**34 ^{th} Annual Good Ideas in Teaching Precalculus And . . .**

**. . . Algebra, Calculus, Geometry, Discrete Mathematics,
and Probability & Statistics, with Technology**

**Rutgers University – Busch Campus – New Brunswick
**

**Friday, March 20, 2020**

8:30 a.m. – 2:30 p.m.

8:30 a.m. – 2:30 p.m.

## Abstracts

(arranged alphabetically by presenter’s last name)

**Angelique Bender and Caroline Bennett – Summit High School abender@summit.k12.nj.us and cbennett@wmrhsd.org
**

**Hands-On Activities for Transformation of Functions**

After a brief presentation on how to do transformations effectively, we will be presenting a few activities that use the concepts in a more interactive way.

**Eric Berkowitz – Parsippany Hills High School
eberkowitz@pthsd.net
**

**DESMOStration 2.0**

After a quick recap of several features Desmos can do easily, we will work through a step-by-step lesson on how to put together a classroom activity that can be used for informal assessment or homework. You will learn how to construct different activities, use interactive graphs, and how to look at students’ answers to get a summary of their understanding. Bring your laptop or Chromebook with you.

**Charles (Chuck) Biehl – Consultant
lchuckbiehl@gmail.com
**

**Computational Geometry: Algorithmic Reasoning in Precalculus**

Algorithms play a large part in the learning of precalculus mathematics. This session examines three non-routine, interesting problems that require heuristic algorithms for problem-solving methods, as well as skills learned from algebra and geometry.

**Kathleen Carter – North Hunterdon High School
kcarter@nhvweb.net
**

**The Desmos Card Sort**

The Desmos teacher dashboard has a Card Sort Lab available to teachers to customize a matching activity for any topic. From factoring to functions, a Card Sort can quickly assess student understanding, help make connections and create lively classroom discourse. Come make your first Card Sort using the Desmos Activity Builder!

**Michael E. Cedrone – Hopewell Valley Regional School District
mcedrone3@gmail.com
Fractals for the Uninitiated
**If you’ve ever asked, “What are fractals and how can I use them in the classroom?” then this is for you. If you’ve ever asked, “What is the relationship between the bifurcation locus of the quadratic family and the Mandelbrot Set?” then we shall have to meet another day. For the uninitiated, come discover what fractals are, how they can be used in classrooms from middle school through Calculus, and what their connections are to nature, art, and Hollywood. A sample activity will be provided to start your adventure into this fascinating world.

**Ken Collins – Charlotte (NC) Latin School
kcollins@charlottelatin.org
Composite Functions and Continuity
**A recent AP Calculus Exam question asked students to demonstrate that a composite function was continuous at a particular point. Many students had difficulties with this question. How can we prepare our pre-calculus students to understand these concepts? Our discussion will include classroom ready examples.

**Neil Cooperman – Millburn High School (Ret.)
NCoop@att.net
Socratic Teaching: Ask, Don’t Tell!
**88% of our students who graduate from college will not have majored in a STEM related field. Our overarching goal in our instruction should be to teach critical-thinking and self-reliance, not how to memorize processes to achieve answers. Turn your classroom into a mine of inquiry where students can excavate and discover their own mathematics.

**Stephanie Cooperman – William Paterson University
shc.amtnj@gmail.com
Using Desmos to Analyze Student Photographs and Artwork
**Many students enjoy taking photographs or are budding artists. Participants will learn how to use the Desmos Calculator to analyze the underlying linear equations of original images. Sequential lesson plans and activities will be presented for students of varying abilities, to further develop their understanding of the graphs of linear equations. Explorations by “guess and check” will provide participants with a method for determining the angle of the line (slope) and its location as determined by where the line crosses the y axis (y intercept). Participants will also learn how to construct tables of points, which will allow them to derive the attributes of the general equation, y = mx + b. Relating the mathematics to original photographs and artwork by students allows them to see the application of mathematical concepts to their daily lives and personal interests.

**Nick Corrado and Kelsey Wilton – Middletown Public Schools
corradon@middletownk12.org and wiltonk@middletownk12.org
Effective Use of Choice Boards in the Classroom
**This session is on the effective use of choice boards in the mathematics classroom. Choice boards are applicable to any content area and promote student voice and choice in the classroom. They also integrate technology and differentiation to support best practices in the classroom.

**Fred Decovsky – Consultant, Teachers Teaching with Technology
fdecovsky@aol.com
A Couple of My Most Favorite Hands-On Modeling Activities Using Technology
**Explore how to use technology and standards-based lessons to improve students’ understanding, create an engaging classroom environment and gain hands-on experience with data modeling topics in the high school mathematics curriculum.

**Angelo DeMattia – Kean University
angelomdemattia@gmail.com
The Monte Hall (Let’s Make a Deal) Problem and other Fun/Challenging Problems from Marilyn vos Savant (Author/Columnist)
**How do we know our intuition has failed us? We are aware that some problems just don’t follow our brains’ wiring, but how do we know when this happens? Let’s explore some fun, puzzling, and challenging problems in order to begin the process of fine-tuning our reasoning and language skills – such as the Monte Hall Problem – that simultaneously connect to the NJSLA-M and the Big Ideas in Math – including Number Sense, Probability, and other topical areas of mathematics. The use of visuals will be the primary tools that help us efficiently uncover solutions as well as make sense of the relationships.

**Meghan DeVaney – North Hunterdon High School
mdevaney@nhvweb.net
Polynomial Division: Developing Conceptual Understanding
**Students struggle with manipulation of polynomials. This session will share strategies that help students develop a better understanding of polynomial division for algebra to precalculus. Students will develop greater flexibility in their procedural thinking instead of simply “memorizing” the steps and can make connections to rational functions.

**Hannah Gibbs – Middletown Public Schools
gibbsh@middletownk12.org
Teaching in an Active Learning Classroom
**The purpose of my presentation would be to share my experience of teaching Geometry and Algebra 2 in an active learning classroom with other math teachers. The active learning classroom is designed for student collaboration. The walls of the classroom and lined with large whiteboards and the desks are chairs that roll with an attached surface to write on. In my presentation, I will talk about how I use different teaching methods to motivate students in the classroom and help them engage with the lessons. The active learning classroom encourages flexible seating to motivate students to work together to solve practice problems. The whiteboard walls provide a convenient platform for students to solve problems and check each other’s work. Using the design of the classroom, I have created collaborative activities that motivate students and help them engage in the class to solve practice problems, which I would like to share with other teachers in my presentation.

**Iftikhar Husain – University High School, Newark
husains4ever@gmail.com
Delivering Mathematics Concepts Visually
**One goal of teaching math is a deeper understanding of its concepts. The visual approach delivers concepts in a natural way and made an easy access for students on “Twitter”.

**David Hyman – Livingston High School
dhyman@livingston.org
It’s All Fun and Games… Even After They Realize It’s Math
**Using various non-traditional strategy games to stimulate critical thinking amongst the students.

**Joyce Leslie – Columbia High School, South Orange/Maplewood
joyce.leslie@gmail.com
We Broke Calculus
**It is interesting how students can be surprised by something that they already know (albeit, a little bit). My experience in teaching many different subjects in mathematics is that students need many opportunities to discover ideas, and this includes ideas they have already learned. In this talk I will explain these phenomena and show two examples in teaching integral calculus that engaged and surprised students. In both cases the discovery learning experience became a reference experience for students, experiences they would return to many times as new ideas in integral calculus emerged. These discoveries include a “big idea” conjecture and an algebraic exploration. This talk will connect to previous talks on Teaching Calculus to Multi-Level Students. In this earlier talk I focused upon a stronger connection between student’s understanding of slope and the development of differential calculus.

**Jacquelyn McSheene – Lawrence Township Public Schools
jmcsheene@ltps.org
Using Desmos for Discovery and Assessment
**Desmos is an incredible (free!) tool, but it’s more than just an online graphing calculator. This session will explore how you can use Desmos in your classroom for discovery lessons and investigations, as well as how to use it as a form of alternative formative or summative assessment. Time will be spent exploring the available activities on Desmos as well as learning how to create and modify your own activities. If you’re ready to step beyond the basics of Desmos and find out how to utilize it more regularly in your classrooms, this is the session for you!

**Afroze Naqvi – Saskatchewan Polytechnic
naqvia@saskpolytech.ca
Cardano’s Formulas for Solving Cubic Equations of the Form: x**The goal of this presentation is to use a modern approach, using some basic concepts from calculus, to develop the Cardano’s formulas for solving the cubic equations of the form x

^{3 }+ ax

^{2}+ bx + c = 0

^{3 }+ ax

^{2}+ bx + c = 0. Some cubic equations will be solved using these formulas.

**Robin O’Callaghan – Educational Testing Service
rocallaghan@ets.org
What’s the Best Way to Ask a Question?
**Are student-produced response questions always better than multiple choice? Can innovative formats in testing improve students’ ability to show what they know and are able to do? How can both classroom and standardized tests be more effective? Come hear the answers to these questions and more.

**Dawn Recentio – Lawrence Township Public Schools and Anna Panova Cicchino – Montgomery Township School District
drecentio@ltps.org and apanova@mtsd.k12.nj.us
Rational Function Investigation
**Rational functions can be a difficult topic for students when first introduced. Rational function graphs have characteristics that are new to students, such as discontinuity. This investigation is scaffolded for students to make connections. The students will be able to identify similar characteristics amongst their graphs, find the relationship between the graph and their respective equations. Introducing the topic with this investigation has provided students with a deeper understanding of rational functions, written in different forms, and their graphs.

**Joe Rosenstein – Rutgers University (ret.)
joer@dimacs.rutgers.edu
Approval Voting: As Maine Goes, So Goes the Nation
**Ranked Choice Voting. New York City voters approved on November 5 the use of “ranked choice voting” for all city offices starting in 2021. What is “ranked choice voting,” how does it work, what are its advantages and disadvantages, and what paradoxical situations can arise? In this workshop, we will consider the mathematics of ranked choice voting, and how you can discuss it in your math classroom, with students at any grade or ability level.

**Dana Rubin – Montclair Public Schools
drubin@montclair.k12.nj.us
Calculus: The Good, the Bad and the Ugly
**Best Practices in Teaching Calculus. In this session, teachers will learn new ways to teach various concepts to their students both with and without using technology in the classroom. What is done after the AP Exam? See what other teachers are doing!

**Mark Russo – Pascack Valley Regional High School District
mrusso@pascack.org
Trig Functions as Line Segments
**The Unit Circle is a critical tool for helping students extend their knowledge of right triangle trigonometry to angles of any size. In addition to the useful insights that students can develop about the sine and cosine functions, consideration of the six special line segments in the unit circle can help students make connections about the other four trigonometric functions as well. In this session, participants will consider how a thorough analysis of line segments in the unit circle can lead to a deeper understanding of the six trigonometric functions.

**Ahmed Salama – STEM Innovation Academy
salamamath@yahoo.com
Connecting Mathematics with Work and Life is an Effective Way to Succeed in STEM Program
**The purpose of this presentation is to view good ideas in teaching STEM students Mathematics effectively. In our mathematics classes, we should include projects that connect with science, engineering and technology Integration. I will represent five sample Math projects that have been done with STEM Students.

**Jay Schiffman – Rowan University
schiffman@rowan.edu
Generating Pythagorean Triples in Three Different Ways
**This hands-on presentation illustrates three methods to generate Pythagorean Triples. All PPT’s (Primitive Pythagorean Triples) can be generated using the standard generating equations. In addition, we show how Fibonacci-like sequences where one considers any four consecutive terms in such a sequence and performs three simple tasks generates some (but not all) Pythagorean Triples. Finally, we consider the sum of two unit fractions with consecutive even denominators or consecutive odd denominators and demonstrate how this process likewise generates some (but not all) Pythagorean Triples. Please join us to view the marriage of discrete mathematics, history and number theory in one tidy package.

**Anita Schuloff – Paramus Catholic High School
aschuloff@paramus-catholic.org
Using a Two-Dimensional Method for Solving Volumes of Revolution
**Some calculus students are shaky regarding volumes of revolution. This session will demonstrate an easy way to distinguish between those problems using the disk method and those needing the ring method.

**Robin Schwartz – Math Confidence/College of Mt. St. Vincent and Natalie Perez –
mathconfidence@aol.com and nbperezclan1@gmail.com
SAT for Graduation, College Readiness and Street Cred
**As the SAT will be accepted in 2020 as an exit exam, teaching the content helps students graduate and avoid remediation. Consider preparing for and taking the SAT yourself to know the test and gain some street cred with your students. The presenters will share their 2019 and 2020 test experiences.

**Charleen Schwartzman – Pascack Valley Public Schools
cschwartzman@pascack.org
Care to “Jazz Up” your Algebra 2 Class??? Analyzing Position and Velocity Functions In Algebra 2
**As the SAT will be accepted in 2020 as an exit exam, teaching the content helps students Algebra 2 is much more than skills based! Let’s get some Calculus in there by exploring slopes of secant & tangent lines in functions Algebra 2. We will discuss slope between 2 points as an average rate of change (avg velocity and avg acceleration) and determine instantaneous rate of change by choosing points closer and closer together and conjecturing the instantaneous rate of change. From there we can segue into the conceptual understanding of how you move through a position graph and how that translates to a number line, sketch the velocity graph from the position graph and compare the two functions to see the correlation(s). Your kids will be able to interpret graphs, use their GC’s effectively and learn some Calculus along the way!

**George Soliman – Raritan Valley Community College
george.soliman@raritanval.edu
Play Ball! Hit a Home Run by Incorporating Baseball into your Statistics Class!
**In the age of sabermetrics in baseball, numbers are all over the place in our national pastime. What a perfect opportunity to use these numbers in introductory statistics classes! Check out how traditional statistics concepts can be taught and/or applied using baseball. Turn your singles into home runs with some baseball-related statistics activities!

**Dianna Sopala – Northern Valley Regional High School
diannamsopala@yahoo.com
Personalizing Learning Through the Use of Utilizing Data
**Data collection is a daily occurrence in all industries. As educators, we collect data on our students daily, but what do we do with it? The data can be used to personalize learning, create student-center activities, or a learning-active, technology-infused classroom. This session will provide ideas on how to use the data and how to integrate personalized learning into the participant’s classroom.

**Kara Teehan – Middletown Public Schools
teehank@middletownk12.org
Teaching Mathematics of all Levels using Active Learning Strategies
**The speaker will discuss her experience teaching high school mathematics, including algebra and calculus, in an active learning environment, and the implications of creating opportunities for access to tasks and content in resource room, college prep, and advanced placement settings. Attendees will take away resources for tasks and examples of class structure.

**Elaine Terry – St. Joseph’s University, Philadelphia PA
terry@sju.edu
Using the ROF-1 Method to Solve Precalculus Problems
**ROF is an acronym for Rule of Four, a method that represents mathematics concepts and functions symbolically, numerically, graphically, and verbally. ROF was developed to enhance conceptual understanding of calculus topics by including problems that are non-procedural. This method was later applied to other mathematics courses including pre-calculus. The ROF-1 method is the ROF method applied to pre-calculus that includes a fifth step of illustrating how the pre-calculus concept can be applied to early calculus concepts such as limits, secant lines and rates of change. This presentation will illustrate the ROF-1 Method by way of three pre-calculus concepts.

**Linda Treilman – Mercer County Community College
linda.treilman@gmail.com
Making More with your SMART Board
**Notebook software has added many tools to enable you to create visibly appealing enhancements to your lessons. This session will use the latest Notebook software and include Smart Learning activities with game-based learning and The Smart Learning Space which allows teachers to view students work and comment on it or proceed with a lesson at their own pace.

**Stacy Winters and Aaron Yamamoto – Chatham Public Schools
swinters@chatham-nj.org and ayamamoto@chatham-nj.org
Why You Should Be Using VNPS in Your Classroom
**We often get wrapped up assessing and listening for the right next step or the right answer Come hear how we transformed a classroom into a Thinking Classroom inspired by the research of Peter Liljedahl. Shift the balance of accountability in the classroom from teacher to student to student to student. Provide timely feedback to students and encourage student discourse with this innovative enhancement to any classroom.