Fun with Functions

Fun with FunctionsReva NarasimhanAssociate Professor of Mathematics Kean University, NJwww.mymathspace.net/nctm

(c) 2011 R. Narasimhan For non profit classroom use only

Overview

• Introduction • Why functions?• Challenges in teaching the function concept• Examples of lively applications to connect concepts and

skills• Questions


Why Functions?

• 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


Challenges• Making connections between math

topics• Increasing student interaction

• Balance of skills and technology and applications and …

• Time constraints in covering material


How can applications help?

• Start with an example in a familiar context• Work with the example and obtain new insights• Use the example to introduce a new idea


Time Constraints

• Explore multiple ideas using a single example• Just-in-time introduction of new algebraic skills


Exponential Decay

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 top4. 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.


Making Connections

• Application – Phone plan comparison• Objective – to introduce inequalities and

function notation


Phone plan comparison to introduce linear inequalities

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.


Questions

• 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

• 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?


What next?

• 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?


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 chartYears in the future

Forest acreage(sq km)

0 10000102030405060


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.


What next?

• 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.


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.


Questions

• 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.


Follow up

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.


Balancing Technology

• 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


Pedagogy

• 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


Summary

• 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.


Contact Information

• Email:

[email protected]

• Web:

http://www.mymathspace.net/nctm

Education

Fun with Functions