Page 1 Developing an Intuitive Grasp of Exponential Functions from Real World Examples David M Rheam Buffalo State College This manuscript was competed as a requirement for PHY 690 Masters Project by the State University College of New York at Buffalo Department of Physics under 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 This being said it should not be surprising 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 experiences Unfortunately most of what they see is not exponential It is extremely difficult to change these deep seated beliefs that our students have without several encounters in our classrooms where they experience and analyze exponential growth I describe several possible classroom experiences that help students discover exponential functions and then connect them to some important topics in the realm of 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 This being said it should not be surprising that many of our high school students have serious 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 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 happens Unfortunately most of what students see is not exponential growth and decay but linear The concept of linear growth is then reinforced as students study it in our mathematics curricula as a central theme in algebra Students have countless real life experiences as well as several classroom experiences that all point to the same type of 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 p 640 It is extremely difficult to change these deep seated beliefs that our students have 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 they need to build their own understanding of new concepts Alagic and Palenz 2006 p 636 Real life experiences in our classroom where students can explore exponential functions selecting representations and making connections themselves can make their learning more meaningful Greeno Hall 1997 In this manuscript I describe several possible classroom experiences that can help Page 4 students discover exponential functions and then connect them to some important topics in the realm of physics Exponential Growth When first introducing to the topic of exponential growth I say keep it simple Use only as much mathematics as is needed by restricting your arithmetic to the familiar four 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 about a king and a chessboard According to the legend the game of chess was invented by the Brahmin Sissa to amuse and teach his king Asked by the grateful monarch what he wanted in return the wise man requested that the king place one grain of rice in the first square of the chessboard two in the second four in the third and so on doubling the amount of rice up to the 64th square http www cs berkeley edu vazirani algorithms chap8 pdf para 4 Have students think independently about the strange request and make predictions about how much rice they think it would be It may also be beneficial to ask students whether they think the man would be better off taking 10 000 grains of rice per day or some other linear relationship to compare it to Alagic Palenz 2006 p 644 Then have them start to act out the man s request in groups of three or four having one student in each group recording the data in a spreadsheet Microsoft Excel This spreadsheet should include columns for the square number the number of grains of rice it may be useful for them to write this out with multiplication or as powers of two and the total rice so far If it is possible having students come up with some or all of these categories themselves in the discussion after the story is ideal Using the spreadsheet Page 5 allows students to explore the data quickly and to see a graphical representation of what has been happening with the rice Alagic Palenz 2006 p 643 Also it is essential to encourage students to think hard about finding the equation that relates the number of grains of rice to the number of the square and Excel allows students to compare their equations to their results Doerr 2000 Making a four by four grid of the chessboard is helpful since it is impossible to complete this task For more details on this example see Lesson Plan 1 Exponential Growth Once the students have finished the task it is beneficial to have them reflect back on what they have just done by thinking through some key questions such as Compare the number of grains of rice on the 16th square to how much rice had been used on the previous 15 squares When did you know you were going to run out of rice What do you think would happen if your graph continued How much rice would be used on the 32nd square How about the last square How many squares do you have to move to quadruple the number of rice How many squares do you have to move to have ten times as much rice Using this one activity students have gained knowledge about exponential growth through multiple representations They counted the rice grains and put them on the grid wrote the numbers on the spreadsheet discovered the equation that links the grains of rice to the square that they are on and visualized it in a
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