Polygons are figures created from segments that do not intersect at any points other than their endpoints.


 Anna Martin
 2 years ago
 Views:
Transcription
1 Unit #5 Lesson #1: Polygons and Their Angles. Polygons are figures created from segments that do not intersect at any points other than their endpoints. A polygon is convex if all of the interior angles have measure less than 180. A polygon that is not convex is concave. Naming Polygons Number of Sides Name 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon n ngon Theorem: The sum of the interior angles of a convex polygon with n sides is (n 2)180. Theorem: The sum of the exterior angles of a convex polygon is 360. In a regular polygon all sides are congruent and all angles are congruent. Exploration: 1. Draw a polygon with congruent sides, but not congruent angles. 2. Draw a polygon with congruent angles, but not congruent sides. Examples: 1. Find the measure of the largest angle in the polygon below. (12x + 21) (9x) (13x  10) (5x + 33) (8x  6) (15x) 2. Find the measure of an interior angle of a regular pentagon. 3. Find the measure of an exterior angle in a polygon where the ratio of interior to exterior angles is 3:1.
2
3 Problem Set #1: 1. Find the sum of the interior angles of each polygon described below. b. Hexagon c. Decagon 2. Find the sum of the exterior angles of each polygon described below. b. Pentagon c. Decagon 3. Find the measure of one exterior angle of a regular b. Hexagon c. Octagon 4. Find the measure of one interior angle of a regular b. Hexagon c. Octagon 5. A pentagon has angles measuring 84, 112, 134, and 92. What is the measure of the fifth angle? 6. Find the measure of the smallest angle in a polygon having angles measuring 4x 1, 12x, 9x + 6, and 8x How many sides does a regular polygon have if an exterior angle measures 120? 8. How many sides does a regular polygon have if an interior angle measures 120? 9. How many sides does a polygon have if the sum of the interior angles is 2340? 10. In a regular polygon the measure of an interior angle is five times the measure of an exterior angle. a. What is the measure of an interior angle? b. How many sides does the polygon have? 11. The ratio of interior to exterior angles in a regular polygon is 3:2. a. Find the measure of an exterior angle. b. Name the polygon. 12. A regular polygon s interior and exterior angles have the same measure. What is the most specific name for that polygon?
4 13. Polygon ABCDEF is a regular hexagon. Find the value of x. B A C x F D 14. True or False. If the statement is true, give an example or explain why it is true. If the statement is false, provide a counterexample. a. A triangle that is equiangular must be equilateral. b. A quadrilateral that is equiangular must be equilateral. c. A regular polygon can have an interior angle that measures 100. d. As the number of sides of a polygon increases the number of exterior angles increases. e. If the measure of an interior angle of a regular polygon is equal to the measure of the exterior angle, then the polygon must be a pentagon. f. When a new polygon is formed by joining the consecutive midpoints of the sides of an original polygon, the sum of the measures of the angles of the new polygon is equal to the sum of the measures of the angles of the original polygon. 15. E
5 Questions 20 & 21: Given regular pentagon ABCDE with diagonals AC and BE intersecting at F. 20. Prove AC DE. 21. Prove AFE is isosceles. 22. Prove that the opposite sides of a regular quadrilateral are parallel. 23. Prove that the opposite sides of a regular hexagon are parallel. 24. Complete the table below. # of sides in a polygon # of angles in the polygon a. Graph the points putting the number of sides on the xaxis and the number of angles on the yaxis. b. Write an equation for the line that goes through the points you graphed. c. Graph the line.
6 25. Complete the table below. # of sides in a polygon sum of the interior angles in the polygon a. Graph the points putting the number of sides on the xaxis and the number of angles on the yaxis. b. Write an equation for the line that goes through the points you graphed. c. Graph the line. d. What does the slope of the line represent? e. Does the yintercept have meaning? Why or why not? PS #5: 1. a. 360 b. 720 c a. 360 b. 360 c a. 90 b. 60 c a. 90 b. 120 c a. 150 b a. 36 b. decagon 12. square a. T b. F c. F d. T e. F f. T 15. g = 105 and h = j = 120 and k = a = 116, b = 64, c = 90, d = 82, e = 99, f = 88, g = 150, h = 56, j = 106, k = 74, m = 136, n = 118, p = b. y = x 25. b. y 180x 360 (or y ( x 2)180) d. It is the number of degrees the sum increases by each time the number of sides increases by one.
Angles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More information1.1 Identify Points, Lines, and Planes
1.1 Identify Points, Lines, and Planes Objective: Name and sketch geometric figures. Key Vocabulary Undefined terms  These words do not have formal definitions, but there is agreement aboutwhat they mean.
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationGeorgia Online Formative Assessment Resource (GOFAR) AG geometry domain
AG geometry domain Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 36 ) 1. Amy drew a circle graph to represent
More informationChapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.
Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationSolutions Manual for How to Read and Do Proofs
Solutions Manual for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve
More informationStar and convex regular polyhedra by Origami.
Star and convex regular polyhedra by Origami. Build polyhedra by Origami.] Marcel Morales Alice Morales 2009 E D I T I O N M O R A L E S Polyhedron by Origami I) Table of convex regular Polyhedra... 4
More informationCHAPTER 1. LINES AND PLANES IN SPACE
CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given
More informationGEOMETRIC FIGURES, AREAS, AND VOLUMES
HPTER GEOMETRI FIGURES, RES, N VOLUMES carpenter is building a deck on the back of a house. s he works, he follows a plan that he made in the form of a drawing or blueprint. His blueprint is a model of
More informationSA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid
Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.
More informationGeometric Patterns. Introduction. Geometric Patterns 1 ~ Border Patterns. Geometric Patterns 2 ~ Tile Patterns
Geometric Patterns Geometric Patterns Introduction In working with shapes, whether mathematically or artistically, it is necessary to have a good feeling for how more complex shapes can be made from simpler
More informationFORM 3 MATHEMATICS SCHEME C TIME: 30 minutes Non Calculator Paper INSTRUCTIONS TO CANDIDATES
DIRECTORATE FOR QUALITY AND STANDARDS IN EDUCATION Department for Curriculum Management and elearning Educational Assessment Unit Annual Examinations for Secondary Schools 2011 C FORM 3 MATHEMATICS SCHEME
More informationMEP Pupil Text 12. A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued.
MEP Pupil Text Number Patterns. Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. Worked Example Write down the
More informationObjectives. Cabri Jr. Tools
^Åíáîáíó=NO Objectives To learn how to construct all types of triangles using the Cabri Jr. application To reinforce the difference between a construction and a drawing Cabri Jr. Tools fåíêççìåíáçå `çåëíêìåíáåö
More informationInversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)
Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer
More informationBaltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions
Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.
More informationGeoGebra. 10 lessons. Gerrit Stols
GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter
More informationEVERY DAY COUNTS CALENDAR MATH 2005 correlated to
EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 35 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:
More informationGRADES 7, 8, AND 9 BIG IDEAS
Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More information121 Representations of ThreeDimensional Figures
Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 121 Representations of ThreeDimensional Figures Use isometric dot paper to sketch each prism. 1. triangular
More informationA Review of Four HighSchool Mathematics Programs. Guershon Harel University of California, San Diego harel@math.ucsd.edu
1 A Review of Four HighSchool Mathematics Programs Guershon Harel University of California, San Diego harel@math.ucsd.edu 1. Introduction Recently I was commissioned by Strategic Teaching to review four
More informationElements of Plane Geometry by LK
Elements of Plane Geometry by LK These are notes indicating just some bare essentials of plane geometry and some problems to think about. We give a modified version of the axioms for Euclidean Geometry
More informationKEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007
KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 447) Surveys and
More informationOklahoma School Testing Program
Oklahoma School Testing Program Oklahoma Core Curriculum Tests (OCCT) EndofInstruction ACE Geometry Parent, Student, and Teacher Guide Winter/Trimester 013 01 Oklahoma State Department of Education 70561W
More informationThe Geometry of Piles of Salt Thinking Deeply About Simple Things
The Geometry of Piles of Salt Thinking Deeply About Simple Things PCMI SSTP Tuesday, July 15 th, 2008 By Troy Jones Willowcreek Middle School Important Terms (the word line may be replaced by the word
More informationWelcome to Math 7 Accelerated Courses (Preparation for Algebra in 8 th grade)
Welcome to Math 7 Accelerated Courses (Preparation for Algebra in 8 th grade) Teacher: School Phone: Email: Kim Schnakenberg 402443 3101 kschnakenberg@esu2.org Course Descriptions: Both Concept and Application
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationLyman Memorial High School. PreCalculus Prerequisite Packet. Name:
Lyman Memorial High School PreCalculus Prerequisite Packet Name: Dear PreCalculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These
More informationPossible Stage Two Mathematics Test Topics
Possible Stage Two Mathematics Test Topics The Stage Two Mathematics Test questions are designed to be answerable by a good problemsolver with a strong mathematics background. It is based mainly on material
More informationMA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry. Figure 1: Lines in the Poincaré Disk Model
MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry Put your name here: Score: Instructions: For this lab you will be using the applet, NonEuclid, created by Castellanos, Austin, Darnell,
More informationGeo, Chap 4 Practice Test, EV Ver 1
Class: Date: Geo, Chap 4 Practice Test, EV Ver 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (43) In each pair of triangles, parts are congruent as
More informationStandardsBased Learning Teacher Handbook 20142015
StandardsBased Learning Teacher Handbook 20142015 P.O. Box 819 Ranchester, WY 82839 Phone: 3076559541 Fax: 3076559477 www.sheridan.k12.wy.us Table of Contents INTRODUCTION 3 FREQUENTLY ASKED QUESTIONS
More informationMATH CHEAT SHEET. Basic Math and PreAlgebra Cheat Sheet
1 MATH CHEAT SHEET Basic Math and PreAlgebra Cheat Sheet Groups of Numbers Natural or counting numbers: 1, 2, 3, 4, Whole numbers: 0, 1, 2, 3, Integers: 3, 2, 1, 0, 1, 2, 3, Rational numbers: Integers
More informationGeometry  Semester 2. Mrs. DayBlattner 1/20/2016
Geometry  Semester 2 Mrs. DayBlattner 1/20/2016 Agenda 1/20/2016 1) 20 Question Quiz  20 minutes 2) Jan 15 homework  selfcorrections 3) Spot check sheet Thales Theorem  add to your response 4) Finding
More information12 Surface Area and Volume
12 Surface Area and Volume 12.1 ThreeDimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids
More informationExact Values of the Sine and Cosine Functions in Increments of 3 degrees
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms
More informationGeometric Relationships
Geometric Relationships 8 Lines and ngles 81 83 84 LB Building Blocks of Geometry Measuring and Classifying ngles ngle Relationships Classifying Lines Parallel Line Relationships 8B Polygons LB 85
More informationSAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?
SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationStudy Guide Workbook. Contents Include: 96 worksheets one for each lesson
Workbook Contents Include: 96 worksheets one for each lesson To the Teacher: Answers to each worksheet are found in Glencoe s Geometry: Concepts and Applications Chapter Resource Masters and also in the
More informationLine Segments, Rays, and Lines
HOME LINK Line Segments, Rays, and Lines Family Note Help your child match each name below with the correct drawing of a line, ray, or line segment. Then observe as your child uses a straightedge to draw
More informationHIGH SCHOOL: GEOMETRY (Page 1 of 4)
HIGH SCHOOL: GEOMETRY (Page 1 of 4) Geometry is a complete college preparatory course of plane and solid geometry. It is recommended that there be a strand of algebra review woven throughout the course
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationAdvanced GMAT Math Questions
Advanced GMAT Math Questions Version Quantitative Fractions and Ratios 1. The current ratio of boys to girls at a certain school is to 5. If 1 additional boys were added to the school, the new ratio of
More informationTennessee Mathematics Standards 20092010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes
Tennessee Mathematics Standards 20092010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical
More informationWarmUp 1. 1. What is the least common multiple of 6, 8 and 10?
WarmUp 1 1. What is the least common multiple of 6, 8 and 10? 2. A 16page booklet is made from a stack of four sheets of paper that is folded in half and then joined along the common fold. The 16 pages
More informationIllinois State Standards Alignments Grades Three through Eleven
Illinois State Standards Alignments Grades Three through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY
GEOMETRY Student Name: The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015  /YJ {, ~ h (} l J/VJJJ,O 8:30 to 11:30 a.m., only The possession or use
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationMath BINGO MOST POPULAR. Do you have the lucky card? B I N G O
MOST POPULAR Math BINGO Do you have the lucky card? Your club members will love this MATHCOUNTS reboot of a classic game. With the perfect mix of luck and skill, this is a game that can be enjoyed by students
More informationCOORDINATE GEOMETRY Mathematics 1 MM1G1a,b,c,d,e
Student Learning Map Unit 6 COORDINATE GEOMETRY Mathematics 1 MM1G1a,b,c,d,e Key Learning(s): Unit Essential Question(s): 1. Algebraic formulas can be used to find measures of distance on the coordinate
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationTutorial 1: The Freehand Tools
UNC Charlotte Tutorial 1: The Freehand Tools In this tutorial you ll learn how to draw and construct geometric figures using Sketchpad s freehand construction tools. You ll also learn how to undo your
More informationCatalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni versity.
7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,
More informationWarm Up. Write an equation given the slope and yintercept. Write an equation of the line shown.
Warm Up Write an equation given the slope and yintercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and yintercept From the graph, you can see that the slope is
More informationhttp://schoolmaths.com Gerrit Stols
For more info and downloads go to: http://schoolmaths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It
More informationShapes Bingo. More general matters which apply to the use of this unit are covered on the next page.
Shapes Bingo Shapes Bingo This unit provides the material for practicing some basic shape identification in the context of the wellknown game of Bingo. Directions on how to play Bingo are not given here.
More information1. Graphing Linear Inequalities
Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means
More informationGymnázium, Brno, Slovanské nám. 7, SCHEME OF WORK Mathematics SCHEME OF WORK. http://agb.gymnaslo. cz
SCHEME OF WORK Subject: Mathematics Year: Third grade, 3.X School year:../ List of topics Topics Time period 1. Revision (functions, plane geometry) September 2. Constructive geometry in the plane October
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationSupport Materials for Core Content for Assessment. Mathematics
Support Materials for Core Content for Assessment Version 4.1 Mathematics August 2007 Kentucky Department of Education Introduction to Depth of Knowledge (DOK)  Based on Norman Webb s Model (Karin Hess,
More informationMinistry of Education. The Ontario Curriculum. Mathematics. Mathematics Transfer Course, Grade 9, Applied to Academic
Ministry of Education The Ontario Curriculum Mathematics Mathematics Transfer Course, Grade 9, Applied to Academic 2 0 0 6 Contents Introduction....................................................... 2
More informationYOU CAN COUNT ON NUMBER LINES
Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and
More informationSJO PW  Język angielski ogólnotechniczny, Poziom B2 Opracowanie: I. Zamecznik, M. Witczak, H. Maniecka, A. Hilgier,
GEOMETRY AND MEASUREMENT  Teacher s notes and key to tasks Introduction (5 minutes one week before the actual LESSON): 1. Elicit vocabulary connected with geometric figures ask the students (SS) to look
More informationGED Practice Test #3 (34 Questions  each question is worth 2.94 points)
GED Practice Test #3 (34 Questions  each question is worth 2.94 points) 1) Pickins community will be constructing fourteen identical rectangle gardens below. How many feet of fencing will they need? 1)
More informationQuestion 1a of 14 ( 2 Identifying the roots of a polynomial and their importance 91008 )
Quiz: Factoring by Graphing Question 1a of 14 ( 2 Identifying the roots of a polynomial and their importance 91008 ) (x3)(x6), (x6)(x3), (1x3)(1x6), (1x6)(1x3), (x3)*(x6), (x6)*(x3), (1x 3)*(1x6),
More informationSecondary Mathematics Syllabuses
Secondary Mathematics Syllabuses Copyright 006 Curriculum Planning and Development Division. This publication is not for sale. All rights reserved. No part of this publication may be reproduced without
More informationIn this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
More information1.5 Equations of Lines and Planes in 3D
40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3D Recall that given a point P = (a, b, c), one can draw a vector from
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationChapter 3. Inversion and Applications to Ptolemy and Euler
Chapter 3. Inversion and Applications to Ptolemy and Euler 2 Power of a point with respect to a circle Let A be a point and C a circle (Figure 1). If A is outside C and T is a point of contact of a tangent
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationPROOFS PRODUCED BY SECONDARY SCHOOL STUDENTS LEARNING GEOMETRY IN A DYNAMIC COMPUTER ENVIRONMENT
RAMÓN MARRADES and ÁNGEL GUTIÉRREZ PROOFS PRODUCED BY SECONDARY SCHOOL STUDENTS LEARNING GEOMETRY IN A DYNAMIC COMPUTER ENVIRONMENT ABSTRACT. As a key objective, secondary school mathematics teachers seek
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More information4 th Grade Summer Mathematics Review #1. Name: 1. How many sides does each polygon have? 2. What is the rule for this function machine?
. How many sides does each polygon have? th Grade Summer Mathematics Review #. What is the rule for this function machine? A. Pentagon B. Nonagon C. Octagon D. Quadrilateral. List all of the factors of
More informationEdward B. Burger David J. Chard Earlene J. Hall Paul A. Kennedy Steven J. Leinwand Freddie L. Renfro Dale G. Seymour Bert K. Waits
Edward B. Burger David J. Chard Earlene J. Hall Paul A. Kennedy Steven J. Leinwand Freddie L. Renfro Dale G. Seymour Bert K. Waits Geometry Contents in Brief CHAPTER 1 Foundations for Geometry.........................
More informationRegents Examination in Geometry (Common Core) Sample and Comparison Items Spring 2014
Regents Examination in Geometry (Common Core) Sample and Comparison Items Spring 2014 i May 2014 777777 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234 New York
More informationOptimal 3D Angular Resolution for LowDegree Graphs
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 17, no. 3, pp. 173 200 (2013) DOI: 10.7155/jgaa.00290 Optimal 3D Angular Resolution for LowDegree Graphs David Eppstein 1 Maarten Löffler
More information1 Functions, Graphs and Limits
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xyplane), so this section should serve as a review of it and its
More informationBlue Pelican Alg II First Semester
Blue Pelican Alg II First Semester Teacher Version 1.01 Copyright 2009 by Charles E. Cook; Refugio, Tx (All rights reserved) Alg II Syllabus (First Semester) Unit 1: Solving linear equations and inequalities
More informationTo analyze students geometric thinking, use both formative and summative assessments and move students along the van Hiele model of thought.
van Hiele Rev To analyze students geometric thinking, use both formative and summative assessments and move students along the van Hiele model of thought. M. Lynn Breyfogle and Courtney M. Lynch When you
More informationSolving Equations Involving Parallel and Perpendicular Lines Examples
Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines
More informationSINUS. international. Towards New Teaching in Mathematics. Dynamic Geometry with Polygon Pantographs 8. Wolfgang Neidhardt 8 / 2011.
SINUS international Towards New Teaching in Mathematics Wolfgang Neidhardt Dynamic Geometry with Polygon Pantographs 8 Peter Baptist Carsten Miller 8 / 2011 ISSN 21927596 Dagmar Raab University of Bayreuth
More informationMean Geometry. Zachary R. Abel. July 9, 2007
Mean Geometry Zachary R. bel July 9, 2007 ontents Introduction 3. asic efinitions............................................... 3.2 oint verages................................................ 3.3 Figure
More informationDRAFT. New York State Testing Program Grade 8 Common Core Mathematics Test. Released Questions with Annotations
DRAFT New York State Testing Program Grade 8 Common Core Mathematics Test Released Questions with Annotations August 2014 Developed and published under contract with the New York State Education Department
More informationOverview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: SlopeIntercept Form
Name Date Linear Functions: SlopeIntercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review,
More informationNEW MEXICO Grade 6 MATHEMATICS STANDARDS
PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical
More informationSINUS. international. Towards New Teaching in Mathematics. Three Four Five Many. Peter Baptist & Carsten Miller. Peter Baptist
SINUS international Towards New Teaching in Mathematics 9 Peter Baptist & Carsten Miller Three Four Five Many Peter Baptist Carsten Miller 9 / 2011 ISSN 21927596 Dagmar Raab University of Bayreuth (Eds.)
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationAppendix A Designing High School Mathematics Courses Based on the Common Core Standards
Overview: The Common Core State Standards (CCSS) for Mathematics are organized by grade level in Grades K 8. At the high school level, the standards are organized by strand, showing a logical progression
More informationDesigning instructional tools by Flash MX ActionScript some examples to teach basic geometric concepts Yuan, Yuan Lee, ChunYi
Designing instructional tools by Flash MX ActionScript some examples to teach basic geometric concepts Yuan, Yuan Lee, ChunYi It is believed that technology will transform how we view teaching, and it
More informationThe Clar Structure of Fullerenes
The Clar Structure of Fullerenes Liz Hartung Massachusetts College of Liberal Arts June 12, 2013 Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 1 / 25
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationSample Test Questions
mathematics College Algebra Geometry Trigonometry Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about
More information10 th Grade Math Special Education Course of Study
10 th Grade Math Special Education Course of Study Findlay City Schools 2006 Table of Contents 1. Findlay City Schools Mission Statement 2. 10 th Grade Math Curriculum Map 3. 10 th Grade Math Indicators
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More information8th Grade Texas Mathematics: Unpacked Content
8th Grade Texas Mathematics: Unpacked Content What is the purpose of this document? To increase student achievement by ensuring educators understand specifically what the new standards mean a student must
More information