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IMGALTTAG Volume III: The Great Problems of Mathematics


An Interdisciplinary Unit on Cryptography
Michael Amick
Peabody High School

This document is an interdisciplinary unit on cryptography that is relevant to middle school and high school students. The material focuses on WWII code-breaking methods, and how mathematics was used to alter history. Mathematics, History, English, Physics, and Art came together in a unique way, and the lessons in this unit are designed so that the students will gain an appreciation of the events during this time period.

This document proposes that students will learn mathematics better if the material is taught so that it relates to other subjects. Research is presented to suggest that the current method of teaching material as independent, isolated bodies of knowledge is ineffective. Interdisciplinary lessons will not only be more interesting and relevant, but the students will better retain and apply their knowledge. After presenting this research, I offer a cryptography unit for teachers who want to work together and explore this topic with their students.


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Music in Mathematics
Sharon Birt
Friendship Academy

Whether society realizes it or not, music is all around us. From the songs of the birds to the music on the radio to the classics by Beethoven and Bach, music fills our lives as well as our ears. Every person on the planet likes some sort of music; some people listen to the soothing jazz of Duke Ellington, others enjoy Elvis’ rock and roll, and others still like the hip-hop beats and raps of LL Cool J. What people may not realize, however, is that within each song they listen to is a miniature mathematics lesson.  Music involves many of the basic mathematical skills learned in elementary school, such as counting, pattern identification, addition, division, and fractions. A piece of music can be taken apart and analyzed by its time signature, the rhythmic measurements of each note, and the various repeating patterns that those notes make.  Such an analysis can in fact be used to teach these skills in a mathematics class. A child who normally finds patterns boring would pay attention if s/he had to look for patterns in their favorite song. Students who think that division is hard might benefit from having an easy to understand visual layout, such as music notes. A class full of ten year olds may get restless if lectured to for a long period of time, but if those same students are clapping or hitting a tambourine while learning about rhythm, they are being both engaged and educated. Children enjoy involvement through active participation, which makes music a prime-learning tool for them. Whether they are listening, clapping, playing, or just simply reading the notes, students are taking a physically and mentally active role in the lesson while at the same time learning skills that, if presented in a traditional mathematics class, they may automatically dismiss as unimportant or uninteresting. Due to the reasons stated above, the mathematics of music is a topic worthy of consideration.


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The Mystery of the Prime Number
By A.N. Bennett Blaxter
Fort Pitt Elementary School

What is a prime number? We were taught in elementary mathematics that a prime number is simply a number whose only two factors are one and itself; the opposite of a composite number. It wasn’t until I was a part of class discussions in the Great Problems in Mathematics Seminar that I realized there is so much more to prime numbers. I am truly fascinated by all of the different areas in which prime numbers can be seen in the normalcy of everyday life. Primes are the reason we have our social security numbers, bar codes on the items we purchase at the grocery store, and why it is safe for me to use my credit card when I am shopping on the Internet. This fascination has driven me to prepare a unit on prime numbers and their complexity beyond their very simple definition.


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Playing with Primes
By Barbara I. Lewis
Pittsburgh Gifted Center

Prime numbers have fascinated mathematicians for centuries. Deceptively simple, they can be combined in many different ways and are useful in many different situations. This curriculum unit gives students the opportunity to explore various aspects of prime numbers and number theory. While working on the tasks, they strengthen their understanding not only of primes, but of factors, fractions and divisibility as well. Organized as a series of eight explorations, students work individually, in small groups and as a class to discover some of the unique aspects of the building blocks of the number system.


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Moving Math
Esther Liston
Rogers School for the Creative and Performing Arts

Moving Math is a unit about moving students through mathematical problems and puzzles toward mastery of concepts needed for high school. The lessons are appropriate for middle school students or as introductory pieces for high school students. Students work on creating square, rectangular, and triangular numbers, as well as star, pentagonal, and hexagonal numbers.  Rectangular, triangular, star, and hexagonal numbers appear in the Frogs, Fleas, and Painted Cubes book, part of the Connected Mathematics Project curriculum. The unit includes lab sheets for these number patterns. These lab sheets can be used as cover sheets for weekly homework packets or as stand-alone worksheets. Students explore Pascal’s triangle as a pre-cursor to binomial expansion and generating functions. The narrative encourages teachers to use problem solving, remember the history of math, and reflect on the nature of mathematics.

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Using Quadratics Equations to Find the Parabolas of a Suspension Bridge
Joseph Newkirk
Westinghouse High School

The purpose of this unit is to introduce engineering concepts as an applied learning subject in an algebra and geometry curriculum, at the 9–12 grade level. The goal, is that the students will be able to: 1) create a drawing, - scale model of the suspension bridge; 2) make a three dimensional model of a suspension bridge; and 3) make the connections and application of geometry and algebra using the suspension bridges of Pittsburgh and 4) apply the quadratic equation to determined cables length and suspensions connected to the deck from the cable. Metaphorically, the bridge is used to “bridge the gap”–connects between algebra and geometry. These two courses “come together” as the students learn about the quadratic equations, the design of the bridge using mathematics and an expeditionary exploration of several different brides that would include a walk over a few of the bridges in Pittsburgh.


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Proof as a Learning Tool in Algebra 1
Emily Press
Peabody High School 

Proof is the cornerstone of the mathematical body of knowledge. It is the means by which mathematics is communicated. Reading and constructing mathematical proofs is a requisite component of a rigorous mathematics education, but it has been all but eliminated from our high school curricula. The minimal presence of two column proof in Geometry class is insufficient and often irrelevant to our students. I propose to introduce proof as a learning tool in Algebra 1 classes.

This document is primarily a research paper presenting a brief history of mathematics education and the changing role of proof in secondary math classes. The research is followed by some activities intended for use in a mainstream or honors Algebra 1 course. These activities are designed to introduce elementary proof to ninth grade students as well as to broaden the scope of what students perceive as mathematics. The activities embrace a collaborative, discovery oriented approach to learning mathematics and are intended for use in a student centered classroom.


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Ancient Numerals and Arithmetic
By Paul J. Renne
Oliver High School

We can now solve many problems in the physical universe, but do we really understand the numerals and arithmetic we are using to calculate them? It is the belief of the author that one never completely understands anything, but that understanding is a process that is deepened as one experiments and conjectures. The arithmetic in this unit is different enough than our own, that it will cause one to question and experiment.

The unit examines the numerals and arithmetic done by the ancient Egyptian and Babylonian cultures. It is presented in the context of a brief history of the development of these number systems. Although the processes are simple to perform, the mathematics behind them is deep. In addition to being a worthy pursuit in itself, it is hypothesized that not only will studying these systems enlighten one’s math, it may also enhance our relationship with computers. It appears that several of these ancient mathematical processes are more similar to how a computer would perform the task than our modern mathematical methods. Perhaps students forming their first mathematical ideas could also benefit from these different methods of multiplication and division.


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Connecting Children’s Literature with Mathematics
By Janie Rubin
Fort Pitt Elementary School

The purpose of this unit is to present an overview of ideas that can be incorporated into the classroom to link children’s literature and mathematics. The integration of math and literature provides the ideal opportunity for tapping into the talents of all students no matter what their ability levels may be. Using children’s literature allows students to explore math in real-life and interesting situations. With stories, students have opportunities to use math skills in ways that are meaningful to them. This unit suggests a selection of children’s literature books that can be used in conjunction with the math concepts and problem-solving skills being taught in the classroom to further aid in student understanding. This project is written for a fourth grade classroom, however, it can easily be adapted to fit any grade level.


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Conceptual Understanding of Fractions and Decimals
Taris Washington
Lemington Elementary

This unit will help 5th grade students between the ages of 9 and 10 years of age to conceptually understand fractions and how they are used in everyday life. Student will see how fractions are named and their importance in mathematics. Students will explore alternative strategies to find equivalent fractions and how to compare and order fractions. Students of the 21st century are motivated by the use of technology and manipulating objects. Materials students will use in during instruction ranges from Cuisenaire rods, base ten blocks, graph paper, and any the students choose to assist in the learning process.


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Chatham University
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IMGALTTAG Pittsburgh Teachers Institute
Jointly sponsored by Carnegie Mellon University, Chatham University and the Pittsburgh Public Schools.
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