I will upload here descriptions & .pdf files from my conference presentations.
PREcalculus transformed & iNSPIRED (OCTOBER 28, 2016 – OCTM ANNUAL CONFERENCE, SANDUSKY, OH)
Presentation Description: How many functions can two points define? How do you bend asymptotes and bounce off infinity? Can one function be a parent to every exponential function? How are logistic, rational and trigonometric functions identical? This workshop offers an innovative understanding of pre-calculus concepts through non-standard transformations, allowing functions and concepts to be unified by a handful of underlying mathematical structures. Its approaches dramatically simplify many initially complicated-looking functions and problem.
The central ideas of this workshop are accessible online via Desmos and Wolfram-Alpha, but will be presented partially using the TI-Nspire CAS. With a classroom set of handheld calculators available, all participants will be able to engage in every problem throughout the workshop.
I used the AirSketch app to live project during presentation.
NSPIRED CAS & STATISTICS (october 16, 2015 – OCTM ANNUAL CONFERENCE, Cincinnati, OH)
Presentation Description: The utility of statistical software for student learning is widely accepted. Less well-known is the power of the TI-Nspire to support statistical learning. This session has two goals: review an Nspire document explaining linear regressions without black-box calculus, and explore implications of using the TI Nspire in a statistics classroom, including in some cases using the power of a computer algebra system (CAS) to compute many familiar statistics values. The regressions activity has been successfully used in Algebra II and Statistics courses. The other activities were developed with students in AP and non-AP Statistics classes, helping many of the presenter’s students achieve a deeper understanding of statistics than they would have otherwise.
Participants are encouraged to bring a statistics-capable graphing calculator or computer, ideally a TI-Nspire CAS. All presentation files and .tns Nspire CAS documents will be electronically available.
Powerful Student Proofs from CAS-enhanced classrooms (JULY 18, 2015 – USACAS-9, Cleveland, oh)
Presentation Description: This session outlines proofs established by two students from a CAS-enhanced precalculus class. The first involves a surprising transformation connection in a polar graph of a limaçon. The second establishes an unexpected relationship between the behavior of foci in ellipses and hyperbolas. Time permitting, I will share a third example involving a famous historical calculus result “rediscovered” centuries later by a curious student’s exploration in a CAS-enhanced classroom. While these proofs are beyond typical precalculus courses, the point of the session is to show how the presence of CAS can significantly enhance the abilities of ALL students to explore and create mathematics on their own.
BEND ASYMPTOTES, BOUNCE OFF INFINITY, AND MOVE BEYOND (JULY 18, 2015 – USACAS-9, CLEVELAND, OH)
Session Description: A deep understanding of polynomial behavior and transformations dramatically enhances student understanding of rational function graphs. This session uses CAS as an innovative approach to analyze rational functions via variable transformations far beyond simple stretches and slides. Then, we will leverage the reciprocal and other transformations on exponential, rational and sinusoidal functions to create logistic, other rational, and compound trigonometric functions, unifying several families through a handful of underlying mathematical structures. My goal is to show that many historically complicated functions and concepts are simultaneously richer and simpler than they have traditionally been handled. If time allows, we will conclude with a novel investigation of polar functions. (I’ll include the polar idea in the electronic materials either way.)
NSPIRED CAS & STATISTICS (MARCH 14, 2015 – T^3 International Conference, dallas, tx)
Presentation Description: Dynamic software is ubiquitous in statistics courses. Less well-known is the power of the TI-Nspire CAS to support statistical learning. This session has two goals: review a non-CAS Nspire document explaining linear regressions without black-box calculus, and explore implications of CAS approaches in a AP and non-AP statistics courses. The regressions activity has been successfully used in Algebra II and Statistics courses. The CAS approach helped many of the presenter’s students achieve a deeper understanding of the statistics of binomial & normal distributions, confidence intervals, and minimum sample sizes without ever needing to look up values in probability tables.
finally … a real math app (april 12, 2013 – ett ipad summit, atlanta)
Presentation Description: This session models an Algebra 2 class in which FluidMath creates an environment in which students and teachers explore deep and challenging mathematics with technology. FluidMath reads handwritten math–the way we actually write in a math classroom–while effortlessly creating graphs, adding sliders, and solving equations to help keep users’ attention focused on thinking and mathematics. With FluidMath running in the background, this session will explore some unexpected properties of quadratic, linear, and exponential functions.
to infinity and beyond (november 5, 2012 – GISA 2012)
Co-presented with Nurfatimah Merchant based on ideas from PreCalculus Transformed
Presentation Description: This workshop applies the reciprocal transformation to various function families. It offers an innovative understanding of several pre-calculus topics (including exponential, logistic, rational and trigonometric functions), unifying several families through a handful of underlying mathematical structures. Participants will discover that many historically complicated problems and concepts are both richer and greatly simplified in this process.
RE-ENVISIONING POLAR GRAPHING (OCTOBER 19, 2012 – GCTM & November 5, 2012 – GISA) – by Nurfatimah Merchant
Presentation File: pdf
All the world’s a polynomial (OCTOBER 19, 2012 – GCTM;
NOVEMBER 5, 2012 – GISA)
Presentation Description: Using statistical regressions, you can find polynomial approximations to almost all functions traditionally taught in high school mathematics. Using nothing deeper than Algebra II and basic statistical regressions on a graphing calculator, this session will show how to compute surprisingly accurate polynomial approximations to three fundamental functions. It will conclude with some optional connections to complex numbers, calculus, and the inadequacy of correlation coefficients.
Presentation File: Keynote
INTEGRATING CAS (october 19, 2012 – GCTM 2012)
Presentation Description: Computer Algebra Systems (CAS) have been available in the presenter’s classes for the past decade. This presentation explores how mathematics classes can be dramatically enhanced when a CAS (TI Nspire CAS and/or Geogebra) is available for learning. Examples from a wide range of Algebra II, PreCalculus, and Calculus topics will include, among others, sources from textbooks that recently have incorporated CAS in central and ancillary roles.
Presentation File Formats: Keynote, PowerPoint, pdf
INTEGRATING CAS (September 7, 2012 – MMC – Chicago, IL)
Presentation Description: Mathematics learning is dramatically enhanced when CAS is an integral, omnipresent tool for students. The presentation was designed to showcase problems from Algebra, PreCalculus, and Calculus that are much deeper when CAS is used for exploration.
Fortunately (or unfortunately), I way over-prepared and accomplished only about half of what I had available. Even so, I’ve posted my presentation file in multiple formats along with all of the Nspire and GeoGeogebra files I created in support. Hopefully they’ll serve as an inspiration to you to explore, learn a little you didn’t know before, and ideally start some great conversations about teaching and learning mathematics.
Presentation File Formats: Keynote, PowerPoint, pdf
Nspire Files: LinReg Derivation, QuadSurprises, Quadratic_Forms, QuadAreaSurprises, 3points_rotated, Cubics, Probability, Straightening, CeilingsFloors
GeoGebra Files: 3points, 3points_rotated
PreCalculus Transformed (July 9-11, 2012 – Atlanta, GA)
Co-presented with Nurfatimah Merchant based on ideas from PreCalculus Transformed
Workshop Description: PreCalculus Transformed highlights the under-explored role of non-standard transformations and function composition in learning algebra and precalculus concepts. Families of functions are identified first by immutable distinguishing characteristics and then modified through multiple representations & transformations. Participants will discover that many historically complicated precalculus problems and concepts are both richer and greatly simplified in this process. The course integrates computer algebra system (CAS) technology, but it is certainly possible to use and grasp its concepts without this technology. Potential topics include expanded transformations, polynomials, rational functions, exponentials, logistics, and trigonometric functions. Additional topics may be explored depending on time and participant needs or experience. Textbook is included in the workshop price.
INtegrating CAS (T3 International 2012 – Chicago)
Session Description: This workshop explores how mathematics classes can be dramatically enhanced when CAS is an integral, omnipresent tool for learning.
NOTE: For those who attended my 3/3 session, I’ve repaired the issue with the CeilingsFloors.tns file. It was an issue with me trying to define the same variable twice in one document. Sorry for the session slip up. All works perfectly now.
Session Description: This workshop applies the reciprocal transformation to various function families. It offers an innovative understanding of several pre-calculus topics (including exponential, logistic, rational and trigonometric functions), unifying several families through a handful of underlying mathematical structures. Participants will discover that many historically complicated problems and concepts are both richer and greatly simplified in this process. The session concludes with a novel investigation of polar functions.
Bending Asymptotes and Bouncing off Infinity (GISA 2011)
Session Description: A deep understanding of polynomial behavior and transformations dramatically simplifies the graphing of any rational function (and beyond). This session offers an innovative approach to analyzing rational functions and more, via transformations far beyond simple stretches and slides. It explores ways to use Computer Algebra Systems (CAS) at all levels to facilitate students’ exploration and enhance their understanding of the concepts. While CAS is used in this presentation, the ideas can definitely be taught without it. Primary presenter: Nurfatimah Merchant
All the World’s a Polynomial … (GCTM 2011)
Session Description: Historically, students struggle to understand the utility and origins of Taylor Series. This session makes use of local linearity and statistical regressions to explain tangent lines in a way that is useful to all AP Calculus students before extending the approach to create Taylor Series for AP Calculus BC. This introduction is understandable by both pre-calculus and calculus students. The session will conclude with a student project around a famous Euler problem and techniques for using series to connect circular and hyperbolic trigonometry.
PreCalculus: Transformed & Nspired
Session Description: This workshop offers an innovative understanding of pre-calculus concepts through nonstandard transformations, allowing functions and concepts to be unified by a handful of underlying mathematical structures. It provides approaches that dramatically simplify many initially complicated-looking problems. CAS-enhanced ideas are presented.
Additional notes: Co-presented with Nurfatimah Merchant based on ideas from PreCalculus Transformed
Presentation file: NspiredPrecalculus (blank)
Conics within Conics
Session Description: This session presents the family of conic sections by connecting their algebraic and graphical representations, showing how each section can evolve from the others. The conclusion is a surprisingly elegant conic property and a 9th grader’s proof submitted for publication.
Additional notes: I will upload the proof for the property after the publication has been confirmed.
tns files: Conics within conics
Why the Second Derivative Test is Better (GCTM – Oct 2010)
Session Description: Every single-variable calculus class introduces both the first and second derivative tests as tools to classify function extrema. Unfortunately, we don’t always explore the pros and cons of the two tests, nor do we help students understand when the first derivative test is limited. This session will provide a rationale for why the second derivative test should be presented as the superior test for extrema.
Additional notes: Basically, the FDT is direction-dependent. If you can be certain that you can order the values you plug into your relation, then the FDT can be applied. For all non-Cartesian presentation forms of relations (polar, parametric, differential equations, vector), left-right directionality cannot be guaranteed, so the SDT should be employed.
Presentation file: The Second Derivative Test is Better