# W10

Document Sample

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

Location: Room H111, Nipissing University
Instructor: Daniel H. Jarvis, PhD
Email: danj@nipissingu.ca
Website: http://www.nipissingu.ca/faculty/danj

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
   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
documents.‖
 ―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
 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
Transformations
Starting in
Slides        Flips       Turns

Building
Translations Reflections   Rotations

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)

14
Geometry Carousel of Activities
 Tessellations
 Geoboards
 Isometric Drawings
 Pattern Blocks
Aftermath…
 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