Lively Applications in Precalculus

(c) 2009 Revathi Narasimhan All rights res(c) 2009 Revathi Narasimhan All rights reserved. erved.

Lively Applications to Lively Applications to Introduce College Introduce College Algebra/Precalculus Algebra/Precalculus TopicsTopics

Reva Narasimhan Reva Narasimhan

Kean University, NJKean University, NJ


OverviewOverview

Introduction Introduction Encouraging mathematical Encouraging mathematical

thinkingthinking Challenges in teaching college Challenges in teaching college

algebra and precalculusalgebra and precalculus 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 The formula that ate your 401(k)your 401(k) Gaussian Copula Function – used Gaussian Copula Function – used

for valuation of mortgage backed for valuation of mortgage backed derivativesderivatives


The Megapixel MythThe Megapixel Myth

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

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


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

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

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

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


ChallengesChallenges

Making connections Making connections between between topics topics

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 Start with an example in a

familiar contextfamiliar context Work with the example and Work with the example and

obtain new insightsobtain new insights Use the example to introduce a Use the example to introduce a

new ideanew idea


Making ConnectionsMaking Connections

Application – EbayApplication – Ebay Objective – to introduce Objective – to introduce

piecewise functionspiecewise functions

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


Current Price Bid Increment

$ 0.01 - $ 0.99 $ 0.05

$ 1.00 - $ 4.99 $ 0.25

$ 5.00 - $ 24.99 $ 0.50

$ 25.00 - $ 99.99 $ 1.00

$ 100.00 - $ 249.99 $ 2.50

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


QuestionsQuestions

Explain why the bid increment, Explain why the bid increment, II, is a function , is a function of the price, of 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 Introduce the function notation associated

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

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

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


Time Constraints Time Constraints

Explore multiple ideas using a Explore multiple ideas using a single examplesingle example

Just-in-time introduction of new Just-in-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 – Rainforest declineApplication – Rainforest decline Objective – to introduce Objective – to introduce

exponential functionsexponential functions The total area of the world’s tropical rainforests The total area of the world’s tropical rainforests

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


Fill in the following Fill in the following chartchart

Years in Years in the future the future

Forest acreage(sq km) Forest acreage(sq km)

00 1000010000

1010

2020

3030

4040

5050

6060


QuestionsQuestions

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

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

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

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

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


What next?What next?

Connect the table with symbolic and Connect the table with symbolic and graphical representations of the exponential graphical representations of the exponential function.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 If using technology, incorporate it from the

outset to explore graphs of exponential outset to explore graphs of exponential functions and to find solutions of exponential functions and to find solutions of exponential equations.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 What is the proper role of technology?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



Application – Trigonometric Application – Trigonometric functionsfunctions

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



Application – Curve fitting with Application – Curve fitting with datadata

Objective – to introduce Objective – to introduce piecewise polynomial models and piecewise polynomial models and to see how technology can be to see how technology can be properly integratedproperly integrated

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

Year Attendance

1960 1,443,288

1970 2,297,290

1980 2,000,273

1990 2,823,572

2000 2,838,233


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 x-y Plot the set of points given in the plane on the x-y

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

Find a best fit cubic for the data between 1960 and Find a best fit cubic for the data between 1960 and 1990.1990.

Since the attendance did not change appreciably Since the attendance did not change appreciably between 1990 and 2000, this portion can be modeled between 1990 and 2000, this portion can be modeled by a constant function. How would you choose this by a constant function. How would you choose this constant?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 Ask why just a polynomial model may not be suitable in the long run.not 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 Introduce solutions of equations within this context. How would the within this context. How would the expressions for setting up the expressions for setting up the equations change?equations change?


Creating an Creating an ApplicationApplication

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 from the Internetthe Internet

Input data into spreadsheet and Input data into spreadsheet and graphgraph

Find the line of best fit since the Find the line of best fit since the trend is approximately lineartrend is approximately linear


PedagogyPedagogy

Using functions early and oftenUsing functions early and often Reducing “algebra fatigue”Reducing “algebra fatigue” Multi-step problems pull together Multi-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 interest and get them to connect with the mathematics they are with the mathematics they are learning.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 define mathematical objects the way we do.way we do.


Contact InformationContact Information

Email:Email:

[email protected]@kean.edu

Web: Web:

http://www.mymathspace.nethttp://www.mymathspace.net

Education

Lively Applications in Precalculus