Choose Which: Arithmetic or Algebraic?

By: Al Jupri, M.Sc.*

Let’s say the distance between Jakarta and Semarang is 520 km. Train A departs from Jakarta to Semarang at 7:00 am with an average speed of 80 km/h. On the same day, train B departs from Semarang to Jakarta at 9:00 am with an average speed of 100 km/h. At what time did the two cars cross?

If the above question is given to you, how do I answer it? Of the various possibilities, I suspect there are at least three different ways to answer them. The first way is as follows:

Note that the distance between cities is 520 km.

By 9:00, train A had traveled 80 km + 80 km = 160 km. This means the distance between the two trains is only 520 km – 160km = 360 km.

The total distance between train A and train B in one hour is 80 km + 100 km = 180 km. Therefore, the distance between the two trains at 10:00 is 360 km – 180 km = 180 km.

Since 180 km is the total distance that can be reached by train A and train B in an hour, and the remaining distance of both is 180 km, then the two trains will cross at exactly 11.00 am.

The second way is like the following:

Suppose the total travel time (within hours) required for train A to cross with train B is x.

Since train B departs 2 hours late than train A, the total travel time to cross is x - 2.

Since the distance between the two trains is 520 km, and based on the information in the previous question and initialization, then we can create an equation 80x + 100(x - 2) = 520.

Easily, we get the equation 80x + 100(x - 2) = 520 is x = 4.

Thus, train A will cross with train B at 7 + 4 = 11.

In mathematics, the first way is seen as a way of resolving using the concept of artimetics (numbers); while the second way is seen as a way of resolving with the concept of algebra. The processes used in these two ways are clearly different. The first way, thinking in a world of numbers–is usually easier to digest because it is relatively concrete. Whereas the second way of thinking in the world of symbols is relatively more abstract.

Which is better between the articulate way and the algebraic way? Answering this question is not easy. I think both are just as good. Depending on who answers the alias depends on the insight and knowledge of mathematics or physics that individuals who try to answer the question have.

Taking into account these two ways, we as teachers will probably ask: What level of students or classes above are suitable for students? When to give it: Is it at the beginning, in the middle, or at the end of learning?

I would argue, taking into account the appropriate mathematical topics, the above questions can be given to elementary, junior or high school students. It can be given at the beginning, middle or end of learning. At the beginning of the study, the problem can be used as the starting point of a mathematical concept introduced, the concept of linear equations one variable for example. In the middle of learning, the problem can be used as a “bridge link” between mathematical concepts. And at the end of the study, the problem used to be used as an application or application of mathematical concepts. Of course, the use of the question needs to consider aspects of didactic and pedagogical–which (I suspect) have often been practiced by teachers. Agree?

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Oiya, almost forgot, the third way I suspect is commonly used by those who are “a little lazy” to immediately think seriously. Nothing but guesswork! He he he… :D.

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*PhD student at Utrecht University, The Netherlands.

Note: This article was written in Utrecht, December 30, 2011.