Functions Summary

Monday - Friday, July 12-16, 2004


Project ideas
  • Functions on a plane
  • Parametric curves
  • Graphing calculators to facilitate domain/range or not using graphing calculators
  • Functions used for counting
  • Historical aspects of functions
  • Real world applications (modeling): functions in context
  • Functions related to matrices (linear algebra), transformational geometry, non-Euclidian
  • Domain/range, inverse in real world applications
  • One to one and onto
  • What do teachers need to know about functions?
  • Transformations and functions
  • Input/Output tables to graphs to equations
  • Generating vocabulary for functions from physical experiments
  • Sketchpad
  • Pictures of functions
  • (Original, creative ideas (not duplicating))
  • (Process is important, too, in producing product)


Plan of attack:

Divide up the topics, work on in pairs, and present on Thursday.
Splitting up into groups right away doesn't work.
We should specify which of our ideas from yesterday sound promising.

What would each of us like to pursue as a possible topic? Seth: Pre-calculus functions, better way to approach this unit (domain, range, inverses, composite functions)
Take one of these concepts (inverse, domain/range) and come up with engaging activities, problems
Rani: Engaging problems, one to one and onto, reflectional and rotational matrices, umbrella questions with supporting questions to solve bigger question
Peter: Recursion to explicit with functions to counting, modeling, over arching question
Dick and Celeste: Going from the graph to the rule, equation, and table. Also, domain, range, inverse and how are all of these related?
Kelley: Engaging problems like what Seth and Celeste/Dick suggested.
Lynda: Learn as much as possible about functions at the foundation level.
Marta: How would you teach this concept at different levels (integrated I, II, III)?
Domain/range graph to points and back.
Jim: Domain/range talked about with all functions from the beginning, from informal to formal
Connie: Do something with functions that would work with AP Calculus.
Joyce: Engaging activities with functions to use with AP Calculus students; sketchpad activities, projects.
Donna: Foundations of functions, tiers of levels, how things would work at different levels.

Break into groups of 2-3, work, and report back on Thursday.

Foundation level: Marta, Lynda, Donna
Graph to rules: Dick, Celeste
Pre-Calculus: Seth, Kelley
AP Calculus: Connie, Joyce
Transformations/Matrices: Rani, Peter


Reporting Back

Seth shared 6 examples of different kinds of functions which may or may not be able to be expressed using equations. These expressions provided a wonderful way for us as a group (with help from Jim and Dick) to discuss and learn more about the concepts of one to one and onto. Marta, Lynda, and Donna discussed using a Discovering Algebra type example. Create a lesson or concept to help students understand the relationship between coordinate points, tables and graphs. Connie and Joyce suggested that related rates is a difficult concept in Calculus and would like to come up with a way to make the concept more transparent, engaging, and accessible. Rani: is interested in looking at matrices. She is interested in using matrices for transformations.

Jim's example: Every student (x) and their student number (y)

  • It's a function; it's 1 - to - 1, but it's not onto (grades: post student numbers, but only last 5 digits)
  • Onto: if I make up a student number, there isn't necessarily a student to go with each number (also, like credit card numbers). Restricting the range allows you to make it onto.

If a function has an inverse, it's one to one and onto.
An example of onto but not one to one: a cubic function like y = (x+1)(x+2)(x+3).

Dick suggested the following example: Imagine a set of people in the room and the set of chairs in the room.
The domain is the people, the range is the chairs in the room. If there are not enough chairs, with two to a chair, it's still a function.
If Seth weighed 500 pounds and was sitting on two chairs, it would not be a function. If Rani is up writing on the board, she has no chair, so it's not a function. If Rani leaves or the domain is changed to only those sitting on a chair, it's back to being a function.
One to one: does everyone have a chair? Onto: is every chair taken?


Jim: representing functions (USA weather map - color to represent functions on the plane)
Temperature is a function. (Step function if you have a finite number of colors) Color as a function, you can represent an extra dimension) This allows you to analyze things about the graph (where is it changing the fastest, etc). Also a way to generate a function from the picture.
Lynda: Having kids graph their names (cursive) and have them come up with piece-wise functions to illustrate.
Jim: Drawing things in postscript (Adobe software - how your laser printer draws the letters, etc.) Illustration programs use piecewise quadratic and cubic functions.
Celeste: looking at graphs and trying to come up with a function. Given the graph (one corresponds to C), domain of numbers and range is the numbers. Graph to the table to function.
Jim: some kind of list of things (concepts) that are hard to teach about functions (i.e. trig functions, families of functions, functions as objects, function notation)

Important concepts/difficult concepts

  • families of functions
  • function notation
  • composition of functions
  • trig functions
  • piece-wise functions

Jim would like to (on Monday) look at what happens to functions (linear combination of two linear functions and where this goes) - spend about 15 minutes on Monday with graph paper. Also, we should look at Donna's hand-out.

Current working subgroups. We'd like to trim this down to two or three projects early next week. Activities need to explain the math.

Marta, Celeste, Lynda
Rani, Lynda
Connie, Joyce
Seth, Kelley

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