Mathematics Education
EMTH 215
Chapter 8: Developing Geometric Thinking and Spatial Sense
(Summary Notes)
In our study of geometry we will begin with an exploration of 3-dimensional or space geometry, move into two-dimensional or plane geometry, and then work in the exciting world of transformational geometry. Please brush up on your ability to recognize and create all the polyhedra and polygons appropriate to the elementary mathematics curriculum.
Themes--Major Ideas
Introduction
Is geometry a 'man'-made mathematical invention to describe the way the world is organized or is the world in itself
geometrical?
As one's view of the world changes (from a 'flatland' two-dimensional space, to a parallel view, to a view of bent
'lines' -- e.g., light waves as opposed to straight lines) so does one's way of interpreting the world geometrically
(i.e., from a Euclidean perspective to a non-Euclidean perspective). All present-day geometry can do for us is
to help us in 1999 describe geometrically the shapes around us. New knowledge about outer space, the underwater
world, etc. is changing our view of 'earth' and, consequently, our geometrical thinking.
Learning Objectives
After studying chapter 8, students will be able to:
Chapter Overview
Chapter 8 is about an informal approach to geometry learning which involves explorations, discovery, guessing,
and problem solving. The van Hieles' levels of geometric thinking are presented and briefly discussed. Teachers
are encouraged to engage students in open-ended tasks in order to gain insight into their thinking level.
Aspects of topology, Euclidean geometry, rigid transformations, and coordinate geometry that are part of the elementary
and middle school mathematics program are discussed. Under topology, the ideas presented include things that change
and things that do not change, place and order, mazes and networks, and distortion of figures.
Students learn how to classify three-dimensional shapes and two-dimensional figures. Three-dimensional shapes discussed
include regular and semi-regular polyhedra, cylinders and cones. Suggestions for engaging students in construction
tasks using various mediums are presented.
When studying two-dimensional figures, students learn about open and closed curves, convex and concave polygons,
and diagonals. They also learn to classify different kinds of triangles and quadrilaterals according to structure
and properties.
Activities for developing concepts of line and plane symmetry, congruence and similarity are presented. Other topics
discussed include tessellations and curve stitching.
Aspects of transformation geometry mentioned include three rigid motions: translations, reflections, and rotations.
The development of spatial abilities is an important part of a geometry program. Activities to help students develop
their visualization and orientation skills are presented. These include tasks with tangram pieces, problems with
polyominoes, and dissection motion operations on figures.
Coordinate geometry activities include plotting points from sets of ordered pairs and identifying points on a grid.
A list of good resources for geometric activities is presented in the text at the end of chapter 8.
To Think and Talk About
Relevant NCTM Curriculum Standards
Grades K-4
Standard 9 Geometry and Spatial Sense pages 38-40
The NCTM (1989) Curriculum and Evaluation Standards promote the development of spatial sense. The Standards state that the K-4 math curriculum should include two- and three-dimensional geometry so that students can--
Grades 5-8
Standard 5 Geometry pages 87-90
Implementing the Standards
Battista, M. T. and Clements, D. H. (1990). Research into practice: Constructing
geometric concepts in Logo. Arithmetic Teacher, 38, 3, pp. 15-17.
Battista, M. T. and Clements, D. H. (1991). Research into practice: Using spatial imagery in geometric reasoning.
Arithmetic Teacher, 39, 3, pp. 18-21.
Happs, J. and Mansfield, H. (1992). Research into practice: Estimation and mental-imagery models in geometry. Arithmetic Teacher, 40, 1, 44-46.
Morrow, L. J. (1991). Implementing the Standards: Geometry through the Standards. Arithmetic
Teacher, 38.
8, 21-25.
Owens, D. T. (1990). Research into practice: Spatial abilities. Arithmetic
Teacher, 37,
6, 48-51.
Rowan, T. E. (1990). Implementing the Standards: The geometry Standards in K-8 mathematics. Arithmetic
Teacher, 37, 6, 24-28.
Wheatley, G. H. (1991). Research into practice: Enhancing mathematics learning through imagery. Arithmetic Teacher, 39, 1, 34-36.
Wheatley, G. H. (1992). Research into practice: Spatial sense and the construction of abstract units in tiling.
Arithmetic Teacher, 39, 8, 43-45.
Geometry components dealt with at the elementary and middle levels include:
A. 3-dimensional geometry--the geometry
of solid objects such as cones, rectangular prisms, etc.
B. 2-dimensional geometry--the geometry
of the plane (flatland geometry)--topics like angles, rays, lines; common shapes such as squares, rectangles etc.
C. Motion or transformational geometry--the
geometry of motion, such as translations (slides), reflections (flips), rotations (turns). A transformation of
a plane is any one-to-one correspondence between a plane and itself. A translation is a transformation of a plane
that moves every point of the plane a specified distance in a specific direction along a straight line. Any transformation
that preserves distance is called an isometry or a 'rigid motion.' Thus a translation is an isometry.
In addition to A, B, and C the study of geometry is about developing spatial
sense, "that is the ability to mentally picture objects and to maintain
accurate perception of the objects under different orientations." In elementary school, motion geometry introduces
children to the concepts and language of congruency and similarity; they may initially use informal language to
describe their motion geometry experiences, and gradually, by further immersion in the social and mathematical
world of geometry, their language will become more precise. I personally prefer to always use the correct terminology.
We may find we need to brush up on our geometric mathematical vocabulary
and content knowledge.
DEVELOPMENT OF GEOMETRIC THINKING
The van Hieles (see overhead C8-1)
METHOD OF INSTRUCTION
Geometry learning should be informal, involving explorations, discovery, guessing, and problem solving. The new
elementary math curriculum advocates that problem solving be the central focus of the curriculum. Children can
learn about problem solving (e.g., some strategies) by being involved in geometric problem solving explorations;
they can learn geometric concepts via problem solving (problem solving is used here as a means to teach geometry--the
problem would be carefully selected to address specific geometric principles); they can work together to solve
geometry-type problems mainly to get experience in solving problems (this is teaching/learning for problem solving).
Baroody (1993) suggests we overlap the three approaches in an integrated way--in other words, children get the
opportunity to experience solving problems, by being immersed in a high-impact problem (intended to address particular
geometric concepts), while at the same time being taught some strategies and negotiating others. In keeping with
the Professional Standards, the classroom should be a mathematical community, where children can work in groups,
cooperate, negotiate, collaborate, relate mathematical ideas to each other, make mathematical connections, conjecture,
invent, problem solve and engage in mathematical discussion, reasoning, and argument. The main emphasis to me from
what I have read in professional directives and other literature [e.g., topics like the history of the nature of
mathematical ideas; fallibility (Confrey, Lave), social constructivism (Ernest), is that mathematical ideas originated
in people's minds, were tested against reality, were experimented with, discussed with other people, written down,
held up for public scrutiny, argued and validated, and finally became the clear polished finished products we see
today. Children need also to have the experience of behaving like mathematicians (Fellows, 1992), of testing and
discussing their ideas and their constructions against the constructions and ideas of others, of writing about
their ideas, and being involved in justifying their ideas to others. This kind of mathematical community should
be the norm in all classrooms.
Given, then, that this kind of mathematical community exists, what is the teacher's role? I see the teacher doing
four major things during an 'instructional cycle': (1) determining the students' geometric thinking level; (2)
structuring a high-impact geometric learning experience and engaging the students in 'open-ended' geometric explorations;
(3) interacting by asking 'open-ended' questions and responding by validation (i.e., not giving 'right' answers,
but instead showing appreciation for the contribution and prompting the child to be convinced, to share with others,
to justify, "What did you notice?" "How would you describe this?" "Explain how you arrived
at your answer." "Can you make a different one?" etc.); (4) assessing using alternative assessment
strategies (conferences, open-ended performance tasks, observational records, portfolios, etc.).
An example of an open-ended task might be (text, p. 169):
A task such as described above will enable all students to participate, making the mathematics accessible to each
child. The common experience will result in many different solutions, each child engaging in the experience and
bringing to it his/her particular mathematical/historical background.
CONNECTION TO THE WORLD
A math unit on geometry can be integrated with art, social studies, science, and with other math topics, etc. 3-D
objects can be collected and displayed in class; geometry walk; symmetry of butterflies, etc.
STUDYING ASPECTS OF TOPOLOGY
Things that change and things that do not change--see p. 170--balloon example (figure 8-1) for topological characteristics.
Also see activity 8-1--the Mobius Strip. What is the secret of the Mobius Strip?
A plane surface suggested by a sheet of paper has two sides. In fact, a piece of paper has always two sides, right?
Not always. The remarkable Mobius Strip has only one side! The activity helps us understand this concept, but we
are distracted because our mature view of the world comprises a rigidity and permanence in our perception--it is
therefore hard for us to see and accept that objects can change depending on their perspective or position.
Place and order--the language of topology--see
p. 170.
Mazes and networks--see the rules for
odd and even vertices in Billstein (1993) p. 555. See also Billstein, pp. 553-555 for the Konigsberg Bridge problem.
[Networks: In the 1700s, the people of Konigsberg, Germany, (now Kaliningrad, Russia) used to enjoy walking over
the bridges of the Pregel River. There were two islands in the river and seven bridges, as below. These walks eventually
led to the following problem:
Is it possible to walk across all the bridges so that each bridge is crossed exactly once on the same walk.
Bridges of Konigsberg
The actual town of Konigsberg looked like this:
There is no restriction on where to start the walk or where to finish. Try to find a path. This problem, known as the Konigsberg Bridge Problem, was solved in 1736 by Leonard Euler. He represented the problem in much simpler form by representing the land masses, islands, and bridges in what he called a network, as shown below (the beginning of Network Theory).
The points in a network are vertices, and the curves are arcs. Using a network diagram, we can restate the Konigsberg
Bridge Problem as follows: Is there a path through the network beginning at some vertex and ending at the same
or another vertex such that each arc is traversed exactly once?
A network having such a path is traversable; that is, each arc is passed through exactly once. Networks have 4
major properties--see Billstein p. 555.
There are numerous websites that explore the famnous Konigsberg Bridge problem. Some are as follows:
http://mathforum.org/isaac/problems/bridges1.html
A map of the actual town of Konigsberg can be found at this site http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Mathematical_games.html along with other exciting mathematical games and recreations.
Distortion of figures--see the Mobius strip activity and try the balloon experiment featured in figure 8-1.
STUDYING ASPECTS OF EUCLIDEAN GEOMETRY
Three-dimensional shapes--children should
explore and describe their characteristics. Polyhedra is a category of 3-D shapes with faces made up of polygons
(see pages 171-173). What are the Platonic Solids? (See page 175 and O/H C8-2). Check out Euler's Rule (page 173).
Euler's Rule is V + F - E = 2.
STUDYING 3-D SHAPES
Comparing polyhedra--classification and description of shapes--see activities 8-2 through 8-5 and overheads C8-3
and C8-4.
Constructing 3-D shapes--playdoh, straws & pipe cleaners, etc.--see activity 8-6. Polyhedra can also be made
from nets--see activities 8-7 through 8-12 and figures 8-9 and 8-10. Think of other ways to construct solid polyhedra--e.g.,
playdoh or modeling clay (and cheese cutters to cut the plane) are effective ways to construct 3-D shapes--see
figure 8-11 and activity 8-13.
STUDYING 2-DIMENSIONAL FIGURES
Polygons--the generic name for all simple
closed geometric curves composed of straight line segments; they can be convex or concave--see overhead C8-5. See
Billstein page 525 for hierarchical family tree of polygons--also see below:
Triangles--activities 8-14 through 8-16
demonstrate exploration of different triangles.
Quadrilaterals--parallelogram, rectangle,
rhombus, square--children will recognize properties of shapes according to their geometric thinking level. Activities
8-17 through 8-20 deal with quadrilaterals. See also diagonals.
Circles--see activity 8-22--the LOGO
turtle.
SYMMETRY, CONGRUENCE, AND SIMILARITY
Symmetry--nature, art, familiar objects;
folding--see activities 8-23 through 8-25.
Congruence and similarity.
STUDYING ASPECTS OF TRANSFORMATIONAL GEOMETRY
Rigid transformations (as opposed to topological transformations)--slides, flips, turns--see activities 1-10 on
pages 189-190 and overheads C8, 6-8.
TESSELLATIONS
A tessellated area is a flat region being covered with repetitions of the same figure
without any overlapping (see figure 8-21 and activities 8-26 through 8-29).
Examine some tessellation patterns--with the help of the handout. Use squared paper on which to design a shape
to be tessellated--keeping the area constant. Make a cardboard model, and then use that on large sheets of paper
to continue the pattern.
DEVELOPING SPATIAL SENSE
This involves a visualization and an
orientation factor (Owens, 1990). Many
activities in chapter 8 can contribute to the development of visual perception. Other activities involve: Tangram
puzzles, polyominoes (e.g., pentominoes)--see activities 8-30 through 8-33.
Describing shapes--describe shapes through feeling/partner draw; arrange (for only
your eyes to see) a physical structure with colored blocks, then with different geometric solids and explain to
a partner how this shape is organized--the partner now tries to construct the same shape. Students can also be
given complex figures (e.g., figures 8-24 & 8-25).
Dissection Motion Operations--see activities 8-34 & 8-35.
STUDYING ASPECTS OF COORDINATE GEOMETRY
Ordered pair notation--( e.g., Battleships)--see
activities 8-36 & 8-37.
The students could set up a grid from 0-10 on the X (positive--horizontal) and Y (negative--vertical) axes. One
student could mark something on the grid (e.g., neg. 4, pos. 5) and another student could try to guess the location
of the point by clues such as complete miss (nowhere within 3 squares), near miss (within 3 squares), direct hit
(dead on). We could also do an ordered pair outdoor activity or an in-class activity where the students physically
locate themselves as points on a grid (made with tape on the floor marked into coordinates). They could physically
make a function curve, a parabola, etc. by predicting where they would stand and verifying by the positions of
others.
CURVE STITCHING
The art of string sculpture can provide insights into relationships between lines and curves. It also exemplifies
the aesthetics of geometry.
In class--together--we will do the following activities--and others (listed later):
Station #
Activity
Name
Topic
1. Orthographic views using cuisenaire rods
2. Polygons on the geoboard using geoboards and elastics
3. The mira and symmetry
4. Pattern blocks and rotational symmetry
5. Pentatiles and colored tiles
Independent Geometry Stations
Chapter 8 has 37 activities addressing the different ways of thinking about geometry in the elementary school.
I have enlarged these activities, pasted them onto cards, and organized the cards into 12 groups [note the cards
read 6-1; 6-2 etc. and not 8-1; 8-2. These cards were made from the first edition of the class text]. To do any
of the following activities you will need to collect the materials you need. In class you can spend time going
through as many stations as possible, as these activities will give you a good overview of the chapter. An outline
of the activities is as follows:
1
6-1
Topology
Mobius Strips
2
6-2 to 6-12
Euclidean Geometry
Construction of 3-D Shapes
3
6-6 to 6-12
Euclidean Geometry
Dissection of 3-D Shapes
4
6-13
Euclidean Geometry
Dissection of 3-D Shapes
5
6-14to 6-16
2-D shapes
Triangles
6
6-17 to 6-21
2-D shapes
Polygons
7
6-22
2-D shapes
Polygons and LOGO
8
6-23 to 6-25
Motion Geometry
Symmetry
9
6-26 to 6-29
Motion geometry
Tessellations
10
6-30 to 6-33
Developing Spatial Sense
Polyominoes
11
6-34 to 6-35
Developing Spatial Sense
Dissection
12
6-36 to 6-37
Coordinate Geometry
Coordinate Geometry
Elementary (K-5) Level Geometry Learning
Centres
Pentominoes
Materials Needed:
Procedure:
Rectangles with Pentominoes
Materials Needed:
Procedure:
|
Number of Pentominoes Used |
Area |
Dimensions |
Results |
|
1 |
5 |
1 x 5 |
|
Tangrams
Materials Needed:
| # of Pieces | Square | Triangle | Rectangle | Trapezoid | Parallelogram |
| 3 small traingles | |||||
| 5 small pieces | |||||
| all 7 pieces |
3-D Geometry (or solid geometry or Polyhedra)
Construction of 3-D shapes
Using playdoh (or plasticene or marshmallows) and toothpicks, or straws and pipe
cleaners (or cord or elastic thread) make a skeleton 3-D model. OR, using card, make a net that can be folded to
make a 3-D edge model.
See pages 176-179 in your text.
Note: you can buy blackline masters of 3-D shape nets (then all you do is cut them and fold them), or you can design
the net for your 3-D shape (this is an art in itself and not easy)
Van Hiele's Levels of Geometric Thinking
The Seven-Piece Mosaic Puzzle
Read your text pages 171-172 (Development of Geometric Thinking--the van Hieles
Quickly skim the article "Developing Geometric Thinking Activities that
Begin with Play" (read it more thoroughly later). Go to the section titled "Beginning Geometry and the
Mosaic Puzzle." Van Hiele suggests that we now do the following.
Ask children:
1. What can we do with these pieces? [free play activity]
2. Find all the pieces that can be made from two others.
3. Find one piece that can be made from three others.
4. Use # 5 and # 6 mosaic pieces to make shapes. Make sure that you join the two pieces with sides that match (i.e.,
sides that are the same length). How many different shapes can you make using these two pieces? Try this same activity
with pieces #1 and # 2.
5. Use pieces # 2 and # 4 to make the following shape:
parallelogram
Can you make the same shape with pieces # 1 and # 7? What other two pieces make
this shape? Do they also work if they are flipped over?
6. Can you make the above shape with three pieces? Does it work if these three pieces are flipped over?
7. Do one of the puzzle cards on page 313 of the article.
8. What are some other activities you could do with this mosaic puzzle that would enable children to develop their
geometric thinking?
Geometric Flips
1. Quickly skim through the article "Geometric Flips."
2. On a piece of paper, write the word MOM.
3. Using a MIRA reflect this word over a horizontal line of reflection at the top of the word.
4. Repeat with the line of reflection at the bottom of the word.
5. Try to come up with other words that can be replicated through horizontal reflection.
6. Now look at the word MOM again. Would the reflection of MOM over a vertical line be a real word?
7. Try to come up with other words that can be replicated through vertical reflection.
8. Also try to come up with words that make different (real) words through vertical reflection.
9. Can you think of any words that reflect real words when reflected both horizontally and vertically?
10. Use the following chart to find words that fit one or more of the following categories. Can any one word fit
all categories?
written horizontally
flipped horizontallywritten horizontally
flipped horizontally
written vertically
flipped horizontallywritten vertically
flipped vertically
11. What is the longest word that you can find that will reflect itself when
flipped horizontally? Flipped vertically?
Mira Math
In the tub with the Miras is a four-page handout with Mira activities. Go through
this handout, using the Mira to do the activities.
Think of other geometry activities in which a Mira can be used.
Geoboard Activities
In the tub with the geoboards is a two-page geoboard handout. Go through this
handout using the geoboards to do the activities.
Think of other geometry activities in which the geoboard can be used.
Transformational Geometry
Outline of the day's activities:
especially the resource submitted by Diane Hanson and the one submitted by Vivian Archambault Danielle Desjardins and Terry Wood
TRANSFORMATIONAL GEOMETRY--NOTES
Within this big concept we will explore three sub-concepts, namely:
TRANSLATIONS (glides, slides)
REFLECTIONS (mirror images; flips; symmetry)
ROTATIONS (turns)
We will explore each of the above through making a tessellation (an activity suitable for grades 3-8).
Find the box of pre-cut cardboard squares. Take one of these squares and follow the instructions for Tessellations
by Translation. Notice the different examples of methods for making these tessellation templates. Once you have
made your template, tessellate a sheet of white paper using the template. Color your creation!
If time permits, take another square and follow the instructions for Tessellations by Rotation. When you have made
your template, use it to tessellate the piece of paper.
Notice that in both your templates the final template maintains the same area (is isometric) as the original square.
The original shape has been altered, but not in area.
Other shapes can be tessellated. In the basket are pre-cut samples of hexagons, rectangles, and trapezoids. Using
your knowledge of how to make a square tessellation template, now try your hand at making a hexagon tessellation
template, and so on.
Each of your tessellated sheets will form part of a QUILT. Make sure to put your name on your creation.
Challenge:
Adaptation:
Tessellations by Translation
Polygons that tessellate can be altered to create irregularly shaped pictures
that also tessellate. Using a simple "nibble" technique you can easily create irregularly shaped tiles
that will become tessellating pictures. The translation or slide type of transformation is restricted to polygons
(parallelograms and hexagons) whose opposite sides are parallel and congruent because an operation on one side
always affects the opposite side.
Take one of the cardboard squares and colour one side of the square. This will prevent you from inadvertently flipping
the piece while moving or taping your "nibble." Now cut from one corner of the square to an adjacent
corner.
Now take your newly cut "nibble" and slide (or translate) it across to the congruent and parallel side.
It must match the straight edges and corners before being attached to the side. Tape the "nibble" carefully
and securely in its new home.
Since a square has four sides a second "nibble" can be cut from one of the other pair of parallel sides
and slid to the opposite side, once again matching the straight edges very carefully and then taping it into place.
Remember: no trimming to fit!
Now that you are finished "nibbling" and taping the sides of your square, you are ready to tessellate
with the resulting shape. Take a sheet of 8.5 x 11 white paper and see how many complete replications you can make
from your template. You must be very careful in lining up your shape with the sides of the shapes already traced.
You will be sliding or translating your shape along the plane surface (your 8.5 x 11 sheet of paper). Don't flip
your shape over or rotate it while tracing. Keeping the coloured side up at all times should help.
Once you have completed tessellating your plane surface colour your tessellation to highlight your creation. Notice
that the tessellation template that you made has maintained the same area as the original square. This means that
the square and the template are isometric.
Tessellations by Rotation
This transformation is restricted to polygons (triangles, parallelograms, and
hexagons) with adjacent sides that are congruent. You should have mastered the slide technique before trying rotations
or turns.
Begin again with a square of cardboard and colour one side so as to distinguish front and back. Using the "nibble"
technique cut out a "nibble" from corner to corner of your square, but this time rotate the "nibble"
at its endpoint to an adjacent side of your square, not an opposite side. Again tape the piece securely into place
after carefully matching the straight edges..
Alter another side of your square and rotate this "nibble" from its endpoint to the adjacent side of
the square and tape. It's a good idea to alter each side differently to avoid confusion when tessellating.
You can now rotate or turn your tessellation template as you move and trace the template to cover the plane (another
8.5 x 11 sheet of white paper).
You may also want to rotate through the midpoint of a line rather than at the end. If so, mark the midpoint of
each of the 4 sides of your square and then proceed to make a "nibble" from one corner to the marked
midpoint of a side of the square. Rotate this piece about the side's midpoint onto the remaining half of this same
side, then tape. Repeat this procedure for all four sides of the square.
Check out the examples on the poster board. See also the overhead examples, created using a computer program.
Take a look at some of the tessellation resource materials, the plastic templates, the coloring books, etc.
Also check out the Escher materials, as Eschers are similar to tessellations, but very sophisticated ones. We have
Escher scarves, Escher calendars, Escher cut outs, coloring books, and so on.
1. Solid Geometry Websites
The following sites are some that I have visited that are useful for teaching solid or 3-D geometry.
For every good 3-D geometry web-site I found there were about 10 that were not good. If you are having your students
go to websites for math you need to be sure it is a credible site and will offer a worthwhile learning experience
for the students. I find it useful to collect good site around a very precise topic. Then I could plan a lesson
or activity that takes the students to more than one site. I would give children a worksheet like the following,
but I would be very specific regarding what I would want them to do on each site -- or I would ask questions that
would necessitate visiting a number of the following sites.
The purpose for you folks today is simply to learn about solid shapes or polyhedra. What are the regular polyhedra-called
the Platonic Solids and what are the others called? You should be able to name and describe some polyhedra that
you have never seen or heard of before. You should also consider how you would teach the concepts of polyhedra
to your students and especially how you would employ either web technology and/or software in your polyhedra learning
environment.
Geometry Forum: Constructing Geometry on the Internet http://mathforum.org/sum95/projects.html
For Gr. 6-12, contains a collection of math projects on polyhedra, etc
Java 3-D Engine: Multi-Format Viewer http://www.frontiernet.net/~imaging/java-3d-engine.html
http://www.frontiernet.net/~imaging/java_vrml.html
http://www.frontiernet.net/~imaging/java3dviewer.html
These sites display geometric shapes, and music etc. in motion, using java scripting.
http://www.georgehart.com/virtual-polyhedra/vp.html
As the title suggests, here you will find a variety of (virtual) polyhedra
http://www.mathconsult.ch/showroom/unipoly/
This site shows a number of more advanced polyhedra and their properties.
Zome System
http://www.zometool.com/edu/index.html
This is an education site displaying geometry materials for sale; it has some interesting ideas.
http://www.georgehart.com/pavilion.html
Some fascinating 3-D constructions.
Polyhedra Collection
http://www.physics.orst.edu/~bulatov/polyhedra/index.html
Lots of examples of polyhedra.
http://www.wcsu.ctstateu.edu/~gurkewitz/homepage.html
Polyhedra and links to other similar sites.
Paper Models of Polyhedra
http://www.geocities.com/model-world/indexe.html
Paper models, pictures and cutting advice-a great resource for your students
Mathematics Encyclopedia http://www.mathacademy.com/platonic_realms/encyclop/articles/platsol.html
Platonic Solids with Shockwave
http://www.venuemedia.com/mediaband/collins/cube.html
Math Forum Platonic Solid Applets
http://mathforum.com/alejandre/applet.polyhedra.html
Platonic Realms
http://www.mathacademy.com/pr/index.asp
A fascinating site with information on the platonics and more.
Hands-on Math--Models and photo gallery
Geodisic Gumdrops
http://www.exploratorium.edu/science_explorer/geo_gumdrops.html
A List of Polyhedra Links
http://britton.disted.camosun.bc.ca/jbpolyhedra.htm
Unfolding polyhedra
http://www.cs.mcgill.ca/~sqrt/unfold/unfolding.htmlnfolding polyhedra
Polyhedra Animations
http://www.fc.up.pt/atractor/mat/Polied/poliedros-e.htm
Easter Egg Making (Pysanky)
http://www3.ns.sympatico.ca/amorash/ukregg.html
2. Two-Dimensional Geometry Websites
Geometer's Sketchpad
You will want to explore the Geometer's Sketchpad software.
The ideal would be to have the software. But if you don't have it, don't be dismayed. You can download, for a limited
time, free demos from the web-which you can then run on your computer, and/or you can use demonstration examples
on the web. Go now to the following site and explore dynamic two-dimensional geometry.
http://www.keypress.com/catalog/products/software/Prod_GSP.html
Go to the Java portion of the above
sketchpad site at http://www.keypress.com/sketchpad/java_gsp/gallery.html
Explore the dynamic shapes in the JavaSketchpad DR3 Gallery. Below the gallery visit
some of the links. There are some amazing JavaSketchpad files at http://math2.math.nthu.edu.tw/jcchuan/java-sketchpad/jsp.html
(they're mostly for high school students, but you might like to look at some of them)
Geometer's Sketchpad is perhaps your
foremost computer resource (software or web-based) for inclusion in a rich classroom environment, structured to
enable your students to learn basic concepts in two-dimensional geometry. There are two other similar computer
resources, available as software and also as web-based downloads or demos. They are Cabri and Cinderella.
Go to the Cinderella site at http://www.cinderella.de/ for
information regarding free downloads and some interactive examples.
The following very long website is a collection of JAVA-based educational objects for teaching/learning mathematics.
http://www.eoe.org/FMPro?-db=Objects.fp3&-token=library&-format=/library/objectlist.htm&-error=/library/NoRecordsFound.htm&CategoryNumbers=General_Mathematics&class=JavaApplet&-sortfield=_HasComments&-sortorder=descend&-sortfield=_HasEducatorComments&-sortorder=descend&-sortfield=Name&-sortfield=LastName&-max=10&-find
The Geometry Center at the University
of Minnesota at http://www.geom.umn.edu/ has some good links for elementary level geometry. Click on overview and again on "interactive
Web Applications" or "Interactive JAVA Applications," or "Graphics Archive." Some very
interesting applets or digital images.
Geometry puzzles can be found at http://www.cut-the-knot.com/geometry.html
Again this site has more high-school related puzzles than elementary, but
you'll find some "gems."
The National Council of Teachers of Mathematics (NCTM) has some interactive online math investigations. Go here http://illuminations.nctm.org/imath/index.html Also see the web resources section.
Erich's Puzzle Palace has some excellent puzzles that you can use with your students at almost any level. See it here: http://www.stetson.edu/~efriedma/puzzle.html
You'll find an online geoboard at the following site:
http://standards.nctm.org/document/eexamples/chap4/4.2/index.htm
Some general primary links can be found at:
http://www.norman.k12.ok.us/160/primarylinks3.html
3. Transformational Geometry Websites and other Interesting Geometry-Related Sites
Go first to this site of Jill Britton's. She's done a lot of the work of finding and orgnanizing great sites for you.
http://britton.disted.camosun.bc.ca/jbsymteslk.htm
Origami at http://library.thinkquest.org/28923/frame.html
The following site has some interactive games for fractal geometry. It's called the Java
Gallery of Interactive Geometry http://www.geom.umn.edu/java/
Tessellations at http://library.thinkquest.org/16661/
Suzanne Alejandre's Work on tessellations,
fractals, etc.
Tessellation tutorials can be found at http://forum.swarthmore.edu/sum95/suzanne/tess.intro.html This
site offers a step by step walk through of what a tessellation is and how to make one. Follow the links to the
Geometer's Sketchpad and Tessellations and then click on the JAVAsketchpad version (make sure you're running Explorer
and not Netscape)
From the main site you may want to visit some M. C. Escher sites. Go to http://www.WorldOfEscher.com/ While at this site visit the gallery of Escher drawings.
There are many excellent links from the Tessellation Tutorial site. One for younger children is the pattern block link at http://www.best.com/~ejad/java/patterns/patterns_j.shtml
Also from Suzanne's site you must visit Arcytech,
the home of four (mathematical) educational JAVA applets, this time elementary students as well as older students.
The site can be found at http://www.arcytech.org/
Away from Suzanne's site now and on to a ceramic tile site-which has fascinating tile patterns. You could have your kids plan a tile design for their
bathroom floor. They could draw the design on graph paper, color it etc. They could also work with precut paper
tiles of various sizes and shapes (that you would have measured ahead of time to ensure they would "fit together").
They could then arrange the precut tiles to form a repeating pattern. Have a look at this site. http://www.bedrosians.com/flrpatrn.htm This is an excellent
site to link geometry with measurement. Older kids could determine the ratio between the planning tiles and real
tiles and do a cost analysis of the actual cost to tile a floor, visit a tiling shop (or Home Depot) and discuss
costs, different kinds of tiles, different colors, different materials etc. and maybe even observe tiles being
laid.
More on M. C. Escher can be found at
http://library.thinkquest.org/11750/
From this site you can link to this one http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html where
you will find a detailed biography of Escher, complete with chronologically-appropriate images.
You may want to investigate Pi. Here
is one (of many) websites that you can visit. http://www.ncsa.uiuc.edu/edu/RSE/RSEorange/buttons.html The
activities here are suitable for Grades 5-8. It would be exciting to explore the history of Pi, the importance
of Pi, and the pursuit of Pi NOT being irrational!!
The following site has some K-Grade 3 pattern block activities (note: it uses the Arcytech pattern block JAVA applet) http://mathforum.com/varnelle/index.html
The following site shows some examples of geometry through art, again for young children http://forum.swarthmore.edu/~sarah/shapiro/index.html
Mrs. Glosser's Math Goodies can be found
here. http://www.mathgoodies.com/ This is an interesting site, with interactive math lessons, crosswords, etc. on a variety
of topics including geometry.
Mathematics Lessons that are Fun, Fun, Fun
can be found here. http://math.rice.edu/~lanius/Lessons/
There's a great lesson on online geometry. There are some good worksheets
that can be downloaded and worked with.
http://www.aplusmath.com/ again has some activities for young children, one with geometry ideas. Some games are JAVA
based.
For general math resources go to http://www.enc.org/ and
then click on math topics. There are lots of them.
Jill Britton has another super site--mainly on number theory. Go here http://britton.disted.camosun.bc.ca/jbfunpatt.htm