Fun with Functions and Technology

1. Fun with Functions and Technology Reva Narasimhan Associate Professor of Mathematics Kean University, NJ www.mymathspace.net/presentations

2. Introduction • Why functions? • Challenges in teaching the function concept • Examples of lively applications to connect concepts and skills • Using technology • Questions Overview

3. 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 Common Core NCTM Atlantic Cty October 2011

4. 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 Common Core

5. 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. Functions and the Common Core

8. Start with an example in a familiar context • Work with the example and obtain new insights • Use the example to introduce a new idea How can applications help?

9. Making Connections • Application – Phone plan comparison • Objective – to introduce inequalities and function notation

10. 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 to introduce linear inequalities

11. 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

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?

13. Amazon rainforest - 1975 Source: Google Earth

14. Amazon rainforest - 2009 Source: Google Earth

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)

16. Fill in the following chart

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.

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?

19. 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 Tips in a classroom

20. 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

21. 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 Balancing Technology

22. Free and open source software created by Markus Hohenwarter of Austria www.geogebra.org • A multi-platform dynamicmathematics software that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. GeoGebra

23. Make associations between the algebraic expression of a function and its graph • Add visual meaning to solutions of equations • Dynamic approach GeoGebra

24. Make associations between the symbolic, tabular, and graphical aspects of a function • Powerful tool for solution of problems • Dynamic approach Spreadsheet

25. Free web based computer algebra system • Add visual meaning to solutions of equations • Can be interactive with a plug-in Wolfram|Alpha

26. Make associations between the algebraic expression of a function and its graph • Add visual meaning to solutions of equations • Not dynamic Graphing calculators

27. 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

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

29. 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

30. 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

31. 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 Pedagogy

32. 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

33. Email: rnarasim@kean.edu • Web: http://www.mymathspace.net/presentations Contact Information