Transcript Document

MATH 5005 Geometry
Summer 2006
Highlights
Instructors: Bertha Orona, Don Gilmore, Bill Juraschek
Sketching Geometric Solids
Day 1
•
In the opening activity, participants made “geojackets” and nets for the candy boxes
(Toblerone, Droste) they brought to class. Next,
participants constructed rectangular prisms
using snap cubes and drew sketches of their
constructions.
•
Bertha explains how to use color coding to make
parts of a sketch connect to the real structure.
•
Early student work with sketches.
Cone Problem
Day 2
•
Participants were given a paper cone and asked
to determine its volume and surface area.
•
Two participants present their formal solution to
the cone problem.
•
(The Toblerone container visible in the top
picture was used in our opening activity.)
Introduction to Geometer’s Sketchpad
Day 3
•
Working with geometer’s Sketchpad would be a
primary tool for exploring geometry content, and
Bertha’s seventh graders use it extensively, so
we arranged for them to introduce participants to
its features.
The Painted Cube Problem
Day 4
•
Given a large NxNxN cube constructed of small
cubes. If the outside faces of the large cube are
painted, determine how many of the small cubes
will have 0, 1, 2, or 3 faces painted.
•
Participants struggled in their small groups.
•
Eventually all groups arrived at the results
shown. The formulas show the complexity of
this problem, and the illuminating connections
between geometry and algebra.
Circuits and Instant Insanity
Day 4
•
First we explored Euler and Hamiltonian
Circuits, then we showed students how to model
the Instant Insanity puzzle with circuit diagrams
This was challenging for all, but worthwhile
because of the extended interest is generated.
Volume of Pyramid is One-third Area of Base Times Height
Day 5
•
After calculating the volumes of several prisms,
pyramids, cylinders and cones, many
participants were still wondering about the use
of one-third in the formula for the volume of
pyramids and cones. We devised an activity in
which participants constructed a special pyramid
using straws and duct tape. Three of these
identical pyramids can be placed together to
form a cube.
•
Using the straw pyramids as models,
participants used Sketchpad to construct the net
for the pyramid, printed three copies, taped the
nets together to form pyramids, and then placed
the three pyramids together to form a cube. The
volume of the cube is (Area of Base)x(Height)
where the base is the base of both the pyramid
and cube, and height is the height of both the
pyramid and cube.
Sierpinski Pyramid
Day 5
•
This culminating activity brought together many
of the ideas we had been exploring on Days 1-4.
•
Using marshmallows and toothpicks,
participants first construct a regular tetrahedron
and then use this unit to build larger pyramids.
•
Lyndell presents her group’s findings about
some of the relationships between the length of
a side of the base of the pyramid and its surface
area and volume.
ELL Work with Frayer Model
Day 5
•
Participants chose a geometric concept and
then displayed its attributes, definition, examples
and non-examples on posters.
•
In the top picture, participants explain their work
to the class.
•
In the bottom picture, in response to instructor
feedback, this group has added more
illustrations of trapezoids in order to vary the
irrelevant properties.
Non-Euclidean Geometry
Day 6
•
To do some work at the highest level of the
van Hiele model we explored spherical and
taxicab geometry.
•
Bill used a large rubber ball and elastic bands to
model a triangle in spherical geometry which has
an angle sum greater than 180º.
•
Participants worked through an activity exploring
Taxicab Geometry that was taken from Teaching
Mathematics in the Middle School. One
discovery is that in Taxicab Geometry a “circle”-the set of all points a given distance from a fixed
point--is a Euclidean square.
Using Similarity for Indirect Measurement
Day 6
•
We spent a lot of time on mathematical
similarity. It is an ubiquitous concept that
involves proportional thinking, a kind of thinking
that permeates the middle school curriculum.
•
We have all seen the textbook problem where
one calculates the height of a flagpole by using
shadows and similar right triangles. We had the
participants indirectly measure the heights of
light poles around the school, set up sketches to
show what they did, and use the proportions in
similar triangles to calculate the unknown
heights.
•
We have found that actually doing the
measurement project produces more lasting
understanding of similarity and its role in indirect
measurement.
Transformational Geometry 1
Day 7
•
We began the exploration of transformational
geometry by working with Miras.
•
Participants had to explain how a Mira
reflection/rotation works, mathematically.
Transformational Geometry 2
Day 7
•
Don and Bill showed a giant snowflake created
using rotations and reflections.
•
A participant shows how she converted a
construction done with a Mira into one on
Sketchpad.
The Earth View Problem
Day 8
•
On the morning of Day 8, one of the instructors
read a Wacky Question in the morning paper.
We knew exploring the question and answer
given could lead to some intriguing applications
of geometry, so we planned some activities for
the afternoon.
•
Since we also wanted to do some work with
proofs, we introduced some theorems about
circles, and tangents to circles from an external
point. Participants presented their proofs (which
were critiqued by instructors). These theorems
led to a natural model for exploring the question
on Sketchpad.
The Earth View Problem
Day 8
•
To include some practical pedagogy, we had
participants make booklets by cleverly folding
three sheets of paper to yield a cover page and
five pages on which to put their definitions and
theorems related to circles.
•
The finished booklets, were very popular.
•
Finally, participants used Sketchpad to model
the problem and explore the answers. In the
words of one, “You can see virtually the entire
hemisphere when you are one millimeter from
infinity away.”
Circle Folding
Day 9
•
Start with a circle and fold it to form an
equilateral triangle. Use theorems we have
used to justify that this is an equilateral triangle.
•
After folding the circle into a trapezoid, rhombus
and regular tetrahedron, Bertha received “oohs
and ahhs” for folding a truncated tetrahedron.
She finished by unfolding the circle and asking
for the area of one of the regions formed by the
creases.
Project Presentations
Days 9-10
•
Participants chose a project from the booklet
“101 Project Ideas for Sketchpad.” One of the
most challenging was constructing tangram
pieces that could be rotated and flipped. Julie
and Katherine were proud of their work.
•
Another project involved constructing a
Sketchpad animation to show the path of the
moon around the earth. The participant put in a
lot of work on this challenging project, and
learned a lot about geometry, but had not quite
attained his goal at the end of the class.
Tidbits
•
Co-teaching was common. Don and Derrick
needed an extension of the flip chart paper, and
Bill provided it.
•
Individual attention while working on Sketchpad
was continual.
More Tidbits
•
These students were very pleased with the
intricate snowflakes they constructed using
paper folding. The pleasure of symmetry.
•
Jesselyn had been totally immersed in a
problem, struggling to understand her
teammate’s explanation. Suddenly it made
sense; she knew she had “got it.” Fortunately
for us, her joyful expression lasted long enough
to be captured by the camera, no posing
necessary.