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GEOMETRY MANUAL VOLUME 2 Elementary II Edited by Dr. Pamela Zell Rigg and Jane Wechsler ©Copyright 2004 Montessori Research and Development 16492 Foothill Boulevard, San Leandro, CA 94578-2105 (510) 278-1115 FAX (510) 278-1577 www.MontessoriRD.com Geometry Manual Volume 2 TABLE OF CONTENTS THE STUDY OF GEOMETRY: LEVEL II AN INTRODUCTION ........................................... 1 THE STUDY OF GEOMETRY: LEVEL II FLOW CHART ..................................................... 2 CHAPTER 1: CLASSIFIED NOMENCLATURE BASIC CONCEPTS - POINT, LINE, SURFACE, SOLID ....................................................... 4 STUDY OF THE LINE ................................................................................................... 7 STUDY OF ANGLES ...................................................................................................10 MEASUREMENT OF ANGLES ......................................................................................12 FRACTION/DEGREE/TYPE OF ANGLE CHART ..............................................................17 MEASUREMENT OF ANGLES WITH A REGULAR PROTRACTOR ....................................18 THE FORMATION OF REGIONS - SIMPLE CLOSED CURVE FIGURES AND POLYGONS ...21 STUDY OF THE TRIANGLE .........................................................................................23 STUDY OF TRIANGLES ACCORDING TO SIDES ...........................................................25 STUDY OF TRIANGLES ACCORDING TO ANGLES .......................................................28 STUDY OF TRIANGLES ACCORDING TO ANGLES AND SIDES......................................30 THE RIGHT TRIANGLE ...............................................................................................32 THE EQUILATERAL TRIANGLE ....................................................................................34 THE HEIGHT OR ALTITUDE OF A TRIANGLE ...............................................................36 ORTHOCENTER OF A TRIANGLE ................................................................................39 CONSTRUCT TRIANGLES WITH A COMPASS ..............................................................41 STUDY OF QUADRILATERALS ACCORDING TO SIDES AND ANGLES ...........................43 THE HEIGHT OR ALTITUDE OF A QUADRILATERAL .....................................................48 POLYGONS - REGULAR AND IRREGULAR ................................................................. 52 STUDY OF QUADRILATERALS - EQUIANGULAR AND EQUILATERAL ...........................55 STUDY OF TRIANGLES - EQUIANGULAR AND EQUILATERAL ......................................56 THE DIAGONAL ........................................................................................................57 THE DIAGONAL ........................................................................................................63 THE TRANSVERSAL AND ITS ANGLES .........................................................................64 ALTERNATE ANGLES ARE EQUAL ................................................................................66 INTERIOR ANGLES ON THE SAME SIDE OF THE TRANSVERSAL .................................. 68 EXTERIOR ANGLES ON THE SAME SIDE OF THE TRANSVERSAL .................................. 69 ANGLES ON THE SAME SIDE OF THE TRANSVERSAL ...................................................70 Montessori Research and Development ©2004 Geometry Manual Volume 2 CORRESPONDING ANGLES ..............................................................................72 ADJACENT ANGLES ..........................................................................................74 VERTICAL ANGLES ...........................................................................................77 TWO PARALLEL STRAIGHT LINES CUT BY A TRANSVERSAL ................................81 ERATOSTHENES’ GREAT DISCOVERY: THE CIRCUMFERENCE OF THE EARTH .......83 THE CIRCLE: NOMENCLATURE AND PROPERTIES ............................................................ 85 RELATIONSHIP OF A STRAIGHT LINE AND CIRCUMFERENCE - EXTERNAL ......87 RELATIONSHIP OF A STRAIGHT LINE AND CIRCUMFERENCE - TANGENTIAL ..88 RELATIONSHIP OF A STRAIGHT LINE AND CIRCUMFERENCE - SECANT .........89 RELATIVE POSITION OF TWO CIRCUMFERENCES - EXTERNAL .......................90 RELATIVE POSITION OF TWO CIRCUMFERENCES - INTERNAL .......................91 RELATIVE POSITION OF TWO CIRCUMFERENCES - EXTERNALLY TANGENT ... 92 RELATIVE POSITION OF TWO CIRCUMFERENCES - INTERNALLY TANGENT ... 93 RELATIVE POSITION OF TWO CIRCUMFERENCES - SECANT ...........................94 RELATIVE POSITION OF TWO CIRCUMFERENCES - CONCENTRIC ..................96 CHAPTER 2: STUDY OF CONGRUENCE INTRODUCTION ..............................................................................................98 METAL INSET SQUARES ....................................................................................99 METAL INSET TRIANGLES ................................................................................ 103 CHAPTER 3: STUDY OF SIMILARITIES AND EQUIVALENCE INTRODUCTION ............................................................................................ 106 PROPORTIONALITY WITH TRIANGLES - SENSORIAL PRESENTATION ................ 107 PROPORTIONALITY WITH RECTANGLES - SENSORIAL PRESENTATION ............. 109 PROPORTIONALITY WITH RECTANGLES - MATHEMATICAL PRESENTATION ...... 111 THE STUDY OF EQUIVALENCE: SENSORIAL PRESENTATION ............................. 114 CHAPTER 4: CONSTRUCTIVE TRIANGLES INTRODUCTION ............................................................................................ 117 SEQUENCE OF ACTIVITIES .............................................................................. 118 THE TRIANGLE BOX ....................................................................................... 119 THE GREEN TRIANGLES FROM THE TRIANGLE BOX ........................................ 121 Montessori Research and Development ©2004 Geometry Manual Volume 2 TRANSITIVE QUALITY OF EQUIVALENCIES ...................................................... 123 RELATIONSHIP OF SIDES OF THE TRIANGLE AND SIDES OF THE HALVES ......... 125 LARGE HEXAGON BOX - TRIANGLE AS THE POINT OF REFERENCE .................. 128 RELATIONSHIP OF SIDES OF TRIANGLE AND SIDES OF RHOMBUS .................. 131 LARGE HEXAGON BOX - HEXAGON AS THE POINT OF REFERENCE ................. 132 INSCRIBING A TRIANGLE IN A HEXAGON ....................................................... 133 CIRCUMSCRIBING A TRIANGLE BY A HEXAGON ............................................. 134 RELATIONSHIP OF LINES IN A HEXAGON CONSTRUCTED BY SIX TRIANGLES... 135 THE SMALL HEXAGON ................................................................................... 136 THE SMALL HEXAGON BOX ........................................................................... 138 TERMS OF THE DIFFERENCES AND THE RATIOS BETWEEN THE FIGURES .......... 142 RATIOS BETWEEN THE TWO HEXAGONS: H1 AND H2 .................................... 144 THE DIFFERENCE BETWEEN THE HEXAGONS, H1 AND H2 .............................. 146 EQUIVALENCE OF RED OBTUSE TRIANGLES AND RED EQUILATERAL TRIANGLE148 RATIO BETWEEN AN EQUILATERAL TRIANGLE AND ANOTHER BUILT ON ITS HEIGHT ............................................................................... 151 EQUIVALENCE OF THE GREEN TRAPEZOID AND THE YELLOW TRIANGLE ........ 153 RELATIONSHIP BETWEEN INSCRIBED AND CIRCUMSCRIBED FIGURES ............. 155 THEOREM OF PYTHAGORAS .......................................................................... 157 FIRST EXTENSION OF THE THEOREM OF PYTHAGORAS .................................. 159 THE THEOREM OF PYTHAGORAS ................................................................... 165 FIRST INSET FOR THE THEOREM OF PYTHAGORAS ......................................... 170 SECOND INSET FOR THE THEOREM OF PYTHAGORAS .................................... 172 THIRD INSET FOR THE THEOREM OF PYTHAGORAS ....................................... 173 THE THEOREM OF PYTHAGORAS: THE CONSTRUCTION ................................ 177 CHAPTER 5: EQUIVALENCE INTRODUCTION ............................................................................................ 181 EQUIVALENCE FROM THE TRIANGLE TO THE RECTANGLE .............................. 183 EQUIVALENCE FROM THE RHOMBUS TO THE RECTANGLE ............................. 185 EQUIVALENCE OF THE PARALLELOGRAM AND RECTANGLE ............................ 187 EQUIVALENCE OF THE TRAPEZOID AND RECTANGLE ..................................... 189 REGULAR POLYGONS .................................................................................... 194 STUDY OF APOTHEM ..................................................................................... 201 Montessori Research and Development ©2004