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Volume 2 Shape and Space
Volume 2 Shape and Space Colin Foster Introduction Teachers are busy people, so I’ll be brief. Let me tell you what this book isn’t. • It isn’t a book you have to make time to read; it’s a book that will save you time. Take it into the classroom and use ideas from it straight away. Anything requiring preparation or equipment (e.g., photocopies, scissors, an overhead projector, etc.) begins with the word “NEED” in bold followed by the details. • It isn’t a scheme of work, and it isn’t even arranged by age or pupil “level”. Many of the ideas can be used equally well with pupils at different ages and stages. Instead the items are simply arranged by topic. (There is, however, an index at the back linking the “key objectives” from the Key Stage 3 Framework to the sections in these three volumes.) The three volumes cover Number and Algebra (1), Shape and Space (2) and Probability, Statistics, Numeracy and ICT (3). • It isn’t a book of exercises or worksheets. Although you’re welcome to photocopy anything you wish, photocopying is expensive and very little here needs to be photocopied for pupils. Most of the material is intended to be presented by the teacher orally or on the board. Answers and comments are given on the right side of most of the pages or sometimes on separate pages as explained. This is a book to make notes in. Cross out anything you don’t like or would never use. Add in your own ideas or references to other resources. -
Geometry in Design Geometrical Construction in 3D Forms by Prof
D’source 1 Digital Learning Environment for Design - www.dsource.in Design Course Geometry in Design Geometrical Construction in 3D Forms by Prof. Ravi Mokashi Punekar and Prof. Avinash Shide DoD, IIT Guwahati Source: http://www.dsource.in/course/geometry-design 1. Introduction 2. Golden Ratio 3. Polygon - Classification - 2D 4. Concepts - 3 Dimensional 5. Family of 3 Dimensional 6. References 7. Contact Details D’source 2 Digital Learning Environment for Design - www.dsource.in Design Course Introduction Geometry in Design Geometrical Construction in 3D Forms Geometry is a science that deals with the study of inherent properties of form and space through examining and by understanding relationships of lines, surfaces and solids. These relationships are of several kinds and are seen in Prof. Ravi Mokashi Punekar and forms both natural and man-made. The relationships amongst pure geometric forms possess special properties Prof. Avinash Shide or a certain geometric order by virtue of the inherent configuration of elements that results in various forms DoD, IIT Guwahati of symmetry, proportional systems etc. These configurations have properties that hold irrespective of scale or medium used to express them and can also be arranged in a hierarchy from the totally regular to the amorphous where formal characteristics are lost. The objectives of this course are to study these inherent properties of form and space through understanding relationships of lines, surfaces and solids. This course will enable understanding basic geometric relationships, Source: both 2D and 3D, through a process of exploration and analysis. Concepts are supported with 3Dim visualization http://www.dsource.in/course/geometry-design/in- of models to understand the construction of the family of geometric forms and space interrelationships. -
TOPIC 11 Triangles and Polygons
TOPIC 11 Triangles and polygons We can proceed directly from the results we learned about parallel lines and their associated angles to the ideas concerning triangles. An essential question is a question whose answer is not obvious, but it is a vitally important question that we continue to ponder. Essential question: Why are triangles so important in the study of geometry? We will spend more time studying triangles than any other single topic. Why? Let’s review the definitions: 1. What is a triangle? 2. What is an isosceles triangle? 3. What is an equilateral triangle? 4. What is a scalene triangle? 5. What is a scalene triangle? 6. What is an acute triangle? 7. What is an obtuse triangle? 8. What is a right triangle? Topic 11 (Triangles and Polygons) page 2 What can we discover that is true about a triangle? One well-known result concerns the sum of the angles of a triangle. We will do this with a paper triangle in class … a sort of kinesthetic proof. In a more formal way, we will work through a written proof. Given: Δ ABC Prove: m< 1 + m< 2 + m< 3 = 180 Proof: 1. Draw ECD || AB 1. 2. < 2 ______ 2. 3. < 3 ______ 3. 4. < ____ is supp to < ACD 4. 5. m < ____ + m < ACD = 180 5. Def supp 6. m< 1 +m < ____ = m < ACD 6. angle addition postulate 7. m < 1 + m < ____ + m < ____= 180 7 . 8. m< 1 + m < 2 + m < 3 = 180 8. QED Is the parallel postulate necessary in this proof? (We use this as the parallel postulate: If corresponding angles are congruent, then lines are parallel.) As an implication of this fact, we accept as true the fact that there is only one line through a point which is parallel to a given line. -
CONVERGENCE of the RATIO of PERIMETER of a REGULAR POLYGON to the LENGTH of ITS LONGEST DIAGONAL AS the NUMBER of SIDES of POLYGON APPROACHES to ∞ Pawan Kumar B.K
CONVERGENCE OF THE RATIO OF PERIMETER OF A REGULAR POLYGON TO THE LENGTH OF ITS LONGEST DIAGONAL AS THE NUMBER OF SIDES OF POLYGON APPROACHES TO ∞ Pawan Kumar B.K. Kathmandu, Nepal Corresponding to: Pawan Kumar B.K., email: [email protected] ABSTRACT Regular polygons are planar geometric structures that are used to a great extent in mathematics, engineering and physics. For all size of a regular polygon, the ratio of perimeter to the longest diagonal length is always constant and converges to the value of 휋 as the number of sides of the polygon approaches to ∞. The purpose of this paper is to introduce Bishwakarma Ratio Formulas through mathematical explanations. The Bishwakarma Ratio Formulae calculate the ratio of perimeter of regular polygon to the longest diagonal length for all possible regular polygons. These ratios are called Bishwakarma Ratios- often denoted by short term BK ratios- as they have been obtained via Bishwakarma Ratio Formulae. The result has been shown to be valid by actually calculating the ratio for each polygon by using corresponding formula and geometrical reasoning. Computational calculations of the ratios have also been presented upto 30 and 50 significant figures to validate the convergence. Keywords: Regular Polygon, limit, 휋, convergence INTRODUCTION A regular polygon is a planar geometrical structure with equal sides and equal angles. For a 푛(푛−3) regular polygon of 푛 sides there are diagonals. Each angle of a regular polygon of n sides 2 푛−2 is given by ( ) 180° while the sum of interior angles is (푛 − 2)180°. The angle made at the 푛 center of any polygon by lines from any two consecutive vertices (center angle) of a polygon of 360° sides 푛 is given by . -
Geometry and Mensuration IV.7 11 Geometry and Mensuration
Geometry and Mensuration IV.7 11 Geometry and Mensuration The chapters on geometry and mensuration have their own The following is a comprehensive collection of formulae share of questions in the CAT and other MBA entrance ex- based on two-dimensional figures. The student is advised aminations. For doing well in questions based on this chapter, to remember the formulae in this chapter so that he is able the student should familiarise himself/herself with the to solve all the questions based on this chapter. basic formulae and visualisations of the various shapes of solids and two-dimensional figures based on this chapter. THEORY The following is a comprehensive collection of for- mulae based on two-dimensional and three-dimensional Basic Conversions figures: For the purpose of this chapter we have divided the A. 1 m = 100 cm = 1000 mm B. 1 m = 39.37 inches theory in two parts: 1 km = 1000 m 1 mile = 1760 yd ∑ Part I consists of geometry and mensuration of two- = 5/8 miles = 5280 ft dimensional figures 1 inch = 2.54 cm 1 nautical mile (knot) ∑ Part II consists of mensuration of three-dimensional = 6080 ft figures. C. 100 kg = 1 quintal D. 1 litre = 1000 cc 10 quintal = 1 tonne 1 acre = 100 sq m = 1000 kg PART I: GEOMETRY 1 kg = 2.2 pounds 1 hectare = 10000 sq m (approx.) INTRODUCTION TYPES OF ANGLES Geometry and Mensuration are important areas in the CAT examination. In the Online CAT, the Quantitative Aptitude Basic Definitions section has consisted of an average of 15–20% questions from these chapters. -
Mcgraw-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto
GEOMETRY DEMYSTIFIED STAN GIBILISCO McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto [11:38 6/6/ 69 n:4069 GIBI- ebook_copyright 7.5x9.qxd 9/29/03 11:39 AM Page 1 Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as per- mitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-143389-9 The material in this eBook also appears in the print version of this title: 0-07-141650-1. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate train- ing programs. For more information, please contact George Hoare, Special Sales, at [email protected] or (212) 904- 4069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. -
Summary of Introductory Geometry Terminology
MAT104: Fundamentals of Mathematics II Introductory Geometry Terminology Summary Section 11-1: Basic Notions Undefined Terms: Point; Line; Plane Collinear Points: points that lie on the same line Between[-ness]: exactly what the term implies with regard to points [Line] Segment: part of a line consisting of (and named by) two endpoints and all points between them Closed Segment: segment with two included endpoints Half-Open Segment: segment with one included endpoint (also called Half-Closed Segment) Open Segment: segment with no included endpoints Ray: part of a line consisting of one endpoint, a second unique point on the line, all points between them, and all points that have the aforementioned second point between them and the endpoint Half-Line: part of a line on one side of a point on the line (does not include the point) Coplanar: lying on the same plane (points, lines, or parts of lines) Noncoplanar: not coplanar Intersecting Lines: lines with exactly one point in common Skew Lines: lines that cannot be contained in a single plane Concurrent Lines: three or more lines that intersect in the same point Coinciding Lines: identical lines; lines intersecting at every point Parallel Lines: either: a) nonintersecting coplanar lines; or b) two lines that are actually the same line (any line is parallel to itself; coinciding lines as defined above); symbol is Intersecting Planes: planes with exactly one line in common Coinciding Planes: identical planes; planes intersecting at every line Parallel Planes: either: a) nonintersecting planes; or b) two planes that are actually the same plane (any plane is parallel to itself; coinciding planes as defined above) Parallel Line and Plane: a line and plane that either: a) do not intersect in any point; or b) intersect in every point on the line Half-Plane: part of a plane on one side of a line in the plane (does not include the line) Prof. -
Sixth Grade Mathematics Chapter 9 Geometric Properties
SIXTH GRADE MATHEMATICS CHAPTER 9 GEOMETRIC PROPERTIES TOPICS COVERED : Geometry Vocabulary Digital Picture Treasure Hunt Four Triangle Project Naming, Measuring, and Drawing Angles Classifying Angles Classifying Triangles Classifying Quadrilaterals Classifying Polygons Congruent and Similar Figures Transformations Line of Symmetry Coordinate Grids Geometry is the area of mathematics that deals with the properties of points, lines, surfaces, and solids. It is derived from the Greek “geometra” which literally means earth measurement. Created by Lance Mangham, 6 th grade teacher, Carroll ISD Activity 9-1: Dictionary of Geometry Name: Description Example Symbol/Read A point marks an exact location in Point A space. A A line is a collection of points Line EF along a straight path extending endlessly in both directions. E F EF A line segment is a part of a line Line segment AD between two endpoints. A D AD A ray is a part of a line that has one Ray AC endpoint and extends endlessly in one direction. A C AC B An angle is formed by two rays Angle BAC with a common endpoint called a ∠BAC vertex . Vertex A A C A plane is a flat surface extending J endlessly in all directions. Plane LJK K L A change in position A change in position A change in position resulting from a slide is resulting from a turn is resulting from a flip is called a translation . called a rotation . called a reflection . Created by Lance Mangham, 6 th grade teacher, Carroll ISD Classified by Side Length Classified by Largest Type of Angle Description Example Description Example Equilateral Acute Three sides have the All angles are acute same length (all (less than 90 sides congruent). -
Sample Look Book Pages
LEVEL ONE SAMPLE PAGES A Deep and Complex Look at Geometry: Polygons Details Rules Which details are more important than others, and how do you know they Describe a rule, law, or order related to this topic. are important? Polygon definition: many-sided closed figure made by Polygons are 2-dimensional, closed figures made from joining line segments, “regular” polygon has equal sides. line segments. Standards Parts, Facts, Features, Processes Laws What would happen if you didn’t follow this rule. Who are the important people, places, events? Methods chilliagon - 1,000 sides hexagon: honeycomb If the shape is open, it is not a polygon. megagon - 1,000,000 sides tangram: paper cut into If the sides are curved, it is not a polygon. List the important parts, polygons List facts and opinions facts, features, and processes about the topic. of your topic. triangle FACTS: It is easier to square Draw and label all important parts. draw a triangle than it Draw a picture that would show someone how to follow this rule. rectangle is to draw an octagon. pentagon hexagon octagon Large number of OPINIONS: sides: Hexagons are more 17-gon, 18-gon interesting than pentagons. Intentional Teaching Considerations: Details & Rules with Polygons Most elementary students might be able to draw and identify simple triangles and squares, but they often can’t explain why a polygon is different from other geometric shapes. By taking a deeper look at the Details and Rules that define polygons, students will be able to identify more complex forms of polygons in the world around them. -
Name: Date: WORKSHEET : Polygons a Polygon Is a Closed, Planar
Name: Date: WORKSHEET : Polygons A polygon is a closed, planar shape connected by straight lines. Polygons can be categorized as regular or irregular. Polygons can be categorized as convex or concave. Polygons can be categorized as simple or complex. WORKSHEET : Polygons Interior/Exterior Angles ANSWERS : Polygons Interior/Exterior Angles KEY CONCEPTS: Definition of polygons and interior/exterior angle measures. 1. A polygon is a closed, planar shape connected by straight lines. a. Not closed ≠ polygon b. Not all straight lines ≠ polygon c. Planar means it is 2 dimensional; on an x-y coordinate plane for example d. Definitions of terms i. Vertex = The point where two lines meet on the polygon ii. Interior Angle = The angle inside the polygon between adjacent sides iii. Exterior Angle = If a line is extended from one side of the polygon past the vertex then the exterior angle is the angle between that line and the next adjacent side. 2. Names of polygons depend on the number of sides or interior angles (the same value). The smallest number of sides for a two dimensional polygon is 3. As n approaches infinity the polygon approaches a circle, but a circle is not a polygon due to its curves. Number of Sides, (n) Polygon Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Septagon(or Heptagon) 8 Octagon 10 Decagon 12 Dodecagon 15 Pentadecagon 100 Hecatgon 1,000 Chiliagon 1,000,000 Megagon 10100 Googolgon 3. The sum of interior angles of any n-sided polygon is... Sum Interior Angles = 180(n - 2) expressed in degrees e.g. -
Parole Per La Matematica Inglese - Italiano
MATeXp Parole per la matematica inglese - italiano I rinvii interni mandano: con i primi tre caratteri ai capitoli accessibili in http://www.mi.imati.cnr.it/∼alberto , con il quarto carattere alle sezioni e con quinto e sesto ai paragrafi. σ-compact = σ-compatto .:: p-group = p-gruppo .:: p-subgroup = p-sottogruppo .:: q-analogue = q-analogo .:: t-design = t-disegno .:: abacus = abaco .:: aban == :: abbandonare = to discard :: abbassare = to lower :: abbastanza = enough :: abbondante = abundant :: abbondanza = abundance , abundancy :: abbreviare = to shorten :: abbreviation = abbreviazione :: abbreviazione = abbreviation , shortening :: Abelian = abeliano , commutativo :: Abelianization = abelianizzazione .:: abeliano = Abelian , commutative :: abile = able , skilled , skilful :: abilit`a= ability , skill :: abituale = usual :: able = abile :: aboard = a bordo :: above (av, ag) = di sopra , suddetto , surriportato :: above pre) = sopra , oltre :: above prn) = il suddetto , il sunnominato , il surriportato :: abscissa = ascissa : absis = abside .:: absolute = assoluto :: absolutely = assolutamente :: absorption = assorbimento :: abstract / to = astrarre :: abstract = astratto :: abstraction = astrazione :: abstractly = astrattamente :: abstractness = astrattezza :: absurdity = assurdit`a, assurdo :: abundance = abbondanza :: abundancy = abbondanza :: 2015-11-16 011: Parole per la matematica inglese - italiano 1 Alberto Marini abundant = abbondante :: abuse = abuso :: accedere = to access = accedere :: accelerated = accelerato .:: acceleration -
Developing Mathematical Knowledge and Skills Through the Awareness Approach of Teaching and Learning
Journal of Education and Practice www.iiste.org ISSN 2222-1735 (Paper) ISSN 2222-288X (Online) Vol.8, No.13, 2017 Developing Mathematical Knowledge and Skills through the Awareness Approach of Teaching and Learning Abour H. Cherif, Ph.D.* President Emeritus, American Association of University Administrators Associate Editor, Journal of Higher Education Management 728 W. Jachson Blvd., Unit 407. Chicago, Illinois, U.S.A. [email protected] Stefanos Gialamas, Ph.D. President, American Community Schools of Athens 129 Aghias Paraskevis. 15234 Halandri. Athens, Greece [email protected] Angeliki Stamati, Ph.D. Faculty of Mathematics, American Community Schools of Athens 129 Aghias Paraskevis. 15234 Halandri. Athens, Greece [email protected] Abstract Every object we think of or encounter, whether a natural or human-made, has a regular or irregular shape. In its own intrinsic conceptual design, it has elements of mathematics, science, engineering, and arts, etc., which are part of the object’s geometric shape, form and structure. Geometry is not only an important part of mathematics, but it is also an important part of daily life. However, geometry is challenging for some students, even high- achieving students. One way to help students understand geometry and its relevance in life is to engage students to discover them cognitively, then to research and identify them in real world examples and then to relate them to past, present, and future innovations that improved our way of thinking about ourselves and the world around us. This interdisciplinary activity uses the Developmental Awareness Approach of Teaching and Learning (DAATL) to help students discover principles, acquire knowledge, and learn mathematical concepts including surface area, volume, dimensions, regular and irregular plane figures, solid polygons (regular polygons and polyhedra), thinking design, and graph making, etc.