fun with functions and technology reva narasimhan associate professor of mathematics kean...

Fun with Functions and TechnologyReva NarasimhanAssociate Professor of Mathematics Kean University, NJwww.mymathspace.net/presentations

Overview

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

skills• Using technology• Questions

Common Core

• 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

NCTM Atlantic Cty October 2011

Common Core

• 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

Functions and the 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.

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

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?

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.

Tips in a classroom

• 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

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

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

GeoGebra

• 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.

•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 dynamicGraphing 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.

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.

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:

rnarasim@kean.edu

• Web:

http://www.mymathspace.net/presentations