Lively Applications to Lively Applications to Introduce College Introduce College Algebra/Algebra/PrecalculusPrecalculusTopicsTopics

Reva Narasimhan Reva Narasimhan Kean University, NJKean University, NJIMACC, March 2009IMACC, March 2009

OverviewOverview

Introduction Introduction Encouraging mathematical thinkingEncouraging mathematical thinking Challenges in teaching college algebra Challenges in teaching college algebra

and and precalculusprecalculus Examples of lively applications to Examples of lively applications to

connect concepts and skillsconnect concepts and skills QuestionsQuestions

Mathematical ThinkingMathematical Thinking

How are problems approached? How are problems approached? Why is mathematical thinking Why is mathematical thinking

important?important?

The formula that ate your The formula that ate your 401(k)401(k) Gaussian Copula Function Gaussian Copula Function –– used for used for

valuation of mortgage backed valuation of mortgage backed derivativesderivatives

The The MegapixelMegapixel MythMyth

Does a 10MP camera give you a picture that Does a 10MP camera give you a picture that is twice the length and width of a picture is twice the length and width of a picture from a 5MP camera?from a 5MP camera?

No. The No. The megapixelsmegapixels refer to the amount of refer to the amount of area in the picture. So, while the area is area in the picture. So, while the area is doubled, the length and width are not. doubled, the length and width are not.

Where are the extra Where are the extra MPMP’’s?s? 5 MP: area of 1944 x 5 MP: area of 1944 x

2592 pixels. Printed at 2592 pixels. Printed at 180 dots per inch, 180 dots per inch, thatthat’’s about 11 by 14 s about 11 by 14 inches. inches.

10MP: area 2736 x 10MP: area 2736 x 3648 pixels. An 1803648 pixels. An 180--dpi print thatdpi print that’’s about s about 15 by 20 inches15 by 20 inches——under three inches under three inches more on each margin more on each margin

(Source: David Pogue, (Source: David Pogue, The The New York TimesNew York Times))

ChallengesChallenges

Making connections between Making connections between topicstopics

Increasing student interactionIncreasing student interaction

Balance of skills and technology and Balance of skills and technology and applications and applications and ……

Time constraints in covering materialTime constraints in covering material

How can applications How can applications help?help? Start with an example in a familiar Start with an example in a familiar

contextcontext Work with the example and obtain Work with the example and obtain

new insightsnew insights Use the example to introduce a new Use the example to introduce a new

ideaidea

Making ConnectionsMaking Connections

Application Application –– EbayEbay Objective Objective –– to introduce piecewise to introduce piecewise

functionsfunctions

On the online auction site Ebay, the next highest amount On the online auction site Ebay, the next highest amount that one may bid is based on the current price of the item that one may bid is based on the current price of the item according to this table. The according to this table. The bid incrementbid increment is the amount is the amount by which a bid will be raised each time the current bid is by which a bid will be raised each time the current bid is outdoneoutdone

$ 2.50$ 100.00 - $ 249.99

$ 1.00$ 25.00 - $ 99.99

$ 0.50$ 5.00 - $ 24.99

$ 0.25$ 1.00 - $ 4.99

$ 0.05$ 0.01 - $ 0.99

Bid IncrementCurrent Price

For example, if the current price of an item is $7.50, then the For example, if the current price of an item is $7.50, then the next bid must be at least $0.50 higher.next bid must be at least $0.50 higher.

QuestionsQuestions

Explain why the bid increment, Explain why the bid increment, II, is a function of , is a function of the price, the price, pp..

Find I(2.50) and interpret it.Find I(2.50) and interpret it. Find I(175) and interpret it.Find I(175) and interpret it. What is the domain and range of the function What is the domain and range of the function I I ?? Graph this function. What do you observe?Graph this function. What do you observe? The function The function II is given in tabular form. Is it is given in tabular form. Is it

possible to find just one expression for I which possible to find just one expression for I which will work for all values of the price will work for all values of the price pp? Explain.? Explain.

This gives the entry way to define the function This gives the entry way to define the function notation for piecewise functions.notation for piecewise functions.

Follow upFollow up

What next?What next? Introduce the idea of piecewise functions.Introduce the idea of piecewise functions. Introduce the function notation associated with Introduce the function notation associated with

piecewise functions. Use a simple case first, and piecewise functions. Use a simple case first, and then extend. Relate back to the tabular form of then extend. Relate back to the tabular form of functions.functions.

Practice the symbolic form of piecewise Practice the symbolic form of piecewise functions.functions.

Graph more piecewise functions. Relate to the Graph more piecewise functions. Relate to the table and symbolic form for piecewise functions.table and symbolic form for piecewise functions.

Time Constraints Time Constraints

Explore multiple ideas using a single Explore multiple ideas using a single exampleexample

JustJust--inin--time introduction of new time introduction of new algebraic skills algebraic skills

Amazon rainforest Amazon rainforest -- 19751975

Source: Google Earth

Amazon rainforest Amazon rainforest -- 20092009

Source: Google Earth

Making ConnectionsMaking Connections

Application Application –– Rainforest declineRainforest decline Objective Objective –– to introduce exponential to introduce exponential

functionsfunctions

The total area of the worldThe total area of the world’’s tropical rainforests have been s tropical rainforests have been declining at a rate of approximately 8% every ten years. declining at a rate of approximately 8% every ten years. Put another way, 92% of the total area of rainforests will Put another way, 92% of the total area of rainforests will be retained 10 years from now. For illustration, consider a be retained 10 years from now. For illustration, consider a 10000 square kilometer area of rainforest. (Source: World 10000 square kilometer area of rainforest. (Source: World Resources Institute)Resources Institute)

Fill in the following chartFill in the following chart

606050504040303020201010

100001000000

Forest Forest acreage(sqacreage(sq km) km) Years in the Years in the future future

QuestionsQuestions

Assume that the given trend will continue. Fill in the table to Assume that the given trend will continue. Fill in the table to see how much of this rainforest will remain in 90 years.see how much of this rainforest will remain in 90 years.

Plot the points in the table above, using the number of years inPlot the points in the table above, using the number of years inthe horizontal axis and the total acreage in the vertical axis. the horizontal axis and the total acreage in the vertical axis. What do you observe?What do you observe?

From your table, approximately how long will it take for the From your table, approximately how long will it take for the acreage of the given region to decline to half its original sizeacreage of the given region to decline to half its original size??

Can you give an expression for the total acreage of rainforest Can you give an expression for the total acreage of rainforest after after tt years? (Hint: Think of years? (Hint: Think of tt in multiples of 10.)in multiples of 10.)

Use this as the entry to give a short introduction to exponentiaUse this as the entry to give a short introduction to exponential l functions.functions.

What next?What next?

Connect the table with symbolic and graphical Connect the table with symbolic and graphical representations of the exponential function.representations of the exponential function.

Discuss exponential growth and decay, with Discuss exponential growth and decay, with particular attention to the effect of the base.particular attention to the effect of the base.

Discuss why the decay can never reach zero.Discuss why the decay can never reach zero. Expand problem to introduce techniques for Expand problem to introduce techniques for

solutions of exponential equations.solutions of exponential equations. If using technology, incorporate it from the outset If using technology, incorporate it from the outset

to explore graphs of exponential functions and to to explore graphs of exponential functions and to find solutions of exponential equations.find solutions of exponential equations.

Video Games and Dot Video Games and Dot ProductsProducts

7.6.9

Source: R. Narasimhan, Precalculus

Balancing TechnologyBalancing Technology

What is the proper role of technology?What is the proper role of technology? Explore the nature of functionsExplore the nature of functions Enhance conceptsEnhance concepts Aid in visualizationAid in visualization Attempt problem of a scope not Attempt problem of a scope not

possible with pencil and paper possible with pencil and paper techniquestechniques

Balancing TechnologyBalancing Technology

Application Application –– Trigonometric Trigonometric functionsfunctions

Objective Objective –– to examine the role to examine the role of period and amplitude by of period and amplitude by studying the studying the length of daylight length of daylight in various locations using an in various locations using an ExcelExcel

Balancing TechnologyBalancing Technology

Application Application –– Curve fitting with dataCurve fitting with data Objective Objective –– to introduce piecewise to introduce piecewise

polynomial models and to see how polynomial models and to see how technology can be properly technology can be properly integratedintegrated

You have introduced polynomials and would like to introduce You have introduced polynomials and would like to introduce using curve fitting for polynomial models. This application using curve fitting for polynomial models. This application is written in that context. is written in that context.

0.00.0

0.50.5

1.01.0

1.51.5

2.02.0

2.52.5

3.03.0

2000200019901990198019801970197019601960

Attendance in Yellowstone National Park

(in millions)

2,838,2332000

2,823,5721990

2,000,2731980

2,297,2901970

1,443,2881960

AttendanceYear

QuestionsQuestions

Explain why a linear or a quadratic function would not Explain why a linear or a quadratic function would not model this data set well.model this data set well.

Describe the trend in the data. What do you observe?Describe the trend in the data. What do you observe? Plot the set of points given in the plane on the Plot the set of points given in the plane on the xx--yy

coordinate plane. Connect the points with a smooth curve. coordinate plane. Connect the points with a smooth curve. What do you observe?What do you observe?

Find a best fit cubic for the data between 1960 and 1990.Find a best fit cubic for the data between 1960 and 1990. Since the attendance did not change appreciably between Since the attendance did not change appreciably between

1990 and 2000, this portion can be modeled by a constant 1990 and 2000, this portion can be modeled by a constant function. How would you choose this constant?function. How would you choose this constant?

Write down the expression for a piecewise function Write down the expression for a piecewise function modeling this data set from 1960 to 2000.modeling this data set from 1960 to 2000.

Follow upFollow up

Ask why just a polynomial model may not Ask why just a polynomial model may not be suitable in the long run.be suitable in the long run.

Ask whether the choice of model is Ask whether the choice of model is unique.unique.

Ask to compare values from the fitted Ask to compare values from the fitted function to the actual values to judge function to the actual values to judge effectiveness of the model.effectiveness of the model.

Introduce solutions of equations within Introduce solutions of equations within this context. How would the expressions this context. How would the expressions for setting up the equations change?for setting up the equations change?

Creating an ApplicationCreating an Application

Examine the steps involved in creating Examine the steps involved in creating the relative humidity problemthe relative humidity problem

Gather data and information Gather data and information from the from the InternetInternet

Input data into spreadsheet and graphInput data into spreadsheet and graph Find the line of best fit since the trend Find the line of best fit since the trend

is approximately linearis approximately linear

PedagogyPedagogy

Using functions early and oftenUsing functions early and often Reducing Reducing ““algebra fatiguealgebra fatigue”” MultiMulti--step problems pull together step problems pull together

various concepts and skills in one various concepts and skills in one settingsetting

A simple idea is built upon and A simple idea is built upon and extendedextended

SummarySummary

Lively applications hold student Lively applications hold student interest and get them to connect with interest and get them to connect with the mathematics they are learning.the mathematics they are learning.

New algebraic skills that are New algebraic skills that are introduced are now in some context. introduced are now in some context.

Gives some rationale for why we Gives some rationale for why we define mathematical objects the way define mathematical objects the way we do.we do.

Contact InformationContact Information

Email:Email:rnarasim@kean.edurnarasim@kean.edu

Web: Web: http://http://www.collegemath.infowww.collegemath.info