Mathematical MoMAnts

at the Museum of Modern Art

Ron Lancaster Lecturer, Mathematics Education

35 Oak Knoll Drive Ontario Institute for Studies

in Education of

Hamilton, ON The University of Toronto

L8S 4C2

[email protected]

[email protected]

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Introduction

Welcome to another Math for America Math Trail. The location for this Trail is the

Museum of Modern Art in New York City, a beautiful place to visit while wearing a pair

of mathematical glasses.

The idea behind a Math Trail as it is now commonly called is quite simple: students

follow a planned route and answer or create mathematical questions related to what they

encounter along the path. Through this experience students are given the chance to

connect the mathematics curriculum to many subjects including art, design, architecture,

science, geography and history.

During the weekend of February 16-17, 2008 I visited MoMA, selected a path to follow

and then wrote a draft of mathematical questions related to art that had caught my eye.

On February 19th a group of Fellows from Math for America joined me for the walk.

Also on hand were Jon Schweig, Program Director for Math for America and Lisa

Mazzola, Associate Educator for School Programs in the Department of Education at

MoMA. I Initially felt like a tour guide, but in short order we were all viewing the

surroundings through a mathematical lens and we had interesting discussions about

mathematical questions related to the artwork in MoMA that could be assigned to

students. It was a great day filled with a number of wonderful mathematical MoMAnts.

From February to early June I studied the works of art that appeared on the path, I spoke

with art teachers about these artists and then I wrote the Trail that is now in your hands.

One major point that needs to be brought to your attention is that it is not going to be

possible for you to follow this Trail in its entirety because many of the works of art that

we viewed are not on permanent display. The good news is that you can still use these

questions with your students. I suggest that you consider buying postcards, prints or

posters of the works of art and display them on the walls of your classroom. These items

are readily available from the MoMA bookstore or through other art museums. You

might also consider having books about the artists in your classroom. These books can

be borrowed from a library or an art teacher or purchased in bookstores or museums.

Another important point for you to be aware of is that many of the questions I have

prepared will not necessarily be geared to the grade level that you teach. I would suggest

that you focus less on which question goes with which grade level and more on the notion

of embedding mathematics within a context, in this case a piece of art.

I would like to thank Jon Schweig and Lisa Mazzola for their help. Jon provided

financial support that allowed me to travel to Manhattan in February 2007 and Lisa

provided me with access to the museum during my trip and she ensured that we were all

treated royally on the day of our visit. Lisa provided us with a conference room, access

to employees, guest passes for a year and a great breakfast and lunch.

At the end of the document you will find a bibliography and information about my work

with Math Trails. I hope you find this Trail to be useful. Happy Trails.

Ron Lancaster, Lecturer in Mathematics Education, OISE/UT

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Empress of India by Frank Stella

Source for Photograph 1 and Gallery label text

http://www.moma.org/collection/browse_results.php?object_id=79806

Photograph 1

Gallery label text

Frank Stella. (American, born 1936). Empress of India. 1965. Metallic powder in

polymer emulsion paint on canvas, 6' 5" x 18' 8" (195.6 x 548.6 cm). Gift of S. I.

Newhouse, Jr. © 2008 Frank Stella / Artists Rights Society (ARS), New York

Part of Stella’s Notched–V series, begun in late 1964, Empress of India comprises four

colored chevron–shaped canvases. Stella deliberately avoided dramatic changes in color

intensity, because, he reasoned, "when you have four vectored V's moving against each

other, if one jumps out, you dislocate the plane and destroy the whole thing entirely." The

lines parallel to the canvas edges, painted in metallic browns and ochers, contribute to

Stella’s perceptual play of pushing parts of the whole forward and back. Although it is

unclear why Stella chose the phrase "Empress of India" for this painting, the grandly

scaled work shares the title taken by Queen Victoria when India was incorporated into the

British Empire.

Question 1

(a) Describe several ways in which the chevron-shaped canvas at the far left of

Empress of India (Photograph 1) could be transformed to fill the space occupied

by the adjacent chevron-shaped canvas.

(b) Empress of India is an irregular 16-gon, yet the dimensions are stated as if it is a

rectangle. How are these dimensions related to the work of art?

(c) Figure 1 shows the outline of Empress of India. Let the length of AB be 1 unit.

Find the area and perimeter of the 16-gon ABCDEFGHIJKLMNOPQRSTUV.

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S RQ POML

TU C K

V N

JI

ABDE FGH

Figure 1

(d) Two regular 16-gons appear in Figures 2 and 3. Which one of these has the same

perimeter as the irregular 16-gon ABCDEFGHIJKLMNOPQRSTUV? Which

one has the same area?

(e) Find the length of the side of a regular 16-gon that has the same perimeter as the

irregular 16-gon ABCDEFGHIJKLMNOPQRSTUV.

(f) Find the length of the side of a regular 16-gon that has the same area as the

irregular 16-gon ABCDEFGHIJKLMNOPQRSTUV.

(g) Use your results for parts (e) and (f) to determine if the choices you made in part

(d) are correct.

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Figure 2

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Figure 3

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Wall Drawing #1144 by Sol LeWitt

Source for Photograph 2 and accompanying text

http://www.moma.org/collection/browse_results.php?object_id=94278

Photograph 2

Sol LeWitt. (American, 1928-2007). Wall Drawing #1144, Broken bands of color in four

directions. 2004. Synthetic polymer paint, 8' x 37' (243.8 x 1127.8 cm). Given

Anonymously. © 2008 Sol LeWitt / Artists Rights Society (ARS), New York

Question 2

(a) Count the total number of horizontal bands of color in the second square from the

left side of the image and the total number of upward sloping bands of color in the

second square from the right side of the image. Are these totals the same? Did

you expect them to be the same?

(b) If there were 100 horizontal bands of color, calculate the total number of upward

sloping bands of color (assuming of course that all bands have the same height).

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Wall Drawing #260 by Sol LeWitt

Source for Photograph 3 and accompanying text

http://www.moma.org/collection/browse_results.php?object_id=79898

Photograph 3

Sol LeWitt. (American, 1928-2007). Wall Drawing #260, On Black Walls, All Two-Part

Combinations of White Arcs from Corners and Sides, and White Straight, Not-Straight,

and Broken Lines. 1975. Chalk on painted wall, Dimensions variable. Gift of an

anonymous donor. © 2008 Sol LeWitt/Artists Rights Society (ARS), New York

Question 3

(a) LeWitt used the 20 designs shown in the middle of Photograph 3 as building

blocks to create Wall Drawing #260. Read the above text and check that there are

exactly 20 such designs.

(b) Lewitt created Wall Drawing #260 by assembling the list of every way in which

two of the 20 designs can be selected. How many ways are there of choosing 2 of

the 20 designs?

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Occupancy by more than 582 persons is dangerous and unlawful

Sign posted in The Donald B. and Catherine C. Marron Atrium

Question 4

(a) In order to estimate the size of a crowd, police often work under the assumption

that on average a person occupies an area of two square feet in a crowd. Use this

figure to calculate the area of a space that could accommodate a crowd of 582

people.

(b) Decide if the area of The Donald B. and Catherine C. Marron Atrium would be

the same, more or less than the area of the space discussed in part (a).

(c) In a setting outside of MoMA, use 12 meter sticks to form a 3 meter by 3 meter

square. Have a group of people stand in the square as if they are in a crowded

area. Count the number of people in the square and calculate the average amount

of space occupied by a single person. Compare your result to part (a).

Turning Point >80< (Wende >80<) by Hanne Darboven

Source for Gallery label text and accompanying text

http://www.moma.org/collection/browse_results.php?object_id=71833

Gallery label text

Hanne Darboven. (German, born 1941). Turning Point >80< (Wende >80<). 1980-81.

Portfolio of 416 lithographs, offset printed, eleven long play vinyl records, plus one

duplicate lithograph, composition: 13 3/4 x 9 3/4" (34.9 x 24.8 cm); sheet: 13 3/4 x 9 3/4"

(34.9 x 24.8 cm). Publisher: the artist. Printer: Druck Sost & Co., Hamburg. Edition: 250.

Gift of Angela Westwater and Hanne Darboven. © 2008 Hanne Darboven

A pioneer of Conceptual art who also trained as a pianist, Darboven created a notational

visual language based primarily on numbers and indecipherable script. Here, for the first

time, musical notations dominate her project. The scores are derived from a set of

correspondences the artist established between numbers and notes; working with a

composer, she orchestrated and recorded them. Visitors can hear her compositions

alongside their visual equivalents.

There are no specific mathematical questions for this particular piece of art by Darboven,

but questions have been provided for "Evolution >86<", that Darboven created in 1986.

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Question 5

Read the following excerpt from an article posted at http://gassergrunert.net/test/?cat=23

For the first day of the year, Hanne Darboven postulates the following digit addition

equation: 1 + 1 + 8 + 6 = 16. This contrasts with 57 at the opposite end of the scale (31 +

12 + 8 + 6). All of the digit sums for this year are located somewhere between these two

poles. Although the artist has broken down the year into it’s two digits, she refrains from

doing this with the individual dates, an idiosyncrasy she has retained since the ‘60’s’.

(a) What is a digit sum? What is the digit sum for February 10, 1986?

(b) Why are all the digit sums for 1986 between 16 and 57 as claimed in the above

excerpt?

(c) For the year 1986, does every number between 16 and 57 correspond to a given

day of the year? Do any numbers get missed as being a digit sum?

(d) For the year 1986, which days have a digit sum of 50?

(e) What is the lowest and highest digit sums for the year 2008?

The Shapes Project by Allan McCollum

Source for text

http://www.moma.org/collection/browse_results.php?object_id=107106

Allan McCollum. (American, born 1944). The Shapes Project. 2006. Three sets of one

hundred forty-four unique framed digital prints, each: 5 7/8 x 4 5/16 x 3/8" (15 x 11 x 1

cm). Publisher and printer: the artist, New York. Edition: 54 sets of 144 works from a

projected 31 billion unique variants. John B. Turner Fund and Jacqueline Brody Fund. ©

2008 Allan McCollum

McCollum explores duplication in his art, investigating how objects achieve personal and

social significance in a society driven by mass production. Here he uses the software

program Adobe Illustrator with the goal of producing a unique artwork for every person

on Earth. His system has the capacity to produce thirty-one billion shapes, far more than

peak–population estimates for the mid–twenty–first century, when a decline is predicted

to set in.

Question 6

(a) What percentage of the entire collection of prints that McCollum intends to have

produced are on display at MoMA.

(b) How much wall space would be required if all variants were exhibited? Assume

that the framed prints are arranged with no space between them.

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(c) If all the variants were lined up side-by-side, would the collection wrap around

the earth? If they were stacked on top of each other, would the pile reach the

moon?

The SHAPES Project, 2005/06 by Allan McCollum

Source for Photograph 4 and the accompanying text

http://home.att.net/~amcnet2/album/shapes/intro.html

Photograph 4

Allan McCollum. The SHAPES Project, 2005/06. 7,056 SHAPES Monoprints, each

unique. Framed digital prints, 4.25 x 5.5 inches each. Installation: Friedrich Petzel

Gallery, New York, 2006.

Working over the past few years, I've designed a new system to produce unique two-

dimensional "shapes." This system allows me to make enough unique shapes for every

person on the planet to have one of their own. It also allows me to keep track of the

shapes, so as to insure that no two will ever be alike.

Following the present rate of birth, it is generally estimated that the world population will

"peak" sometime during the middle of the present century, and then possibly begin to

decline. How many people will be alive at this peak are estimated at between 8 billion

and 20 billion people, depending upon what factors are considered and who is doing the

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considering. The most recent estimate published by the United Nations puts the figure at

around 9.1 billion in the year 2050.

Question 7

(a) Suppose that this collection were set up in rows with an equal number of prints

per row (it is hard to tell from the photograph if this is really the case). Judging

by the photo and taking into account the total number of prints, determine the

number of rows and the number of prints per row.

(b) Use the above text to sketch a possible graph of the population of the world

versus time from now to the year 2060.

Macula series A&L by Tobias Putrih

Source for Photograph 5 and accompanying text.

http://www.moma.org/collection/browse_results.php?object_id=102487

Photograph 5

Tobias Putrih. (Slovenian, born 1972). Macula series A&L. 2005. Cardboard,

Dimensions variable. Fund for the Twenty-First Century. © 2008 Tobias Putrih

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Putrih, who studied physics before turning to art, draws inspiration from the pure

geometric forms of modernist architecture. But among the challenges he sets for himself

is "to make an object that expresses its own self–doubt, that questions its own existence."

Question 8

(a) This sculpture consists of a large number of pieces of cardboard (not all the same

size) stacked on top of each other. How could an estimate for the volume of this

three dimensional object be obtained from the pieces of cardboard?

(b) Suggest a way of obtaining a more accurate estimate for the volume of the object.

(c) Suppose that 100 circular sheets of cardboard, each 1 cm thick, were stacked to

form a cone whose height and radius are each 100 cm. Use the sheets of

cardboard to estimate the volume of the cone.

A, C And D From Group/And by Dorothea Rockburne

Source for Photograph 6 and accompanying text

http://www.moma.org/collection/browse_results.php?object_id=78588

Photograph 6

Dorothea Rockburne. (American, born Canada, 1932). A, C And D From Group/And.

1970. Paper, chipboard, nails, and graphite, Overall dimensions variable, approximately

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13' 10 1/2" x 21' 1/2" x 44 1/8" (422.8 x 641.3 x 112 cm). Given anonymously. © 2008

Dorothea Rockburne / Artists Rights Society (ARS), New York

At the experimental Black Mountain College, Rockburne was introduced to mathematics

by a professor who saw its principles as a defining force in nature. Mathematical ideas—

from set theory to the golden section—have been among her tools ever since. Discussing

her personal approach, Rockburne has said, "When I'm working well and am totally

focused, I feel as though the work is making itself." This piece, meant to be site–specific,

can be expanded or contracted through manipulation of the rolls of paper.

There are no specific mathematical questions for this particular piece of art by

Rockburne, but questions have been provided for two other pieces.

Golden Section Painting: Parallelogram with 2 Small Triangles by Dorothea Rockburne

Source for Photograph 7

http://www.dorothearockburne.com/

Click on the link for 1973-1978

Photograph 7

Question 9

(a) Where does the golden section appear in this painting?

(b) Without doing any calculations, estimate if the area of the six-sided figure is less,

more or equal to the combined area of the two triangles.

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(c) Measure the lengths of whatever segments you deem to be necessary to calculate

the area of the six-sided figure and the two triangles. Find the combined are of

the two triangles and compare it with the area of the six-sided figure. How does

this compare with your estimate made in part (b)?

Golden Section Painting: Triangle, Rectangle, Square by Dorothea Rockburne

Source for Photograph 8

http://www.dorothearockburne.com/

Click on the link for 1973-1978

Photograph 8

Question 10

Where does the golden section appear in this painting?

Golden Section Painting: Triangle, Rectangle, Square by Dorothea Rockburne

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Source for Photograph 9

http://www.2x4.org/

Photograph 9

Question 11

(a) For the purposes of this and questions that follow, imagine that the back and seat

of each chair consists of three straight pieces of plastic. Work right on the photo

and measure the lengths of the combined seats/backs for the five most recent

chairs at the end of the collection. How do these lengths change from one case to

the next? Do these lengths form an arithmetic or geometric sequence.

(b) Use your result from part (a) to estimate the length of the combined seat/back for

the next chair in this sequence.

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Constellation with Red Object by Alexander Calder

Source for Photograph 10 and accompanying text

http://www.moma.org/collection/browse_results.php?object_id=81834

Photograph 10

Alexander Calder. (American, 1898-1976). Constellation with Red Object. Roxbury,

Connecticut, 1943. Painted wood and steel wire, 24 1/2 x 15 1/4 x 9 1/2" (62.2 x 38.7 x

24.1 cm). James Thrall Soby Fund. © 2008 Estate of Alexander Calder / Artists Rights

Society (ARS), New York

Members of the NCTM who receive the journal Mathematics Teaching in the Middle

School will be interested in using the activities from the article Connecting the Mobiles

of Alexander Calder to Linear Equations in the April 2007 issue (Vol. 12, NO. 8).

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Places with no street by Mario Merz

Source for Photograph 11

http://www.flickr.com/photos/31345340%40N00/2404882258

Photograph 11

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Source for text

Mario Merz: Italian artist who used 'poor' materials and was fascinated by the geometry

of nature

The Guardian, Thursday November 13 2003

http://www.guardian.co.uk/news/2003/nov/13/guardianobituaries.italy

Mario Merz, who has died in his native Milan aged 78, was a leading member of the

Italian artistic movement known as Arte Povera. Merz and his colleagues, who included

his wife Marisa, used ordinary, "poor" materials, both natural and manufactured, to create

the most poetic, extraordinary effects. Their work gained international prominence in the

late 1960s, partly thanks to the efforts of the indefatigable critic Germano Celant. Merz

developed a repertoire of imagery on which he was to draw throughout the rest of his

career, culminating this year in the installation of his trademark neon tubes among the

ruins of the Imperial Forums in Rome.

Arranged in spiralling patterns derived from the Fibonacci number sequence, the lights

expressed Merz's almost mystical views of the universe's underlying structure - the

geometry governing both the natural and human worlds. By this time Merz had himself

become something of an artistic monument, notably as the subject of an atmospheric

film, Mario Merz (2002), by the British artist Tacita Dean.

Question 12

(a) Merz used 15 numbers in Places with no street. Only a few of them are visible in

Photograph 11. Use the information provided above to form a list of these 15

numbers.

(b) How many even numbers are there in this set of 15 numbers? If Merz had used

100 numbers, how many of them would have been even?

(c) House numbers on a street are usually assigned in such a way that the even

numbers are on one side and the odd numbers are on the other side. Would

Merz's 15 numbers be suitable for the addresses of 15 houses (all about the same

size) on a street? Why or why not?

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Spectrum Colors Arranged by Chance II by Ellsworth Kelly

Source for Photograph 12 and accompanying text

http://www.moma.org/collection/browse_results.php?object_id=37202

Photograph 12

Ellsworth Kelly. (American, born 1923). Spectrum Colors Arranged by Chance II. 1951.

Cut-and-pasted color-coated paper and pencil on four sheets of paper, 38 1/4 x 38 1/4"

(97.2 x 97.2 cm). Purchased with funds given by Jo Carole and Ronald S. Lauder. ©

2008 Ellsworth Kelly

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Question 13

Suppose that Kelly created Spectrum Colors Arranged by Chance II by rolling a pair of

dice and painting the individual squares according to the sum of the numbers showing on

the dice (i.e. black if the sum is a 2, red if the sum is a 3 and so on).

(a) For a 49 x 49 grid determine the total number of paintings that could be created

through the use of this method.

(b) If the dimensions of the grid were doubled, would the total number of paintings

that could be made also double in number?

(c) Suppose that you were to create a similar work by mentally selecting numbers at

random without the use of dice. Could someone with a good mathematical

background quickly be able to tell that you made up these numbers as you went

along without the use of a pair of dice? How?

Line 1000 Meters Long by Piero Manzoni

Source for Photograph 13 and accompanying text

http://www.moma.org/collection/browse_results.php?object_id=80955

Photograph 13

Piero Manzoni. (Italian, 1933-1963). Line 1000 Meters Long. July 24, 1961. Chrome-

plated metal drum containing a roll of paper with an ink line drawn along its 1000-meter

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length, 20 1/4" (51.2 cm) x 15 3/8" (38.8 cm) diameter. Gift of Fratelli Fabbri Editori and

purchase. © 2008 Artists Rights Society (ARS), New York / SIAE, Rome

Question 14

Source for text

http://en.wikipedia.org/wiki/Paper_density

The paper density of a type of paper or cardboard is the mass of the product per unit of

area. Two ways of expressing paper density are commonly used:

Expressed in grams per square metre g/m2 , paper density is also known as grammage.

This is the measure used in most parts of the world. Expressed in terms of the mass (in

pounds) of a ream of 500 (or in some cases 1000) sheets of a given (raw, still uncut) basis

size, paper density is known as basis weight. The base size and area used here depends on

the product type. This convention is used in the United States, and (to a lesser degree) in

a very small number of other countries that use United States paper sizes. Japanese paper

is expressed as the weight in kg of 1000 sheets.

Typical office paper has 80 g/m2 .

(a) If Piero Manzoni used typical office paper for his work Line 1000 Meters Long,

how much would the paper weigh?

(b) Piero Manzoni has created other works of art similar to this one with different

lengths of paper. Find the diameter of the chrome-plated metal drum that could

house a similar work called Line 2000 Meters Long.

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Campbell's Soup Cans by Andy Warhol

Source for Photograph 14 and Gallery text label

http://www.moma.org/collection/browse_results.php?object_id=79809

Photograph 14

Andy Warhol. (American, 1928-1987). Campbell's Soup Cans. 1962. Synthetic polymer

paint on thirty-two canvases, Each canvas 20 x 16" (50.8 x 40.6 cm). Gift of Irving Blum;

Nelson A. Rockefeller Bequest, gift of Mr. and Mrs. William A. M. Burden, Abby

Aldrich Rockefeller Fund, gift of Nina and Gordon Bunshaft in honor of Henry Moore,

Lillie P. Bliss Bequest, Philip Johnson Fund, Frances Keech Bequest, gift of Mrs. Bliss

Parkinson, and Florence B. Wesley Bequest (all by exchange). © 2008 Andy Warhol

Foundation / ARS, NY / TM Licensed by Campbell's Soup Co. All rights reserved.

Gallery label text, 2006

When Warhol first exhibited these thirty–two canvases in 1962, each one simultaneously

hung from the wall like a painting and rested on a shelf like groceries in a store. The

number of canvases corresponds to the varieties of soup then sold by the Campbell Soup

Company. Warhol assigned a different flavor to each painting, referring to a product list

supplied by Campbell's. There is no evidence that Warhol envisioned the canvases in a

particular sequence. Here, they are arranged in rows that reflect the chronological order

in which they were introduced, beginning with "Tomato" in the upper left, which debuted

in 1897.

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Question 15

(a) Given that Warhol was not concerned about the order of the canvases, how many

different ways are there of displaying them in the 8 x 4 grid?

(b) Use the size of the canvases provided below Photograph 14 to estimate the size of

one of the painted cans of soup. If this painted can were manufactured and filled

with soup, how much would it weigh? How much would it cost?

NOTE: In order to answer this question you will need to measure the radius and

height of an actual can of Campbell's soup. If such a can is not in your home,

visit a grocery store and bring a tape measure. You will also need to know the

price of the can (which will of course vary somewhat from one store to the next).

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Fez (2) by Frank Stella

Source for Photograph 15 and accompanying text

http://www.moma.org/collection/browse_results.php?criteria=O%3AAD%3AE%3A5640

&page_number=3&template_id=1&sort_order=1

Photograph 15

Frank Stella. (American, born 1936). Fez (2). 1964. Fluorescent alkyd on canvas, 6' 7

1/8" x 6' 7 1/8" (195.6 x 195.6 cm). Gift of Lita Hornick. © 2008 Frank Stella / Artists

Rights Society (ARS), New York

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Question 16

Figure 4 shows a drawing of the upper left square of Fez (2).

(a) Let a1 be the area of triangle ABF, a2 be the area of trapezoid BCGF, a3 be the

area of trapezoid CDHG, a4 be the area of trapezoid DEIH, ... , a17 be the area of

trapezoid JKQR, a18 be the area of trapezoid KLPQ, a19 be the area of trapezoid

LMOP, and a20 be the area of triangle MNO. Find a1 , a2 , a3 , ... a20 .

(b) Graph an versus n . What types of functions might be used to generate this

graph?

(c) Find an equation for an in terms of n .

n

(d) Let tn ai . Find a formula for tn in terms of n and then graph tn versus n .

i 1

FGH I

A

B

C

D

E

R

Q

P

O

N

J K LM

Figure 4

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1/2 W Series (1968) by Robert Mangold

Source for Photograph 16 and accompanying text

http://www.moma.org/collection/browse_results.php?criteria=O%3AAD%3AE%3A3723

&page_number=1&template_id=1&sort_order=1

Photograph 16

Robert Mangold. (American, born 1937). 1/2 W Series. 1968. Synthetic polymer paint on

composition board, in two parts, Overall 48 1/4" x 8' 1/2" (122.5 x 245.1 cm). Larry

Aldrich Foundation Fund. © 2008 Robert Mangold/Artists Rights Society (ARS), New

York

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Question 17

Figure 5 shows a drawing of Mangold's work.

AG F E D

BC

Figure 5

(a) Let F be the center of the semicircle with AD as the diameter. Let G and E be the

midpoints of AF and FD respectively. Decide through estimation which of the

following is closest to the combined area of triangles AFB and FDC expressed as

a percentage of the area of the semicircle.

(i) 0% (ii) 10%

(iii) 20% (iv) 30%

(v) 40% (vi) 50%

(vii) 60% (viii) 70%

(ix) 80% (x) 90%

(xi) 100%

(b) Calculate the combined area of triangles AFB and FDC and the area of the

semicircle. Express the combined area of triangles AFB and FDC as a percentage

of the area of the semicircle. Compare this answer with your estimate from part

(a).

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Vinculum II by Eva Hesse

Source for Photograph 17

http://www.insecula.com/oeuvre/O0026316.html

Source for accompanying text

http://www.moma.org/collection/browse_results.php?criteria=O%3AAD%3AE%3A2623

&page_number=16&template_id=1&sort_order=1

Question 18

(a) What types of functions might be used to generate this graph?

(b) Set up a coordinate system directly on Photograph 18 provided on the next page,

place a number of points along the curve, then estimate the coordinates of these

points. Find an equation that passes through these points.

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Photograph 17

Eva Hesse. (American, born Germany. 1936-1970). Vinculum II. 1969. Latex on metal

screening stapled to wire with vinyl tubing, Overall 9' 9" x 2 1/2" x 9' 7 1/2" (297.2 x 5.8

x 293.5 cm). The Gilman Foundation Fund. © 2008 Estate of Eva Hesse. Galerie Hauser

& Wirth, Zurich

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Movement in Squares by Bridget Riley

Source for Photograph 18

http://www.mishabittleston.com/artists/bridget_riley/

Question 19

(a) Starting with the square in the upper left corner of the painting and ending with

the square in the upper right corner, measure directly on the photo the widths of

squares in the top row. How do these widths change from one square to the next?

Do the widths form an increasing then decreasing arithmetic sequence or

geometric one?

(b) Make a version of Movement of Squares using your own sequence of widths and

see if you can create one that has the same physical effect on the eye.

Photograph 18

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Bibliography

Websites

The National Math Trail

www.nationalmathtrail.org

The Canadian Math Trail

http://www.brocku.ca/cmt/English/index.htm

Vancouver 2010 Olympic Math Trail

http://www.brocku.ca/cmt/upload/1070004370.8764/index.html

Math Trails

http://www.cmste.uregina.ca/MathTrails/

Math Trails

http://www.saskschools.ca/~otl_el/grassroots/mathtrail/index.html

Welcome to the Welland Canal Math Trail

http://spartan.ac.brocku.ca/~emuller/mathtrail/wcmt/wcmtstart.html

NCTM Journal Articles

"Designing Math Trails for the Elementary School". Kim Margaret Richardson, Teaching

Children Mathematics, August 2004

"Making Mathematics Real: The Boston Math Trail". M Rosenthal and Clement K

Ampadu, Mathematics Teaching in the Middle School, November 1999

"A Mathematics Trail at Phillips Exeter Academy". Vince Delisi and Ron Lancaster, The

Mathematics Teacher, March 1997

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Tales about my Trails by Ron Lancaster

I created my first Math Trail on Toronto's Centre Island in the summer of 1985 for

teachers who were enrolled in an Additional Qualifications (AQ) course I taught at the

Faculty of Education of the University of Toronto. The suggestion to go to Centre Island

came from Al Fleming, a well known mathematics teacher, who had developed a

tradition of going to the island for the last class of his AQ courses. In the early 1990s I

began to create Trails for teachers who attended the Phillips Exeter Academy

Mathematics, Science and Technology Conference in Exeter, New Hampshire. I

developed these Trails with Vince Delisi, a good friend of mine, who now works for

Texas Instruments. Vince and I created Trails on the campus of the school, in the

downtown area and in stores including Whirlygigs Toys and Exeter Candy.

In 1996 I began to develop Trails for my own students. At the time I was teaching at St.

Mildred’s, an all-girls school in Oakville, and I was looking for new ways of making the

study of mathematics more attractive to the young women in my classes. The first Trail

that I created was set in the Financial District of Toronto and the day was a major

success. The students loved the active nature of the project, they enjoyed being able to

work collaboratively and they were stunned at how much mathematics there is in the

world around us. The following year my grades 7 and 8 students participated in three

Trails. The first Trail took place in Toronto and I used it as a way of getting to know my

students and as a way for my students to be exposed to the big ideas of the course. The

second Trail was on the campus of the school and I used it to help students practice what

they had learned to date. The third Trail was also in Toronto and this one was used to

help students review everything we had done during the year. The third Trail also

contained questions from the curriculum for the next year in an effort to give students a

sense for what was to come.

Prior to joining the faculty of OISE I worked as an independent mathematics consultant

for six years and through my work I ended up creating Trails in many locations

throughout North America and Asia. The settings for my Trails have literally been all

over the map and have included shopping malls, parks, art museums, schools, farms,

downtown areas of big and small cities and even in a church in Kilgore Texas. Many of

the teachers I have worked with have created their own Trails and several have used the

idea of a Trail as the basis of their Masters thesis. By now I have seen a tremendous

amount of evidence that the idea works well for students in all grades, at all levels of

learning and in all settings (co-ed, all-boys, all-girls schools).

I would be delighted to discuss any issues with you regarding the logistics of organizing a

Trail along with the assessment of student work. With regard to logistics, I have a few

points to share with you. Set up the Trail so that it forms a large circuit with the end of

the Trail near the start. Put your students into very small groups (three or four students

per group) and have the groups start the Trail in different locations. If they all move in

the same direction around the circuit, there will never be a point when too many groups

are in the same place at the same time. For supervision, ask parents and other teachers to

join you. The best idea of all is to involve senior students as the guides and mentors. If

you are designing a Trail for students in say grade 9, have students from grade 12 help

out. Be sure to check with officials at every site on the Trail. Ask for permission to be

on the property and make sure that everyone knows what is going on.

35

There have been other people around the world who have developed their own Trails.

Kay Toliver is one of the best known Trail blazers and information about her work can be

found at www.nationalmathtrail.com. Several years ago I worked with Kay in Los

Angeles and New Orleans and she was a pleasure to be with. Geoff Kavanagh, an

Instructor at OISE, was a real pioneer in the development of Math Trails and he was out

with his students in downtown Toronto many years ago. Eric Muller, a Mathematics

Professor at Brock University in Canada was another early Trail blazer. Many years ago

he developed Math Trails in Niagara Falls and at the Welland Canal.

Ron Lancaster

Lecturer in Mathematics Education, OISE/UT