lipschitz Surfaces- In this lecture we will discuss geometrically meaningful ways of mapping a surface S into a hyperbolic manifold

More information

Angles, Polygons, Circles

Angles, Polygons, Circles Page 1 of 5 Part One Last week we learned about the angle properties of circles and used them to solve a simple puzzle. This week brings a new puzzle that will make us use our algebra a bit more. But first,

More information

And Now From a New Angle Special Angles and Postulates LEARNING GOALS

And Now From a New Angle Special Angles and Postulates LEARNING GOALS And Now From a New Angle Special Angles and Postulates LEARNING GOALS KEY TERMS. In this lesson, you will: Calculate the complement and supplement of an angle. Classify adjacent angles, linear pairs, and

More information

The Geometry of Solids

The Geometry of Solids CONDENSED LESSON 10.1 The Geometry of Solids In this lesson you will Learn about polyhedrons, including prisms and pyramids Learn about solids with curved surfaces, including cylinders, cones, and spheres

More information

Amarillo ISD Math Curriculum

Amarillo ISD Math Curriculum Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board

More information

Worksheet 3.5: Triple Integrals in Spherical Coordinates. Warm-Up: Spherical Coordinates (ρ, φ, θ)

Worksheet 3.5: Triple Integrals in Spherical Coordinates. Warm-Up: Spherical Coordinates (ρ, φ, θ) Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find

More information

Basic and Intermediate Math Vocabulary Spring 2017 Semester

Basic and Intermediate Math Vocabulary Spring 2017 Semester Digit A symbol for a number (1-9) Whole Number A number without fractions or decimals. Place Value The value of a digit that depends on the position in the number. Even number A natural number that is

More information

High School Geometry. Correlation of the ALEKS course High School Geometry to the ACT College Readiness Standards for Mathematics

High School Geometry. Correlation of the ALEKS course High School Geometry to the ACT College Readiness Standards for Mathematics High School Geometry Correlation of the ALEKS course High School Geometry to the ACT College Readiness Standards for Mathematics Standard 5 : Graphical Representations = ALEKS course topic that addresses

More information

SOL Chapter Due Date

SOL Chapter Due Date Name: Block: Date: Geometry SOL Review SOL Chapter Due Date G.1 2.2-2.4 G.2 3.1-3.5 G.3 1.3, 4.8, 6.7, 9 G.4 N/A G.5 5.5 G.6 4.1-4.7 G.7 6.1-6.6 G.8 7.1-7.7 G.9 8.2-8.6 G.10 1.6, 8.1 G.11 10.1-10.6, 11.5,

More information

Geometric structures on manifolds

Geometric structures on manifolds CHAPTER 3 Geometric structures on manifolds In this chapter, we give our first examples of hyperbolic manifolds, combining ideas from the previous two chapters. 3.1. Geometric structures 3.1.1. Introductory

More information

6.3 Poincare's Theorem

6.3 Poincare's Theorem Figure 6.5: The second cut. for some g 0. 6.3 Poincare's Theorem Theorem 6.3.1 (Poincare). Let D be a polygon diagram drawn in the hyperbolic plane such that the lengths of its edges and the interior angles

More information

1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral

1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral 1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral Show your working and give your answer correct to three decimal places. 2 2.5 3 3.5 4 When When When When When

More information

3 Identify shapes as two-dimensional (lying in a plane, flat ) or three-dimensional ( solid ).

3 Identify shapes as two-dimensional (lying in a plane, flat ) or three-dimensional ( solid ). Geometry Kindergarten Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). 1 Describe objects in the environment using names of shapes,

More information

Light: Geometric Optics

Light: Geometric Optics Light: Geometric Optics 23.1 The Ray Model of Light Light very often travels in straight lines. We represent light using rays, which are straight lines emanating from an object. This is an idealization,

More information

9-1 GCSE Maths. GCSE Mathematics has a Foundation tier (Grades 1 5) and a Higher tier (Grades 4 9).

9-1 GCSE Maths. GCSE Mathematics has a Foundation tier (Grades 1 5) and a Higher tier (Grades 4 9). 9-1 GCSE Maths GCSE Mathematics has a Foundation tier (Grades 1 5) and a Higher tier (Grades 4 9). In each tier, there are three exams taken at the end of Year 11. Any topic may be assessed on each of

More information

Chapter 23. Geometrical Optics: Mirrors and Lenses and other Instruments

Chapter 23. Geometrical Optics: Mirrors and Lenses and other Instruments Chapter 23 Geometrical Optics: Mirrors and Lenses and other Instruments HITT1 A small underwater pool light is 1 m below the surface of a swimming pool. What is the radius of the circle of light on the

More information

1 Euler characteristics

1 Euler characteristics Tutorials: MA342: Tutorial Problems 2014-15 Tuesday, 1-2pm, Venue = AC214 Wednesday, 2-3pm, Venue = AC201 Tutor: Adib Makroon 1 Euler characteristics 1. Draw a graph on a sphere S 2 PROBLEMS in such a

More information

Glossary of dictionary terms in the AP geometry units

Glossary of dictionary terms in the AP geometry units Glossary of dictionary terms in the AP geometry units affine linear equation: an equation in which both sides are sums of terms that are either a number times y or a number times x or just a number [SlL2-D5]

More information

Teaching diary. Francis Bonahon University of Southern California

Teaching diary. Francis Bonahon University of Southern California Teaching diary In the Fall 2010, I used the book Low-dimensional geometry: from euclidean surfaces to hyperbolic knots as the textbook in the class Math 434, Geometry and Transformations, at USC. Most

More information

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions Calculus III Math Spring 7 In-term exam April th. Suggested solutions This exam contains sixteen problems numbered through 6. Problems 5 are multiple choice problems, which each count 5% of your total

More information

MATHEMATICS SYLLABUS SECONDARY 4th YEAR

MATHEMATICS SYLLABUS SECONDARY 4th YEAR European Schools Office of the Secretary-General Pedagogical Development Unit Ref.: 2010-D-581-en-2 Orig.: EN MATHEMATICS SYLLABUS SECONDARY 4th YEAR 4 period/week course APPROVED BY THE JOINT TEACHING

More information

Escher s Circle Limit Anneke Bart Saint Louis University Introduction

Escher s Circle Limit Anneke Bart Saint Louis University  Introduction Escher s Circle Limit Anneke Bart Saint Louis University http://math.slu.edu/escher/ Introduction What are some of the most fundamental things we do in geometry? In the beginning we mainly look at lines,

More information

Virginia Geometry, Semester A

Virginia Geometry, Semester A Syllabus Virginia Geometry, Semester A Course Overview Virginia Geometry, Semester A, provides an in-depth discussion of the basic concepts of geometry. In the first unit, you ll examine the transformation

More information

A simple problem that has a solution that is far deeper than expected!

A simple problem that has a solution that is far deeper than expected! The Water, Gas, Electricity Problem A simple problem that has a solution that is far deeper than expected! Consider the diagram below of three houses and three utilities: water, gas, and electricity. Each

More information

The radius for a regular polygon is the same as the radius of the circumscribed circle.

The radius for a regular polygon is the same as the radius of the circumscribed circle. Perimeter and Area The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area.

More information

Geometry Unit 5 Geometric and Algebraic Connections. Table of Contents

Geometry Unit 5 Geometric and Algebraic Connections. Table of Contents Geometry Unit 5 Geometric and Algebraic Connections Table of Contents Lesson 5 1 Lesson 5 2 Distance.p. 2-3 Midpoint p. 3-4 Partitioning a Directed Line. p. 5-6 Slope. p.7-8 Lesson 5 3 Revisit: Graphing

More information

Cambridge University Press Hyperbolic Geometry from a Local Viewpoint Linda Keen and Nikola Lakic Excerpt More information

Cambridge University Press Hyperbolic Geometry from a Local Viewpoint Linda Keen and Nikola Lakic Excerpt More information Introduction Geometry is the study of spatial relationships, such as the familiar assertion from elementary plane Euclidean geometry that, if two triangles have sides of the same lengths, then they are

More information

Amarillo ISD Math Curriculum

Amarillo ISD Math Curriculum Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board

More information

Course: Geometry PAP Prosper ISD Course Map Grade Level: Estimated Time Frame 6-7 Block Days. Unit Title

Course: Geometry PAP Prosper ISD Course Map Grade Level: Estimated Time Frame 6-7 Block Days. Unit Title Unit Title Unit 1: Geometric Structure Estimated Time Frame 6-7 Block 1 st 9 weeks Description of What Students will Focus on on the terms and statements that are the basis for geometry. able to use terms

More information

Geometry Definitions, Postulates, and Theorems. Chapter 3: Parallel and Perpendicular Lines. Section 3.1: Identify Pairs of Lines and Angles.

Geometry Definitions, Postulates, and Theorems. Chapter 3: Parallel and Perpendicular Lines. Section 3.1: Identify Pairs of Lines and Angles. Geometry Definitions, Postulates, and Theorems Chapter : Parallel and Perpendicular Lines Section.1: Identify Pairs of Lines and Angles Standards: Prepare for 7.0 Students prove and use theorems involving

More information

Geometry Tutor Worksheet 4 Intersecting Lines

Geometry Tutor Worksheet 4 Intersecting Lines Geometry Tutor Worksheet 4 Intersecting Lines 1 Geometry Tutor - Worksheet 4 Intersecting Lines 1. What is the measure of the angle that is formed when two perpendicular lines intersect? 2. What is the

More information

Assignment List. Chapter 1 Essentials of Geometry. Chapter 2 Reasoning and Proof. Chapter 3 Parallel and Perpendicular Lines

Assignment List. Chapter 1 Essentials of Geometry. Chapter 2 Reasoning and Proof. Chapter 3 Parallel and Perpendicular Lines Geometry Assignment List Chapter 1 Essentials of Geometry 1.1 Identify Points, Lines, and Planes 5 #1, 4-38 even, 44-58 even 27 1.2 Use Segments and Congruence 12 #4-36 even, 37-45 all 26 1.3 Use Midpoint

More information

1 Reasoning with Shapes

1 Reasoning with Shapes 1 Reasoning with Shapes Topic 1: Using a Rectangular Coordinate System Lines, Rays, Segments, and Angles Naming Lines, Rays, Segments, and Angles Working with Measures of Segments and Angles Students practice

More information

Geometric structures on manifolds

Geometric structures on manifolds CHAPTER 3 Geometric structures on manifolds In this chapter, we give our first examples of hyperbolic manifolds, combining ideas from the previous two chapters. 3.1. Geometric structures 3.1.1. Introductory

More information

GEOMETRY CURRICULUM MAP

GEOMETRY CURRICULUM MAP 2017-2018 MATHEMATICS GEOMETRY CURRICULUM MAP Department of Curriculum and Instruction RCCSD Congruence Understand congruence in terms of rigid motions Prove geometric theorems Common Core Major Emphasis

More information

Use throughout the course: for example, Parallel and Perpendicular Lines Proving Lines Parallel. Polygons and Parallelograms Parallelograms

Use throughout the course: for example, Parallel and Perpendicular Lines Proving Lines Parallel. Polygons and Parallelograms Parallelograms Geometry Correlated to the Texas Essential Knowledge and Skills TEKS Units Lessons G.1 Mathematical Process Standards The student uses mathematical processes to acquire and demonstrate mathematical understanding.

More information

Introduction to Transformations. In Geometry

Introduction to Transformations. In Geometry + Introduction to Transformations In Geometry + What is a transformation? A transformation is a copy of a geometric figure, where the copy holds certain properties. Example: copy/paste a picture on your

More information

CURRICULUM CATALOG. Geometry ( ) TX

CURRICULUM CATALOG. Geometry ( ) TX 2018-19 CURRICULUM CATALOG Table of Contents GEOMETRY (03100700) TX COURSE OVERVIEW... 1 UNIT 1: INTRODUCTION... 1 UNIT 2: LOGIC... 1 UNIT 3: ANGLES AND PARALLELS... 2 UNIT 4: CONGRUENT TRIANGLES AND QUADRILATERALS...

More information

Unit 3 Higher topic list

Unit 3 Higher topic list This is a comprehensive list of the topics to be studied for the Edexcel unit 3 modular exam. Beside the topics listed are the relevant tasks on www.mymaths.co.uk that students can use to practice. Logon

More information

= f (a, b) + (hf x + kf y ) (a,b) +

= f (a, b) + (hf x + kf y ) (a,b) + Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals

More information

3rd Quarter MATHEMATICS Pointers to Review S.Y

3rd Quarter MATHEMATICS Pointers to Review S.Y Grade 1 Grouping Count groups of equal quantity using concrete objects up to 50 and writes an equivalent expression. e.g. 2 groups of 5 Visualizes, represents, and separates objects into groups of equal

More information

Geometry Workbook WALCH PUBLISHING

Geometry Workbook WALCH PUBLISHING Geometry Workbook WALCH PUBLISHING Table of Contents To the Student..............................vii Unit 1: Lines and Triangles Activity 1 Dimensions............................. 1 Activity 2 Parallel

More information

, Geometry, Quarter 1

, Geometry, Quarter 1 2017.18, Geometry, Quarter 1 The following Practice Standards and Literacy Skills will be used throughout the course: Standards for Mathematical Practice Literacy Skills for Mathematical Proficiency 1.

More information

Geometry Geometry Grade Grade Grade

Geometry Geometry Grade Grade Grade Grade Grade Grade 6.G.1 Find the area of right triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the

More information

An experienced mathematics teacher guides students through the use of a number of very important skills in coordinate geometry.

An experienced mathematics teacher guides students through the use of a number of very important skills in coordinate geometry. Mathematics Stills from our new series Coordinates An experienced mathematics teacher guides students through the use of a number of very important skills in coordinate geometry. Distance between Two Points

More information

Achievement Level Descriptors Geometry

Achievement Level Descriptors Geometry Achievement Level Descriptors Geometry ALD Stard Level 2 Level 3 Level 4 Level 5 Policy MAFS Students at this level demonstrate a below satisfactory level of success with the challenging Students at this

More information

Review 1. Richard Koch. April 23, 2005

Review 1. Richard Koch. April 23, 2005 Review Richard Koch April 3, 5 Curves From the chapter on curves, you should know. the formula for arc length in section.;. the definition of T (s), κ(s), N(s), B(s) in section.4. 3. the fact that κ =

More information

UC Davis MAT 012, Summer Session II, Midterm Examination

UC Davis MAT 012, Summer Session II, Midterm Examination UC Davis MAT 012, Summer Session II, 2018 Midterm Examination Name: Student ID: DATE: August 24, 2018 TIME ALLOWED: 100 minutes INSTRUCTIONS 1. This examination paper contains SEVEN (7) questions and comprises

More information

Prime Time (Factors and Multiples)

Prime Time (Factors and Multiples) CONFIDENCE LEVEL: Prime Time Knowledge Map for 6 th Grade Math Prime Time (Factors and Multiples). A factor is a whole numbers that is multiplied by another whole number to get a product. (Ex: x 5 = ;

More information

Light: Geometric Optics (Chapter 23)

Light: Geometric Optics (Chapter 23) Light: Geometric Optics (Chapter 23) Units of Chapter 23 The Ray Model of Light Reflection; Image Formed by a Plane Mirror Formation of Images by Spherical Index of Refraction Refraction: Snell s Law 1

More information

Mathematics High School Geometry An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts

Mathematics High School Geometry An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts Mathematics High School Geometry An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts interpreting a schematic drawing, estimating the amount of

More information

Unit 6: Connecting Algebra and Geometry Through Coordinates

Unit 6: Connecting Algebra and Geometry Through Coordinates Unit 6: Connecting Algebra and Geometry Through Coordinates The focus of this unit is to have students analyze and prove geometric properties by applying algebraic concepts and skills on a coordinate plane.

More information