Transcript Fun with Functions and Technology

Fun with Functions and Technology
Reva Narasimhan
Associate Professor of Mathematics
Kean University, NJ
www.mymathspace.net/presentations
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Introduction
Why functions?
Challenges in teaching the function concept
Examples of lively applications to connect concepts and
skills
• Using technology
• Questions
Overview
• Functions Overview
• Interpreting Functions
• Understand the concept of a function and use function notation
• Interpret functions that arise in applications in terms of the context
• Analyze functions using different representations
• Building Functions
• Build a function that models a relationship between two quantities
• Build new functions from existing functions
Common Core
NCTM Atlantic Cty October 2011
• Linear, Quadratic, and Exponential Models
• Construct and compare linear and exponential models and
solve problems
• Interpret expressions for functions in terms of the situation
they model
• Trigonometric Functions
• Extend the domain of trigonometric functions using the unit
circle
• Model periodic phenomena with trigonometric functions
• Prove and apply trigonometric identities
Common Core
• Sample curriculum documents - These documents
represent how the concepts and skills described in the
Common Core State Standards for Mathematics might be
developed across the course of a school year.
• Functions and the common core - Various animations
show the increasing complexity of the functions strand.
• Sample assessment - Algebra assessments through the
Common Core, Grades 6-12. Note the level of scaffodling
present in the given examples.
Functions and the
Common Core
• Start with an example in a familiar context
• Work with the example and obtain new insights
• Use the example to introduce a new idea
How can applications
help?
Making Connections
• Application – Phone plan comparison
• Objective – to introduce inequalities and
function notation
The Verizon phone company in New Jersey has two plans
for local toll calls:
• Plan A charges $4.00 per month plus 8 cents per minute
for every local toll minute used per month.
• Plan B charges a flat rate of $20 per month regardless of
the number of minutes used per month.
Your task is to figure out which plan is more economical
and under what conditions.
Phone plan comparison to
introduce linear inequalities
• Write an expression for the monthly cost for Plan A,
using the number of minutes as the input variable.
• What kind of function did you obtain?
• What is the y-intercept of the graph of this function and
what does it signify?
• What is the slope of this function and what does it
signify?
Questions
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Introduce new algebraic skills to proceed further.
Practice algebraic skills
Revisit problem and finish up
Develop other what-if scenarios which build on this
model.
• Discuss limitation of model
• If technology is used, how would it be incorporated
within this unit?
What next?
Amazon rainforest - 1975
Source: Google Earth
Amazon rainforest - 2009
Source: Google Earth
Making Connections
• Application – Rainforest decline
• Objective – to introduce exponential
functions
The total area of the world’s tropical rainforests have been
declining at a rate of approximately 8% every ten years. Put
another way, 92% of the total area of rainforests will be retained
ten years from now. For illustration, consider a 10000 square
kilometer area of rainforest. (Source: World Resources Institute)
Fill in the following chart
Years in Forest acreage(sq km)
the
future
0
10
20
30
40
50
60
10000
Questions
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Assume that the given trend will continue. Fill in the table to see how
much of this rainforest will remain in 90 years.
Plot the points in the table above, using the number of years in the
horizontal axis and the total acreage in the vertical axis. What do you
observe?
From your table, approximately how long will it take for the acreage
of the given region to decline to half its original size?
Can you give an expression for the total acreage of rainforest after t
years? (Hint: Think of t in multiples of 10.)
Use this as the entry to give a short introduction to exponential functions.
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Connect the table with symbolic and graphical
representations of the exponential function.
Discuss exponential growth and decay, with particular
attention to the effect of the base.
Discuss why the decay can never reach zero.
Expand problem to introduce techniques for solutions
of exponential equations.
If using technology, incorporate it from the outset to
explore graphs of exponential functions and to find
solutions of exponential equations.
What next?
• Emphasize “Just-in-time” algebraic skills – quick
factoring review to be followed by unit on quadratic
functions
• Common core standards on algebra go hand-in-hand with
the function standards
• Discuss word problems from text in class using the
multiple representational approach
• Whenever possible, use tables, graphs in addition to
symbolic manipulation
Tips in a classroom
Concepts and Connections
• Expressions, equations, functions
• Algebra and function : solutions of
equations are zeros of a related function
• Fluency in terminology – e.g. one does
not “solve” a function
• Working through function concepts such
as zeros, intercepts, asymptotes etc.
require algebraic skills
• Skills and concepts are not separate
entities
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What is the proper role of technology?
Explore the nature of functions
Enhance concepts
Aid in visualization
Attempt problem of a scope not possible with pencil and
paper techniques
Balancing Technology
• Free and open source software created by Markus
Hohenwarter of Austria
www.geogebra.org
• A multi-platform dynamic mathematics software that
joins geometry, algebra, tables, graphing, statistics
and calculus in one easy-to-use package.
GeoGebra
• Make associations between
the algebraic expression of
a function and its graph
• Add visual meaning to
solutions of equations
• Dynamic approach
GeoGebra
• Make associations between
the symbolic, tabular, and
graphical aspects of a function
• Powerful tool for solution of
problems
• Dynamic approach
Spreadsheet
• Free web based computer
algebra system
• Add visual meaning to
solutions of equations
• Can be interactive with a
plug-in
Wolfram|Alpha
• Make associations between
the algebraic expression of
a function and its graph
• Add visual meaning to
solutions of equations
• Not dynamic
Graphing calculators
Making Connections
• Application – Ebay
• Objective – to introduce piecewise
functions
On the online auction site Ebay, the next highest amount that one
may bid is based on the current price of the item according to
this table. The bid increment is the amount by which a bid will
be raised each time the current bid is outdone
Ebay minimum bid
increments
Current Price
Minimum Bid
Increment
$ 0.01 - $ 0.99
$ 0.05
$ 1.00 - $ 4.99
$ 0.25
$ 5.00 - $ 24.99
$ 0.50
For example, if the current price of an item is $7.50, then the
next bid must be at least $0.50 higher.
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Explain why the bid increment, I, is a function of the
price, p.
Find I(2.50) and interpret it.
Find I(175) and interpret it.
What is the domain and range of the function I ?
Graph this function. What do you observe?
The function I is given in tabular form. Is it possible
to find just one expression for I which will work for all
values of the price p? Explain.
This gives the entry way to define the function notation
for piecewise functions.
Questions
What next?
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Introduce the idea of piecewise functions.
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Introduce the function notation associated with
piecewise functions. Use a simple case first, and then
extend. Relate back to the tabular form of functions.
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Practice the symbolic form of piecewise functions.
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Graph more piecewise functions. Relate to the table
and symbolic form for piecewise functions.
Follow up
• Using functions early and often
• Reducing “algebra fatigue”
• Multi-step problems pull together various concepts and
skills in one setting
• A simple idea is built upon and extended
Pedagogy
• Lively applications hold student interest and get them to
connect with the mathematics they are learning.
• New algebraic skills that are introduced are now in some
context.
• Gives some rationale for why we define mathematical
objects the way we do.
Summary
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