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MATH 5005 GeometrySummer 2006

- Highlights
- Instructors Bertha Orona, Don Gilmore, Bill

Juraschek

Sketching Geometric SolidsDay 1

- In the opening activity, participants made

geo-jackets 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 ProblemDay 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 Geometers SketchpadDay 3

- Working with geometers Sketchpad would be a

primary tool for exploring geometry content, and

Berthas seventh graders use it extensively, so

we arranged for them to introduce participants to

its features.

The Painted Cube ProblemDay 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 InsanityDay 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

HeightDay 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 PyramidDay 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 groups 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 ModelDay 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 GeometryDay 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 MeasurementDay 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 1Day 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 2Day 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 FoldingDay 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 PresentationsDays 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 teammates

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.

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