Teaching Dilemmas
In this section, we feature teacher reflections related to mathematics topics challenging to teach. These reflections include some common student difficulties and errors as well as ways of addressing these challenges If you have such a topic you wish to share with other practitioners in the field, please respond to the call for manuscripts.
A Math Lesson that Answers
“When Are We Going to Use This?”
Carrie Case
We all know that mathematics are vital to one’s life. One of the reasons is “it provides the ordinary citizen with quantitative tools needed to function competently in today’s complicated economies, essentially helping each citizen to make sound decisions.” (Gates & Yistro-Yu, 2003, pp.46) I believe that it is my job, as an educator, to show students the importance of the math that I am teaching whenever I can. Time constraints often prohibit a full “when are we going to use this” lesson. In these situations, I may show an example from a news article or from an advertisement that was presented in a misleading way or how a topic I have taught can help them to understand what is presented in the news, a journal article in their field, or a novel. It is this belief, along with a personal experience, that made me craft my “Let’s buy a car” lesson. In 1999, I went to buy a car. After negotiating the price down, I went to the finance office to finalize the loan. Since I am familiar with loan payment formulas and I can calculate a monthly payment, I knew what to expect for my monthly payment. However, the financier quoted a monthly payment that was too high and I informed him that his quote was too high. He responded that “it can’t be wrong because I put it in my spreadsheet.” I asked for a piece of paper and quickly showed him the math. To this day I do not know if the mistake in that company’s spreadsheet was there intentionally or just an honest mistake, but I do know that I paid exactly the price that I had negotiated plus the 0.9% annual interest. This is an example of effectively using math in a real life situation. I found the experience empowering and was able to save money by exposing the error in the spreadsheet!
At one point or another in our lives, most of us will purchase a car using a loan. In my car buying lesson, students use order of operations, rules of exponents, and find solutions to multi-step algebraic equations. They are able to answer questions about how much their monthly payments will be, how much they will pay over the life of a loan, and how much they will pay in interest over the life of the loan. There are multiple parts to this lesson, which also includes the impact of credit scores on interest rates.
Relevant SOLs and Prior Knowledge
In general, this lesson can be used when teaching about exponential functions, which may also include exponential growth or decay. Prior to teaching this lesson, I recommend that students are introduced to simple interest, compound interest, and the basic formulas for each. The basic formulas for simple interest, compound interest, and the monthly payments for a loan, are shown below.
Simple Interest
P = principal or amount deposited into an account
I = interest earned on the account
r = interest rate in decimal form (e.g. 1.5% = .015)
t = the time, in years, the money remains in the account
Compound Interest
A = Amount in the account after a specified time
with the compounded interest added
P = the principal initially deposited into the account
n = the number of times the interest is compounded (calculated) per year
t = time in years
r = annual interest rate in decimal form (e.g. 6.8% = 0.068)
Loan
PMT = payment
A = amount borrowed
r = annual interest rate in decimal form (e.g. 6.8% = 0.068)
n = number of annual compounding periods (monthly, bi-annually…)
t = time in years
I see this lesson aligning with the following high school courses and their subsequent SOLs.
Algebra II & Trigonometry
AII.1 The student, given rational, radical, or polynomial expressions, will
b) add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents;
AII/T.4 The student will solve, algebraically and graphically,
c) equations containing rational algebraic expressions
Math Analysis
MA.9 The student will investigate and identify the characteristics of exponential and logarithmic functions in order to graph these functions and solve equations and real-world problems. This will include the role of e, natural and common logarithms, laws of exponents and logarithms, and the solution of logarithmic and exponential equations.
Discrete Mathematics
DM.10 The student will use the recursive process and difference equations with the aid of appropriate technology to generate
a) compound interest;
The Assignment (requires approximately 50 minutes if cars are selected in advance)
You wish to buy a car and today you are going to do just that—well, at least you’re going to calculate the math to buy that car! (As part of the lead-in, I share my own experience from 1999 to illustrate to the students that they really will use this.)
Part 1
Using a newspaper ad or your laptop, find the new car of your choice. Use the manufacturer’s suggested retail price, MSRP, to determine the cost of your car or, if you can find a quoted lower price, you may use that. (If you are a decent negotiator, you will get this cost down on many models when actually purchasing.) Use this information and an average car annual loan rate of 6.5% compounded monthly—assuming you put no money down and finance the entire purchase to determine:
· How much your monthly payment will be on a 5 year loan?
· How much you will pay over the life of the 5 year loan?
· How much will you pay in interest on the 5 year loan?
· How much your monthly payment will be on a 3 year loan?
· How much you will pay over the life of the 3 year loan?
· How much will you pay in interest on the 3 year loan?
· What is the difference in the amount paid between the 5 and 3 year loans?
Part 2
When you entered the room you were handed a colored piece of paper. This colored paper indicates your credit score. You REALLY want a good credit score—the higher the better. Interest rates were valid at a local credit union as of March 1, 2016.
Table 1. Examples of credit scores assigned to students at the beginning of class |
||
Credit Score |
Interest Rate |
Color |
0-549 |
18% |
Red |
550-599 |
15.5% |
Gray |
600-639 |
12.25% |
Green |
640-679 |
8% |
Orange |
680-719 |
6.5% |
Pink |
720-850 |
5.75% |
Yellow |
You are purchasing a 2015 Chevy Cruze (base model). This car has an MSRP of $16,170.
Using the interest rate you get based on your credit score, determine:
· Your monthly payment for a 5 year loan
· How much you will pay over the life of the loan
· How much you pay in interest over the life of the loan
Part 3:
You know that you need to buy a car and that you want a 5 year loan. You also know that you need to keep your monthly payments below $300. If you were to qualify for a 6.5% interest rate, what is the most you can spend on your car?
General Information about the Assignment & Reflection on Lessons Learned
Historically, I had students shop for their cars at home prior to our lesson to save time. For them it was a fun homework assignment and one I never had to worry they would complete. I always had a few newspapers for students to shop from if they hadn’t shopped in advance but never had to use them. The 6.5% interest is the interest rate that was an average car loan rate for the last time I taught the class as quoted by the local credit union. This is also true for the interest rates based on credit scores. Each semester, I call a local credit union and ask for an average car loan rate as well as rates based on average credit scores. I had previously called a number of local banks, but only the credit union was willing to provide this detailed information after I explained why I was asking for it. You may not want to do this, but I prefer to use accurate and current loan information. Additionally, when handing out the colored papers that indicated the interest rate in part 2, I shuffle the papers and randomly hand them out as students walk into class.
Part 1
Students practice with the formula in part 1. An example solution is provided below for a 2015 Ford Mustang which has a base model price of $23,098.
How much your monthly payment will be on a 5 year loan:
· How much you will pay over the life of the 5 year loan:
· How much will you pay in interest on the 5 year loan:
· How much your monthly payment will be on a 3 year loan: $707.93
· How much you will pay over the life of the 3 year loan: $25485.48
· How much will you pay in interest on the 3 year loan: $2387.48
· What is the difference in the amount paid between the 5 and 3 year loans? $1630.92
The students become engaged in conversation when we look at the differences between the total amount paid when they have a three year loan or a five year loan. When we examine the monthly payments, students notice that a three year loan is often prohibitively expensive for many people. Students are also surprised at how much less they would pay in total for the car when the same interest is applied to a three year loan instead of a five year loan. This was always a great source of conversation among my students. I do not try to guide the students in this conversation, but instead just listen and let them come to their own conclusions about which aspect was more important to them at the time: a lower monthly payment or a lower total cost. Often, students are shocked by the amount of the interest they would have to pay and more often they realize that this is a huge monthly expense. They also see the direct application of various math topics they are learning.
Part 2
Once again, students are asked to compute the payment they will make per month, the total paid and the amount of interest paid over the life of the loan. At this point, the lesson turns to comparing all of the different interest rates and how a larger interest rate quickly raises the overall cost of the vehicle, which is a direct variation. See Table 2.
Table 2. Example of Student’s Comparison of Interest and Loan Payments |
|||||
Color |
Interest Rate |
Car Cost |
Monthly Payment |
Total over Loan |
Paid in interest |
Yellow |
0.0575 |
16,170.00 |
310.74 |
18,644.11 |
2,474.11 |
Pink |
0.065 |
16,170.00 |
316.38 |
18,983.08 |
2,813.08 |
Orange |
0.08 |
16,170.00 |
327.87 |
19,672.16 |
3,502.16 |
Green |
0.1225 |
16,170.00 |
361.74 |
21,704.33 |
5,534.33 |
Gray |
0.155 |
16,170.00 |
388.94 |
23,336.41 |
7,166.41 |
Red |
0.18 |
16,170.00 |
410.61 |
24,636.70 |
8,466.70 |
Students are surprised again, but this time, by the difference higher interest rates make in not only the monthly payment, but also the amount of interest paid over the life of the loan. Frequently, I heard things comments such as, “do you know what I could do with that $8000—it’s half of another car!”
We then use the table of student answers to create a scatterplot of interest rate vs. total amount paid over the life of the loan, which clearly shows this direct variation relationship.
The scatter plot of interest rate by either total cost of the car over the life of the loan, or the monthly payment, or the amount paid in interest also shows direct variation. The amount paid in interest over the life of the loan is a powerful message to students about how quickly interest adds up.
Part 3:
Finally, in part three, students learn how to work backwards or use constraints. If they know the monthly payment they can afford, then they can determine how much they can pay for the car.
In this example, the interest rate is set, but you could certainly use the interest rates from part two in order to show the impact that different interest rates have on the monthly payments, which determines the maximum price of a car they can afford. Below, I show the constraint of the monthly payment to calculate the car price they can afford.
By the time we get to this third portion of the lesson, students have a very good idea of what their car can cost and still keep it within the $300 monthly payment. This is a nice bonus from a teacher perspective, because I often see order of operation errors or algebraic errors when students solve this problem. Students often have a good intuition when their answer is unreasonable, if they calculate a price of $90,000 or $1200. A separate error often occurs here and that is the problem of rounding too soon in a problem—a magnitude problem. Students will often try to simplify the math in the monthly payment (PMT) formula by rounding during the calculation process. In the end, they find that their answer is off by a couple hundred dollars. Again, this is a great place to have a conversation with students about carrying as many decimal places as possible until the end.
“Research has shown that there is a relationship between pupils’ learning outcomes in mathematics and their beliefs about mathematics. Beliefs do not only influence how pupils learn but may also form an obstacle for effective learning, which means that pupils, who hold negative beliefs about mathematics, become more passive learners.” (Kislenko, 2009, pp. 143) I have used this lesson many times over the years to engage students and they love it—even students who do not love math. I repeatedly receive positive comments in my teacher evaluations about this lesson. For example, some say it was “fun to buy a car” or they “learned a lot about interest and how quickly it adds up.” In addition to active learning, I also want students to see the importance of math to life. We as math educators know that “the perceived usefulness and the acknowledgement of importance of mathematics is one of the most cited properties in learning mathematics.” (Kislenko, 2009) This lesson allows students to experience exponential functions in a real life situation and many proclaim that “they are going to use this when they go buy their first car.” This is always the statement that makes me proud—it’s not just about the math, but my students’ connection to it outside the classroom.
An Extension to Other Grades
This same type of lesson can be used in teaching compound interest which would be more applicable to a lower level of math class—specifically Algebra I and Grade 8. The application of some simple and compound interest lessons that highlight exponential growth and decay are:
· Use the same interest rate in both simple and compound interest formulas to have students determine which of these is most beneficial in an interest bearing savings account.
· Use the same interest rate compounding both daily and monthly to have students determine which of these is most beneficial in an interest bearing savings account.
References
Gates, P., & Yistro-Yu, C. P. (2003). Is Mathematics for All? In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick & F. K. S. Leung (Eds.), Second International Handbook of Mathematics Education (Vol. Part 1, pp. 31-73). Dordrecht: Kluwer Academic Publishers.
Kislenko, K. (2009). "Mathematics is a Bit Difficult But You Need it a Lot": Estonian Pupils' Beliefs About Mathematics. In J. Maaß & W. Schlӧglmann (Eds.), Beliefs and Attitudes in Mathematics Education (pp. 143-163). Rotterdam: Sense Publishers.
Carrie Case
Instructor
Radford University
cscase@radford.edu