Teacher's Guide - Diwa Learning Systems

Third Year
ARTICLE 1: THE MANY FACES OF A CRYSTAL
I. LEARNING OBJECTIVES
A. To define terms related to the polyhedron
B. To discover the relationship between the number
of edges, vertices, and faces of a polyhedron and
determine one given the others
C. To present a rough proof of why there are only five
platonic solids
D. To find the dual of a given polyhedron
II. SUBJECT MATTER
3D-Geometric Relations
III. MATERIALS
Real polyhedral objects or pictures of crystals
Pair of scissors, pieces of cardboard, paste
IV. REFERENCES
Brown and Montgomery, Plane and Solid Geometry
Bennett and Nelson, Mathematics an Informal Approach
Encyclopedia Britannica 2003 (CD Rom)
Tatsulok. Third Year, Vol. 13, Issue 2, SY 2007-2008
V. LEARNING STRATEGIES
A. Opening Activity
• Ask your students if they have jewelry made up
of precious stones. Let them describe the shapes
of the stones embedded in their jewelry.
• Say that these precious stones are examples of
crystals, which are formed by the solidification
of a chemical element, a compound, or a
mixture and has a regularly repeating internal
arrangement of its atoms and often external
plane faces.
B. Development Activity
• Relate these crystals to the polyhedra. Show
some polyhedral objects like the cube, tetrahedron,
prism, etc. Show the edges, faces, and
vertices of the polyhedra and guide the students
in coming up with their own definition of these
terms.
• Starting with the tetrahedron, guide the students
in determining the relationship among
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Teacher’s Guide
Vol. 13 No. 2 SY 2007-2008
the number of vertices, faces, and edges of the
polyhedra. Let them verify this relationship by
counting the vertices, faces, and edges of other
polyhedra. As stated in the article “Crystals and
the Platonic Solids,” this relationship is now
stated as the Euler’s formula. Guide them in
completing the table for the vertices, faces, and
edges of several polyhedra given in the article.
Give more examples to emphasize the application
of Euler’s formula.
• Introduce the platonic solids as special polyhedra.
Say that Euclid was the first to prove
that there are only five such polyhedra. Give a
brief background of how Euclid proved it and
challenge them to try proving it on their own
(they may prove this by constructing regular
polygons and by trying to assemble some of
these polygons to form solids–-they must be
able to show that regular polygons with more
than five sides cannot be assembled to form a
solid with properties of the platonic ones).
• PROOF OF THE PLATONIC SOLIDS
The proof that there are only five regular
convex polyhedra (the Platonic solids) and the
determination of what these five are made use
of “exhausting the possibilities”.
From the definition of regularity (equal
sides and angles), it is easy to deduce that all
the faces of a Platonic solid must be congruent
regular n-gons for a suitable n, and that all the
vertices must belong to the same number j of
n-gons. Because the sum of the face angles at
a vertex of a convex polyhedron is less than 2π
(360°), and because each angle of the n-gon
( )
is n − 2 À ( n − 2) 180 °
(or ) , it follows that
n
n
j n ( − 2)
À
< 2À,
or (j – 2)(n – 2) < 4. Therefore,
n
the only possibilities for the pair (j, n) are (3, 3),
(3, 4), (3, 5), (4, 3), and (5, 3). It may be verified
that each of these pairs actually corresponds to a
Platonic solid, namely, to the tetrahedron (with
3 equilateral triangles at each vertex), the cube
(3 squares at each vertex), the dodecahedron (3
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egular pentagons at each vertex), the octahedron
(4 equilateral triangles at each vertex), and
the icosahedron (5 equilateral triangles at each
vertex), respectively.
• Introduce next, the duals of polyhedra. Show
them first how the duals are determined and then
let them determine the duals of some polyhedra.
• Guide them in constructing models of the platonic
solids. Let them do the activity page of the
article.
C. Generalization/Summary
• A polyhedron is formed by the union of polygons
intersecting at their sides and vertices.
• Crystals are natural representations of polyhedra.
• Euler related the number of faces (f), vertices (v),
and edges (e) of any polyhedron in the equation
f + v = e + 2. If two of these are given, one can
be determined.
• There are only five platonic solids---they are the
tetrahedron (triangular pyramid), tetrahedron
(cube), octahedron, dodecahedron, and the icosahedron.
The key observation is that the interior
angles of the polygons meeting at a vertex of a
polyhedron add to less than 360 degrees. This
is because if such polygons met in a plane, the
interior angles of all the polygons meeting at a
vertex would add to exactly 360 degrees.
o Triangles. The interior angle of an equilateral
triangle is 60 degrees. Thus on a
regular polyhedron, only three, four, or five
triangles can meet a vertex. If there were
more than six, their angles would add up
to at least 360 degrees which they can’t.
Consider the possibilities:
♦ Three triangles meet at each vertex.
This gives rise to a tetrahedron.
♦ Four triangles meet at each vertex. This
gives rise to an octahedron.
♦ Five triangles meet at each vertex. This
gives rise to an icosahedron
o Squares. Since the interior angle of a
square is 90 degrees, at most three squares
can meet at a vertex. This is indeed possible
and it gives rise to a hexahedron or cube.
o Pentagons. As in the case of cubes, the only
possibility is that three pentagons meet at a
vertex. This gives rise to a dodecahedron.
o Hexagons or regular polygons with more
than six sides cannot form the faces of
a regular polyhedron since their interior
angles are at least 120 degrees.
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D. Assignment
Do the following:
1. Find the duals of the platonic solids.
2. Make models of these duals using bendable
plastic (e.g. transparent plastic folders).
Answers:
1. The dual of a tetrahedron is another tetrahedron.
The cube and the octahedron are duals of each
other while the dodecahedron and the icosahedron
are also duals of each other.
2. They may copy the patterns given in the article
for their models.
VI. EVALUATION
Let the students answer the worksheet on page 15
of the magazine.
VII. QUESTIONNAIRE
Guide questions:
1. How are polyhedra formed? (They are formed by
the union of polygons intersecting at their edges or
sides and vertices.)
2. How are the number of faces (f), vertices (v) and
edges (e) of any polyhedron related? (The equation
f + v = e + 2 relates the three.)
3. How are the duals of polyhedra formed? (They are
formed by connecting the center points of the faces
of the polyhedra.)
VIII. OTHER TEACHING IDEAS
The Elements Linked to the Platonic Solids
Plato associates four of the Platonic Solids with the
four elements. He writes:
We must proceed to distribute the figures [the solids]
we have just described between fire, earth, water,
and air. . .
Let us assign the cube to earth, for it is the most
immobile of the four bodies and most retentive of shape
the least mobile of the remaining figures, icosahedron,
to water, the most mobile, tetrahedron, to fire, the intermediate,
octahedron, to air.
(Note that Earth is associated with the cube, with its
six square faces. This lent support to the notion of the
“foursquaredness” of the Earth.)
But there are five regular polyhedra and only four
elements. Plato writes:
“There still remained a fifth construction, which
the god used for embroidering the constellations on the
whole heaven.”
Plato’s statement is vague, and he gives no further
explanation. Later Greek philosophers assign the dodecahedron
to the ether or heaven or the cosmos. The
dodecahedron has 12 faces, and our number symbolism
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associates 12 with the zodiac. This might be Plato’s meaning
when he writes of “embroidering the constellations”
on the dodecahedron.
(You may let your students research about Sacred
Geometry to learn more about the Platonic Solids linked
to the elements of the universe.)
ARTICLE 2: SKILLFULLY COVERED
I. LEARNING OBJECTIVES
A. To define tilings and tessellations
B. To find polygons that will tessellate
C. To create their own tessellations
II. SUBJECT MATTER
Tilings and Tessellations
III. MATERIALS
Pictures of tiled floor, ceilings, walls, etc.
Sample pictures of tessellations by M.C. Escher (or any
other tessellations if these are not available)
Pieces of cardboard, pair of scissors, masking tape, clean
bond paper or construction paper, coloring materials
IV. REFERENCES
Brown and Montgomery, Plane and Solid Geometry
Bennett and Nelson, Mathematics an Informal Approach
Tatsulok. Third Year, Vol. 13, Issue 2, SY 2007-2008
V. LEARNING STRATEGIES
A. Opening Activity
• Show the pictures of tiled floor, walls, or ceilings,
or you may let your students look for patterns
like these, around them. Let them describe
how these are made.
• Also show the sample pictures of tessellations
by Escher. Ask the students to describe the pictures
as to what they are composed of. (They
must see that these pictures are composed of
repeating patterns or figures.)
• Say that they will be creating their own tilings
and tessellations later.
B. Development Activity
• Introduce the words tilings and tessellations. Say
that tessellations are special kinds of tilings.
• Discuss which polygons may tile up. Guide them
in discovering that the sum of the measures of
the angles of the polygons that meet at one point
must be 360 degrees, otherwise, they cannot be
made to tile up. Ask them which of the regular
polygons can be used to tile or cover a plane. Let
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them show some tilings by cutting out regular
polygons and arranging them.
• Discuss how tessellations are made. Demonstrate
the steps presented in the article and let
your students do the same. Challenge them to
create their own tessellations, applying the steps
presented in the article.
C. Generalization/Summary
• Tiling refers to any arrangement of non-overlapping
polygons that can be placed together to
cover a region.
• Tessellations are special kinds of tilings where
irregularly shaped polygons (which where created
by transforming convex polygons) are used
repeatedly to cover a region.
• For polygons to tile up or tessellate, the sum
of their angles that meet at a point must be 360
degrees. The square, equilateral triangle and
regular hexagon are the only regular polygons
that tessellate by themselves. However, combinations
of some regular polygons like the square
and the regular octagon may also tessellate.
• To create a tessellation, start with a polygon,
transform it by cutting and then translating, rotating
or reflecting the cut part, bearing in mind
that no part of the polygon must be wasted. To
complete the tessellation, trace the transformed
polygons on a clean sheet of paper, making sure
that no part of the paper is left uncovered. Apply
designs and colors to the tessellated pieces to
make them more attractive.
D. Assignment
Do the following:
1. Create a tiling using combinations of regular
polygons.
Answers here will vary. But here is a sample
tiling using the regular triangle and the regular
dodecagon.
2. Create a tessellation by transforming the
square.
Answers here will vary.
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VI. EVALUATION
Let the students accomplish the worksheet on page
16 of the magazine.
VII. QUESTIONNAIRE
1. How do tilings and tessellations differ? (The two do
not really differ. The first is the term which generally
refers to the use of repeating polygons to cover
a plane while the second is a special type under the
first.)
2. What is the most important property to consider in
finding polygons that will tile up or tessellate? (Their
interior angles-–when arranged about a point must
sum up to 360 degrees.)
VIII. OTHER TEACHING IDEAS
DEMIREGULAR TILINGS AND TESSELLATIONS
These are tessellations made of three or more regular
polygons combined. Here is a sample of these tessellations.
You may let your students create their own.
ARTICLE 3: THE ART OF DISTORTION
I. LEARNING OBJECTIVES
A. To define topology
B. To explain why the coffee cup and the donut are
equivalent
C. To create figures that are topologically equivalent
II. SUBJECT MATTER
TOPOLOGY
III. MATERIALS
Ball of clay
Pieces of string
The letters of the alphabet
IV. REFERENCES
Brown and Montgomery, Plane and Solid Geometry
Bennett and Nelson, Mathematics: An Informal Approach
Tatsulok. Third Year, Vol. 13, Issue 2, SY 2007-2008
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V. LEARNING STRATEGIES
A. Opening Activity
• Pose this question: Did you know that the coffee
cup is equivalent to a donut? (Anticipate different
reactions from your students.) Ask them to
figure out why if they can.
B. Development Activity
• Introduce the word topology. Ask one to look
up its meaning in a dictionary and explain what
he understood about it.
• Discuss its relation to geometry as presented
in the article. Show examples of topologically
equivalent things by demonstrating them thru
the ball of clay (transform it to any form without
tearing)
• After the discussion, challenge the students to
explain why the coffee cup is equivalent to a
donut.
• Present the letter of the alphabet and then let
them figure out which letters are topologically
equivalent.
• Let them answer the activity sheet after the
article.
C. Generalization/Summary
Topology is an extension of geometry. It involves
analyzing or studying figures and surfaces that
are formed into another figure or surface by twisting,
bending, stretching, shrinking or any motion that
does not allow tearing parts apart or removing any
part from the bunch of stuff. It is also sometimes
referred to as the Rubber Sheet Geometry because
topological transformations can be performed as if
the original object is a sheet of rubber.
D. Assignment
List as many pairs of things around which you
think are topologically equivalent. Explain your
samples.
(Answers here will vary but you may give this
as an example: the plate and the spoon are topologically
equivalent because they contain no holes and
that they may have been formed by transforming a
solid ball into their shapes now.)
VI. EVALUATION
Let the students accomplish the worksheet on page
17 of the magazine.
VII. QUESTIONNAIRE
1. Why is the coffee cup equivalent to the donut?
(Both of them have one hole-–the donut has a hole
at the center while the coffee cup has a hole that
separates the handle and the body of the cup. Had
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the donut been a ball of clay, it can be bent twisted
and stretched to form the coffee cup)
2. Why are D and P equivalent? (They both have one
hole each.) Which other letters of the alphabet are
topologically equivalent? (C E F G H I J K L M N S
T U V W X Y Z are topologically equivalent to each
other, A D O P Q R are also topologically equivalent
and B is the only letter which has no topological
equivalent in the alphabet.)
VIII. OTHER TEACHING IDEAS
SCULPTURE AND TOPOLOGY
Sculpture as an art form goes back to prehistoric
times. Most Stone Age statuettes were made of soft stones
and clay. Let your students relate sculpture to topology.
To let them appreciate topology, let them create their
own sculpture using clay.
ANSWERS TO THIS ISSUE’S ACTIVITIES
Article 1: The Many Faces of a Crystal
Problem Buster (Article 1)
Students should be able to construct 5 polyhedra using the
cutouts.
Worksheet 1 (PAGE 15)
The five platonic solids are as follows:
1. zircon crystal
12 faces, 10 vertices, and 20 edges
2. dual of the zircon crystal
10 faces, 12 vertices, and 20 edges
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3. boracite crystal
26 faces, 24 vertices, and 48 edges
4. 24 faces, 26 vertices, and 48 edges
5. A semi-regular polyhedron with 8 regular hexagon edges,
6 square faces, and 24 vertices
Article 2: SKILLFULLY COVERED
Problem Buster (Article 2)
Answers will vary.
Worksheet 2 (PAGE 16)
Answer Key:
A.
1.
2.
cut and
rotated
cut and
rotated
cut and
translated
cut and
rotated
cut and
translated
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3.
cut and translated
3
cut, translated,
and rotated
B. Answers here will vary but here is a sample tessellation
of the first figure.
Article 3: The Art of Distortion
Problem Buster (Article 3)
A.
Worksheet 3 (PAGE 17)
A.
1. Figure 1
2. Figure 2
3. Figure 3
4. Figure 2
5. Figure 1
6. Figure 1
7. Figure 3
8. Figure 1
9. Figure 2
10. Figure 3
B.
Category 1: Letters that are topologically equivalent to a line
segment:
C E F G H I J K L M N S T U V W X Y Z
Category 2: 1-holed letters:
A D O P Q R
Category 3: 2-holed letters:
B
FUN AND MATH
What Letters?
Answer: Y, X, and Z
Figure Operations
Answer:
Pass around the Candies Please . . .
Answer: 6 children
SUDOKU PUZZLE
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