The document discusses teaching functions in mathematics. It provides examples of using real-world applications to introduce functions and related concepts in an engaging way. Some challenges in teaching functions are balancing skills practice with conceptual understanding and addressing time constraints. The document also discusses using technology to enhance function lessons without replacing pencil-and-paper work. Overall, it advocates for an application-driven approach to teaching functions.

1. Fun with Functions
Reva Narasimhan
Associate Professor of Mathematics
Kean University, NJ
www.mymathspace.net/nctm

2. • Introduction
• Why functions?
• Challenges in teaching the function concept
• Examples of lively applications to connect concepts and
skills
• Questions
Overview
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3. • Serves as a unifying theme in understanding concepts in
algebra through calculus and beyond
• The language of functions is extremely useful in
applications
• Has a graphical, symbolic, tabular and verbal components
Why Functions?
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4. Challenges
• Making connections between math
topics
• Increasing student interaction
• Balance of skills and technology and
applications and …
• Time constraints in covering material
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5. • 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?
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6. • Explore multiple ideas using a single example
• Just-in-time introduction of new algebraic skills
Time Constraints
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7. 1. Count how many total M&M’s there are in your
packet. This is the initial value.
2. Shake up M&M’s and drop the candies on a piece of
paper.
3. Count how many have “m” on top
4. Record on paper.
5. Remove the candies with no “m” on top and repeat
Steps 2-4 until no more candies are left.
6. Record results in the given table and model the
behavior.
Exponential Decay
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8. Making Connections
• Application – Phone plan comparison
• Objective – to introduce inequalities and
function notation
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9. 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
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10. • 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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11. • Write an expression for the monthly cost for Plan
B, 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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12. • 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?
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13. Amazon rainforest - 1975
Source: Google Earth
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14. Amazon rainforest - 2009
Source: Google Earth
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15. 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)
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16. Fill in the following chart
Years in the Forest acreage(sq km)
future
0 10000
10
20
30
40
50
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17. Questions
• 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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18. • 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?
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19. 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
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20. Ebay minimum bid
increments
Minimum Bid
Current Price
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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21. • 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
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22. What next?
• Introduce the idea of piecewise functions.
• Introduce the function notation associated with
piecewise functions. Use a simple case first, and then
extend. Relate back to the tabular form of functions.
• Practice the symbolic form of piecewise functions.
• Graph more piecewise functions. Relate to the table
and symbolic form for piecewise functions.
Follow up
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23. • 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
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24. • 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
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25. • 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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26. • Email:
rnarasim@kean.edu
• Web:
http://www.mymathspace.net/nctm
Contact Information
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