Page 1 Developing an Intuitive Grasp of Exponential Functions from Real World Examples David M Rheam Department Of Physics SUNY Buffalo State College 1300 Elmwood Avenue Buffalo NY 14222 teamrheam gmail com This manuscript was completed as a requirement for PHY 690 Masters Project at SUNYBuffalo State College Department of Physics under the supervision of Dr Dan MacIsaac Dr David Abbott also contributed comments and insights Page 2 Abstract Albert A Bartlett 1976 stated The greatest shortcoming of the human race is our inability to understand the exponential function p 394 It is unsurprising that many of our high school students have difficulty grasping concepts involving exponentials Weber 2002 One main reason for this is that students enter our classrooms with their own ideas about the way things grow and decay from what they have seen in their own life experience Unfortunately most of what they see is not exponential It is extremely difficult to change deep seated student beliefs Students cannot simply be taught a formula or shown a graph dealing with exponential growth and decay and be expected to understand how it works Rather students need to build their own understanding of new concepts Alagic and Palenz 2006 Real life experiences in our classrooms in which students can explore exponential functions select representations and make connections can make their learning more meaningful Greeno Hall 1997 Not only are exponential functions essential to mathematics they are also embedded in the sciences and provide a model for representing growth and decay in real world phenomena Strom 2006 Here I describe several possible classroom experiences that can help students discover exponential functions and then connect them to some important topics in the realm of physics I also take an in depth look at compound interest as a physical example of e which is the essential link to move from understanding discrete exponential functions in our classroom examples to grasping the continuous exponential functions that are explored in physics Page 3 Developing an Intuitive Grasp of Exponential Functions from Real World Examples Introduction Albert A Bartlett 1976 stated The greatest shortcoming of the human race is our inability to understand the exponential function p 394 It is unsurprising that many high school students have serious difficulty grasping concepts involving exponentials Weber 2002 One main reason for this is that students enter the classroom with their own extensive na ve observations and ideas about the way things grow and decay from their own life experiences Students see the way they themselves develop or plants and animals around them grow or watch the way a candle shrinks as it burns and they formulate ideas about how growth and decay happen Unfortunately most of what students see is not exponential but linear growth and decay instead The concept of linear growth is then reinforced at school as students study linear relationships as a central theme in algebra Students have many real life and classroom experiences with linear growth and because of this many students still revert back to linear representations when they first start to deal with exponential growth Alagic Palenz 2006 It is extremely difficult to change deep seated student beliefs Students cannot simply be taught a formula or shown a graph dealing with exponential growth and decay and be expected to understand how it works Rather students need to build their own understanding of new concepts Alagic and Palenz 2006 Conventional lessons on functions use a correspondence approach that begins by establishing a rule that connects x values and y values usually in the form an equation However research shows it is often more powerful to use a covariational approach where students first work to fill in the table of x values and y values by an operation they create using the context of a real life problem Confrey and Smith 1994 These real life experiences in classrooms in which our students can explore exponential functions select Page 4 representations and make connections can make their learning more meaningful Greeno Hall 1997 The Nation Council of Teachers of Mathematics Principles and Standards has recognized and emphasized the importance of function development using real world examples as well NCTM 2000 Not only are exponential functions essential to mathematics they are also embedded in the sciences and provide a model for representing growth and decay in real world phenomena Strom 2006 Here I describe several possible classroom experiences that can help students discover exponential functions and then connect them to some important topics in the realm of physics I also take an in depth look at compound interest as a physical example of e which is the essential link to move from understanding discrete exponential functions in our classroom examples to grasping the continuous exponential functions that are explored in physics Classroom Examples of Exponential Growth Chessboards and Rice Paper Folding When introducing the topic of exponential growth a simple approach is best by restricting arithmetic to the familiar multiplication division addition and subtraction Goldberg Shuman 1984 p 344 With this in mind one of this simplest ways to think of exponential growth is something that has a constant doubling period A good introduction to the idea of a doubling period and the power of exponentials is a lesson on the famous story of The King s Chessboard Birch 1988 Here a man requests as a reward to have one grain of rice for the first square and then asks the king to double it for each of the squares on the chessboard A short version of this story can be found online at http www cs berkeley edu vazirani algorithms chap8 pdf paragraph 4 Have students think independently about the strange request and whiteboard predictions about how much rice they think it would be Ask students whether they think the man would be Page 5 better off taking 10 000 grains of rice per day or some other linear relationship instead Alagic Palenz 2006 p 644 Then have students start to act out the man s request in groups of three or four having one student in each group record the data in a spreadsheet figure 1 The spreadsheet allows students to explore the data quickly and to see a graphical representation of the grains of rice figure 2 Alagic Palenz 2006 p 643 It is essential that students think hard about finding an equation that relates the
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