Exploring Functions ...
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Summary & Vitae  | Complete Description |  Handout |

Title:     Exploring Functions through the Use of Manipulatives
50-Word Summary
    The audience will use manipulatives to consider a "parent function," f(x).   Using translation and reflection techniques,
consider related functions, functions of the form y = af(x - h) + k
and a possible inverse.
    Solve equations and systems.
    Discuss and compare functions.
    Explain and sketch rational and envelope functions.
Brief Vitae of Presenter
  Agnes (A2) Azzolino, recipient of a 1997 AMATYC Award for Teaching Excellence, was president of MATYCNJ in both '86-'87 and '95-'96.
    Though a full time two-year college math teacher, she is also president of Mathematical Concepts, inc. She created the Graphing with Manipulatives line of products including the Parent Function Pack, the Demo Pack, Slopemeters and Isolator, Absolute Value Dilators, and the Teacher Pack, all distributed by Mathematical Concepts.
    Her publications include: Math Games for the Young Child, How to Use Writing to Teach Mathematics, Math Spoken Here!, Exploring Functions through the Use of Manipulatives. and NCTM's classic Mathematics and Humor.
    She is a Life Member of NCTM and AMATYC. She has presented minicourses at AMATYC and MAA-AMS national meetings and papers at these as well as at NCTM and NADE national meetings and at ICME-7. Agnes earned BA and MA degrees in Mathematics Education from Montclair State College.
Brief description of presentation style
    The audience will work in workshop format with manipulatives.
    The presenter will demonstrate each item and lecture briefly at the start and end of presentation.
Complete Description of Proposed Session
    The audience will use manipulatives to graph f(x), parent functions. The audience will given f(x), consider:
  • f(x) + k,
  • f(x - h),
  • -f(x),
  • f(-x),
  • f(x - h) + k,
  • f(y),
  • 1/f(x), and
  • f-1(x).
    Describe a function.
    Compare input to output.
    Compare two functions.
    Generalize about a family of expressions.
    Solve traditional and nontraditional equations and systems (as a end in itself and as a precursor to calculator or computer solution).
    Examine functions of the form y = a(x)f(x), power, rational, and envelope functions.
    Consider the implications and manipulative/concrete presentation techniques.

  • Manipulatives for Function Exploration:
            Slopemeter, Isolator, Absolute Value Dilator
  • The Possibilities with Concrete Thought
A Closer Look at Functions
  • Speaking, Hearing, and Writing about Functions and their Graphs
  • Visualizing Functions and Graphs
  • The Slope
    -- of Curves,
    -- of a Curve at a Point
  • Reflections on Y = X
  • Inversely So: Finding the Potential Inverse Functions
  • Considering the Slope of Inverse Functions
  • Solving Systems
Families of Functions: Translations, Reflections
  • Parent Functions:
            x, -x, 1/x, x, x2,
            |x|, exp(x), ln(x), sin(x), cos(x), tan(x),
  • Early Examples of the Addition of Functions: y = f(x) + k
  • Early Examples of the Product of Functions: y = af(x)
  • Early Examples of the Composition of Functions: y = g(f(x))
  • More on the Composition of Functions: y = g(f(x))
  • Math Exercises
  • Food for Thought:Dilation
    Why does a cubic look like a cubic?
    Why does the tangent function behave the way it does?
    The Unit Circle Manipulatively
    Why does a rational function look have the properties it has?
    Dilation by a Constant Functions: y = af(x)
    More on the Addition of Functions: y = f(x) + g(x)
    More on the Product of Functions: y = a(x)f(x)
  • Rules of Manipulative Use
  • Strategies for Change
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