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Teaching Math Through Art
Recognizing
Symmetry etc.
--Randi Waxman
EdM
University
of Phoenix on-line
Summary
A
Math Textual & Pictorial Journal with drawings, pictures or visual
features that the student finds which relate daily to math and the other
subjects studied in school, will be utilized. Math symbols and symbols used to
write English will be used. Lessons
on a/symmetry will be implemented by using student or verbal description of an
item or topic that will be translated into pictures.
Patterns are a part of math and by writing a poem with a pattern we
explore various visual and verbal aspects of rhythm and rhyme.
During its 3-dimensional construction, an Icosahedron will challenge
several areas of abilities and biological systems, such as small and large
muscular motor skills. Also a fun
way of learning multiplication will be done with String Art and lines and
circles.
This
is aimed at use for fifth grade students but may be used in higher grades also.
Since everyone is different, different aspects of this unit will speak to
different people differently.
Each lesson will take from one to several 20-40 minute periods. The journal will be maintained throughout the school year and will touch on a wide scope of topics, conceivably all secular school
disciplines, including Math, Science, Social Studies- history, cultures, geography; English Language- literature, writing, reading, spelling, punctuation; even Physical Fitness, Gym or Phys Ed will
be food for creative expression and learning via these math through art lessons.
Unit
Goals
The underlying goal for the following lessons will be to recognize symmetry and asymmetry in art through doing art, math, construction projects, writing poetry and descriptive or narrative
writing. Working with Cubes to learn about different kinds of spatial arrangements i.e., symmetry, asymmetry and rotational symmetry is the first unit. . Drawing ability changes as our brain
develops and we ‘learn to see’. Sometimes with our fertile imaginations we can create meaning by drawing. This may be called creative seeing and increasing this ability is a goal of this lesson.
Another goal is to improve communication abilities by setting up a consistent, organized format in a Math Art Journal. Unit 2 is Describing A/Symmetry in Words by Drawing from Someone’s
Description. The goal of this lesson is to strengthen decision-making abilities by using everyday experiences and using the focus areas fabricated by the teachers, causing students to improve
their ability to listen and express themselves graphically and verbally. Various formats, English, Mathematical Language, Pictorial ‘Vocabulary’ will be used to expand student’s knowledge base.
Additionally, making connections between creative seeing and pragmatic definitions may also strengthen decision-making abilities. For instance, a student may claim to know what parallel is.
Here she can prove it by describing and drawing some parallel lines.
Unit
3 is Poetic Pattern. “What
is the point of writing A Poem in the Shape of a Lamp?” one may ask.
Commonly known as a Japanese lantern, the goal is to make an uncommon
unique work of art for its own sake. Additionally
we will be able to answer if it is symmetrical; determine what makes it good;
and learn some things about the areas of math fellow students are currently
learning. Unit 4 is String Art.
How is making a pretty design part of Math?
For one thing, the goal of achievement of rote memorization can
improve calculation and problem solving skills in the future.
This can be attained while students are, in a sense, distracted doing
art. By observing and analyzing the patterns created, symmetrical
aspects of shapes and numbers will be learned.
Unit 5, A Multifaceted Activity is Literally an Icosahedron.
Concepts in science, such as mechanics, optics and motion will also be
studied from observations of student’s creation.
Rationale
A
certain population of students, very frequently “verbally smart” girls claim
they hate math, which becomes a self-fulfilling prophecy.
They see a problem, or a page of information that makes no sense to them,
and from their experience, they immediately give up trying.
These art activities help the student see how so much of math is already
a part of their experience and they can use these activities to broaden and
deepen their math knowledge and find connections between curriculum areas they
did not know existed. The more they
work with it, the more it becomes their own, the fear lessens and confidence
builds confidence, and good self esteem.
More
insights shared by author of Arts with the Brain in Mind, Eric Jensen are
as follows. He asks, “How do the
arts stack up as a Major Discipline?” 1) They have minimal risk involved [when
it is well supervised]. (All brackets are my own thoughts.) 2) They are
inclusive [because no one is forced to do something at a level they cannot or do
not want to]. 3) They are culturally necessary [because our limited human
capabilities (or differential potentials) cross all cultural lines]. 4) They are
brain-based and they may promote self-discipline and motivation. 5) They are
wide ranging. 6) They have survival value [when the skills taught are not
distracting from any serious or more necessary ones]. 7) They are assessable.
[They can be measured or analyzed to provide feedback, not necessarily
correction.]
That sums it up. However, other researchers, Kellah M. Edens and Ellen F. Potter of the University of South Carolina, sought to obtain evidence that would strengthen Art’s position as a
Discipline and to enhance its perceived value. They examined an isolated descriptive drawing task and their findings suggest that descriptive drawing provides a viable way for students to learn
scientific concepts and supports the processes of selection, organization, and integration that underpin the cognitive processes necessary for meaningful learning. Art is not just a fun filler
activity when investigated within the cognitive model of learning, an approach that considers the learner’s cognitive system. Furthermore, they maintain it is possible to promote conceptual
understanding with specific types of drawing and any kind of drawing can be an instructional strategy to facilitate learning. Discussion and critiques of the drawings will allow students to make
connections across symbol systems
and build meaning. (Edens, Potter 2001)
There is yet another aspect of arts in education to consider.
According to Phoenix University’s Instructor
of MED522, Sonia McKenzie, who posted this on a class forum, “Effective
integrated instruction blends, harmonizes, coordinates, and unifies concepts to
lead to more authentic real tasks. Also,
integrated instruction allows students to develop understanding and find
connections to what they know and value. As a result of an integrated curriculum
students are more likely to understand and feel confident in their learning.”
Unit
1 Title: Making
Cubes to Learn about Symmetry (& other mathematical Principals such as
Volume)
Grade
level: 4+
Curriculum
concepts: Form,
a three-dimensional object; space, the empty area between, around, above,
below, or within an object-- are Elements of Design, some of the building blocks
of visual art. Balance, the
arrangement of equal parts, stable; contrast, the difference between two
or more things; repetition, the parts used over and over in a pattern; proportion,
the relation of one part to another; unity, all parts working together--
are Principals of Design, how the blocks are used or put together.
Math/Art Topics
involved in this lesson are: Polyhedra,
Space forms, Symmetrical designs, Geometric sculptures, Rotational symmetry,
Constructions, Artists who use math, Technical drawings
Corresponds
with National Art Standards:
Understanding and applying media, techniques, and processes; using knowledge of
structures and functions (elements and principles of art); choosing and
evaluating a range of subject matters, symbols, and ideas; reflecting upon and
assessing the characteristics and merits of their work and the work of others;
making connections between visual arts and other disciplines
Corresponds
with National Math Standards:
Geometry and Spatial Sense, Measurement, Patterns and Relationships, Number
Sense and Numeration, Mathematics as Problem Solving, Communication, Reasoning,
and Mathematical Connections are several National Math Standards involved in
this lesson.
Lesson
objectives:
Students will learn about symmetry of a cube, balance and weight, volume of a
cube and be able to recognize them in their surroundings.
Students will learn computer and computer navigational skills by looking
for cubes on especially, http://randisart.com/miscellaneous.htm
and randisart.com/boxtemplate.htm
Background
information and motivation: The
symmetry and volume of a cube can be depicted in a schematic type drawing.
But it does not necessarily translate in a student’s mind to the
3-dimensional structure it is meant to convey.
Students will explore different ways of representing a cube, making a
cube and making measurements and making a mobile to demonstrate and increase
their mathematical knowledge and artistic creativity.
Integration
areas: Math-Geometry,
Art, Science Language Arts
Materials:
paper,
pencils or crayons, sticks, string, scissors, Math Art Journal, tape, foam
board, card board, clear plastic
Vocabulary:
2
dimensional objects 3-dimensional, cube, diagonal, bisect, rhombus, vertex or
vertices, rotational symmetry, symmetrical, asymmetrical, area, volume
Procedure:
1. Warm-up: Draw, shade (consider directions of light source)
and color squares and then cubes on paper or in journal. Glue “good” ones into Journal.
Identify what makes them “good cubes”, i.e., symmetrical, six faces,
parallel sides, perpendicular lines, right (90 degree) angles.
Use randisart.com/boxtemplate.htm
2.
Record Process in Math Art Journal like a Laboratory Experiment with following
components:
Purpose or Goal: Find out all about cubes, how many ways we can make a cube and do it.
Materials: paper, foam board, cardboard, clear plastic, writing and coloring
utensils, scissors, Math Art Journal, tape
Procedure: 1.Copy possible shapes needed to make a cube onto paper, cardboard,
foam board and plastic. 2.Cut them
out and decorate with 30, 60 90 degree lines or other geometric shapes.
3. Tape sides together to make cubes.
Record all results and answer questions.
Results (or data and drawings) Record all information in Math Art Journal:
Make a table with Characteristics of Cubes with headings:
1. face shape, 2. number of faces, 3. number of edges, 4. number of
vertices, 5-8. volume and 9-12. weight of one inch3 piece of paper,
piece of cardboard, piece of foam board, plastic.
Conclusion: Explain how many ways there are to make a cube and what difficulties,
if any were encountered and how they were overcome. What are the characteristics of a cube and how are they
different from another polyhedra such as a tetrahedron or a icosahedron?
3.
Make cubes out of paper by folding them origami style:
Make a box by folding paper by starting with a square piece of paper. (1st
challenge: How can we make a square out of a 8.5x11” piece of rectangular
paper?). Fold diagonally twice.
What shape do we get and what is a diagonal line?
Does it bisect the angle? Make a tent. (2nd challenge--see
Instructor for help) Has anyone
ever made a real tent? What were
its parts? How did you do it? What
did you use? Fold up all four
‘legs’. Fold 4 corners into
center. Fold down 4 flaps. Tuck in
four flaps (just like putting children to bed-not always easy).
Crease top and bottom edges or you will get a rhomboidal shape, not a
cube. (A rhombus is a diamond)
Hold it, spread out gently in both hands. Blow into hole to make box.
Answer questions in Math Art Journal.
4.
Add the cubes to the Mobile with at least two cubes, either in a balanced,
symmetrical aesthetically pleasing way; in a line or attached together to make
an animal or other object yet to be known, perhaps an object sold in student’s
Store, such as jewelry box or a bar of soap. Decide if it is symmetrical,
asymmetrical or has rotational symmetry. (yes, no, yes) If a line can be drawn
and you see the same thing on both sides, it is symmetrical, if not, it is
asymmetrical. If you can spin it on
a point and see the same thing, it has rotational symmetry. Where else do you
see these shapes? Answer these
questions in Math Art Journal.
5.
Follow directions on drawing a square, then a square as it would look not
straight on, from various angles. Use
a square object as a model.
Draw a cube several times, shade them and decorate them or make item
labels out of them. For
example, say, Draw a 1-inch square. Use
a ruler or straight edge. Rotate it
about 450. Give it depth
(or height) by drawing lines down, also 1 inch long.
Now it looks like a table. How
would you make it look like a solid (or see through) cube?
What is its volume? How much
space does it take up? How many
cubic inches or in3 does it have?
Answer questions in Math Art Journal or write it like a Laboratory
Experiment with the components of a lab exercise in mind (Purpose, Material,
Procedure or steps, Results or data and Conclusion or further questions.
Assessment
and/or evaluation: Students
complete procedures listed, problems on volume in textbook, workbooks and
teacher’s worksheets. Explain, write in Math Art Journal, or type what the
project involved and what they learned and how they could or did make it better.
(also see Culmination)
Optional
variations: 1. Combine cubes with the Icosahedra Mobile. 2. Use
the drawings and measurements of the cubes to improve Store Drawings. 3. Make
measurements that may be realistic and label dimensions on drawings of Stores.
4. Make a separate mobile with just cubes, 2-d, 3-d colored or plain. 5.
Design a cover for a cube shaped student’s Store commodity (soap, perfume,
jewelry box). 6. Include identifying features that are found on real items such
as weight. 7. Innovate and add dimensions to label. 8. Describe it in Math Art
Journal. 9. Add on shapes or thematic decorations to the boxes according to the
Jewish Holiday or topic currently studying. 10. Make boxes out of clay.
References:
teacher’s ideas,
Unit
2 Title:
Symmetrical or Asymmetrical Drawing Through a Verbal Description
Grade
level:
4 +
Curriculum
concepts:
Geometry, Symmetry, Communication through Description, Objects (nouns, materials
or tools) studied in Science, Social Studies or other school subjects.
Corresponds
with National Art Standards:
reflecting upon and assessing the characteristics and merits of their work and
the work of others
Corresponds with National Math Standards: Geometry and Spatial
Sense, Measurement, Patterns and Relationships, Communication, Reasoning, and
Mathematical Connections are the most prominent National Math Standards involved
in this lesson.
Lesson
objectives:
Improve students abilities and confidence in communicating in
mathematical language, assessing characteristics, making connections, spatial
sense, measurement and patterns and relationships (between things and people).
And students will learn about symmetry.
Background
information and motivation:
There are many mathematical abilities, which can be expressed verbally.
When Language Arts is a student’s strong point or when writing and presenting
their written work for other students and parents is part of the lessons, Math
and Art can be integrated into the curriculum with this activity.
Integration
areas:
Math, Art, English Language, Literature
Materials:
Paper pencils, pens, objects to describe and draw.
Vocabulary:
parallel lines, perpendicular lines, 3- dimensional, 2 dimensional, special
sense, pattern relationship, relative size, precise or exact size, descriptive
writing, noun, pronoun, adjective
Procedure:
One student in a pair or group describes some object without telling the
others what exactly it is (unless it is absolutely necessary to name it in order
to come close to the correct object).
She can write it out, type it or describe it orally, using as many or as
few words as needed; at least one 4 –5 sentence paragraph.
More information may be better than not enough information, for the
person drawing to get a nearly accurate idea of what it is.
The one describing the object has to state, among other features, whether
it is symmetrical or not.
If the one describing the object does not give enough information then
the Artist can ask specific questions in order to get a good enough visual
image, to create a drawing.
Changes can be made to the drawing as necessary, for the one describing
to be satisfied that her description was understood.
Switch roles so everyone gets to be the Artist and the Writer.
Assessment
and/or evaluation:
Discuss and write in math Art Journal whether it was an easy task or
difficult and why.
Was the communication between the pairs smooth?
Did the object need to be identified to get it right? Was it classified
as symmetrical or asymmetrical correctly? How my adjectives were used to get it
done? (also see Culmination)
Optional
variations:
Limit the items being described to certain objects in the room or
geometric shapes or 2-dimensional objects or 3-dimensional objects:
Shorten or extend the time; make it a homework assignment; limit the
materials to a drawing pencil and nothing else; extend it to something they have
never actually seen:
Make it a character in a book, an object or tool used in Science or
Social Studies:
Place the drawing in a cartoon or time-line depiction.
References:
Created
by the Instructor.
Examples: One teacher had the 4th grade girls do a Treasures
Project. They described a
favorite family item for Language Arts. They
wanted a picture of it in the pamphlet. The
student described it to me. I drew
it. The child’s Mother was amazed
at the “real communication between” us. I did not think that it was such an amazing thing.
How different can a ‘round, brown teapot with silver base and silver
ball shaped handle on the top and swirley handle’ come out? My Mother later
reminded me, “For you it was easy but it is not easy for everyone.”
Drawing
from a description (below)
|
Unit
3 Title: Poetic
Pattern
Grade
level: 5th+
Curriculum
concepts: geometric shapes, repetition of motif to create
pattern (where the motif is the rhythm or shape of the poem), 2-d space
forms, symmetrical designs
Corresponds
with National Art Standards:
Making connections between visual arts and other disciplines;
Understanding the visual arts in relation to history and cultures;
Using knowledge of structures and functions (elements and principles of
art-shape, space, balance, etc.)
Corresponds
with National Math Standards:
Mathematical connections, Geometry and spatial sense, Measurement,
Patterns and relationships
Lesson
objectives: Students will find words to arrange into a designated pattern
or poem and learn about symmetry.
Background
information and motivation:
Choosing and evaluating a range of subject matters, symbols, and ideas is
an exercise that is sometimes considered a Visual Standard.
This lesson stretches to accomplish that task because it has more to do
with Language Arts than Visual Arts, however, combining ideas about math and
molding one’s writing into quantified or measured steps, as is required in
order to make lines consisting of a certain number of syllables, can challenge
the math and linguistic areas of talent. A lantern is a Japanese poem written in the shape of a
lantern. Lantern poems have a
pattern, which resembles the profile of the kind of lantern that gives off
light.
The pattern is:
Line 1-one syllable
X
Line 2-two
syllables XX
Line 3-three syllables
XXX
Line 4- four syllables
XXXX
Line 5- one syllable
X
Integration
areas: Language Arts; writing, syllable counting; Mathematics;
mathematical concept, number or geometric shape, Art: symmetry, pattern
Materials: Paper, writing utensils or computer
Vocabulary: Lantern, shape, syllable
Procedure: Brainstorm with students for ideas about what to write about,
such as geometric shapes, then write a lantern about a mathematical concept, a
number, or a geometric shape. Add
the result and the steps taken to accomplish this to Math Art Diary in a
narrative description. Keep track
of syllables with tally marks. Add
pictures.
Assessment
and/or evaluation: Objective assessment: Completion
of a Lantern to the specified dimensions, meaningful use of words.
Subjective Assessment: Positive
attitude, joy and contentment with creation; rhythm and tone aesthetically
pleasing. (and see Culmination)
Optional
variations: Write a lantern about any other topic touched upon in
Language Arts, Social Studies or Science class:
Compose and perform it to a melody.
In order to help students brainstorm, do these two activities:
Symmetry
fun: Fold a piece of graph
paper in half. The fold line is the line of symmetry, and each side is a mirror
image of the other. Open the paper.
On one side of the line, color several squares to make a pattern.
Another student copies the other side of the line. Fold the paper with
the patterns inside. Do the squares
of the same color cover each other? If
so, you have created a symmetric design. Describe
how you could create a design with more than one line of symmetry.
Size
it up: On a sheet of graph paper, draw the x-axis near the bottom of
the paper and the y-axis near the left-hand side of the paper to show a quadrant
I grid. Graph the following ordered
pairs, connecting the first point to the second point, then continue as each
point is plotted: (2,1),(1,2),(3,2),
(3,8),(4,8),(3,9),(4,10),(5,9)(4,8),(5,8),(5,2),(7,2),(6,1),(2,1). Double each
number in the ordered pairs, and graph each new point on another sheet of graph
paper, again using quadrant I. What
happened to the drawing? These
drawings are similar. They have the
same shape and proportional dimensions. Create
a simple, small drawing and write a list of ordered pairs that could be plotted
to copy your drawing. “Scale”
the drawing to a larger size by multiplying each number in the order pairs by
the same factor. What scale would
your object be if you divided the ordered pair number by the same factor? (Half)
Record exercises in Math Art Journal.
References:
modified
from Gayle Cloke, Nola Ewing, Dory Stevens. (2001). The
fine art of mathematics, Teaching Children Mathematics, 8(2), 108-110.
Examples:
Lantern
by Mrs. Waxman
Cube
/
Three d
//
6 faces
///
top, bottom,
sides////
squares
/
______________________________________________________________________________
Unit
4 Title: String
Art to Learn About Symmetry (and
other Math Facts like Multiplication)
Grade
level: 5+
Curriculum
concepts: Math/Art Topics involved in this lesson are: Polyhedra, Space forms, Symmetrical designs, Geometric
sculptures, Rotational symmetry, Constructions, Artists who use math, Technical
drawings and see unit 1
Corresponds
with National Art Standards:
Form, a three-dimensional object; space, the empty area
between, around, above, below, or within an object-- are Elements of Design,
some of the building blocks of visual art.
Balance, the arrangement of equal parts, stable; contrast,
the difference between two or more things; repetition, the parts used
over and over in a pattern; proportion, the relation of one part to
another; unity, all parts working together-- are Principals of Design,
how the blocks are used or put together.
Corresponds
with National Math Standards:
Mathematics as Problem Solving: How
will we get it to look how we want it to look? We may have to add, subtract,
multiply or divide and we will have to decide what to do with those
calculations, where to apply our answers.
Mathematics as Communication: We
will be talking to each other as we work, about the math-artwork, using the
necessary math and art vocabulary. Mathematics
as Reasoning: For something to be
done in an organized way we will use reasonable methods which logic, and some
call math as a part. Mathematical Connections:
We will discuss and log where these shapes are seen besides here and what
else in the universe they look like. Measurement
and Estimations of length of yarns and lines will be made.
Geometry and Spatial Sense and Patterns and Relationships, Number Sense
and Numeration, Concepts of Whole Number Operations, Whole Number
Computation will be standards addressed in the Procedure or Optional
Variations.
Of
the National Council Teachers Math Association Standards, Statistics and
probability and Fractions and decimals may be the only ones not touched on in
this lesson design. Changing the
scales to fractions and decimals would take care of that, leaving only
Statistics and probability which, with some thinking we can include also, if we
stay innovative with the lesson.
Lesson
objectives:
Students will learn about patterns (Mathematics
has been called “the science of patterns”).
Students will learn how to use patterns to make beautiful line art while
practicing number and operation sense, geometry, measurement estimation, whole
number operations and computation.
Background
information and motivation: Students
may be surprised at the dramatic artwork they can create if they measure
carefully and follow a pattern and it can be fun to put the separate parts of
something together, hang it on the ceiling or wall to make a moving sculpture or
a mobile. Multiplication can be demonstrated to themselves by making squares or
four sided polygons from crossed yarns lines. The more free form activity is the Procedure while the
more structured ones are Optional Variations.
Integration
areas: Geometry,
number sense, computation, Art
Materials:
paper,
pencils or crayons, sticks, string, scissors, hole punch or needle, foam board
or clear plastic sheets, Math Art Journal
Vocabulary:
2
dimensional objects 3-dimensional, cube, angles, right angle, obtuse, acute,
equivalent triangles, scalene, isosceles, intersecting lines, symmetrical,
asymmetrical, rotational symmetry, polygon
Procedure:
1.
Draw two line segments (or use the edges of foam board or clear plastic sheets)
that meet at a right angles or cut small, equally spaced marks on the edge,
about 3″ line segments with 1/4″ marks.
2.
Sequentially connect the pairs of marks with straight lines, starting at the
first mark on each segment so that the lines cross as they are shown or, wrap
colored yarns to create different patterns. Students can design their own arrangements of line segments
similar to these (see Instructor for help):
.
3.
Demonstrate, identify and define the vocabulary words in the artwork you are
doing: 2-dimensional objects
3-dimensional, angles, right angle, obtuse, acute, equivalent triangles,
scalene, isosceles, intersecting lines, symmetrical, asymmetrical, rotational
symmetry, polygon. Discuss where
you see them and write all about it in Math Art Journal.
3.
Add line art or string art pieces to the Icosahedra Mobile or Cube structure in
an aesthetically pleasing way, balanced by color or shape or in a the shape of
and animal or other object yet to be known, perhaps an object in ‘Student’s
Store’, such as combs or hair brushes.
Assessment
and/or evaluation: Students
complete multiplication problems in Math Journal. Explain, write or type what
the project involved. (also see
Culmination)
Optional
Variations: A digit circle is a circle with digits 0-9 equally spaced
around the outside. Patterns can be
created while practicing number operations.
Use a black line master from www.TeachingK8.com
or draw your own circle and add numbers 1-9 as shown.
Multiplication:
Choose a multiplying number, then multiply each of the digits from 0-9 by
that multiplier. Draw an arrow from that digit to the number, which is the last
digit of the product on the circle. For
example, suppose your multiplier (multiplying number) is 7.
1x7=7, so draw an arrow from 1 to 7.
2x7=14,
so draw an arrow from 2 to 4 (4 is the last digit of 14).
3x7=21
so 3 connects to 1,
4x7=28
so 4 connects to 8, and so on.
Make
designs for each of the multipliers, 0-9, and then compare the designs and look
for connections. There are similarities between pairs of designs of multipliers
that sum to 10 and some can come out pretty. Use different colors. This is good
for practicing multiplication facts.
Fill
in a Times Table chart: (refer to calculator or rear cover of Math Art Journal
where it is pre-printed with a Conversion Table and Grammar Rules)
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Addition:
Do this with all digits from 0-9, and then compare
the designs to look for similarities and differences. There are some striking
patterns that emerge; for example, the designs are identical for pairs that add
to 10, so 1 and 9 make the same design, as do 2 and 8, 3 and 7 and 4 and 6. The
designs are created in the opposite direction, though. Ask your students to come
up with ideas as to why this might be. One way to think about it is that adding
3 gives the same last digit as subtracting 7 and vice versa.
Problem: If there are four people at a party and everyone shakes hands
with everyone else, how many handshakes are there? What if there are five people? Six people?
100 People? A string art
picture like the one shown can help solve this problem:
Some
patterns students may find: With six people, the first person needs to shake
five hands. The second person shakes four new hands (they already shook with
person #1), the third person shakes three new hands, the fourth shakes two and
the fifth shakes one hand. Everyone
will have shaken hands with the sixth person.
The total number 5+4+3+2+1+0=15. With
100 people, the toal is 99+98+97+…+1+1+0+_?
Another way is to see that in a group of six people, each person shakes
hands with five others for a total of 30 hands shaken. But "a
handshake" is two people shaking hands, so 30 is exactly two times too
many, 30 ÷ 2 = 15 handshakes. So with 100 it should be 99 x 100 ÷ 2
handshakes. Can you spot the pattern?
References:
Adapted from Naylor, M. (2006, March). Do you see a pattern? Teaching Prek-8,
36(6), 38.
Examples: see www.randisart.com/pottery/String_Art_mobile_orange_sample.jpg
_______________________
Unit
5 Title:
A Multifaceted Activity, A Polyhedra Icosahedron
(20 sided figure)
Grade
level: 5+
Curriculum
concepts:
see unit 4
Corresponds
with National Art Standards:
see unit 1
Corresponds
with National Math Standards:
see unit 1
Lesson
objectives: Students will learn the basic geometric shape of equilateral
triangles, icosahedron, tetrahedron; they will be able to identify functional
aspects of them and their unique mechanical property of strength; learn about
rotational symmetry; learn about graphic design by creating horizontal, vertical
and alternating patterns and study their motion and how distortions occur; learn
that art is made from shapes and that some shapes occur naturally, are invented
by humans, and have specific names and sometimes, purposes.
Background
information and motivation: It
is commonly thought the Principal of Twisting and Release was first used by
ancient Greeks to power catapults, which tossed heavy stones great distances.
(Weisman 2006)
Although
this is made of one piece of paper, when properly constructed, it can support a
heavy book without being crushed. An
icosahedron’s structural qualities are demonstrated by triangulation.
The triangle is a shape used to make things (like bridges and buildings)
that need to withstand a lot of weight or force.
They spread out the force so it is not focused at one point, causing
something to break or fracture.
The
zoetrope is one of several animation toys that were invented in the 19th
century. They have the property of
causing the images to appear thinner than their actual sizes when viewed in
motion through the slits and were precursors to animation and films.
This multifaceted project incorporates elements from several academic areas. It requires varied tasks and satisfies artistic, technical and hands-on personal preferences while providing success
for
students of all artistic skill levels
Integration
areas: This project combines geometry, structure, physical
science, graphic design, animation, motion, mechanical free-hand drawing with
catapult mechanics. It is an icosahedron,
a geometric figure with 20 triangles made of equilateral triangles, therefore,
it is a multifaceted lesson. It
has three distinct surface areas consisting of five triangles and a central band
of ten triangles.
Materials:
18" x 24" Paper,
ruler, a sharp edged instrument or scissors, coloring utensils-crayons, markers,
etc., tape, string, glue, pipe cleaner, T-square and a 30-60 degree triangle (optional) to make 60 0
triangles.
Vocabulary:
Polyhedron,
icosahedron, zoetrope, rotational symmetry, torsion, static, stationary,
tetrahedron
Procedure:
1.
Trace the notched template pattern of triangles with 3” sides.
Cut it out being careful to leave the hems. The hems will not be seen and are not decorated.
2.
Design, draw and color the surfaces, possibly with the form of motion in mind
since the static drawing will look different in motion.
One end will have generally vertical lines or alternating color circling
the structure, the other will have horizontal lines that waver. The middle can
be designed freely by the student, using simple geometric and free forms, or
elaborate representational drawings. Drawing
skill is not a necessity and the outcome is a mystery until the icosahedron is
in motion.
3.
Score the edges: Hold ruler on line. Hold
the knife like a pencil. Press with
sharp edge along lines (or teacher will do it to ensure sharp, crisp straight
lines). Fold edges to make a creased form.
The ends are assembles first. Starting
at one end, each hem is glued to its neighbor from the inside. The form begins to take shape as the ends come together.
The center follows automatically. The
last two hems of each end should be left unattached. This will also leave two unattached hems in the center
creating three openings that are connected end to end. A pipe cleaner axle with
looped ends (bend and twist the loops around a pencil) is inserted into this
opening. The three edges of the opening are then glued together.
Assessment
and/or evaluation: Students
write an essay on the project, including the physical science and historical
information learned, in Math Art Journal. Students
read it to students or parents; demonstrates the properties discussed, such as
strength, by putting a heavy book on it. (see also Culmination)
Optional
variations: 1.
Add catapult torsion using string
threaded through each loop of the pipe cleaner. Tie ends together. Hold the
ends, stretch the string and spin the icosahedron. As it spins, the string loops
twist around themselves. Pull gently and release, the string will unwind and
rewind. Each pull and release keeps the icosahedron in motion, animating the
surface designs. The horizontal lines move up and down the surface, the colors
in the vertical pattern optically mix and the shapes and colors in the center
mix and move. 2. Hang the Polyhedra from the ceiling.
3. Leave out the string or coloring.
4. Prepare the shape ahead of time or have the student actually use the
template. 5. Make a tetrahedron
with four triangles 6. Attach other shapes from the other lessons to
Icosahedron.
References:
Adapted from Strazdin,
R. (2000, May), Icosahedrons: A
multifaceted project. Arts& Activities, 127 (4), 38.
Examples:
see http://www.randisart.com/kid's%20page.htm
Culmination
Units 1-5: Invite Parents. Everyone
views the display of drawings and projects, Icosahedron Mobile (or Combination
of Shapes Mobile) in a gallery style exhibit in the classroom or school’s
hallways. Students may share their
poems by reading them out loud or matting them on a nice background to post them
on the wall; add pictures or decorations with crayons, colored pencils or pens.
References
Adejumo, Christopher O.
2002. Vol. 55, Iss. 5; pg. 6, 6 pgs
Baicker,
K.
Kellah
M Edens, Ellen F Potter.
Gayle Cloke, Nola Ewing, Dory Stevens.
(2001). The Fine Art of Mathematics, Teaching
Children Mathematics, 8(2), 108-110. Retrieved August 8, 2008,
from Research Library database. (Document ID: 83776531).
Jensen, Eric.
Naylor, M.
The Owl at Perdue < http://owl.english.purdue.edu/owl/resource/560/01/>
Weissman, Rabbi Moshe.
Strazdin, R.
_________________________________________________________________________________________________________________________________
click
on pictures to enlarge
'Centagon', a hundred side figure/Dragon 
box
template sketch
mvrlpw@yahoo.com
