· Activity
Name: Tetrahedron Kites
· Objectives:
During
the making of the kites, the students will learn about the tetrahedron,
networks, 3-dimensional models and geometric vocabulary. After the kite is built, they will explore
the relationships of length, area, surface area, and volume. The students will also explore some of the
patterns in the tetrahedral kite including the packing of space with the basic
units involved in the kite.
· EALR/Standards:
· Materials:
Straws (24 per group to build two levels,
60 per group to build three levels)
Ribbon or yarn (4 or 10
pieces per group each approximately 6 feet long)
Tissue paper (4 or 10
pieces of 10” by 15” per group)
Scissors
· Teacher
Notes
o
Prerequisites for the learner:
Depends on the use of the project. See teacher hints for more information.
o
Teacher hints for the activity:
This is an activity that can
be used as an introduction to the school year for vocabulary building, review,
and an introduction to Geometry. It can
also be used later in the year or with advanced classes to teach relationships
between linear, area, and volume calculations, or for regression models and
finite differences.
Before starting this project,
the instructor needs to build a tetrahedral, an octahedral, and a tetrahedron
that is sheared.
o
Introductory questions: For the building of the kites
Holding up a tetrahedron model:
What
is this made of material wise and piece wise?
What
will you need to build one of these?
Can
one of these be built by putting the ribbon/string through each straw only
once.
o
Solutions:
Introductory questions: Straws and ribbons, 6 straws, approximately 4 feet ribbon
The
straws and ribbon.
No
– there are 4 odd nodes. (Do not give
this answer until the end of the building)
o
Assessment suggestions:
Does
it fly (checking for team work and direction following)
Given
a model of cubes build in the same fashion, redo the worksheets.
· The
Activity:
Begin by showing the students a
tetrahedron. Explain that they each
need to build one. Do not give a lot of
instructions on the how, just let them experiment and try. It is important that they help each other in
their groups.
If
you want to teach networks and
traversability, introduce this by asking students to try to go through each
straw only once with the ribbon. It is
also important that they do not cut the ribbon. (They will hopefully discover that this is impossible.)
Once
the group has produced four tetrahedrons, have them brainstorm vocabulary words that they can see on
the tetrahedron. (Examples: face, edge, segment, circle, dihedral
angle.) Groups should be able to find
at least 25 words. After brainstorming
in groups, have the class build a vocabulary list on the board. Have students show where the words can be
seen on their tetrahedrons, especially those words that are questionable.
To
finish the kites, have students place three of the tetrahedrons together so
that their bases form a large triangle.
Use short pieces of ribbon to tie these three tetrahedral together. Place the fourth tetrahedron on top of the
bottom three to form a large tetrahedron.
Use ribbon to tie this in place.
While two students are tying these together, the other two should be
cutting the tissue paper according to the pattern. When the cutting is complete, the students need to tape the
tissue paper onto the kite according to the diagram. Finally, they need to attach the bridal.
For
homework or for those students who finish early, each group needs to build an
addition 6 tetrahedra and one octahedron.
For extra credit or as an optional, have each student take 12 straws,
place them on a string with the end straws taped to the ribbon end, and then
fold the line of straws into an octahedron.
This shows students that octahedra are traversable networks.
After the kites are completed, begin working through
the worksheets and questions on the student pages.
· Assessment
material:
· Extensions:
Two level kite: Four tetrahedra
TETRAHEDRON
WORKSHEET
After completing the building of your two level tetrahedron
kite, answer the following question.
1) Fill in the chart using the given information:
First
level tetrahedron Second
level tetrahedron
Edge length One
Unit ______Unit(s)
Area of one Face One
Square Unit ______Square
Unit(s)
***Are the units of the edge length and the face area the
same? (I.e. Inches for both or
Centimeters for both) Answer with
complete sentences and give an explanation.
A picture might be helpful.
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Total surface area ______Square
Units ______Square
Units
Volume One
Cubic Unit ______Cubic Unit(s)
Build a third level tetrahedron
2) Give the dimension for each of the following: (i.e. 1,2 or 3 dimensional)
a)
Length _____________ b)
Area __________ c) Volume
_____________
3) Using the chart and the information from question 2,
imagine placing a fourth and fifth tier on your tetrahedron. Fill in the following chart.
One
Tetrahedron Third Tier Fourth Tier Fifth Tier
Edge Length One
Unit _____Unit(s) _____Unit(s) _____Unit(s)
Area of One Face One
Sq. Unit _____Sq. U(s) _____Sq.
U(s)_____Sq. U(s)
Surface Area _____Sq.
U(s) _____Sq. U(s) _____Sq. U(s)_____Sq. U(s)
Volume One
Cubic Unit _____Cu. Unit(s) _____Cu. U(s)_____Cu. U(s)
4) Describe a pattern for comparing areas and volumes of
different tetrahedron tiers.
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5) What is the volume relationship between your second
tiered tetrahedron and the octahedron built with the same straws? How about the single tetrahedron and a
square pyramid built with the same straws?
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6) Fill in the following chart
Level Up Tetrahedra Octahedra Down Tetrahedra
0 0 0 0
1
1 ____ ____
2
4 ____ ____
3
_____ ____ ____
4
_____ ____ ____
5
_____ ____ ____
7) Using the information in question six, write an equation
to find the number of up tetrahedra given n-level.
8) Using the
information in questions six and seven, write an equation for the number of
down tetrahedra given n-level.
9) Using the information in question six, write an equation
for the number of octahedra given n-level.
10) What is the volume for each level of your tetrahedral
kite?
Level Volume
0
0
1
1
2
_____
3
_____
4
_____
5
_____
n _____
11) What is the ratio of “level-one” tetrahedra to octahedra in an n-level tetrahedral
kite?
12) If n goes to infinity, what happens to the ratio found
in 11)? What does this mean for the
tiling of space with tetrahedra and octahedra?
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Extra Thoughts and
Extensions
1) If you had several squares which grew in size by taking
the smallest edge length and doubling it for the second square, tripling it for
the third and so on, would you find a pattern in the areas and volumes? Explain why or why not. Give the pattern if you think yes.
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***Draw a set of six squares on graph paper following the
idea presented in question 6.
2) Using your graph paper squares, make and fill in a chart
that shows edge lengths, area, and volume*.
*For
volume, Use the square as one side of a cube
3) Was your answer to question 1) correct? _______ Do you think this pattern will work
for other shapes? Give an explanation
and possibly examples.
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****More thoughts and questions:
Do the shapes have to be identical except for edge length?
What happens if you consider the fourth dimension or even
higher dimensions?
Can you draw or build models of shapes that fit the pattern
and ones that do not fit the pattern?
Why does the kite fly?
Could you get an octahedron to fly?
Can you find relationships in the areas and volumes of cubes
and tetrahedrons?