Interest Mathematics
Alma Wass
The author acknowledges the many current and historical problems caused by usurious interest rate practices; this book will not cover the very interesting and important debate about whether current interest practices make sense and whether or not interest should exist at all. The fact is that if you want to borrow or loan money through an institutional method (a bank, payday loan company, credit card, or similar structure), there is going to be an interest calculation involved. So we will endeavor to understand some things about these calculations.
A loan accrues simple interest, if the total interest is the product of the initial loaned amount known as the principal, a fixed percent known as an interest rate, and the length of the loan known as the term. The balance is the total amount in the account at the end of the term, whose length is a time \(t\) in years. We use the formulas:
This will give us a quick way to find the balance of a loan -- that is, the total amount that is owed -- if we know the rate, initial amount, and the length of the loan. Let's see an example of how these formulas are used.
First of all: if someone knows where to find a loan like this, please contact the author immediately! Unfortunately this is not a realistic situation due to the low interest rate and the simplicity of its structure; rather, it is meant to illustrate the use of the formula using relatively nice numbers. We will see more realistic situations later on.
In the previous section, we studied simple interest. While simple interest is relatively straightforward to compute, it is not the type of interest that is typically used in most actual loans. Instead, most loan accounts use some form of compound interest.
A loan accrues compound interest if the balance is computed as a repeated simple interest calculation on a periodic schedule. The balance on a compound interest loan can be expressed as a function of the time of the loan. This function is given by: \[B(t) = P\left(1+\fracrn\right)^nt \]
If you stare at these for a few minutes, you will likely see some similarities. On the left side, there is the balance. (In the second equation, we happened to write the balance as a function of time, but it's still standing for the same thing.) On the right side, there is the principal \(P\), multiplied by a factor that involves the interest rate. However, in the compound interest equation, the variable \(t\) is in the exponent. For this reason, compound interest is an exponential function.
In the previous definition, we are familiar with all of the variables besides \(n\) from the simple interest formulas. The idea behind \(n\) is that it counts the number of times per year the interest is calculated. Since the interest rate is annual, we take that rate, \(r\), and divide it by the number of times per year the interest is calculated. This evenly distributes the percent interest calculation throughout the year. However, since the interest is being calculated on a higher and higher balance each time, the amount of interest continues to grow over time.
In general, you can determine what \(n\) is by looking for a keyword that indicates the compounding schedule. Here is a table that shows the most common values of \(n\) and their corresponding keywords. Remember that \(n\) is the number of times per year interest is calculated, so these values are obtained by simply counting the number of a given period in a year.
Leanne would like to purchase an iPad Pro using her credit card. Her credit card has \(24\%\) interest rate compounded monthly. Assuming that she does not make any payments on the purchase, how much will she owe after \(2\) years? Compare this with a simple interest rate for the same rate and time period.
In order to solve this question, we must note the word "compounded" in the question. That tells us that this is a compound interest question. That tells us to use the formula: \[B(t) = P\left(1+\fracrn\right)^nt \]
Take a moment to note that this has the same format as the exponential functions we saw in the previous section. We will use the same techniques to calculate it, and we find that: \[B(2) = 649(1.02)^24 = 1043.86\]
That is, the same loan with a simple interest structure would be \(\$960.52\), as compared to the \(\$1043.86\) price tag of compound interest. The reason is that compound interest was calculated multiple times over the course of the loan; even though it was a smaller percent interest each time, the exponential growth caused it to grow larger over time.
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The existence of a significant percentage of low-achieving students is probably due to teacher-led instruction, which still dominates mathematics classrooms in most Asian countries. It should be noted that students in every classroom possess different abilities and hence demonstrate different achievements. Unfortunately, in teacher-led instruction, all the students are required to learn from the teacher in the same way at the same pace (Hwang et al. 2012). Low-achieving students, without sufficient time, are forced to receive knowledge passively. Barr and Tagg (1995) pointed out that it is urgent for low-achieving students to have more opportunities to learn mathematics at their own pace. Researchers suggested one-to-one technology (Chan et al. 2006) through which every student is equipped with a device to learn in school or at home seamlessly. Furthermore, they can receive immediate feedback from Math-Island, which supports their individualized learning actively and productively. Thus, this may provide more opportunities for helping low-achieving students improve their achievement.
Third, strategic competence refers to mathematical problem-solving ability, in particular, word problem-solving in elementary education. Some researchers developed multilevel computer-based scaffolds to help students translate word problems to equations step by step (e.g., Gonzlez-Calero et al. 2014), while other researchers noticed the problem of over-scaffolding. Specifically, students could be too scaffolded and have little space to develop their abilities. To avoid this situation, many researchers proposed allowing students to seek help during word problem-solving (Chase and Abrahamson 2015; Roll et al. 2014). For example, Cheng et al. (2015) designed a Scaffolding Seeking system to encourage elementary students to solve word problems by themselves by expressing their thinking first, instead of receiving and potentially abusing scaffolds.
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Second, the sightseeing mechanism provides students with a social stage to show other students how well their Math-Islands have been built. This mechanism is implemented as a public space, where other students play the role of tourists who visit Math-Island. In other words, this sightseeing mechanism harnesses social interaction to improve individual learning. As shown in Fig. 4, because students can construct different areas or roads, their islands may have different appearances. When students visit a well-developed Math-Island, they might have a positive impression, which may facilitate their self-reflection. Accordingly, they may be willing to expend more effort to improve their island. On the other hand, the student who owns the island may also be encouraged to develop their island better. Furthermore, when students see that they have a completely constructed building on a road, they may perceive that they are good at these concepts. Conversely, if their buildings are small, the students may realize their weaknesses or difficulties in these concepts. Accordingly, they may be willing to make more effort for improvement. On the other hand, the student who owns the island may also be encouraged to develop their island better. In a word, the visualization may play the role of stimulators, so that students may be motivated to improve their learning status.
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