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Good Ideas in Teaching Precalculus And...

... Algebra, Calculus, Discrete Mathematics, Probability & Statistics, with Technology

Rutgers University - Busch Campus - New Brunswick
Friday, March 20, 2009
8:30 a.m. - 3:00 p.m.


Siham Alfred:
Relating Related Rates to Students

Many students have difficulty with related rates simply because the function of time is not given explicitly or implied and not given at all. I intend to share a simple approach that will encourage students to understand rates of change. I will also share some good related rates problems.

Aggie Azzolino – mathnstuff.com
Term Tiles & Tokens for Algebra I & II

Move pictures (tokens) on paper or in digital form on a spreadsheet, to represent algebraic ideas and perform operations. It's algebra made concrete. Complete integer and algebraic computations. Write, simplify, and evaluate algebraic expressions and equations. Factor and distribute. Solve linear equations, inequalities and quadratics. Examine binomial sums and differences and their squares and cubes. Examine the Binomial Expansion and Pascal's Triangle.

Carrie Baker and Lindsay Segreto:
Geometry Projects using Geometer’s Sketchpad

Hands-on and interactive Geometry projects that can be implemented using Geometer’s Sketchpad. Project topics include symmetry, transformations and 3-D perspective drawings. If Sketchpad is unavailable to you or your school district, many of these projects can be done using geometry tools. Need a way for students to remember what they learned last week? Last month? Last year? Why review only one topic at a time? Don't give students a chance to forget anything. Come hear some suggestions on how to pose questions that pull from different areas of the curriculum simultaneously. This idea can be used at any level, but the examples presented will focus on advanced algebra through calculus.

Eric Berkowitz – Parsippany Hills High School:
Mix it Up: Posing Questions that Draw on Multiple Topics

Need a way for students to remember what they learned last week? Last month? Last year? Why review only one topic at a time? Don't give students a chance to forget anything. Come hear some suggestions on how to pose questions that pull from different areas of the curriculum simultaneously. This idea can be used at any level, but the examples presented will focus on advanced algebra through calculus.

L. Charles “Chuck” Biehl – The Charter School of Wilmington (DE):
Network Design and Special Right Triangles

In the search for interesting applications, special right triangles can play an important role. In the design of what is called a “Steiner Network”, minimum direct distances between sites are not always the best connections to make. The initial problem was raised by Delta Airlines in the 1960’s, in designing a private communication network. Designing similar and ever larger networks has been of special interest ever since. The presentation even contains the outline of a proof that was published in the New York Times.

Brother Patrick Carney – DePaul Catholic High School (Wayne):
Putting the InTrigue Back in Trig

Unfortunately trigonometry is sometimes taught as if it were a dull subject that has to be committed to rote memory to be used for some vague purpose in the distant future. This presentation will try to go through some techniques which can help overcome this tendency and making it more lively and thus better understood.

Ihor Charischak – DMCpress.org (Stevens Inst. Of Technology, retired):
Back to the Future: Teaching and Learning Math with Web 2.0 Tools

The Internet (i.e. Web 2.0) is changing the way we communicate and learn in exciting new ways. It’s also beginning to make inroads in how we teach and learn math. This session will showcase examples of how Web 2.0 and dynamic software are transforming math learning and teaching. (Grade level: 9-11)

Neil Cooperman – Millburn High School:
What Is Alternative Assessment? It Is Not Just Assigning Projects!

Many people think that Alternative Assessment just means assigning projects. It does not! Creating projects, doing a performance assessment, solving “Real-World Problems”, or explaining one’s understanding in written or verbal form may be part of this process, but simply assigning any of these does not necessarily constitute Alternative Assessment. The primary goal is to find methods of measuring student progress with validity and reliability in lieu of paper and pencil tests. This session will try to address what Alternative Assessment truly is, and how you might create some of your own without making yourself crazy in the process.

Fred Decovsky:
Introduction to Basic Graphing on the TI-Nspire.

In this session we will first learn some basic graphing techniques on the TI-Nspire handheld. We will then apply these techniques in a problem solving situation. Finally we will explore ways to use the handheld device improve students’ understanding and create an engaging classroom environment.

Angelo DeMattia – Consultant:
Big Ideas in Probability and Statistics

Do you need a bank of fun activities that get at the Big Ideas of Probability and Data Analysis as well as create unforgettable connections within and outside the traditional math content? This session will help you to build that bank. Since Standard 4 has increased in worldwide relevance [it now represents 30% of the score value on the HSPA], more connections to all math topics have become "standardized" in curricula materials. In addition to the use of hands-on materials, participants will also experience graphing calculators to help solidify that understanding of the Big Ideas and the related connections.

David Glatzer – Consultant:
A Six-Pack of Ideas for your Geometry Class.

Explore six (perhaps 7 or 8) activities to motivate, connect, and enhance the learning of geometry. Several questioning techniques will be included along with the following topics: coordinate geometry, area and perimeter, the Pythagorean Theorem, discrete mathematics, and more. Join in on the fun and you will be ready to use these activities with your students.

Bonnie Gold:
Talking the Talk:

Helping students learn to use the language of college-level mathematics Abstract: The “Introduction to Mathematical Reasoning” course at Monmouth University, taken by freshman mathematics majors, introduces students to the standard method of determining truth in mathematics: proofs and co unter-examples. To do this, they need to learn to use mathematical language correctly. This involves both the logical structure as well as learning standard mathematical conventions. This would be appropriate for a high school discrete mathematics course, and somewhat for a precalculus course, as well as for introduction to proof courses at the college level. I will discuss difficulties our students have with mathematical language and how I have worked on remedying them.

John Hammett:
Writing to Learn Precalculus: Sharing Ideas

Ever read some student work and wonder, “What WERE they thinking?!” Teachers of Precalculus and other math subjects can answer that question and others by incorporating writing to learn techniques into their classes. Your students can share their ideas with you in a number of ways, including journal writing, creative writing, and problem posing. Participants will explore and model selected strategies.

John Hanna:
Discovering interesting Mathematics using TI_Nspire CAS.

TI-Nspire CAS opens new doors to the discovery of mathematical phenomena. Come see two such developments that are appropriate for your Algebra II, Precalculus, and Calculus classes.

Brian Hopkins - Saint Peter's College:
Understanding Pascal and Fibonacci with Cuisenaire Rods

A popular elementary education manipulative, Cuisenaire rods can also be used to understand proofs of many combinatorial numbers, including the binomial coefficients of Pascal's Triangle and the Fibonacci numbers. In this session, participants will have an opportunity to work with Cuisenaire rods to "see" identities that are typically proven with induction or algebraic manipulations. Combinatorial proofs give a nice supplement to more common techniques and encourage deeper understanding.

Iftikhar Husain – University High School (Newark):
Visual Mathematics

: Participants will receive an activity CD to be able to appreciate the power of Geometer's Sketchpad. The presenter has integrated the technology into mathematics as a visual learning tool to enhance, expand and embrace the existing curriculum. In order to make Visual Mathematics more useful to all types of learners, the author has a collection of activities ready for the classroom for basic Algebra, Algebra I, Geometry, Algebra II, Precalculus, and Trigonometry.

Joyce Leslie:
Teaching Algebra to Future Calculus Students

After several years of teachings honors pre-calc, calc and AP calc, I am teaching several sections of Algebra 1 this year. Right away, I noticed that my students (non-honors classes) were asking some questions that I asked calculus students: For example: ""If y> x, how much greater than x does y have to be?" In this presentation, I will discuss how I teach slope, lines, graphs and inequalities with an explicit goal of creating a foundation for precalculus and calculus. Some of the ideas I use are effective in helping students become interested in the algebra 1 topics and some of them just naturally arise as we discuss a particular problem? I will provide various examples of lessons (or lesson components) and the student questions they generated on the subject of slope, straightness of a line, continuous functions, distance-time graphs, closed intervals, and inequalities. Do calculus ideas regularly arise in algebra 1 class? I hope we can discuss this at the presentation. Precalculus concepts become clear using multiple representations. The new technology (TI-Nspire) allows for a real variety of approaches (numeric, graphic, symbolic, and verbal) that appeals to all learners. Come check it out.

Liz Marquez and Paul Westbrook:
Teaching for Financial Literacy in PreCalculus

Even our most accomplished students lack the basic financial skills crucial to success in life, yet they all take math and are all interested in money. We will demonstrate how to tap that money interest and help kids become mathematically and financially savvy by infusing money applications into pre-calculus. Some of the applications that will be covered are: linear regression and the cost of owning a car; measures of central tendency and buying stock; sequences and future value; and series and annuities. We will also address how the proposed 2.5 credits of financial literacy education can be met in math class

Kevin Merges:
Pascal’s Triangle and Tetrahedron

This session will focus on the types of problems that can be solved using Pascal's triangle or Pascal's tetrahedron. Pascal's tetrahedron is a three-dimensional version of Pascal's triangle that is constructed using the same principle: each entry is the sum of the entries immediately above it. Many of the problems discussed will be useful in lessons using combinatorics, binomial theorem, and trinomial theorem.

Robin O’Callaghan and Brian O’Reilly – The College Board:
rocallaghan@collegeboard.org, boreilly@collegeboard.org
Equity and the SAT Mathematics Test

How do the developers of the SAT make sure that the test is fair and equitable? What role does the SAT Test Development Committee take in ensuring equity? What part does the calculator policy play? Come hear the answers to these questions and more.

Larry Ottman:
An Hour with the Tower: Using Multiple Approaches to Investigate the Tower of Hanoi Problem

Participants will explore this rich and interesting problem with various approaches that touch on patterns, recursion, binary numbers, and probability using everything from technology to paper-folding.

Mark Richman:
Just Let Me Survive Today.

Through a unique combination of games, puzzles, rewards and incentives, a structured system of rules, lots of humor (including “Math Dancing” and singing, among much else), “brain - based” study strategies as well as traditional techniques, attendees will learn how to motivate and manage their students so the pupils will enjoy class and improve their exam results. Pupils will be provided with opportunities for success and the building of confidence in a framework of fun and excitement. Mr. Richman will supply you with a blueprint for successful classroom management via procedures that cover nearly every situation that could arise in your class.

Bob Riehs – New Jersey Department of Education:
The New New Jersey Mathematics Standards for Algebra I and Geometry

The session will include soon-to-be adopted draft revisions to the high school math standards, a clarification (including sample assessment items) of some of the individual Cumulative Process Indicators (CPIs), and an opportunity for participants to suggest further refinements before the revisions are formally adopted by the State Board of Education.

Joe Rosenstein – Rutgers University:
The New New Jersey Mathematics Standards for Grades K-8.

The Department of Education scrapped the draft math standards for Grades K-8 prepared by its own writing team and replaced it by a hurriedly written cut-and-paste document. We will compare (parts of) the two documents and discuss the efforts currently underway (see http://sites.google.com/site/cmeofnj) to restore the process and the product of the writing team.

Ahmed Salama - PANTHER Academy, Paterson:
The “Art” of Transforming Functions.

By doing various transformations to functions, students will be able to use graphing calculators (TI-84) to obtain art figures that are relevant to geometry, Algebra 2 and other SAT problems. Mathematics connects Art. Come find out more.

Jay Schiffman – Rowan University:
Addressing Student Misconceptions about Calculus

Many students arrive in calculus courses at the collegiate level having experienced a high school version. Unfortunately, the deep understanding of the rich content based knowledge we desire may be severely lacking. In this presentation, I will furnish a variety of student misconceptions experienced in freshman level calculus and discuss strategies to enhance understanding and stimulate interest in the discipline. Among the topics discussed will be the appropriate use of technological tools such as graphing calculators equipped with CAS (Computer Algebra System) capability, problem solving via multiple representations, and a selection of problems that help students acquire new mathematical insights through the use of a discovery based approach in which they formulate and test their conjectures. Finally, the important role played by several vital theorems and the need to assure that all of the hypotheses of the theorems are fulfilled to guarantee the conclusion will be stressed. Please join us and discover some new insights that hopefully will result in a more positive experience for both teacher and students.

Anita Schuloff – Paramus Catholic High School & Montclair State University Gifted/Talented Program
See the Derivative Come Alive!

If you find that your students are having trouble visualizing the fact that the derivative of a function represents the slope of the function, then it’s time to make use of the Advanced Toolkit Option in Geometer’s SketchpadTM. In this presentation, you will be shown how to attach a tangent line to a function, find its slope, superimpose the graph of the derivative onto the original function and show that the y-values of the derivative perfectly match the slope of the tangent line at each x-value.

Charles Schwartz – Rider University
Collect All Four

Imagine that you go to the grocery store with a child. The cereal boxes proclaim: “Free, inside specially marked packages: One of four presidential coins. Collect All Four!” Your child begs “Please, Please, Please. I want to get this cereal so I can get all four coins.” The problem: how many boxes of cereal would you need to buy, on average, to get at least one of each kind of coin? In this workshop, we will introduce (or review) the concept of the expected value of a random variable, develop the formula for the expected value of a geometric random variable, and apply this formula to find the answer to the coin problem.

Doug Smith – A.P. Schalick High School (Pittsgrove):
Nothing compares to mu. . . (stats with pennies)

A sampling of what I do in my stats class. I won't be mean, but I'll be confident. With your help, it will all make cents.

Michael Thayer:
Preparing Precalculus Students for Physics

Many students who take precalculus will either be taking physics in the following year, or perhaps during the same year. There are several topics that physics teachers would like to have their students be more aware of – and many of these can be easily incorporated into the precalculus curriculum (if they're not there already!). We will discuss some of these topics, as well as applications and problem types that can be easily added into homework assignments or given as projects. Relevant differences between "regular" and the Advanced Placement physics curricula will also be discussed.

Linda Treilman:
Use a SMART Board™ To Teach Precalculus Topics

A demonstration of ways to engage learners, appeal to a variety of learning styles and aid students in the understanding of mathematical concepts. Topics covered will include transformations, functions, real world examples, tools for creating lessons and more. Other software integrates easily into the latest version 10 notebook software and the Lesson Activity Toolkit and enhances lessons. TI-SmartView™ and Autograph, a dynamic graphing program will be included in the program.

Michael Weingart – Rutgers University:
The Monty Hall Problems

This talk will discuss the notorious "Monty Hall problem", a thought provoking and at one time very controversial problem in conditional probability. In the television game show “Let’s Make a Deal” in the 1960s-1980s, a prize lies behind one of three doors, the contestant picks a door, and the host opens another door revealing no prize behind it. Should the contestant stick with the originally selected door, or switch to the other? Or does it not matter? The solution is accessible to high school students. The talk will include an experimental approach to the problem which is illuminating and very suitable for the classroom.

Cathleen Zucco-Teveloff – Rowan University:
Using Web-Based Activities to Give Statistics an Environmental Slant

Today more environmental science is being infused into algebra as modeling applications. I will explain how I used this same type of approach in an elementary statistics course. I will share a variety of environmental application problems to which I exposed my students when teaching descriptive statistics, probability and regression.

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