Algebraic Analysis of Huzita s Origami
|
|
- Kristopher Johns
- 6 years ago
- Views:
Transcription
1 1 / 19 Algebraic Analysis of Huzita s Origami Origami Operations and their Extensions Fadoua Ghourabi, Asem Kasem, Cezary Kaliszyk University of Tsukuba, Japan. Yarmouk Private University, Syria University of Innsbruck, Austria ADG, 17 September 2012
2 2 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
3 3 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
4 4 / 19 What is Origami Ancient Japanese art of paper folding Representation of objects using paper folds traditionally no cutting or glue Used in education, sometimes even daily life Tool for geometrical constructions Instead of ruler and compass in Euclidean geometry Start with a square surface Paper folds describe new lines Intersections of lines give rise to new points
5 5 / 19 Computational Origami Scientific discipline studying mathematical and computational properties of origami. Mathematical theories of paper folding Modelling origami by algebraic and symbolic methods Analyzing origami with computers Simulating origami constructions Proving properties of constructions Correspondence between geometry and algebra Expressed logically Axiom system for origami operations (complete)
6 6 / 19 Advancements in Origami Making origami more formal Larger community effort More problem specifications System of Operations and their power Computational Geometry Methods
7 6 / 19 Advancements in Origami Making origami more formal Larger community effort More problem specifications System of Operations and their power Computational Geometry Methods Origami in Education HS Students are shown the basic operations and given simple tasks Dividing a segment into n equal sub-segments Constructing a square, regular hexagon, equilateral triangle Bi-secting an angle Properties of constructions shown on the blackboard Haga theorem (2/3 with one fold) Various methods of angle trisection
8 7 / 19 Eos System for visualizing origami constructions and proving their properties with the help of Mathematica [Ida et al] Visualizing constructions based on Huzita s axioms Analysing the origami folds algebraically Showing properties of the constructions Algorithmic translation of folds into algebraic properties Usage of Gröbner bases or CAD on whole formulas Flat constructions (every fold followed by unfold) Trisection of an angle Maximum equilateral triangle Regular Heptagon Morley s Triangle Layers and sides Crane WebEos
9 8 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
10 9 / 19 Origami operations (trad. axioms) (O1) Given two points P and Q on the origami O, we can fold O along the line that passes through P and Q. (O2) Given two points P and Q on the origami O, we can fold O to superpose P and Q. (O3) Given two lines m and n which pass through the origami O, we can fold O to superpose m and n. (O4) Given a point P and a line m passing through the origami O, we can fold O along the line that is perpendicular to m and passes through P. (O5) Given a line m and two points P and Q, which the origami O is passing through, it is decidable whether we can fold O to superpose P and m along the line that passes through Q. (O6) Given two points P and Q and two lines m and n, which the origami O is passing through, it is decidable whether we can fold O to superpose P and m, and Q and n, simultaneously. (O7) Given two lines m and n and a point P, which the origami O is passing through, it is decidable whether we can fold O to superpose P and m along the line perpendicular to n.
11 10 / 19 Superposition Superposition pair, s-pair, (α, β) Point-point superposition distinct points P and Q (P, Q) defines a unique fold line that superposes P and Q perpendicular bisector (P Q) Line-line superposition When lines m and n are equal, infinite set of fold lines (with m excluded) Otherwise two distinct lines Point-line superposition (P, m) If P m tangent to the parabola with focus P and directrix m. Otherwise any line perpendicular to n or passing by P. Defines the set Γ(P, m)
12 11 / 19 Superpositions in Huzita s fold principle Table: Superpositions in Huzita s fold principle operation s-pairs degeneracy incidence (1) (P, P), (Q, Q) P = Q (2) (P, Q) P = Q (3) (m, n) m = n (4) (m, m), (P, P) (5) (P, m), (Q, Q) P m (6) (P, m), (Q, n) P = Q m = n P m Q n (7) (P, m), (n, n) P m We can reformulate Huzita s fold principle using the concept of superposition!
13 12 / 19 General Origami Principle (G) Given two points P and Q and two lines m and n, fold O along a line to superpose P and m, and Q and n. Can do all the rest of fold operations? Yes, but we need to carefully analyze the degenerate and incident cases. We analyze the origami constructible s-pairs. details in the paper
14 13 / 19 Principle G related to HO incidence degeneracy operation movement P m, Q n m n B(m) (, ) (m n) P = Q I(P) (, ) (m n) P Q (O1) (, ) (m n) P Q (O4) (, ) (m n) P Q (O4) (, ) P m, Q n (O5) (, ) (O7) (, ) P m, Q n (O5) (, ) (O7) (, ) P m, Q n m = n P = Q Γ(P, m) (, ) (m = n P = Q) (O6) (, ) Table: (G) to perform (O1), (O4) - (O7)
15 14 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
16 15 / 19 Increasing the power of Fold Compass Simplifies constructions But the power is same (equations can be reduced) Multi-fold Arbitrary degree of equations But not feasible by hand Conic sections! Superposition of a point and a conic on the origami
17 16 / 19 General conic fold operation Abstract from the method used to draw a conic on the origami pins, strings, pencil and straightedge Add general fold operation: Given two points P and Q, a line m and a conic section C, where P is not on C and Q is not on m, fold O along a line to superpose P and m, and Q and C. Analyzing the equation of the fold like we get the result: Then the slope of the fold line satisfies a polynomial equation of degree six over the field of origami constructible numbers
18 17 / 19 Example Fold lines k 1,, k 6 whose slopes are the six distinct real solutions of the equation 16t 6 78t t t 3 66t 2 + t + 8 = 0 Mathematica gives 6 approximate solutions to the equation By sliding the operation can be performed by hand
19 18 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
20 19 / 19 Conclusion and Future work Traditional Origami Operations Reformulation using s-pairs Precise degeneracy and incidence conditions General fold operation Conditions for reducing it to the reformulated operations Fold with conic sections Gives rise to 6 possible fold lines
21 19 / 19 Conclusion and Future work Traditional Origami Operations Reformulation using s-pairs Precise degeneracy and incidence conditions General fold operation Conditions for reducing it to the reformulated operations Fold with conic sections Gives rise to 6 possible fold lines Degenerate and incident cases? What equations of degree 6 can be solved? Formalized origami theory
Three connections between origami and mathematics. May 8, 2011
Three connections between origami and mathematics May 8, 2011 What is origami? From Japanese: oro, meaning to fold, and kami, meaning paper A form of visual/sculptural representation that is defined primarily
More informationOrigami Folds in Higher-dimension
EPiC Series in Computing Volume 45, 2017, Pages 83 95 SCSS 2017. The 8th International Symposium on Symbolic Computation in Software Science 2017 Origami Folds in Higher-dimension Tetsuo Ida 1 and Stephen
More informationSFUSD Mathematics Core Curriculum Development Project
1 SFUSD Mathematics Core Curriculum Development Project 2014 2015 Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own
More informationConstructions. Unit 9 Lesson 7
Constructions Unit 9 Lesson 7 CONSTRUCTIONS Students will be able to: Understand the meanings of Constructions Key Vocabulary: Constructions Tools of Constructions Basic geometric constructions CONSTRUCTIONS
More informationWhat role does the central angle play in helping us find lengths of arcs and areas of regions within the circle?
Middletown Public Schools Mathematics Unit Planning Organizer Subject Geometry Grade/Course 10 Unit 5 Circles and other Conic Sections Duration 16 instructional + 4 days for reteaching/enrichment Big Idea
More information*Unit 1 Constructions and Transformations
*Unit 1 Constructions and Transformations Content Area: Mathematics Course(s): Geometry CP, Geometry Honors Time Period: September Length: 10 blocks Status: Published Transfer Skills Previous coursework:
More informationComputational Origami Construction of a Regular Heptagon with Automated Proof of Its Correctness
Computational Origami Construction of a Regular Heptagon with Automated Proof of Its Correctness Judit Robu 1,TetsuoIda 2,DorinŢepeneu 2, Hidekazu Takahashi 3, and Bruno Buchberger 4, 1 Babeş-Bolyai University,
More informationCircles Assignment Answer the following questions.
Answer the following questions. 1. Define constructions. 2. What are the basic tools that are used to draw geometric constructions? 3. What is the use of constructions? 4. What is Compass? 5. What is Straight
More informationChallenges from Ancient Greece
Challenges from ncient Greece Mathematical goals Make formal geometric constructions with a variety of tools and methods. Use congruent triangles to justify geometric constructions. Common Core State Standards
More information8.2 Slippery Slopes. A Solidify Understanding Task
7 8.2 Slippery Slopes A Solidify Understanding Task CC BY https://flic.kr/p/kfus4x While working on Is It Right? in the previous module you looked at several examples that lead to the conclusion that the
More informationFolding Activity 1. Colored paper Tape or glue stick
Folding Activity 1 We ll do this first activity as a class, and I will model the steps with the document camera. Part 1 You ll need: Patty paper Ruler Sharpie Colored paper Tape or glue stick As you do
More informationComputational Construction of a Maximum Equilateral Triangle Inscribed in an Origami
omputational onstruction of a Maximum quilateral Triangle Inscribed in an Origami Tetsuo Ida, Hidekazu Takahashi, Mircea Marin, adoua hourabi, and sem Kasem epartment of omputer Science University of Tsukuba,
More informationMathematics and Origami: The Ancient Arts Unite
Mathematics and Origami: The Ancient Arts Unite Jaema L. Krier Spring 2007 Abstract Mathematics and origami are both considered to be ancient arts, but until the 1960 s the two were considered to be as
More informationConstruction Junction, What s your Function?
Construction Junction, What s your Function? Brian Shay Teacher and Department Chair Canyon Crest Academy Brian.Shay@sduhsd.net @MrBrianShay Session Goals Familiarize ourselves with CCSS and the GSE Geometry
More information(Geometry) Academic Standard: TLW use appropriate tools to perform basic geometric constructions.
Seventh Grade Mathematics Assessments page 1 (Geometry) Academic Standard: TLW use appropriate tools to perform basic geometric constructions. A. TLW use tools to draw squares, rectangles, triangles and
More informationConstructing π Via Origami
Constructing π Via Origami Thomas C. Hull Merrimack College May 5, 2007 Abstract We present an argument for the constructibility of the transcendental number π by paper folding, provided that curved creases
More informationUnit 1 Foundations of Geometry: Vocabulary, Reasoning and Tools
Number of Days: 34 9/5/17-10/20/17 Unit Goals Stage 1 Unit Description: Using building blocks from Algebra 1, students will use a variety of tools and techniques to construct, understand, and prove geometric
More informationFolding Activity 3. Compass Colored paper Tape or glue stick
Folding Activity 3 Part 1 You re not done until everyone in your group is done! If you finish before someone else, help them finish before starting on the next part. You ll need: Patty paper Ruler Sharpie
More information8.2 Slippery Slopes. A Solidify Understanding Task
SECONDARY MATH I // MODULE 8 7 8.2 Slippery Slopes A Solidify Understanding Task CC BY https://flic.kr/p/kfus4x While working on Is It Right? in the previous module you looked at several examples that
More informationJMG. Review Module 1 Lessons 1-20 for Mid-Module. Prepare for Endof-Unit Assessment. Assessment. Module 1. End-of-Unit Assessment.
Lesson Plans Lesson Plan WEEK 161 December 5- December 9 Subject to change 2016-2017 Mrs. Whitman 1 st 2 nd Period 3 rd Period 4 th Period 5 th Period 6 th Period H S Mathematics Period Prep Geometry Math
More informationSec Geometry - Constructions
Sec 2.2 - Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB. A B C **Using a ruler measure the two lengths to make sure they have
More informationBig Ideas Math: A Common Core Curriculum Geometry 2015 Correlated to Common Core State Standards for High School Geometry
Common Core State s for High School Geometry Conceptual Category: Geometry Domain: The Number System G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
More informationStep 2: Extend the compass from the chosen endpoint so that the width of the compass is more than half the distance between the two points.
Student Name: Teacher: Date: District: Miami-Dade County Public Schools Test: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the given
More informationName Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines
Name Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines Two lines are if they are coplanar and do not intersect. Skew lines. Two
More informationE-Origami System Eos
23 2006 1 -Origami System os sem Kasem, etsuo Ida, idekazu akahashi, ircea arin and adoua hourabi e are developing a system called os (-Origami System). os does what a human origamist would do with a piece
More informationSlopes of Lines Notes What is slope?
Slopes of Lines Notes What is slope? Find the slope of each line. 1 Find the slope of each line. Find the slope of the line containing the given points. 6, 2!!"#! 3, 5 4, 2!!"#! 4, 3 Find the slope of
More informationUNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 2: Constructing Lines, Segments, and Angles Instruction
Prerequisite Skills This lesson requires the use of the following skills: using a compass understanding the geometry terms line, segment, ray, and angle Introduction Two basic instruments used in geometry
More informationTHE FOLDED SHAPE RESTORATION AND THE RENDERING METHOD OF ORIGAMI FROM THE CREASE PATTERN
PROCEEDINGS 13th INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS August 4-8, 2008, Dresden (Germany) ISBN: 978-3-86780-042-6 THE FOLDED SHAPE RESTORATION AND THE RENDERING METHOD OF ORIGAMI FROM THE
More informationSection V.1.Appendix. Ruler and Compass Constructions
V.1.Appendix. Ruler and Compass Constructions 1 Section V.1.Appendix. Ruler and Compass Constructions Note. In this section, we explore straight edge and compass constructions. Hungerford s expression
More informationObjective: Use a compass and straight edge to construct congruent segments and angles.
CONSTRUCTIONS Objective: Use a compass and straight edge to construct congruent segments and angles. Introduction to Constructions Constructions: The drawing of various shapes using only a pair of compasses
More information9.3 Properties of Chords
9.3. Properties of Chords www.ck12.org 9.3 Properties of Chords Learning Objectives Find the lengths of chords in a circle. Discover properties of chords and arcs. Review Queue 1. Draw a chord in a circle.
More informationObjective: Use a compass and straight edge to construct congruent segments and angles.
CONSTRUCTIONS Objective: Use a compass and straight edge to construct congruent segments and angles. Oct 1 8:33 AM Oct 2 7:42 AM 1 Introduction to Constructions Constructions: The drawing of various shapes
More informationThe Folded Rectangle Construction
The Folded Rectangle Construction Name(s): With nothing more than a sheet of paper and a single point on the page, you can create a parabola. No rulers and no measuring required! Constructing a Physical
More informationGeometry Unit 3 Note Sheets Date Name of Lesson. Slopes of Lines. Partitioning a Segment. Equations of Lines. Quiz
Date Name of Lesson Slopes of Lines Partitioning a Segment Equations of Lines Quiz Introduction to Parallel and Perpendicular Lines Slopes and Parallel Lines Slopes and Perpendicular Lines Perpendicular
More informationElementary Geometric Drawings Angles. Angle Bisector. Perpendicular Bisector
Lessons and Activities GEOMETRY Elementary Geometric Drawings Angles Angle Bisector Perpendicular Bisector 1 Lessons and Activities POLYGONS are PLANE SHAPES (figures) with at least 3 STRAIGHT sides and
More informationConstructing Perpendicular and Parallel Lines. Adapted from Walch Education
Constructing Perpendicular and Adapted from Walch Education Perpendicular Lines and Bisectors Perpendicular lines are two lines that intersect at a right angle (90 ). A perpendicular line can be constructed
More informationDesign Your Own Dream Home! Michael Daniels Olive Grove Charter School Grade Levels: 9-12 Subject: Mathematics
Design Your Own Dream Home! Michael Daniels Olive Grove Charter School Grade Levels: 9-12 Subject: Mathematics Project Summary: Using Free CAD, a computer aided drafting software program, students design
More informationUNIT PLAN. Grade Level: Unit #: 7 Unit Name: Circles
UNIT PLAN Subject: Geometry Grade Level: 10-12 Unit #: 7 Unit Name: Circles Big Idea/Theme: The understanding of properties of circles, the lines that intersect them, and the use of their special segments
More informationThe Magic Circle Basic Lesson. Developed by The Alexandria Seaport Foundation
The Magic Circle Basic Lesson Developed by The Alexandria Seaport Foundation The Tools Needed Compass Straightedge Pencil Paper (not graph paper, 8.5 x 11 is fine) Your Brain (the most important tool!)
More informationAGS Math Algebra 2 Correlated to Kentucky Academic Expectations for Mathematics Grades 6 High School
AGS Math Algebra 2 Correlated to Kentucky Academic Expectations for Mathematics Grades 6 High School Copyright 2008 Pearson Education, Inc. or its affiliate(s). All rights reserved AGS Math Algebra 2 Grade
More informationName: Date: Chapter 2 Quiz Geometry. Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Date: Chapter 2 Quiz Geometry Multiple Choice Identify the choice that best completes the statement or answers the question. 1. What is the value of x? Identify the missing justifications.,, and.
More informationCopying a Line Segment
Copying a Line Segment Steps 1 4 below show you how to copy a line segment. Step 1 You are given line segment AB to copy. A B Step 2 Draw a line segment that is longer than line segment AB. Label one of
More informationMITOCW watch?v=3jzqchtwv6o
MITOCW watch?v=3jzqchtwv6o PROFESSOR: All right, so lecture 10 was about two main things, I guess. We had the conversion from folding states to folding motions, talked briefly about that. And then the
More information3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm.
1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify
More informationParallel and Perpendicular Lines on the Coordinate Plane
Did You Find a Parking Space? Parallel and Perpendicular Lines on the Coordinate Plane 1.5 Learning Goals Key Term In this lesson, you will: Determine whether lines are parallel. Identify and write the
More informationGrade 6 Math Circles. Math Jeopardy
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 6 Math Circles November 28/29, 2017 Math Jeopardy Centre for Education in Mathematics and Computing This lessons covers all of the material
More informationGeometry Vocabulary Book
Geometry Vocabulary Book Units 2-4 Page 1 Unit 2 General Geometry Point Characteristics: Line Characteristics: Plane Characteristics: RELATED POSTULATES: Through any two points there exists exactly one
More informationUNIT 1 GEOMETRY. (revision from 1 st ESO) Unit 8 in our books
UNIT 1 GEOMETRY (revision from 1 st ESO) Unit 8 in our books WHAT'S GEOMETRY? Geometry is the study of the size, shape and position of 2 dimensional shapes and 3 dimensional figures. In geometry, one explores
More informationMATHEMATICS LEVEL: (B - Γ Λυκείου)
MATHEMATICS LEVEL: 11 12 (B - Γ Λυκείου) 10:00 11:00, 20 March 2010 THALES FOUNDATION 1 3 points 1. Using the picture to the right we can observe that 1+3+5+7 = 4 x 4. What is the value of 1 + 3 + 5 +
More informationFind the coordinates of the midpoint of a segment having the given endpoints.
G.(2) Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and two-dimensional coordinate systems to
More informationUNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 3: Constructing Polygons Instruction
rerequisite Skills This lesson requires the use of the following skills: using a compass copying and bisecting line segments constructing perpendicular lines constructing circles of a given radius Introduction
More informationAssignment. Visiting Washington, D.C. Transversals and Parallel Lines
Assignment Assignment for Lesson.1 Name Date Visiting Washington, D.C. Transversals and Parallel Lines Do not use a protractor in this assignment. Rely only on the measurements given in each problem. 1.
More informationMathematical Construction
Mathematical Construction Full illustrated instructions for the two bisectors: Perpendicular bisector Angle bisector Full illustrated instructions for the three triangles: ASA SAS SSS Note: These documents
More informationThis early Greek study was largely concerned with the geometric properties of conics.
4.3. Conics Objectives Recognize the four basic conics: circle, ellipse, parabola, and hyperbola. Recognize, graph, and write equations of parabolas (vertex at origin). Recognize, graph, and write equations
More informationWhere s the Math in Origami?
Where s the Math in Origami? Origami may not seem like it involves very much mathematics. Yes, origami involves symmetry. If we build a polyhedron then, sure, we encounter a shape from geometry. Is that
More informationAnalytic Geometry/ Trigonometry
Analytic Geometry/ Trigonometry Course Numbers 1206330, 1211300 Lake County School Curriculum Map Released 2010-2011 Page 1 of 33 PREFACE Teams of Lake County teachers created the curriculum maps in order
More informationS. Stirling Page 1 of 14
3.1 Duplicating Segments and ngles [and riangles] hese notes replace pages 144 146 in the book. You can read these pages for extra clarifications. Instructions for making geometric figures: You can sketch
More informationMathematics Success Grade 8
Mathematics Success Grade 8 T429 [OBJECTIVE] The student will solve systems of equations by graphing. [PREREQUISITE SKILLS] solving equations [MATERIALS] Student pages S207 S220 Rulers [ESSENTIAL QUESTIONS]
More information1. Construct the perpendicular bisector of a line segment. Or, construct the midpoint of a line segment. 1. Begin with line segment XY.
1. onstruct the perpendicular bisector of a line segment. Or, construct the midpoint of a line segment. 1. egin with line segment. 2. lace the compass at point. djust the compass radius so that it is more
More informationSession 1 What Is Geometry?
Key Terms for This Session Session 1 What Is Geometry? New in This Session altitude angle bisector concurrent line line segment median midline perpendicular bisector plane point ray Introduction In this
More information10.1 Curves defined by parametric equations
Outline Section 1: Parametric Equations and Polar Coordinates 1.1 Curves defined by parametric equations 1.2 Calculus with Parametric Curves 1.3 Polar Coordinates 1.4 Areas and Lengths in Polar Coordinates
More information0810ge. Geometry Regents Exam y # (x $ 3) 2 % 4 y # 2x $ 5 1) (0,%4) 2) (%4,0) 3) (%4,%3) and (0,5) 4) (%3,%4) and (5,0)
0810ge 1 In the diagram below, ABC! XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements
More informationStandards of Learning Guided Practice Suggestions. For use with the Mathematics Tools Practice in TestNav TM 8
Standards of Learning Guided Practice Suggestions For use with the Mathematics Tools Practice in TestNav TM 8 Table of Contents Change Log... 2 Introduction to TestNav TM 8: MC/TEI Document... 3 Guided
More informationHomi Bhabha Centre for Science Education Tata Institute of Fundamental Research
Homi Bhabha Centre for Science Education Tata Institute of Fundamental Research Mathematics Activity Manual Prepared as a Part of an Internship Project Prepared by Ekta Shokeen Edited By Shweta Naik Internship
More informationACT Coordinate Geometry Review
ACT Coordinate Geometry Review Here is a brief review of the coordinate geometry concepts tested on the ACT. Note: there is no review of how to graph an equation on this worksheet. Questions testing this
More informationWorksheet 10 Memorandum: Construction of Geometric Figures. Grade 9 Mathematics
Worksheet 10 Memorandum: Construction of Geometric Figures Grade 9 Mathematics For each of the answers below, we give the steps to complete the task given. We ve used the following resources if you would
More informationTeacher Lesson Pack Lines and Angles. Suitable for Gr. 6-9
Teacher Lesson Pack Lines and Angles Suitable for Gr. 6-9 1 2 Sir Cumference and the Great Knight of Angleland By: Cindy Neuschwander, Charlsebridge Publishing, ISBN: 1570911525 Read the book to the students.
More informationFolding Tetrahedra and Four-Dimensional Origamis
Original Paper Forma, 15, 49 56, 2000 Folding Tetrahedra and Four-Dimensional Origamis Keimei KAINO Sendai National College of Technology, Aobaku, Sendai 989-3124, Japan E-mail: kaino@cc.sendai-ct.ac.jp
More informationLearning how to axiomatise through paperfolding
Learning how to axiomatise through paperfolding D. Nedrenco Abstract: Mathematical paperfolding can be seen as a helpful tool to start a course with some axiomatisation issues of a mathematical theory.
More informationTHINGS TO DO WITH A GEOBOARD
THINGS TO DO WITH A GEOBOARD The following list of suggestions is indicative of exercises and examples that can be worked on the geoboard. Simpler, as well as, more difficult suggestions can easily be
More informationMATHEMATICS GEOMETRY HONORS. OPTIONS FOR NEXT COURSE Algebra II, Algebra II/Trigonometry, or Algebra, Functions, and Data Analysis
Parent / Student Course Information MATHEMATICS GEOMETRY HONORS Counselors are available to assist parents and students with course selections and career planning. Parents may arrange to meet with the
More informationFrom Flapping Birds to Space Telescopes: The Modern Science of Origami
From Flapping Birds to Space Telescopes: The Modern Science of Origami Robert J. Lang Notes by Radoslav Vuchkov and Samantha Fairchild Abstract This is a summary of the presentation given by Robert Lang
More informationand Transitional Comprehensive Curriculum. Geometry Unit 3: Parallel and Perpendicular Relationships
Geometry Unit 3: Parallel and Perpendicular Relationships Time Frame: Approximately three weeks Unit Description This unit demonstrates the basic role played by Euclid s fifth postulate in geometry. Euclid
More informationEngineering Graphics, Class 5 Geometric Construction. Mohammad I. Kilani. Mechanical Engineering Department University of Jordan
Engineering Graphics, Class 5 Geometric Construction Mohammad I. Kilani Mechanical Engineering Department University of Jordan Conic Sections A cone is generated by a straight line moving in contact with
More informationFoundations of Projective Geometry
C H T E 15 Foundations of rojective Geometry What a delightful thing this perspective is! aolo Uccello (1379-1475) Italian ainter and Mathematician 15.1 XIMS F JECTIVE GEMETY In section 9.3 of Chapter
More informationChapter 2 Review WS Period: Date:
Geometry Name: Chapter 2 Review WS Period: Date:. A transversal intersects two parallel lines. The measures of a pair of alternate interior angles are 5v and 2w. The measures of a pair of same-side exterior
More informationProperties of Chords
Properties of Chords Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org
More informationDeriving the General Equation of a Circle
Deriving the General Equation of a Circle Standard Addressed in this Task MGSE9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square
More informationHands-On Explorations of Plane Transformations
Hands-On Explorations of Plane Transformations James King University of Washington Department of Mathematics king@uw.edu http://www.math.washington.edu/~king The Plan In this session, we will explore exploring.
More informationLocus Locus. Remarks
4 4. The locus of a point is the path traced out by the point moving under given geometrical condition (or conditions). lternatively, the locus is the set of all those points which satisfy the given geometrical
More informationPAPER. Connecting the dots. Giovanna Roda Vienna, Austria
PAPER Connecting the dots Giovanna Roda Vienna, Austria giovanna.roda@gmail.com Abstract Symbolic Computation is an area of computer science that after 20 years of initial research had its acme in the
More informationProblem of the Month What s Your Angle?
Problem of the Month What s Your Angle? Overview: In the Problem of the Month What s Your Angle?, students use geometric reasoning to solve problems involving two dimensional objects and angle measurements.
More informationStereometry Day #1. Stereometry Day #2
8 th Grade Stereometry and Loci Lesson Plans February 2008 Comments: Stereometry is the study of 3-D solids, which includes the Platonic and Archimedean solids. Loci is the study of 2-D curves, which includes
More informationFrom Rabbit Ears to Origami Flowers: Triangle Centers and the Concept of Function
Bridges 2017 Conference Proceedings From Rabbit Ears to Origami Flowers: Triangle Centers and the Concept of Function Alan Russell Department of Mathematics and Statistics Elon University 2320 Campus Box
More informationAnthony Chan. September, Georgia Adult Education Conference
Anthony Chan September, 2018 1 2018 Georgia Adult Education Conference Attendees will be able to: Make difficult math concepts simple and help their students discover math principles on their own. This
More informationChapter 4: The Ellipse
Chapter 4: The Ellipse SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza Chapter 4: The Ellipse Lecture 1: Introduction to Ellipse Lecture 13: Converting
More informationThe 7* Basic Constructions Guided Notes
Name: The 7* asic Constructions Guided Notes Included: 1. Given an segment, construct a 2 nd segment congruent to the original. (ctually not included!) 2. Given an angle, construct a 2 nd angle congruent
More informationJune 2016 Regents GEOMETRY COMMON CORE
1 A student has a rectangular postcard that he folds in half lengthwise. Next, he rotates it continuously about the folded edge. Which three-dimensional object below is generated by this rotation? 4) 2
More informationUnit 6 Task 2: The Focus is the Foci: ELLIPSES
Unit 6 Task 2: The Focus is the Foci: ELLIPSES Name: Date: Period: Ellipses and their Foci The first type of quadratic relation we want to discuss is an ellipse. In terms of its conic definition, you can
More informationThe Basics: Geometric Structure
Trinity University Digital Commons @ Trinity Understanding by Design: Complete Collection Understanding by Design Summer 6-2015 The Basics: Geometric Structure Danielle Kendrick Trinity University Follow
More informationTIalgebra.com Algebra 1
Perpendicular Slopes ID: 8973 Time required 45 minutes Topic: Linear Functions Graph lines whose slopes are negative reciprocals and measure the angles to verify they are perpendicular. Activity Overview
More informationFollow each step of the procedure to fold a strip of 10 equilateral triangles into a flexagon with 3 faces.
Assignment 1 Start with an arbitrary folding line on your paper roll. Do action Folding Up (U) to create a new folding line Do action Folding down (D) to create a new folding line Repeat this (doing U,
More informationSpecial Geometry Exam, Fall 2008, W. Stephen Wilson. Mathematics Department, Johns Hopkins University
Special eometry xam, all 008, W. Stephen Wilson. Mathematics epartment, Johns opkins University I agree to complete this exam without unauthorized assistance from any person, materials or device. Name
More informationPaper 2. Mathematics test. Calculator allowed. First name. Last name. School KEY STAGE TIER
Ma KEY STAGE 3 TIER 6 8 2004 Mathematics test Paper 2 Calculator allowed Please read this page, but do not open your booklet until your teacher tells you to start. Write your name and the name of your
More informationLet s Get This Started!
Lesson 1.1 Assignment 1 Name Date Let s Get This Started! Points, Lines, Planes, Rays, and Line Segments 1. Identify each of the following in the figure shown. a. Name all points. W X p b. Name all lines.
More information6.1 Justifying Constructions
Name lass ate 6.1 Justifying onstructions Essential Question: How can you be sure that the result of a construction is valid? Resource Locker Explore 1 Using a Reflective evice to onstruct a erpendicular
More informationProblem of the Month: Between the Lines
Problem of the Month: Between the Lines Overview: In the Problem of the Month Between the Lines, students use polygons to solve problems involving area. The mathematical topics that underlie this POM are
More informationGeometric Constructions
Geometry Name: Part 1: What are Geometric Constructions? Geometric Constructions Go to http://www.mathopenref.com/constructions.html. Answer the following questions. 1. What is a construction? 2. What
More information!"#$ %&& ' ( ) * ' ) * !"#$!%&&'
!"#$ %&& ' ( ) * ' ) *!"#$!%&&' (+'* ',, '!-.,!!! #,,!,.!! -!, '!*!!,,,!!-. *!'*,-!-,./ From an article written by J.J. O'Connor and E.F. Robertson located at: http://www-history.mcs.st-andrews.ac.uk/mathematicians/hippocrates.html
More informationActivity: Fold Four Boxes
ctivity: Fold Four Boxes 1. Cut out your copy of the crease pattern for the square-base twist box but only cut along the solid lines. 2. Look at this key: mountain crease valley crease When folded, a mountain
More information