Page 1Developing an Intuitive Grasp of Exponential Functions from Real World ExamplesDavid M. RheamBuffalo State CollegeThis 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 2AbstractAlbert 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 3Developing an Intuitive Grasp of Exponential Functions from Real World ExamplesIntroductionAlbert 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 helpPage 4students discover exponential functions and then connect them to some important topics in the realm of physics. Exponential GrowthWhen first introducing to the topic of exponential growth “I say keep it simple! Use onlyas 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 amuseand 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 spreadsheetPage 5allows students to explore the data quickly and to see a graphical representation of what has beenhappening 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 whatthey 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 yourgraph 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
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