Over the past 20 years and under several different NSF and NY State Education grants almost 40 workshops were designed for the elementary and secondary school teachers in the Syracuse and the surrounding suburb’s school districts. Many of the workshops were also presented at AMTNYS (Association of Mathematics Teachers of New York State) and NCTM (National Council of Teachers of Mathematics) meetings. Also under these grants, several student enrichment projects were designed for use by teachers in their classrooms. At the end of the last grant it was decided that these workshops and enrichment projects should be made available for general use. They are slowly being prepared for general use. Originally written in LaTex, we are reformatted them in Microsoft Word so that they may be easily altered. Anyone wishing to use these materials should feel free to modify them to fit their needs. The enrichment projects and the workshops differ only in length and scope. Some enrichment projects have been used as short teacher workshops and most of the workshops could be used as extended enrichment projects for high school students. Since most workshops were written and rewritten several times over the years, no authorship has been designated for the individual projects and workshops. Below we have listed all of the individuals that have been involved in the writing and presenting of these workshops as well as a complete list of the grants under which they were written.
Authors: Thomas Bleier, Jack Graver, Christina Graves, Steven Graves, Elizabeth Hartung, Larry Lardy, Jeff Meyer, Yvette Monachino, Moises Venouziou, all were members of the Mathematics Department of Syracuse University at the time they worked on these materials. In addition, the following people had significant input into selection of topics for these workshops and projects and into the subsequent revisions of these materials. From the Syracuse University School of Education: Jennifer Adler, Brian Cohen, Alexander Currin, Christine Mathews, Janet O’Flynn, Patti Owens, Nichole Schmidt, Nancy Sellmeyer, Barbara Shelly, Patricia Tinto, Brian Young. Nancy Zarach, from the Syracuse School District, Nancy Starke from the Chittenango School District, Preety Tripathi from SUNY Oswego and Susana Davidenko from SUNY Cortland.
1992-97. Mathematics Teacher/Researchers Collaborating for Collaboration in the Classroom, National Science Foundation.
1997-2012. NY State Teacher Leader Quality Partnership
2009-2012. Learning the mathematics needed to teach young children mathematics. Partnership with Syracuse City School District and
Chittenango School District together with the SU Department of Mathematics.
2004-07. NCLB/ Title II B Mathematics Science Partnership (MSP): Beyond Access for Mathematics Achievement (BAMA); core partners Department of Mathematics, School of Education, and Syracuse City School District.
2007-10. NCLB/ Title II B Mathematics Science Partnership (MSP); core partners Department of Mathematics, School of Education, and Syracuse City School District. [Extended through spring 2011 to put workshops online.]
Enrichment Project #1 SYMMETRIES OF THE SQUARE. This project will help students develop their geometric intuition about the symmetry of certain objects. Students can manipulate a square to explore and find its symmetries. After students have found all possible symmetries of the square they will learn how to compose two or more symmetries to obtain another symmetry. There are also extensions to less symmetric objects.
Enrichment Project #2 FRIEZE PATTERNS. Transformations are used to identify all possible structures for one-dimensional repeated patterns. One of the main purposes of this and our other geometric enrichment projects is to strengthen and extend the student’s geometric intuition.
Enrichment Project #3 A SPECIAL SOCIAL SECURITY NUMBER. A United States Social Security number is a nine-digit number. The problem is to find a nine digit (SS#) that uses each of the digits 1 to 9 exactly once so that the number consisting of the first k digits is divisible by k for all k. It is designed to develop problem solving skills and a deeper understanding of the divisibility properties of numbers.
Enrichment Project #4 THE TRIANGLE AND TETRAHEDRAL NUMBERS. In this activity students will work with the triangle and tetrahedral numbers. This should develop their numeric and algebraic reasoning. Constructing tables and looking at successive differences, students should be able to understand the degree of polynomials that model the triangle and tetrahedral numbers and then find the formulas for those numbers.
Enrichment Project #5 TAX INCLUDED PRICES. This enrichment project helps build awareness of how sales tax works. You can use this to build number sense, particularly with changing between percentages and decimals. Students will have to reason numerically in order to formalize patterns into algebraic expressions.
Enrichment Project #6 DISCOVERING FORMULAS. This project focuses on the students’ ability to formalize patterns into algebraic notation. Students will see visual representations of the sum of the first even and odd integers. This leads nicely into the sum of the first n integers and a discussion about the famous mathematician Gauss will arise naturally.
Enrichment Project #7 GEOMETRIC DIFFERENTATION AND INTEGRATION. This project is intended to have students work geometrically with the ideas of function of slopes of a curve (the derivative) and the area function (integral).
Enrichment Project #8 MARGINAL FUNCTIONS AND SUCCESSIVE DIFFERENCES. The method of successive differences for find the formula for the nth term of a sequence is developed and eventually justified. In addition, these activities will help students see the relationship between the degree of a polynomial and the differences of that polynomial. Building a strong understanding of rates of change will help students later on in Calculus.
WORKSHOP #1 The Mathematics of Personal Finance. All of the basic topics in the mathematics of personal Finance are introduced here: compound interest, loans, annuities, IRAs and credit cards. Most realistic computations cannot be carried out by hand and are often too time consuming to be worked out on a standard scientific calculator. Hence we rely heavily on the financial package on the TI84.
WORKSHOP #2 The Complex Numbers. In this workshop we explore the arithmetic, geometry and algebra of complex numbers with a little trigonometry thrown in. We then use the linear and conjugate linear functions of a complex variable to investigate and classify the similarities and congruences of the Euclidean plane.
WORKSHOP #3 Functions and Numbers. This workshop was designed for elementary and middle school teachers. Its purpose is to develop a deeper understanding of the fundamental concepts of a function and a number. It starts at a very elementary level with the definition of a function and the elementary properties of functions. The abstract concept of cardinality is then developed using one to one, onto functions.
WORKSHOP #4 Fractals. The concept of iteration is introduced in a historical context as an algebraic method for approximating square roots. Geometric iteration is then used to investigate the Cantor set, the Sierpinski Gasket, the Devil’s Staircase and the Koch Snowflake. At that point the workshop participants move to the computer lab to investigate the many fractal resources on the web.
WORKSHOP #5 Algebra Counts. The algebra that we teach our students in middle school and high school has many applications. One of the more interesting and important applications is the topic of “enumeration,” or simply counting. The goal of this workshop is to introduce teachers and students to this application. The binomial theorem is our point of departure. It is reviewed as a counting theorem and then generalized to more complicated counting problems. This workshop has no natural ending; one can simply continue to consider more and more complicated counting problems. The workshop participants need only be familiar with 9th grade algebra.
WORKSHOP #6 Linear Functions. Students often identify a function with the “rule” which describes it. Thus functions become purely algebraic objects. The concept of a function as a mapping of one set onto another or onto itself is often not well understood by our students. The mapping properties of linear functions are particularly easy to describe geometrically and hence particularly easy to work with. The goals of this workshop are: to develop a deeper understanding of linear functions as functions and to strengthen the skill of geometric reasoning. This geometric understanding of linear functions will then be used to investigate linear difference equations and their applications.