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# Fun with Functions

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Class activities to model and understand functions

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### Fun with Functions

1. 1. Fun with FunctionsReva NarasimhanAssociate Professor of MathematicsKean University, NJwww.mymathspace.net/nctm
2. 2. • Introduction• Why functions?• Challenges in teaching the function concept• Examples of lively applications to connect concepts and skills• QuestionsOverview(c) 2011 R. Narasimhan For non profit classroom use only
3. 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 componentsWhy Functions?(c) 2011 R. Narasimhan For non profit classroom use only
4. 4. Challenges• Making connections between math topics• Increasing student interaction• Balance of skills and technology and applications and …• Time constraints in covering material (c) 2011 R. Narasimhan For non profit classroom use only
5. 5. • Start with an example in a familiar context• Work with the example and obtain new insights• Use the example to introduce a new ideaHow can applicationshelp?(c) 2011 R. Narasimhan For non profit classroom use only
6. 6. • Explore multiple ideas using a single example • Just-in-time introduction of new algebraic skillsTime Constraints(c) 2011 R. Narasimhan For non profit classroom use only
7. 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 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. Exponential Decay(c) 2011 R. Narasimhan For non profit classroom use only
8. 8. Making Connections• Application – Phone plan comparison• Objective – to introduce inequalities and function notation (c) 2011 R. Narasimhan For non profit classroom use only
9. 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 tointroduce linear inequalities(c) 2011 R. Narasimhan For non profit classroom use only
10. 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(c) 2011 R. Narasimhan For non profit classroom use only
11. 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(c) 2011 R. Narasimhan For non profit classroom use only
12. 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?(c) 2011 R. Narasimhan For non profit classroom use only
13. 13. Amazon rainforest - 1975 Source: Google Earth (c) 2011 R. Narasimhan For non profit classroom use only
14. 14. Amazon rainforest - 2009 Source: Google Earth (c) 2011 R. Narasimhan For non profit classroom use only
15. 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) (c) 2011 R. Narasimhan For non profit classroom use only
16. 16. Fill in the following chartYears in the Forest acreage(sq km)future0 10000102030405060 (c) 2011 R. Narasimhan For non profit classroom use only
17. 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. (c) 2011 R. Narasimhan For non profit classroom use only
18. 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?(c) 2011 R. Narasimhan For non profit classroom use only
19. 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 (c) 2011 R. Narasimhan For non profit classroom use only
20. 20. Ebay minimum bidincrements 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.(c) 2011 R. Narasimhan For non profit classroom use only
21. 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(c) 2011 R. Narasimhan For non profit classroom use only
22. 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(c) 2011 R. Narasimhan For non profit classroom use only
23. 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 techniquesBalancing Technology(c) 2011 R. Narasimhan For non profit classroom use only
24. 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 extendedPedagogy(c) 2011 R. Narasimhan For non profit classroom use only
25. 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(c) 2011 R. Narasimhan For non profit classroom use only
26. 26. • Email:rnarasim@kean.edu• Web:http://www.mymathspace.net/nctmContact Information(c) 2011 R. Narasimhan For non profit classroom use only