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Beyond Numbers: Why study math?
Where is math apart from math class and the minds of crazy geniuses?

 Heather Jenkins 

Academic Setting 

School Setting 

Harrison Middle School is a public school located in a semi-rural region of Albuquerque. Although the region is designated semi-rural, the school and its population share many of the characteristics of inner-city schools. The majority of the student population belongs to a low level of the socio-economic echelon; during the 1999-2000 school year, 89% of the students qualified for free and reduced meals. Scores on standardized tests are low, particularly in the areas of math and literacy. Enrollment for the 2000-20001 school year was 672. Ethnicity statistics show that of those students, 85.1% are Hispanic, 10.3% Anglo, 3.3% Native American, .7% African American, 0.3% Asian and 0.3% other. Many of the students are labeled PHLOTE (primary home language other than English) and qualify to receive bilingual services.  Because the school has failed to meet the state board of education’s standards for student achievement, Harrison has been deemed a school-in-need-of-improvement (SINOI), and is on probationary status. In order not be taken over by the state, the school must implement a plan for student success and demonstrate improvement within two years. 

To reach its goals of higher levels of literacy and mathematical proficiency for students, the school has implemented a balanced literacy program, mandatory for all students. As of the 2001-2002 school year, Harrison also switched to an alternating block schedule, in which most classes meet every other day for 108 minutes at a time. Two years ago, the math department adopted the Math in Context (MiC) program, which is inquiry-driven rather than skill based.  

            The MiC program has been met, both by teachers and students, with mixed reactions. Although teachers appreciate the problem-solving aspects of the program, because of our students’ low reading skills and lack of exposure to math as problem solving, students have not yet had great success with the program. Also, it is often necessary to tailor the program to our students’ needs, as many of the references and much of the content have to do with experiences unfamiliar to our students. As a bilingual math teacher, a particular challenge to me has been to present such language-based math in an accessible, meaningful way. 

Class Setting 

During the 2002-2003 school year, I will be teaching 2 blocks each of sixth grade bilingual math and seventh grade bilingual math. The students’ levels of language proficiency as well as mathematical abilities vary along a wide spectrum. Some students are lacking basic skills while others desperately need challenges. Typically, my students are generally quite unenthusiastic about math. Often parents expect their children to be working on basic, rote skills rather than out of problem-based texts with which they themselves are unfamiliar. My students tend to be quite social in nature, as are most middle school students. Although they are all capable, many students already decide by the time they enter middle school that math is something they cannot “get” or have no reason to learn.  

For whatever reason, math seems to be viewed as something different from other subjects learned in school. While it is not commonly heard that a person has “social studies anxiety” or “science phobia” those phrases are often heard with regard to math, and my students have been no exception. Many of those students who do not fear math often, regrettably, view it as useless in their lives. Indeed, we probably do not actively, directly utilize much of the knowledge we learned as students in the formal educational system. Is that, however, sufficient rationale for not learning any of? Might learning itself—regardless of utility—be one of the aims of education? I would not presume to answer for all people that never-ending debate about the philosophy of education but, as far as mathematics education is concerned, I side with Harold Jacobs (xii) in that “the significance of mathematics does not rest only on its practical value…mathematics has its own kind of beauty and appeal to those who are willing to look.” 

Goals and Objectives 

The goal of this unit, therefore, is not to attempt to solve all problems regarding math education but to expose students to math as another means, much like social studies or science, of learning about the world. Like other subjects, math is rooted in social contexts, although it is not often presented to students in such a manner. I want my students to see that math is not just about manipulating numbers in ways created by evil deities, but, rather, that it is deeply embedded in many aspects of life in ways students might not have realized. Through problem-solving and hands-on activities, students will investigate math as an aspect of culture, as a way of thinking and organizing the world, as molding our perceptions of beauty, as present in nature and as a means of entertainment. Math extends beyond numbers and arithmetic and, with any success, students will find that sometimes people do math just because they can and want, occasionally without even realizing it. 

            Stated in its simplest form, the objective of the unit is for students to experience math as it occurs in the world. They will develop a deeper understanding of math as a way of perceiving of and existing in the world. While I do expect students to achieve more concrete objectives, most crucially I hope they will gain a greater appreciation for math and its manifestations in the world. More specific objectives are included with each lesson. 

Context and Background 

Math is often presented to young (and not so young) students as though it is an objective, fundamental truth of life whose nature and origins are not to be questioned. Such, however, is not the case. “What we term mathematical ideas are, in every instance, part of the web that constitutes a culture” (Ascher 186). Whether or not we believe mathematics to exist independent of mankind, it is seen and expressed through the work of people and is, therefore, a cultural endeavor. Just as staples of food differ among cultures, so do staples of mathematics. What may be emphasized in one culture has little significance in another. 

Even the number system, a mathematical idea so intrinsic to our way of viewing the world that there is a tendency to view it as an undeniable certainty, has been (and probably still is) tailored to the needs and beliefs of different cultures. Numerous number systems exist in order to meet the needs of the cultures in which they originate. 

The ancient Egyptians, for example, used a hieroglyphic system of numerals with a base of ten and powers of ten (Zaslavsky 75). The hieroglyphics used depicted objects and beings, such as frogs, familiar in their world. Quantities could be represented without concern for order, as the Egyptian system did not include place value. To write the number 432, for example, nine symbols were used: four hundred symbols, three ten and two one symbols. The Egyptians had more than one system of fractions, according to the Ah-mose papyrus, from which stems much of our knowledge about ancient Egyptian math. Fractions were denoted by placing the symbol of a mouth, which meant “part,” above the quantity. Although the Egyptians used a system of  “unit fractions,” in which the numerator was almost always one, they also had separate hieroglyphs for the quantities of 2/3 and ¾ (O’Connor and Robertson).  Most fractions were written as a combination of other fractions, rather than given a new hieroglyph. For instance, 5/8 was depicted as ½ and 1/8. Grain, because of its importance to the Egyptians, had its own system of fractions, based on the power of two Eye of Horus hieroglyphics.  Though legends of the origin vary, the six hieroglyphs for ½, ¼…1/64 represent six different parts of the eye of Horus, the falcon god.  

Unlike the Egyptian hieroglyphic system, the Maya of Mesoamerica used a number system based on fives and twenties, a vigesimal system, that did include place value, with the smallest value represented at the bottom of the column. The base of twenty and importance of five presumably arose from the act of counting on fingers and toes (O’Connor and Robertson). Their system used three symbols: a bar, a dot and a symbol resembling a shell. The symbols may have grown from their use of sticks and pebbles for smaller calculations. With those three symbols, the Maya were able to do complex astronomical calculations (though their procedures are not known) and to record significant events in their history.  

The ancient Hebrew number system provides a clear illustration of how intertwined people’s beliefs are with the mathematics they use. Both ancient Hebrew and Greek numerals were represented by the letters of their alphabets. The first ten letters stood for the numbers one through ten, respectively, and then successive letters represented multiples of tens and hundreds. In Hebrew, however, an alternate representation for the number fifteen originated out of a religious restriction. Ordinarily, fifteen would have been written using together the letters yod for ten and heh for five, but, because that spelled the name of Jehovah—a forbidden name—fifteen was written with the letters for nine and six together (Zaslavsky 86). Thus, at least for ancient Jews, mathematical concepts clearly did not find expression clear of human touch.  

As the ancient Egyptians used fractions for grain measurement, the Mayans for astronomical calculations and the ancient Jews for numerology or Gematria, mathematical ideas are used as ways of thinking about and organizing the world. “Math is first and foremost a way of thinking, rather than a body of facts” (Burns 3).  If energy tends toward chaos, mathematics is an endeavor at times employed to reign in and make sense of that chaos. That said, in contrast to what is frequently implied by our educational system, all of math does not necessarily have to be approached in a linear, sequential manner of thinking. “Mathematics is a living, breathing, changing organism with many facets to its personality. It is creative, powerful, and even artistic. Mathematics uses penetrating techniques of thought that we can all use to solve problems, analyze situations, and sharpen the way we look at our world” (Burger and Starbird x). 

Much like the old assertion that the world was flat and could not be round, until recently it was commonly believed that Euclidean, planar geometry was the true geometry describing spatial relations. Euclid’s axioms, postulates and theorems defined and prescribed order in the spatial world. Mathematicians did not dare contradict Euclidean geometry, as it was so ingrained and revered in society. In The Mathematical Experience, Davis and Hersh refer to this mathematical ideal as “the Euclid myth.” Using the axiomatic methods of Euclid meant knowing logic and having the ability to reason. Though Euclidean geometry is still valid and thriving, it is not anymore the one and only way of understanding space. 

            Euclid’s fifth postulate, having to do with parallelism, was always subject to some degree of scrutiny. The fifth postulate states: “If a straight line falling on two straight lines makes the interior angles on the same side together less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than two right angles” (Eves 34). Lacking self-evidence, it reads much more like a theorem in need of proof than a postulate one naturally assumes to be true, and is much lengthier and more cumbersome than Euclid’s other four postulates. Although the word parallel is absent, the postulate can be restated to read that through a given point, with respect to a given line, there exists only one line parallel to the given line. Unease with the fifth postulate led to much research and, eventually, to new mathematical and world perspectives.    

In the former half of the eighteenth century Friedrich Gauss and the slightly younger Johann Bolyai did “conceive of a non-Euclidean geometry” (Eves 62) but their discoveries did not garner a great deal of attention, and not until 1829-1830 was a methodical paper published on the subject by Lobachevsky, a Russian mathematician. Still, in his day, despite publishing his work in several languages, Lobachevsky’s brand of non-Euclidean geometry—known as hyperbolic—did not receive much recognition. Unlike Euclidean, planar geometry, hyperbolic geometry occurs in spaces with negative curvatures, such as the seat of a riding saddle (Burger and Starbird 296). In such spaces Euclid’s fifth postulate does not hold true, as there are at least two parallel lines through a point with respect to a given line.  

In 1854 Bernhard Riemann presented a paper on a second type —elliptic—of non-Euclidean geometry which received a wider audience and has served as the basis for a great deal of research. In spaces where there is positive curvature, there is elliptic geometry. Elliptical geometry differs from planar and hyperbolic, regarding the fifth postulate, in that there are no parallel lines through a given point with respect to a given line. The discovery of elliptical (also called spherical) and hyperbolic geometries changed our understanding of space and opened up other ways of viewing the world and, literally, the universe. Indeed, with hyperbolic and elliptic geometries among their tools, scientists are attempting to discover the shape of the universe. 

         Inextricably intertwined with how we make sense of the world, perhaps another reason that math is a mandatory subject in our educational system is that mathematical ideas affect our perceptions of beauty and art. Western culture seems to perceive as beautiful that which is symmetric and ordered. Many cultures, in fact, incorporate symmetry into their aesthetics and, arguably, objects and artifacts can sometimes be associated with a particular culture by the symmetry patterns they display. 

The Maori, for example, have master craftsmen whose responsibility it is to carve strip patterns on the walls and ceilings of marae, or meeting houses. Following traditional observances, the carvers use no sketches or blueprints for their designs, yet carve intricate symmetries often involving scrolls. It is likely that the structures of the marae reflect structures of the Maori belief system; complementarity (differences uniting pairs) and symmetry are underlying features of the social system (Ascher 171).  

For the Incas as well, symmetry and precision played a significant role in the adornment of buildings and objects, and particularly of pottery. Unlike in the Maori strip patterns, color in Inca pottery tended to follow the symmetry of the design rather than itself creating a new symmetry. Ascher (180) states that “many terms [such as methodical, conservative, concerned with detail and precision of fit] that have been used to describe the Inca are exemplified by their strip patterns.” Most likely neither the Inca nor the Maori used (or use) our Western system of thinking about or classifying their respective strip patterns, yet in some terms—mathematical or not—most cultures display an appreciation for the concept of symmetry and pattern. 

Nature too has an abundance of mathematical displays and, in fact, often capitalizes on mathematical concepts. Bees, for example, long ago mastered a sort of mathematical efficiency without, no doubt, struggling over theorems or calculations. Painstakingly arranging wax to form hexagonally structured honeycombs, bees minimize labor by creating regions of equal area with minimal total perimeter (Peterson 60). Not only is this arrangement important for bees, but it has implications for humans as well. From more mundane applications such as the arrangement of fruit at the grocery store, the “honeycomb conjecture” recently proven by mathematician Thomas Hales may also be of interest to researchers of the behavior of, according to Peterson, fluids, bubbles, foams, crystals and other biological structures.  

            Also of much interest recently is the subject of fractals. Seeming to merge or blur, depending on one’s perspective, the line between chaos and order, fractals are sets that exhibit self-similarity at all scales (Burger and Starbird 430). According to “the father of fractals,” Benoit Mandelbrot, the definition of a fractal is a shape, such as that of a cloud or tree, in which “small parts are the same as big parts” (Goldsmith). Mandelbrot began studying what later became known as fractals around the middle of the twentieth century while working at IBM. He attributes the fruition of his mathematical ideas not only to his antecedents and vision, but, also, to the advent of computers. Without computers, he believes, there would be no tool to exactingly create or study fractals.  

Mathematical computations notwithstanding, fractals can be formed by creating what Burger and Starbird (435) refer to as “repeated-image collages.” One selects an image, reproduces reduced versions of that image, positions the reproduced images in specified locations, and continues the process infinitely. Appropriately, Mandelbrot coined the term fractal from the Latin word fractus, meaning broken, and its counterpart, the verb frangere, meaning to break (Burger and Starbird 428). 

            “Broken” is probably not the first adjective one thinks of when looking at broccoli but, nevertheless, it is a commonly cited example of fractal geometry in nature. Each floret is similar to each stalk, which is similar to the head, etc. Ferns, clouds, and coastlines are other naturally occurring fractals. Though fractal geometry is (and has been) prevalent in nature, it is only recently that its applications are being recognized. Researchers are applying fractal geometry to topics as diverse as population growth, the stock market, DNA, metabolic rates and antenna design (Musser 84). According to an article in U.S. News and World Report, even “evolution appears to have put a fractal stamp on biology, right down to its DNA, that mimics the chaotic physical processes that shape the Earth, weather and stars.” 

Despite the omnipresence of mathematical phenomena and ideas in the world, one aspect often overlooked (especially by “math phobics” and perhaps middle school students) is that math forms the basis of many forms of entertainment (and perhaps addictions). The field of probability, in fact, owes its origins to games of chance, which people have been playing for many centuries. One of those people, who lived in the seventeenth century and was a catalyst for the field, was a French dice player named Antoine Gombauld. Driven by curiosity about betting outcomes, he wrote to Blaise Pascal about the problem. Pascal then conferred with Pierre de Fermat. Together they answered Gombauld’s questions and thus the field of probability theory was born (Burger and Starbird 524). They were not the first to delve into the field though, as Geralamo Cardano wrote The Book on Games of Chance, an early book about probability theory, in the sixteenth century (Jacobs 448). Until the beginning of the nineteenth century, the focus of probability theory was on games of chance. Pierre de Laplace, however, began using probability theory to study other sorts of problems and today its applications are widespread (Apostol).  

Like probability theory, much of mathematical inquiry arises from curiosity and observation. Often, newly discovered branches of math are thought to have little use, but later prove to be quite applicable to real world situations. Fortunately, math is often useful, but regardless of utility, it is ubiquitous in our lives. Without need for complex calculations or theorems but only an open mind, sometimes it is even fun and beautiful too. 

Implementation

The lessons in this unit are not necessarily intended to form a single, comprehensive unit; rather, they may be presented as an introduction to some of the ways math exists in the world or they may be used as supplementary material to pertinent lessons. While covering diverse topics, the lessons also may form one cohesive of which the unifying theme is that math extends beyond numbers and beyond the classroom. The order of the lessons is somewhat arbitrary, although the lesson on symmetry and strip patterns allows the teacher an opportunity to get to know the students’ perceptions of themselves and may be more beneficial early in the year. Though written for middle school students, the lessons may be adapted to different grade levels.

            CS and PS, respectively, denote the Albuquerque Public Schools (APS) content standards and performance standards (from grades six and seven) met by each lesson.

Lesson Plans

Lesson 1: Symmetry— math as part of culture and ideas of beauty (2 blocks)

CS: The student demonstrates an understanding of concepts, properties, and relationships of geometry and measurement through experiences with meaningful mathematical problems that focus on identifying, describing, classifying, visualizing, comparing, estimating, and measuring various aspects of shapes and objects.

PS: Identifies and explains the reflectional symmetry (mirror image) of familiar objects.

Objectives: Students will understand and apply vocabulary of symmetry.
            Students will construct strip patterns.
            Students will analyze connections between math (symmetry) and culture. 

Materials: examples of strip patterns from various cultures, grid paper for constructing strip patterns (optional), art materials for strip patterns (optional) 

Procedure:

-Journal/ Essay question: Is math the same everywhere around the world? Explain your thinking. (Think about how and why people use math.)
-Discuss students’ answers and reasoning.
-Introduce idea that math may be present where we do not always think of it. Show various examples of strip patterns/symmetry in art/artifacts from around the world. Do not yet formally discuss symmetry until students have had an opportunity to discuss patterns that they see.
-Introduce students to terms symmetry, rotation, reflection, translations, and motif.
-Have students complete exercises (working with letters of alphabet) to ensure understanding and application of terms.
-Have students revisit the strip patterns from earlier and, after modeling, have them address the following two statements for each different group of patterns:

1)      Describe mathematically how the pattern is constructed. (Ex. The motif is rotated 90°.)

2)      Describe, based on your own experiences and ideas, what you think the patterns convey about the people who created them. (For example, are they a colorful, complicated group of people? Do they value nature and order? Is there a particular shape that is important to them?) Be careful to stress that this only speculation, and discussion should be positive.

-Discuss students’ responses, particularly to the second question.
-Have students create their own strip patterns, representative of themselves. (Explain that perhaps they should choose motifs representative of their beliefs or interests.) They should also answer the above two questions about their own patterns.
-If class is willing to share, students’ strip patterns could be displayed anonymously and class could try to guess the creators of each pattern.

Assessment: Students’ answers to questions and own symmetry creations. 

Lesson 2: Logic—math as a way of thinking/organizing (1-2 blocks) 

CS: The student understands and uses mathematical processes.

PS: Develops resourcefulness and perseverance in problem solving in mathematics and other disciplines.

Works in teams to share ideas, to develop and coordinate group approaches to problems, and to share from each other in communicating findings.

Communicates mathematical thinking coherently and clearly to others.

Objectives: Students will work cooperatively to solve problems.
Students will present their strategies for problem solving.
Students will analyze the effectiveness of different strategies.

Materials: laminated logic puzzles (one set per group of four) such as those in the book Háganlo Juntos or its English equivalent, Get It Together 

Procedures:

Journal question/starter: Does math require you to think a lot? If so, in what ways? How do you solve mathematical problems? Explain.

-Instruct students that they will be using problem-solving skills to solve puzzles. Explain to students that they need to record the processes they use to solve the puzzles as well as their solutions, which they will be presenting to the class.
-Pass out each group’s puzzle/problem.
-Each student should read his/her own piece of puzzle.
-When finished, have students present puzzles and solutions. Depending on time, groups might also exchange puzzles and then, when presenting, compare strategies.
-Discuss strategies and explain to students the importance of logic in math.

Assessment: Presentations of group work.

Lesson 3: Honeycombs, Tessellations, and Perimeter—math as a presence in nature
(1 block)

CS: See lesson 1.

PS: Develops and tests strategies for finding perimeters and areas.

Draws and explains congruent 2-dimensional figures using mathematical terminology.

Objectives: Students will identify and review polygons.

Students will construct tessellations.
            Students will evaluate predictions about perimeter of tessellations.
            Students will justify their ideas for which polygon is most efficient.

Materials: cardstock (or other sturdy material) cutout polygons—all of which have sides of the same length (1 in.) for tracing, pre-made empty grids of 3in. by 3 in. (several for each student), colored pencils

Procedures:

-Journal question/starter: What are some of the ways you see math in nature?
-Discuss students’ responses to journal question.
-Ask students why they think bees make their nests the way they do. (Make sure everyone knows that honeycombs are hexagonally based, whether or not they know the word hexagon.) Encourage them to think also about how sodas and fruit are packed at the store.
-Ask students to name as many polygons (explain the term if necessary) as they can, and have students draw and label the shapes on the board for everyone’s benefit. (Students could race in teams to make that more fun.) If students did not already think of or know them, add to list pentagon, hexagon, octagon, decagon, dodecagon, etc.
-Practice vocabulary with students by holding up a shape and having them name it.
-Explain to students what a tessellation is, and model one with a square.
-Ask students to predict which of the other listed shapes will tessellate, and record their answers for later discussion.
-Distribute cut-out shapes and have students attempt to tessellate them –in pencil—by tracing in their pre-made grids, starting at one corner. If the shape does tessellate, have them fill in the entire square grid.
-Discuss results.
-Have students make a table with one column representing polygons that actually tessellated and the other column representing the perimeter of the grid.
-Explain to students that they should imagine that they are insects and must construct a home using the least material possible. Their homes can be constructed of any one of the polygons that tessellates. Now they must decide which polygon to use so they will “re-construct” (trace again with colored pencil) their tessellation in the grid, this time keeping track of how much material they have used. Once they have used a side, they do not need to count it again as new material. (This will be much easier if the sides of the polygons are something easily measurable and countable.) Model this for the students, using the same polygon as before.
-Have students complete their tables. Instruct them to use colored pencils to mark a side once they have used it. (This way they’re just re-using the tessellations they already made.)
-Discuss results.
- Have students write an essay or paragraph using the following prompt: If you had to build a house with many rooms, but you had to use the least amount of material possible (because the house is for a crazy math teacher who likes bees) what shape would you make the rooms, and why? Explain with as much detail as possible, including why you wouldn’t use certain shapes. You may include illustrations.

Assessment: Discussions, tables and students’ writing.

Lesson 4: Fractals—math in nature and as art (1- 2 blocks)

CS: The student understands and uses mathematical processes.

PS: Develops a logical sequence of arguments leading to a valid conclusion or solution to a problem (statement/reasons, proof, informal proof and algebraic steps).

Uses representation to model and interpret physical, social, and mathematical phenomena.

Objectives: Students will construct a fractal, the Koch snowflake.
            Students will create original fractals.
            Students will write instructions for making their fractals.

Materials: pictures of and/or models of fractals (broccoli, ferns, coast lines, the Mandelbrot set, the Sierpinski triangle, etc.), triangular grid paper, overhead with grids for modeling Sierpinski triangle, overhead of triangular grid paper, other grid paper, Quaker Oats container, mirrors, colored pencils

Procedure:
-Journal/starter question: (Show students a Quaker Oats container.) What does this container have to do with math? Describe all the ways you can think of.
-Show class other examples of fractals and have them attempt to deduce what the examples have in common. Guide a discussion.
-Introduce the terms self-similarity, iteration, scale, and fractal, using the examples to explain.
-If available and appropriate, have students take turns facing two mirrors toward each other and looking at the fractal images produced.
-Explain to students that there are many ways to construct fractals. Show students a black colored in grid (stage or iteration 0) of the Sierpinski triangle, and a later stage of the triangle. Ask students if they can imagine a way to get from the black square to the triangle. Model the construction of the triangle on the overhead, using grids. Remember to reinforce vocabulary.
-Explain that students will now be constructing another famous fractal, the Koch snowflake. Distribute triangular grid paper. Instruct students to use a colored pencil to construct a 9 by 9 triangle near the top of their paper for the first iteration. Then instruct students to use a different colored pencil to cut each side of that triangle into three pieces, with the middle piece being another triangle. (They should now have a six-sided star). Begin students on the next iteration, using a different color, which is to cut each new side of the triangle into three pieces. Model a few sides and then let students finish. Encourage them to go through five iterations, using a different color for each.
-Discuss any difficulties and observations. Perhaps, depending on students’ level, ask students if they think the snowflake is growing much larger, and discuss area and perimeter. Make sure that everyone understands what a fractal is.
-Explain to students that now they are going to construct their own, original fractals. If necessary, model again. Have various types of grid and dot paper available. Instruct students to name their fractals and to write clear instructions for making them, incorporating the vocabulary they have learned.
-If student are interested, they could attempt to use each other’s instructions to construct other fractals.

Assessment: Koch snowflakes, students’ fractals and instructions.

Lesson 5: Poison—math as entertainment (1 block)
CS: The student understands and uses mathematical processes.
PS: Develops and uses strategies for solving given problems.
Monitors, discusses, and reflects on the process of problem solving.

Objectives: Students will play the game poison.
            Students will deduct and write winning strategies for poison.

 Materials: none 

Procedure:
-Journal question/starter: Describe your favorite non-sport games. Why do you like those games? What strategies do you use while playing them?
-Challenge class to a game. Explain that it will be the students’ job to figure out how to win the game.
-Explain the rules of poison. (Write the numbers 1 through 13 vertically on the board. Explain that two players take turns removing or crossing out numbers, and that the one who has to cross out the last number, 13, loses the game. Players can cross out either one or two numbers at a time.)
-Have a volunteer play the game at the board, and then a second volunteer.
-Discuss what is happening, and what strategies might ensure that one wins each game.
-Have students work in pairs, playing the game with each other, and trying to figure out a winning strategy. Each player should cross out his or her numbers in a different way so they will be able to analyze the games and look for patterns.
-Have students explain, in a paragraph or essay, the game of poison to someone who had never played it. They must explain the rules as well as a strategy to win the game. They must explain why that strategy will be successful.
-If necessary, bring class back for discussion and guide them toward a strategy.
-If students very quickly discover a winning strategy, challenge them to find strategies for the same game but with a different number (for example 17 instead of 13). Have them explain their strategy and why it’s applicable.

Assessment: Students’ writing.

Lesson 6: That’s Not Fair, Or Is It?—math as entertainment (1 block) 

CS: The student identifies patterns and special features of data and events of chance through experiences with meaningful mathematical problems that focus on comparing, predicting, representing data, and making decisions to communicate mathematical understanding.

PS: Determines simple probability in experimental and theoretical situations. 

Objectives: Students will record results of game.
            Students will analyze results to determine fairness of game.
            Students will predict fairness of dice tossing.
            Students will experiment to conclude fairness of dice tossing.

Materials: pair of dice for every 2 or 3 students

Procedure:
-Journal question/starter: Would you play a game if you thought it was not fair? What do you think makes a game fair or unfair?
-Discuss whether students think rock, scissors, paper is fair or not, and why. (Make sure everyone is familiar with the game.)
-Play a few rounds with volunteers, or have volunteers play in front of the class.
-Instruct students now, in pairs or groups of three with one person as the recorder, to play twenty games and to record the winning result for each round. (Students can do this by writing the three possible results and writing tick marks next to each winning result).
-On the board, tally the results for each pair or group. Discuss the results and the sample size to determine whether or not the game is fair. If appropriate, discuss probability in terms of fractions between 0 and 1, and that each outcome must be equally likely for the game to be fair.
-Explain to students that they are going to conduct another experiment to see if another situation is fair. Discuss casino games such as roulette or other games with which students might be familiar. Ask students whether they think all possible outcomes of the roll of one die are fair or equally likely. Discuss answers. Then ask if they think the same is true of two dice. Discuss what the possible outcomes are when rolling two dice, and write results on the board.
-Instruct students, again in pairs or threes, to roll the dice thirty times and record the results.
-On the board, tally all the results, and note the sample size. Discuss what all the possible outcomes are and how those different outcomes can be achieved.
-Present a common scenario to the students. For example, two students want to run an errand but the teacher only will allow one to go. She tells one student he can go if, after ten dice rolls, he has rolled more sixes than the other student has twelves. In a paragraph or essay, have the students explain the problem in their own words. Then have students explain why that sort of dice tossing is or is not fair. Students should refer to the class’ tally.

Assessment: Class discussions and students’ writing.

Unit conclusion/ assessment: If the unit was presented as a series of consecutive lessons, discuss with students whether or not any of the lessons changed the way they view math. Students might write about what they learned from the unit and what could have been done differently, etc.

Documentation

Bibliography

Allman, William F. “The Mathematics of Human Life.” U.S News & World Report 114 (1993):              84-85.

           Fairly brief article about the chaos and fractal nature of the human body. Interesting definition             of chaos.

 Apostol, Tom M. Calculus, Volume II. New York: John Wiley and Sons, 1969. 

            University math text. 

Ascher, Marcia. Ethnomathematics: A Multicultural View of Mathematical Ideas.  Boca Raton:               Chapman and Hall/CRC, 1991.

Presents various topics in math in a socio-cultural context. Useful for teachers.

Burger, Edward and Michael Starbird. The Heart of Mathematics. Emeryville: Key College               Publishing, 2000.

            An introductory college text with a lot of humor. Presents a variety of topics in a way so as               to be accessible to a wide audience. Also includes many puzzles.  

Davis, Philip and Reuben Hersh The Mathematical Experience. Boston: Houghton Mifflin
            Company, 1981.    

A very readable, survey-like book on topics of interest to both laymen and students of math.               Not too technical.

Erickson, Tim. Háganlo Juntos. Berkeley: Lawrence Hall of Science, 1997. Spanish language                version of Get It Together.

Collection of reproducible cooperative learning, problem-solving activities for students in grades 4-12. Includes many logic activities.

Eves, Howard. Foundations and Fundamental Concepts of Mathematics. Mineola: Dover             Publications, 1958.

          University text covering various mathematical topics. Useful reference for teachers although             the reading is somewhat dry.

 Goldsmith, Jeffrey. “The Geometric Dreams of Benoit Mandelbrot.” Wired Magazine Aug. 1994. 

            Brief interview with Mandelbrot in which he discusses how computers have aided his work.

Jacobs, Harold R. Mathematics: A Human Endeavor. New York: W. H. Freeman and Company,              1970.

Very accessible high school and introductory college text. Good refresher or background material for teachers.

 

Musser, George. “Practical Fractals.” Scientific American. Jul. 1999: 38-39. 

            Brief article about use fractal design for antennae.

Nelson, David, George Gheverghese Joseph, and Julian Williams. Multicultural
            Mathematics. Oxford: University Press, 1993.    

Selection of topics presented in essay-like format. Useful for teachers. Lots of                            
            illustrations, including some strip patterns.

 O’Connor, J.J. and E.F. Robertson. “Benoit Mandelbrot.” The MacTutor History of  Math              Archive.1999. University of St. Andrews. 27. Jul. 2002.                 
           <http://www-gap.dcs.st-and.ac.uk/~history/>

 ---. “Egyptian Numerals.” The MacTutor History of Math Archive. 2000.
          University of St. Andrews. 20 Jun. 2002 <http://www-gap.dcs.st-and.ac.uk/~history/>

Brief summary of Egyptian number system, including illustrations. Useful for teachers and students.

 Peterson, Ivars. “The Honeycomb Conjecture.” Science News 156 (1999): 60-61.            

Article about Thomas Hales’ proof that hexagons are the most efficient building structure in terms of perimeter.

Zaslavsky, Claudia. The Multicultural Math Classroom: Bringing in the World. Portsmouth:              Heinemann, 1996.

Written for teachers. Background of and activities appropriate for elementary and middle school students.