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Siham Alfred – Raritan Valley Community College
salfred@raritanval.edu
Functions with Easily Computed Arc Lengths

When writing exam questions for arc length in Calculus II, sometimes the integral is difficult or impossible to evaluate. In this 50 minute presentation, a family of functions is constructed for which the arc length formula yields an easy integrable power function. This problem is for teachers of Calculus or can also be given as an innovative project for students.

Mani Arguelles – Columbia High School , Maplewood
marguell@comcast.net
Non-AP Calculus: Teaching Kids Who Don't Think They're Smart in Math

This session looks at the challenges associated with teaching Calculus to students who, though bright and capable, aren't necessarily confident in math, let alone interested in pursuing math-related careers such as science or engineering. A variety of approaches aimed at making Calculus less intimidating, more engaging and more relevant to the non-AP Calculus student are presented and discussed.

Eric Berkowitz – Parsippany High School
eberkowitz@pthsd.k12.nj.us
Racecars and Mirrors

Students always ask why they are learning math and when they will have to use it. Mathematics is involved in many scientific situations, as well as in some everyday ones. Teaching math in terms of these applications can make learning more interesting to students, as well as give concrete examples of the mathematical theory in action. This presentation will suggest ways to bring applications of various topics into the math classroom.

L. Charles (Chuck) Biehl – The Charter School of Wilmington (DE)
cbiehl@charterschool.org
Voronoi Diagrams and Empty Circles

This session includes a set of classroom activities that use geometric construction, coordinate geometry, algebra, and (optional) graphing calculator technology, to investigate the partitioning of a set of sites into "service regions". These regions might be delivery areas for pizza parlors, ATM's, hospitals, or even nuclear waste dumps. Prerequisite knowledge is limited to solving systems of linear equations and basic properties from Euclidean geometry. The activities are suitable for students enrolled in algebra, geometry, precalculus, or discrete math. Bring any TI-83 or 84 to obtain a program which supports more complex problem situations.

Amy Cohen-Corwin – Rutgers University
acc@mathpc530.rutgers.edu
Some Favorite Examples and Test Items from Precalculus

Rutgers began introducing some aspects of "reform calculus" in the early 1990's. After several years expecting students to make connections between algebraic and graphical understanding of functions and their derivatives -- and expecting students to use their calculators when hand computation was too messy -- we began to restructure our Precalculus course to prepare for our new point of view about calculus itself. This talk will describe some lecture examples and test items which illustrate these intentions, discuss student responses, and attempt to draw some wisdom from the experience. One example used compound interest computations to reason about the exponential function. Another discusses "good linear fit" to small data sets, both by eye and by calculator.

Neil Cooperman – Millburn High School
ncoop@worldnet.att.net
Don't Just Solve for “x”, Solve for “Why”!

Did you ever wonder why students never seem to remember what they have learned? This workshop will look at some of the reasons behind the typical “loss of memory” of many of our students. We will also examine some student responses to “Why?” questions on tests. Additionally, we will develop some questions that get to the heart of whether students truly understand the material and are not just memorizing processes designed to get “the answer.”

Suzanne Cranwell, Matt Hanas – Rumson-Fair Haven Regional High School l
scranwell@rfhrhs.org
mhanas@rfhrhs.org
Integrating Technology in the Precalculus Classroom
(Previously presented at 2004 conference)

The introduction of computer algebra systems (CAS) in the math classroom has made it challenging to create activities and assessments that test knowledge and understanding of precalculus topics while supplementing, rather than supplanting, traditional paper-and-pencil techniques. This workshop will introduce a variety of discovery-based activities and assessments that involve the use of the TI-89 calculator as well as Geometer's Sketchpad and Converge software.

Bill Crombie – Relearning by Design (The College of New Jersey )
bcrombie@aol.com
Calculus Using Only High School Algebra and Geometry

On this point the conventional wisdom is simply wrong. It holds that limits are necessary in determining the area under a parabola because this type of region cannot be decomposed into a finite number of polygonal regions. Actually the area under a parabola, or for that matter the area under the graph of any polynomial function, can be determined exactly – without approximations or limits – using nothing more than high school algebra and geometry. In fact the complete differential and integral calculus of polynomial functions can be reconstructed and taught as simply a thematic reorganization of the content of Algebra I, Geometry and Algebra II. In this session we will see how and why this is true and examine the thesis that, consequently, precalculus does not exist.

Stacy Del Vecchio, Susan Van Riper – Chatham High School
sdelvecchio@chatham-nj.org
svanriper@chatham-nj.org
Project Ideas for Algebra, Geometry, Precalculus and Calculus
(An extension of presentation done at 2004 conference)

Come hear about creative ideas for alternative assessments for students in classes from Algebra through Calculus. Examples and sample rubrics will be provided. These projects include daily, multiple day, and marking period projects. These ideas can be adapted to specific levels.

Eileen Edelman – Tenafly High School
eedelman@tenafly.k12.nj.us
Is Love Transitive? (Discrete Math)

Apply mathematical properties and techniques to all sorts of non-mathematical situations! Analyze personal relationships for the reflexive, symmetric and transitive properties. Use graphs and matrices to determine the most influential member of a group. And, the most fun of all, arrange stable marriages on a desert island using a matrix, an algorithm and a sense of humor.

Roy Eismann – Columbia High School , Maplewood
reismann@att.net
Leaping Frogs, Popsicles, and Gummi Bears: Modeling Through Data
(Previously presented at 2004 conference)

This session will have participants doing hands-on activities that are motivating to students. The activities will use ready-made lessons that help prepare students for the NJ HSPA as well as integrate probability and data analysis concepts into the high school curriculum. Some time will be reserved for teachers in the audience to share their own successful lessons.

Jim Fey – University of Maryland
jimfey@umd.edu
New Thinking about Pre-Calculus Mathematics ( PLENARY SESSION)

College preparatory high school mathematics has long been designed with the central goal of preparing students for calculus. But changes in thinking about calculus, the mathematics that serves important client disciplines, and the technology for learning and doing mathematics suggest changes in the ways we should think about the pre-calculus curriculum. This talk will examine challenges and opportunities of new thinking about students' pre-calculus mathematical experiences.

Jim Fey is Professor of Mathematics and Curriculum & Instruction at the University of Maryland. He is a principal investigator and author on the Connected Mathematics and Core-Plus Mathematics projects and director of the Mid-Atlantic Center for Mathematics Teaching and Learning.

Kelly Gaffney, Jeff Herkimer – Rumson-Fair Haven Regional High School
kgaffney@rfhrhs.org
jherkimer@rfhrhs.org
How Geometer's Sketchpad Has Changed the Way We Teach Geometry

Revolutionize and enlarge your geometry curriculum through dynamic investigations, demonstrations, and assessments. A variety of computer generated lessons, projects, and assessments will be presented. Get a new appreciation for the Geometer's Sketchpad potential.

Debra Gulick – East Brunswick Public Schools
dgulick@ebnet.org
Assessment Activities that Promote Learning

Alternative assessments can be valuable tools in any precalculus or calculus course. Activities that can be used to reform instruction will be discussed. Examples of activities that require student reflection will be shared. Grading techniques which minimize the time required to grade will be explored. Discussion will also focus on how, why and when they should be implemented.

David Hyman – Livingston High School
dhyman@livingston.org
How CAN We Do Better? (Making Better Cans)
(Previously presented at 2004 conference)

An exploration via rational functions, as to whether a tin can manufacturing company is using the minimum possible surface area for their can. How "can" you make improvements? The discovery and development of volume and surface area of a cylinder. How the radius and height affect volume and surface area of a cylinder. Participants will actively derive a surface area function as it relates to the volume and radius of the cylinder. Using graphing calculators, participants will determine if their cylinder utilizes the minimum possible surface area for its volume. Using some of the same ideas participants will determine the maximum possible volume of a can given a fixed surface area. Bring in a canned food container and a graphing calculator and see what you "can" do.

Robert Kiessling – Haddonfield Memorial High School
kiess.run@erols.com
Using PowerPoint to Present Precalculus Lessons Clearly and Efficiently (Part I of a 2-Session Presentation)
(Previously presented at 2004 conference)

PowerPoint Presentations are a very efficient way to present math lessons. There is no lost time while an example is erased and a new one written on the board. That lost time is time where students can lose their focus. An overhead is static while PowerPoint Slides can be very dynamic.
PowerPoint can be emailed to absent or homebound students. In the case where the student does not have access to PowerPoint and/or the Internet, slides can be printed out and photocopied.
Using the TI Graphlink an entire sequence of TI-83 Screens can be put on a PowerPoint Slide and brought up screen by screen. PowerPoint Presentations are a lot of work up-front but pay off with a lot less work later. Anyone can learn PowerPoint and teach math using it.

Robert Kiessling – Haddonfield Memorial High School
kiess.run@erols.com
Ideas on Creating PowerPoint Slides for Math Presentations (Part II of a 2-Session Presentation)

This session will show some of the ins and outs of creating a lesson using PowerPoint, the Equation Editor, TI-Graphlink, and even the Geometer's Sketchpad. Find out how you can create your own permanent lessons for any class.

Allison McCulloch – Rutgers University
awmcculloch@hotmail.com
Functions Online: Using Technology to Investigate Functions from Algebra through Precalculus
(Computer lab session: space limited to first 25 participants.)

In this session we will use data collected from the Internet to explore real world relationships. Come discover the fun you can have working with functions of all types (linear, quadratic, exponential, and trigonometric) using data from some great websites. Bring your graphing calculator along and be prepared to walk away with activities to use at all levels from
algebra to precalculus!

Kevin Merges – Rutgers Preparatory School
merges@rutgersprep.org
Fractals in Math and Art

This session will focus on a cross-curricular project used at Rutgers Preparatory School . Students in the Discrete Math class created fractals using transformational geometry, complex planes, Sierpinski's triangle and the TI-83 calculator. Students in the second year Portfolio class designed fractal artwork. The math classes focused on a study of fractal dimension and the art classes focused on naturally existing fractals. Both classes studied Jackson Pollock's works. Mandelbrot's 80 th birthday celebration was part of the class.

Robin O'Callaghan, Brian O'Reilly – The College Board
rocallaghan@collegeboard.org
boreilly@collegeboard.org
The New SAT

With the New SAT being given for the first time on the Saturday before the conference, what could be timelier than hearing from College Board staff about the new math section?  Are you familiar with the expanded content in the math section?  Have your students been taught these concepts by the time they take the SAT?  How were these changes arrived at?  Does research show they'll be effective?  If you review math with students preparing for the new SAT, what materials are available and how can you get them?  This session will answer these questions and many more.

Ralph Pantozzi – Mount Olive Public Schools
rpantozzi@mtoliveboe.org
The First Thing Students Should Learn in Calculus Is…

Newton made a hypothesis – the Fundamental Theorem of Calculus – that he could not prove. This hypothesis, however, guided the development of calculus for centuries. Students can start their year of calculus the same way, building meanings that will guide their symbolic and algebraic actions later on. After they make hypotheses using these Sketchpad-enhanced paper and pencil activities, students will demand – and deeply understand – formal definitions of limit, derivative, antiderivative, and integral.

Manya Raman – Rutgers University
mjraman@rci.rutgers.edu
History of Functions

Much of precalculus centers around the concept of function.  But where does the concept of function come from?  Why is it defined the way it is?  What were some of the motivating questions that shaped the definition?  And what are the implications for our classrooms?

Carol Richards – Rowan University
richards6686@fcc.net
A Tongue in Cheek History of e, i, and pi

As pre-calculus teachers, we devote a lot of time to the uses of pi, e, and sometimes i. Usually it is just taken for granted that our students know all about these fascinating numbers, and for that matter, so do we. This presentation will look at where these numbers truly originated and how we can incorporate their past into the present. In addition, this presentation will provide several in-class activities designed to enhance the meaning of these important numbers.

Mark Richman – Columbia High School ( Maplewood )
srich39661@aol.com
Unique Brain-Based Study Techniques to Enhance the Learning of Mathematics
(Abstract available soon)

Joseph G. Rosenstein – Rutgers University
joer@dimacs.rutgers.edu
The Utilities Problem and Planar Graphs

The Utilities Problem is the question of whether and how you can connect each of three houses A, B, and C to each of the three utilities G(as), E(lectric), and W(ater) so that no two supply lines cross. In this session we will explore the Utilities Problem and the general question of whether a vertex-edge graph can be drawn so that its edges don't cross. That will lead us to planar graphs, Euler's formula, crossing numbers, and (maybe) Platonic solids.

Jay Schiffman – Rowan University
schiffman@rowan.edu
Number Theoretic Activities with the TI-89

The TI-89 serves as an excellent vehicle for explorations in elementary number theory. Participants will marvel at the capability of this hand-held technological tool in factoring large integers as well as determining their primality. In addition, the TI-89 is excellent in illustrating open problems in the field, such as Goldbach's conjecture, twin primes, and the distance between an integer and the next prime. Please join us in partaking of a number of stimulating ideas which have delighted both professional and amateur mathematicians for over two millennia.

Kathleen Shay – Middlesex County College
kathleen_shay@middlesexcc.edu
Exploring Data with Fathom
(Computer lab session: space limited to first 25 participants.)

This hands-on session will provide an introduction to Fathom, the dynamic software that helps students use data to gain a better understanding of mathematics, statistics, and science. We will focus on applications in statistics: graphical displays of data, descriptive statistics, and data analysis.

Doug Smith – A.P. Schalick High School (Pittsgrove)
smithd@pittsgrove.k12.nj.us
WOW! It can do this? (The TI-8x)
(Previously presented at 2004 conference)

Bring your TI-8x calculator and learn some of the tricks I've found out over the years – most of them by complete accident. If time allows, I'll show you why you have to finish all the medicine in your prescription, even though you feel better after a day or two!

Keith Weber – Rutgers University
khweber@rci.rutgers.edu
Teaching and Understanding Trigonometric Functions

This presentation will consist of three parts. First, I will discuss what it means to conceptually understand trigonometric functions and why it is so difficult for students to do so. Second, I will describe a method for teaching trigonometry that can develop students' understanding of trigonometric functions. Third, I will present the results of a study that show how students in a traditional class had little conceptual understanding of the sine and cosine functions, while students who received the novel instruction described in this presentation achieved a sophisticated understanding of these functions.

Peggy Wissler – Tenafly High School
mwissler@tenafly.k12.nj.us
Exploring Triangle Congruence Postulates

Students can grasp concepts more quickly when they build models that enhance their learning. Use paper, glue, and pop-cycle sticks to explore the ideas behind congruent triangles.  Apply similar methods to help understand the triangle inequality.