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					Mathematics Education (Primary/Junior)
            (EDUC 4274)
 Workshop 10: Geometric 2-D and 3-D Shapes

        Location: Room H111, Nipissing University
               North Bay, Ontario, Canada
             Instructor: Daniel H. Jarvis, PhD

                     Office Number: H331
            Office Phone: (705) 474 3461 x 4445
   Office Hours: by appointment (see embedded schedule)
Workshop 15: Exploring 2-D & 3-D Geometry
   Exploring the Towers of Hanoi puzzle
   Some Past Notions and Central Ideas regarding Geometry and
    Spatial Sense Instruction
   Van Hiele Levels of Geometric Understanding
   Angle Measuring Issues: Protractors and Compasses
   3-D Solids: Projective Geometry and 3-D Modelling Software
    (TABS+, Geometer’s Sketchpad, GeoGebra)
   Platonic Solids, Greek Cosmology, & Reciprocity

   Break (10 Minutes)

   Geometry Carousel of 4 Classroom Activities (10 minutes each)
     – Tessellations
     – Geoboards
     – Isometric Drawings
     – Pattern Blocks
   Aftermath: Until We Meet Again
The Towers of Hanoi
    The Tower(s) of Hanoi is a
mathematical game or puzzle. It consists
of three pegs, and a number of disks
(usually 8) of different sizes which can
slide onto any peg. The puzzle starts
with the disks neatly stacked in order of
size on one peg, smallest at the top,
thus making a conical shape.
    The puzzle was invented by the French mathematician Edouard Lucas
in 1883. There is a legend about an Indian temple which contains a large
room with three time-worn posts in it surrounded by 64 golden disks. The
priests of Brahma, acting out the command of an ancient prophecy, have
been moving these disks, in accordance with the rules of the puzzle.
According to the legend, when the last move of the puzzle is completed,
the world will end.
    The objective of the game is to move the entire stack to another peg,
obeying the following rules: (i) Only one disk may be moved at a time; (ii)
Each move consists of taking the upper disk from one of the pegs and
sliding it onto another peg, on top of the other disks that may already be
present on that peg; and, (iii) No disk may be placed on top of a smaller
disk. (Wikipedia, Jan 2007)
Main Ideas for Geometry (MMM)
   Researchers have described a taxonomy of
    geometric development that elaborates growth from
    the least sophisticated to the most sophisticated
    geometric understanding.
   Properties and attributes of geometric shape become
    the object of study in geometry. Many of the same
    properties and attributes that apply to 2-D shapes
    apply to 3-D shapes. The fact that every shape can
    be ―cut up‖ and rearranged into other shapes is a
    fundamental part of this study.
   Many geometric properties and attributes are related
    to measurements, for example, whether two sides
    are the same length, two faces are the same area,
    lines equally distant apart, etc.
Main Ideas for Location/Movement (MMM)

   Although properties of shapes are often the
    focus of attention in geometry, development
    of skills in describing and predicting location
    is also an important aspect of spatial sense.
   There is a continuum of simple to much more
    sophisticated systems to describe location.
    Even the youngest elementary school student
    can use a simple system.
   The study of transformations and the use of
    constructions provide an excellent vehicle for
    exploring geometric properties of both
    position and shape.
Some Past Ideas
 ―We do geometry in June—it’s a fun way
  to end the course.‖
 ―We don’t usually even get to geometry at
  the end of the year—too much to cover.‖
 ―Boys are better than girls at geometry and
  spatial sense.‖
 ―There is little connection to the curriculum
 ―We don’t have resources and software.‖
Central Ideas of Geometry Instruction

 Connects mathematics to the environment
 Language develops over the grades
     square corners  right angles
     corners  vertices
 Connects space: 1D  2D  3D
 Parallels measurement development in
  curriculum (geometry = ―land measure‖)
Van Hiele Levels of Geometric Understanding
                      Stage 5: Rigor
               working in geometric systems
                    Stage 4: Deduction
              developing proofs using axioms

               Stage 3: Informal Deductions
    generalizing by attributes into properties / “if-then”

                     Stage 2: Analysis
        describing the attributes / classes of shapes

                  Stage 1: Visualizations
         recognizing /naming figures / “look like”
Geometry Systems
2-D Space: Polygons (‘many sides’)
   Integrate with measurement curriculum
   ―Edges are straight‖ / ―Faces are flat‖
   Regular and irregular polygons
   Classification of polygons (hierarchy)
     Triangles (―trigons‖): 3 sides
     Quadrilaterals (―tetragons‖): 4 sides
     Pentagons: 5 sides
     Hectagon (―centagon‖): 100 sides
     Chiliagon: 1000 sides
     Myriagon: 10 000 sides                               Art Idea
     Polygons: many (i.e., more than two sides)         Build Mobiles
    (Many books like Burn’s Greedy Triangle available)

      Triangles  Trigons  Trigonometry
 Starting in
                   Slides        Flips       Turns
  Grade 1

                 Translations Reflections   Rotations
to Grade 6

Teaching Aids:

1. Computer Tools (Sketchpad, Cabri, GeoGebra, TABS+)
2. Grid paper transformations
3. Math/Learning Carpet
3-D Geometric Modeling
   TABS+ software (making and exploring nets)
   Tubes using newspapers
   Toothpicks with marshmallows
   Straws/Pipecleaners
   Commercial manipulatives
   Technical Drawing Grid mappings
    – squared paper / isometric paper
   Box Constructions cardstock
Platonic Solids and Connections to
Ancient Greek Cosmology

 Tetrahedron (Fire)
 Hexahedron (Earth)
 Octahedron (Air)
 Icosahedron (Water)
 Dodecahedron (Heavens)
Break (10 minutes)

Geometry Carousel of Activities
 Tessellations
 Geoboards
 Isometric Drawings
 Pattern Blocks
 Read MMM text chapters 15 & 16 on
  Measurement topics for next class
 Check out books on reserve in NU
  library > see Required/Recommended
  Texts webpage for reference codes
 Continue working on your Assignment 2
  with your partner, or individually


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