discovering geometry, chapter 0

CHAPTER

0

My subjects are often

playful....It is, for example, a

pleasure to deliberately mix together

objects of two and of three dimensions,

surface and spatial relationships, and to

make fun of gravity.

M.C.ESCHER

Print Gallery, M.C. Escher, 1956©2002 Cordon Art B.V.–Baarn–Holland.All rights reserved.

Geometric Art

O B J E C T I V E SIn this chapter you will

see examples of geometryin naturestudy geometric art formsof cultures around theworldstudy the symmetry inflowers, crystals, andanimalssee geometry as a way ofthinking and of looking atthe worldpractice using a compassand straightedge

A work by Dutch graphic artistM.C. Escher (1898–1972) openseach chapter in this book. Escherused geometry in creative waysto make his interesting andunusual works of art. As youcome to each new chapter,seewhether you can connect theEscher work to the content ofthe chapter.

© 2008 Key Curriculum Press

Lesson http://acr.keypress.com/KeyPressPortalV3.0/Viewer/Lesson.htm

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2 CHAPTER 0 Geometric Art

There is one art,

no more no less,

To do all things

with artlessness.

PIET HEIN

L E S S O N

0.1Geometry in Natureand in ArtNature displays a seemingly infinite variety of geometric shapes, from tiny atomsto great galaxies. Crystals, honeycombs, snowflakes, spiral shells, spiderwebs, andseed arrangements on sunflowers and pinecones are just a few of nature’s geometricmasterpieces.

Geometry includes the study of the properties of shapes such as circles,hexagons, and pentagons. Outlines of the sun, the moon, and the planetsappear as circles. Snowflakes, honeycombs, and many crystals are

hexagonal (6-sided). Many living things, such as flowers and starfish,are pentagonal (5-sided).

People observe geometric patterns in nature anduse them in a variety of art forms. Basketweavers, woodworkers, and other artisans oftenuse geometric designs to make their worksmore interesting and beautiful. You will learnsome of their techniques in this chapter.



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LESSON 0.1 Geometry in Nature and in Art 3

Artists rely on geometry to show perspective and proportion, and to producecertain optical effects. Using their understanding of lines, artists can give depth totheir drawings. Or they can use lines and curves to create designs that seem to popout of the page. You will create your own optical designs in Lesson 0.4.

Symmetry is a geometric characteristic of both nature and art. You may alreadyknow the two basic types of symmetry, reflectional symmetry and rotationalsymmetry. A design has reflectional symmetry if you can fold it along a lineof symmetry so that all the points on one side of the line exactly coincide with(or match) all the points on the other side of the line.

You can place a mirror on the line of symmetry so that half the figure and itsmirror image re-create the original figure. So, reflectional symmetry is also calledline symmetry or mirror symmetry. Biologists say anorganism with just one line of symmetry, like thehuman body or a butterfly, has bilateralsymmetry. An object with reflectionalsymmetry looks balanced.

A design has rotational symmetry if itlooks the same after you turn it arounda point by less than a full circle. Thenumber of times that the design looksthe same as you turn it through acomplete 360° circle determines the typeof rotational symmetry. The Apache baskethas 3-fold rotational symmetry because itlooks the same after you rotate it 120°(a third of a circle), 240° (two-thirdsof a circle), and 360° (one full circle).



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A starfish has 5-fold symmetry. It looks the same after you rotate it 72°, 144°, 216°,288°, or 360°.

Countries throughout the world use symmetry in their national flags. Notice thatthe Jamaican flag has rotational symmetry in addition to two lines of reflectionalsymmetry. You can rotate the flag 180° without changing its appearance. Theorigami boxes, however, have rotational symmetry, but not reflectional symmetry.(The Apache basket on page 3 almost has reflectional symmetry. Can you see whyit doesn’t?)

Consumer

Many products have eye-catching labels, logos, and designs. Have you everpaid more attention to a product because the geometric design of its logo wasfamiliar or attractive to you?

EXERCISES

1. Name two objects from nature whose shapes are hexagonal. Name two livingorganisms whose shapes have five-fold rotational symmetry.

2. Describe some ways that artists use geometry.

3. Name some objects with only one line of symmetry. What is the name for this typeof symmetry?



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4. Which of these objects have reflectional symmetry (or approximate reflectional symmetry)?A. B. C.

D. E. F.

5. Which of the objects in Exercise 4 have rotational symmetry (or approximaterotational symmetry)?

6. Which of these playing cards have rotational symmetry? Which ones havereflectional symmetry?

7. British artist Andy Goldsworthy (b 1956) uses materials from natureto create beautiful outdoor sculptures. The artful arrangement ofsticks below might appear to have rotational symmetry, but insteadit has one line of reflectional symmetry. Can you find the line ofsymmetry?

If an exercise has an at theend, you can find a hint tohelp you in Hints for SelectedExercises at the back of thebook.

LESSON 0.1 Geometry in Nature and in Art 5© 2008 Key Curriculum Press


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8. Shah Jahan, Mughal emperor of India from 1628 to 1658, had the beautiful Taj Mahalbuilt in memory of his wife, Mumtaz Mahal. Its architect, Ustad Ahmad Lahori,designed it with perfect symmetry. Describe two lines of symmetry in this photo.How does the design of the building’s grounds give this view of the Taj Mahal evenmore symmetry than the building itself has?

9. Create a simple design that has two lines of reflectional symmetry. Does it haverotational symmetry? Next, try to create another design with two lines ofreflectional symmetry, but without rotational symmetry. Any luck?

10. Bring to class an object from nature that shows geometry. Describe the geometrythat you find in the object as well as any symmetry the object has.

11. Bring an object to school or wear something that displays a form of handmade ormanufactured geometric art. Describe any symmetry the object has.

6 CHAPTER 0 Geometric Art © 2008 Key Curriculum Press


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Line DesignsThe symmetry and patterns in geometric designs make themvery appealing. You can make many designs using the basictools of geometry—compass and straightedge.

You’ll use a straightedge to construct straight linesand a compass to construct circles and to mark offequal distances. A straightedge is like a ruler but it has nomarks. You can use the edge of a ruler as a straightedge.The straightedge and the compass are the classicalconstruction tools used by the ancient Greeks, who laidthe foundations of the geometry that you are studying.

LESSON 0.2 Line Designs 7

We especially need

imagination in science.It is

not all mathematics,nor all

logic,but it is somewhat

beauty and poetry.

MARIA MITCHELL

L E S S O N

0.2



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You can create many types of designs using only straight lines. Here are two linedesigns and the steps for creating each one.

EXERCISES

1. What are the classical construction tools of geometry?

2. Create a line design from this lesson. Color your design.

3. Each of these line designs uses straight lines only. Select one design and re-create iton a sheet of paper.

4. Describe the symmetries of the three designs in Exercise 3. For the third design,does color matter?



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5. Many quilt designers create beautiful geometric patterns withreflectional symmetry. One-fourth of a 4-by-4 quilt pattern and itsreflection are shown at right. Copy the designs onto graph paper,and complete the 4-by-4 pattern so that it has two lines ofreflectional symmetry. Color your quilt.

6. Geometric patterns seem to be in motion in a quilt design with rotationalsymmetry. Copy the quilt piece shown in Exercise 5 onto graph paper,and complete the 4-by-4 quilt pattern so that it has 4-fold rotationalsymmetry. Color your quilt.

7. Organic molecules have geometric shapes. How many different lines ofreflectional symmetry does this benzene molecule have? Does it haverotational symmetry? Sketch your answers.

LESSON 0.2 Line Designs 9

Architecture

Frank Lloyd Wright(1867–1959) is often calledAmerica’s favorite architect.He built homes in 36 states—sometimes in unusual settings.

Fallingwater, located inPennsylvania, is a buildingdesigned by Wright that displayshis obvious love of geometry.Can you describe the geometryyou see? Find more informationon Frank Lloyd Wright at

www.keymath.com/DG



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Its where we go,and what

we do when we get there,

that tells us who we are.

JOYCE CAROL OATES

L E S S O N

0.3Circle DesignsPeople have always been fascinated by circles. Circles are used in the design ofmosaics, baskets, and ceramics, as well as in the architectural design of buildings.

You can make circle designs with a compass as your primary tool. For example,here is a design you can make on a square dot grid.Begin with a 7-by-9 square dot grid. Construct three rows of four circles.

Construct two rows of three circles using the points between the first set of circlesas centers. The result is a set of six circles overlapping the original 12 circles.Decorate your design.



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Here is another design that you can make using only a compass. Start byconstructing a circle, then select any point on it. Without changing your compasssetting, swing an arc centered at the selected point. Swing an arc with each of thetwo new points as centers, and so on.

See the Dynamic Geometry Exploration Daisy Designs at

Notice the shape you get by connecting the six petal tips of the daisy. This is aregular hexagon, a 6-sided figure whose sides are the same length and whoseangles are all the same size.

You can do many variations on a daisy design.

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LESSON 0.3 Circle Designs 11

keymath.com/DG

Dynamic Geometry Explorationsatshow geometric concepts visuallyusing interactive sketches.

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EXERCISES

For Exercises 1–5, use your construction tools.

1. Use square dot paper to create a 4-by-5grid of 20 circles, and then make 12 circlesoverlapping them. Color or shade thedesign so that it has reflectionalsymmetry.

2. Use your compass to create a set ofseven identical circles that touch butdo not overlap. Draw a larger circle thatencloses the seven circles. Color or shadeyour design so that it has rotationalsymmetry.

3. Create a 6-petal daisy design and coloror shade it so that it has rotationalsymmetry, but not reflectional symmetry.

4. Make a 12-petal daisy by drawing a second 6-petal daisybetween the petals of the first 6-petal daisy. Color or shadethe design so that it has reflectional symmetry, but notrotational symmetry.

5. Using a 1-inch setting for your compass, construct a centralregular hexagon and six regular hexagons that each shareone side with the original hexagon. Your hexagon designshould look similar to, but larger than, the figure at right.This design is called a tessellation, or tiling, of regularhexagons.

You will need



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Everything is an illusion,

including this notion.

STANISLAW J. LEC

L E S S O N

0.4

LESSON 0.4 Op Art 13

Op ArtOp art, or optical art, is a form of abstract art that uses lines or geometricpatterns to create a special visual effect. The contrasting dark and light regionssometimes appear to be in motion or to represent a change in surface, direction,and dimension. Victor Vasarely was one artist who transformed grids so thatspheres seem to bulge from them. Recall the series Tsiga I, II, III that appearsin Lesson 0.1. Harlequin, shownat right, is a rare Vasarely workthat includes a human form.Still, you can see Vasarely’strademark sphere in the clown’sbulging belly.

Op art is fun and easy to create. To create one kind of op art design, first makea design in outline. Next, draw horizontal or vertical lines, gradually varying thespace between the lines or each pair of lines, as shown below, to create an illusionof hills and valleys. Finally, color in or shade alternating spaces.



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To create the next design, first locate a point on each of the four sides of a square.Each point should be the same distance from a corner, as shown. Your compass is agood tool for measuring equal lengths. Connect these four points to create anothersquare within the first. Repeat the process until the squares appear to converge onthe center. Be careful that you don’t fall in!

Here are some other examples of op art.

You can create any of the designs on this page using just a compass andstraightedge (and doing some careful coloring). Can you figure out how eachof these op art designs was created?

14 CHAPTER 0 Geometric Art © 2008 Key Curriculum Press


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LESSON 0.4 Op Art 15

EXERCISES

1. What is the optical effect in each piece of artin this lesson?

2. Nature creates its own optical art. At firstthe black and white stripes of a zebra appearto work against it, standing out against thegolden brown grasses of the African plain.However, the stripes do provide the zebraswith very effective protection frompredators. When and how?

3. Select one type of op art design fromthis lesson and create your own versionof it.

4. Create an op art design that has reflectionalsymmetry, but not rotational symmetry.

5. Antoni Gaudí (1852–1926) designed theBishop’s Palace in Astorga, Spain. List asmany geometric shapes as you can recognizeon the palace (flat, two-dimensional shapessuch as rectangles as well as solid, three-dimensional shapes such as cylinders).What type of symmetry do you see onthe palace?



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L E S S O N

0.5Knot DesignsKnot designs are geometric designs that appear to weave or to interlace like aknot. Some of the earliest known designs are found in Celtic art from the northernregions of England and Scotland. In their carved stone designs, the artists imitatedthe rich geometric patterns of three-dimensional crafts such as weaving andbasketry. The Book of Kells (8th and 9th centuries) is the most famous sourceof Celtic knot designs.

Here are the steps for creating two examples of knot designs. Look them overbefore you begin the exercises.

You can use a similar approach to create a knot design with rings.

In the old days, a love-sick

sailor might send his

sweetheart a length of

fishline loosely tied in a love

knot. If the knot was returned

pulled tight, it meant the

passion was strong. But if the

knot was returned untied—

ah, matey, time to ship out.

OLD SAILOR’S TALE

Today a very familiar knot design is the set ofinterconnected rings (shown at right) used asthe logo for the Olympic Games.



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LESSON 0.5 Knot Designs 17

Here are some more examples of knot designs.



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n


EXERCISES

1. Name a culture or country whose art uses knot designs.

2. Create a knot design of your own, using only straight lines on graph paper.

3. Create a knot design of your own with rotational symmetry, using a compass or acircle template.

4. Sketch five rings linked together so that you could separate all five by cutting openone ring.

5. The coat of arms of the Borromeo family, who lived during the Italian Renaissance(ca. 15th century), showed a very interesting knot design known as the BorromeanRings. In it, three rings are linked together so that if any one ring is removed theremaining two rings are no longer connected. Got that? Good. Sketch theBorromean Rings.

6. The Chokwe storytellers of northeastern Angola are called Akwa kuta sona (“thosewho know how to draw”). When they sit down to draw and to tell their stories, theyclear the ground and set up a grid of points in the sand with their fingertips, asshown below left. Then they begin to tell a story and, at the same time, trace afinger through the sand to create a lusona design with one smooth, continuousmotion. Try your hand at creating sona (plural of lusona). Begin with the correctnumber of dots. Then, in one motion, re-create one of the sona below. The initialdot grid is shown for the rat.

7. In Greek mythology, the Gordian knot was such a complicated knot that no onecould undo it. Oracles claimed that whoever could undo the knot would becomethe ruler of Gordium. When Alexander the Great (356–323 B.C.E.) came upon theknot, he simply cut it with his sword and claimed he had fulfilled the prophecy, sothe throne was his. The expression “cutting the Gordian knot” is still used today.What do you think it means?

You will need

Science

Mathematician DeWittSumners at Florida StateUniversity and biophysicistSylvia Spenger at theUniversity of California,Berkeley, have discovered thatwhen a virus attacks DNA, itcreates a knot on the DNA.



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LESSON 0.5 Knot Designs 19

8. The square knot and granny knot are very similar but do very different things.Compare their symmetries. Use string to re-create the two knots and explain theirdifferences.

9. Cut a long strip of paper from a sheet of lined paper or graph paper. Tie the strip ofpaper snugly, but without wrinkles, into a simple knot. What shape does the knotcreate? Sketch your knot.

SYMBOLIC ARTJapanese artist Kunito Nagaoka (b 1940) uses geometry in his work. Nagaoka was bornin Nagano, Japan, and was raised near the active volcano Asama. In Japan, heexperienced earthquakes and typhoons as well as the human tragedies of Hiroshima andNagasaki. In 1966, he moved to Berlin, Germany, a city rebuilt in concrete from the ruinsof World War II. These experiences clearly influenced his work.

You can find other examples ofsymbolic art at

Look at the etching shown hereor another piece of symbolic art.Write a paragraph describingwhat you think might havehappened in the scene or whatyou think it might represent.What types of geometric figuresdo you find?Use geometric shapes in yourown sketch or painting to evoke afeeling or to tell a story. Write aone- or two-page story related toyour art.Your project should include

A paragraph describing theshapes found in the art shownhere or in another piece of symbolic art.(If you use a different piece of art, include a copy of it.)Your own symbolic art that uses geometric figures.A one- or two-page story that relates to your art.

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Patience with small details

makes perfect a large work,

like the universe.

JALALUDDIN RUMI

L E S S O N

0.6Islamic Tile DesignsIslamic art is rich in geometric forms. Early Islamic, or Muslim, artists becamefamiliar with geometry through the works of Euclid, Pythagoras, and othermathematicians of antiquity, and they used geometric patterns extensively in theirart and architecture.

Islam forbids the representation of humans or animalsin religious art. So, instead, the artists use intricategeometric patterns.

One striking example of Islamic architecture isthe Alhambra, a Moorish palace in Granada, Spain.Built over 600 years ago by Moors and Spaniards, theAlhambra is filled from floor to ceiling with marvelousgeometric patterns. The designs you see on this page arebut a few of the hundreds of intricate geometric patternsfound in the tile work and the inlaid wood ceilings ofbuildings like the Alhambra and the Dome of the Rock.

Carpets and hand-tooled bronze plates from the Islamicworld also show geometric designs. The patterns oftenelaborate on basic grids of regular hexagons, equilateraltriangles, or squares. These complex Islamic patternswere constructed with no more than a compass and astraightedge. Repeating patterns like these are calledtessellations. You’ll learn more about tessellations inChapter 7.



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The two examples below show how to create one tile in a square-based and ahexagon-based design. The hexagon-based pattern is also a knot design.

LESSON 0.6 Islamic Tile Designs 21© 2008 Key Curriculum Press


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EXERCISES

1. Name two countries where you can find Islamic architecture.

2. What is the name of the famous palace in Granada, Spain, where youcan find beautiful examples of tile patterns?

3. Using tracing paper or transparency film, trace afew tiles from the 8-Pointed Star design. Notice thatyou can slide, or translate, the tracing in a straightline horizontally, vertically, and even diagonally toother positions so that the tracing will fit exactlyonto the tiles again. What is the shortest translationdistance you can find, in centimeters?

4. Notice that when you rotate your tracing fromExercise 3 about certain points in the tessellation,the tracing fits exactly onto the tiles again. Findtwo different points of rotation. (Put your pencilon the point and try rotating the tracing paper ortransparency.) How many times in one rotationcan you make the tiles match up again?

You will need



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LESSON 0.6 Islamic Tile Designs 23

5. Currently the tallest twin towers in theworld are the Petronas Twin Towers in KualaLumpur, Malaysia. Notice that the floorplans of the towers have the shape of Islamicdesigns. Use your compass and straightedgeto re-create the design of the base of thePetronas Twin Towers, shown at right.

6. Use your protractor and ruler to draw a square tile. Use your compass,straightedge, and eraser to modify and decorate it. See the example inthis lesson for ideas, but yours can be different. Be creative!

7. Construct a regular hexagon tile and modify and decorate it. See the example in thislesson for ideas, but yours can be different.

8. Create a tessellation with one of the designs you made in Exercises 6 and 7. Trace orphotocopy several copies and paste them together in a tile pattern. (You can alsocreate your tessellation using geometry software and print out a copy.) Add finishingtouches to your tessellation by adding, erasing, or whiting out lines as desired. If youwant, see if you can interweave a knot design within your tessellation. Color yourtessellation.

Architecture

After studying buildings in other Muslim countries, thearchitect of the Petronas Twin Towers, Cesar Pelli (b 1926),decided that geometric tiling patterns would be key to thedesign. For the floor plan, his team used a very traditionaltile design, the 8-pointed star—two intersecting squares.To add space and connect the design to the traditional“arabesques,” the design team added arcs of circles betweenthe eight points.



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In this chapter, you described the geometric shapes andsymmetries you see in nature, in everyday objects, in art, and inarchitecture. You learned that geometry appears in many types ofart—ancient and modern, from every culture—and you learnedspecific ways in which some cultures use geometry in their art. Youalso used a compass and straightedge to create your own works ofgeometric art.

EXERCISES

1. List three cultures that use geometry in their art.

2. What is the optical effect of the op art design at right?

3. Name the basic tools of geometry you used in this chapterand describe their uses.

4. With a compass, draw a 12-petal daisy.

5. Construction With a compass and straightedge, construct aregular hexagon.

6. List three things in nature that have geometric shapes. Nametheir shapes.

7. Draw an original knot design.

8. Which of the wheels below have reflectional symmetry?How many lines of symmetry does each have?

9. Which of the wheels in Exercise 8 have only rotational symmetry? What kind ofrotational symmetry does each of the four wheels have?

The end of a chapter is a good timeto review and organize your work.Each chapter in this book will endwith a review lesson.

You will need



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CHAPTER 0 REVIEW 25

10. A mandala is a circular design arranged in rings that radiate from the center.(See the Cultural Connection below.) Use your compass and straightedge to create amandala. Draw several circles using the same point as the center. Create a geometricdesign in the center circle, and decorate each ring with a symmetric geometricdesign. Color or decorate your mandala. Two examples are shown below.

11. Create your own personal mandala. You might include your name,cultural symbols, photos of friends and relatives, and symbols thathave personal meaning for you. Color it.

12. Create one mandala that uses techniques from Islamic art, is aknot design, and also has optical effects.

Cultural

The word mandala comes from Sanskrit, the classical language of India, andmeans “circle” or “center.” Hindus use mandala designs for meditation. TheIndian rug at right is an example of a mandala, as is the Aztec calendar stonebelow left. Notice the symbols are arranged symmetrically within each circle.The rose windows in many gothic cathedrals, like the one below right from theChartres Cathedral in France, are also mandalas. Notice all the circles withincircles, each one filled with a design or picture.



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13. Before the Internet, “flags” was the most widely read topic of the World BookEncyclopedia. Research answers to these questions. More information about flags isavailable ata. Is the flag of Puerto Rico symmetric? Explain.b. Does the flag of Kenya have rotational symmetry? Explain.c. Name a country whose flag has both rotational and reflectional symmetry.

Sketch the flag.

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An essential part of learning is being able to show yourself and others how muchyou know and what you can do. Assessment isn’t limited to tests and quizzes.Assessment isn’t even limited to what your teacher sees or what makes up yourgrade. Every piece of art you make and every project or exercise you completegives you a chance to demonstrate to somebody—yourself, at least—what you’recapable of.

BEGIN A PORTFOLIOThis chapter is primarily about art, so you might organizeyour work the way a professional artist does—in a portfolio. A portfolio is differentfrom a notebook, both for an artist and for a geometry student. An artist’snotebook might contain everything from scratch work to practice sketches torandom ideas jotted down. A portfolio is reserved for an artist’s most significant orbest work. It’s his or her portfolio that an artist presents to the world todemonstrate what he or she is capable of doing. The portfolio can also show howan artist’s work has changed over time.

Review all the work you’ve done so far and choose one or more examples of yourbest art projects to include in your portfolio. Write a paragraph or two about eachpiece, addressing these questions:

What is the piece an example of ?Does this piece represent your best work? Why else did you choose it?What mathematics did you learn or apply in this piece?How would you improve the piece if you redid or revised it?

Portfolios are an ongoing and ever-changing display of your work and growth.As you finish each chapter, update your portfolio by adding new work.

This section suggests how you might review, organize, andcommunicate to others what you’ve learned. Whether you followthese suggestions or directions from your teacher, or use studystrategies of your own, be sure to reflect on all you’ve learned.



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