In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Mason Maynard. His topic, from Precalculus: compound interest.
What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)
- A deposit of $3000 earns 2% interest compounded semiannually. How much money is in the bank after for 4 years?
- A deposit of $2150 earns 6% interest compounded quarterly. How much money is in the bank after for 6 years?
- A deposit of $495 earns 3% interest compounded annually. How much money is in the bank after for 3 years?
These word problems are some of the basic compound interest problems that your students learn how to do where you just plug in the correct values for their corresponding variables.
- If you invested $1,000 in an account paying an annual percentage rate (quoted rate) compounded daily (based on a bank year of 360 days) and you wanted to have $2,500 in your account at the end of your investment time, what interest rate would you need if the investment time were 1 year, 10 years, 20 years, 100 years?
- If you invested $500 in an account paying an annual percentage rate compounded quarterly , and you wanted to have $2,500 in your account at the end of your investment time, what interest rate would you need if the investment time were 1 year, 10 years, 20 years, 100 years?
These are the types of problems that get more difficult for the students. You want them to use compound interest to solve but then they must incorporate logs into their solutions because they are looking for time instead of interest.
How does this topic extend what your students should have learned in previous courses?
With compound interest, students first learn about the simple interest formula. The only main difference is that you start to include exponents with compound interests. Then when you introduce your students to compound interest, you start to get into some more complicated problems. After they learn about compound interest and its basic problems, then you transition into logs with your students. This is used in compound interest and instead of just looking for the interest that will be accumulated after a specific amount of time, you then shift the variable around that you are looking for. The most coming type of problem that refers to this is they give you all of the information except for the amount time it takes to get a certain amount of interest. The last thing that leads up to compound interest in Calculus is when you transition from calculating the amount of interested over specific time intervals and a specific amount of times you compound it to calculating it with compounding it continuously over a specific time interval.
How have different cultures throughout time used this topic in their society?
Interest is something you have to pay on a load. Depending on what side you are and how thinks go, you are either getting some more money back that what you invested or you are paying off a massive debt. Some think that the idea behind charging loans on interest came from the early days of neighbors loaning there cattle to one another. What is really unique about this is that the words in the Egyptian, ancient Greek and Sumerian languages is connected to cattle and their offspring. This leads some to believe that interest came about due to the natural increase of the herd that occurred when you loaned out your cattle.
The first evidence that comes of a compound interest problem dates back to 2000-1700 B.C. in Babylon. A clay tablet was found and the unique thing is that the interest rate use to solve it was not written. Some researchers assume that the rate was 20% due to that mainly all the other compound interest problems dating back closer to this used it. What is really crazy is that 20% worked to solve the problem. The only thing that was wrong was that the time was corresponding to the Babylon calendar of 360 days instead of our 365 days.
In 50 B.C. Cicero writes to a friend in Rome. The letter tells that he would not normally recognize more than 12 percent interest on a loan, even though a decree was passed which required money lenders to charge no more than 12 percent. Cicero would then write a few days later that they will pay back the loan in 6 years will 12 percent interest and more money will be added each year.
Compound Interest History: