The activities in the exponential functions unit are based upon students' real life. If they just study the types of functions and don't relate their knowledge in explaining things happening in their life, there is no meaning of learning. In particular, the exponential functions can be a good model for increasing or decreasing functions which are not linear functions. But The activities in this exponential functions unit should follow after having a firm knowledge of linear and quadratic functions and studying a general idea of exponential functions. The examples which can show the forms of the exponential functions are plentiful in life. So the activities will be the combination of the real life based explorations and theoretical examinations with technologies. Growth of an investment, price increases due to inflation, interest owed while repaying loans, population growth, and radioactive decay are some examples of them. This unit is composed of two-day activity . The first day activity is answering some questions of a simple bank business problem and its extensions. The second day activity is examining some exponential functions with Algebra Xpressor. Students are supposed to be familiar with the shapes of different exponential functions through the second day activity. In addition, a logarithmic functions will be introduced by the concept of reflection. This is to offer an informal concept of the logarithmic functions for further study.
Question 1 : Suppose that Christy borrows $3,000 at the begging of every year from 1991 through 1996 at an annual interest rate of 10 %. How much money does she have to pay back at the end of 1996?
Answer : Students will make a table. A graphic calculator or spreadsheet can be used. A teacher is a
facilitator of this class and all activities are done by the students.
If a students use a TI-82 graphic calculator, then he obtains the following ( example 1 ).
3000*1.10 3300 1991
(Ans+3000)*1.10 6930 1992
25461.513 1996 ( TI-82 )
If a student use a spreadsheet, then she optains the following ( example 2 ).
In 1996, Christy owes $25,461.513. Since an algorithmic process is involved in the process of calculation the teacher needs to check the process of how the students can write "(Ans+3000)*1.10" in TI-82 and " =(a2+3000)*1.1" in the spreadsheet.
(example 3) 3000+(3000*0.10)=3000*1.10 in 1992
(3000+(3000*0.10)) *0.10+3000+(3000*1.10) =3000*(1.10) in 1993 ...
That is to say, the process of example 3 should be understood by the students.
Question 2 : Does Christy owe the same amount of money every five years?
Answer : The students have already looked at their graphs and the graphs showed that the money was not increased by the same amount. Therefore, the answer would come up quickly. "No, she does not." But, the answer could be difficult for some students without calculating the money difference since the teacher was asked to find the money in 5 years and the graph might look like a linear function. So the students are encouraged to examine the money in 10 years or more. The spreadsheet work can be effective in this case.
What are some of the characteristics of the graph? The students now can apply their knowledge of the exponential functions to the graph when they want to suggest things that will happen to the money in years 30, and 40?
Question 3 : Now, Christy wants to repay the loan including the interest from 1997. Assume that she repays $3,000 at the beginning of every year. In what year will Christy be free of debt?
In TI-82, (Ans-3000)*1.10 is used. The next data is obtained by the spreadsheet (example 4).
Answer : As the students can show in the graph above, in 16 years Christy is free of debt.
Question 4 : How about changing $3,000 into $1,000 or $2,000? Examine each repayment process after loaning the money for 5 years.
Answer : The amount of money that Christy borrowed does not matter in deciding the years that she needs to repay with the same rate. The students have to find the answer through their activity with the spreadsheet or the graphic calculator. But the use of spreadsheet is recommended (example 5).
Question 4 can cause another question. Then, what if the bank change the rate from 10% into less than or greater than 10%? This question can be raised naturally by the students. The teacher should derive the students if they need a help.
Question 5 : Examine the cases of interest rate 9%, 12%, and 13%.
For the case of 9% interest rate, the students can make a conjecture with it. The lower the interest rate is, the less the year is needed. But for the case of the other two cases, they have to make each table and check the results (example 6).
If Christy was 21 years old in 1991. She will be 60 years old when she repays the money with 12% interest rate. For the case of 13% interest rate, it is very interesting for the students to see the result.
Christy would not be free of debt even though she will be able to live upto 100 years old. The graphs show the fact clearly (example 7).
The students have to understand the dynamic fact of their daily life through the activity 1. The teacher gives a final wrap-up session for his students. The students are given a homework set with a similar type of problem.