# Engaging students: Arithmetic series

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 Eduardo Torres Manzanarez. His topic, from Precalculus: arithmetic series.

A1) What interesting (i.e., uncontrived) word problems using this topic can your students do now?

One interesting word problem to ask students to get them thinking about the idea of an arithmetic series, specifically a finite arithmetic series, is to have students come up with the total sum of the first 100 positive integers larger than 0 (i.e., 1 to 100) without actually adding all the integers up. Students will probably not figure out the total sum without adding the integers up one by one but if students are shown these numbers physically as cards labeled then a few might notice that the numbers taken at each end form pairs that add to the same sum. Turns out that the total sum is the number of pairs multiplied by 101. It can be explained to students that the 101 results from taking the first term and the last term (i.e., 1 and 100) and seeing that the sum is 101. This is true when we add 2 and 99, 3 and 98, 4, and 97, and so on. Hence, we will have 50 pairs since we have 100 numbers and so we have 50*101 as our sum. This problem can be extended to the story Gauss and how he apparently solved this problem as a child relatively fast and the teacher pointed out this question to them because he was apparently lazy. Now, this can be extended to adding all the integers from 1 to 200 and so on and having students come up with a general formula. Students can then think about an odd number of integers and see if that formula holds. Lastly, the connection between adding a number of terms with the same difference between each term is defined as an arithmetic series and so all the problems they have been doing are arithmetic problems in disguise.

B2) How can this topic be used in your students’ future courses in mathematics and science?

This topic is heavily used when discussing convergence in calculus. It provides insight into the validity that every series has a total sum that can be written as a number. Turns out this is true for all series that are finite but when discussing infinite series, it can be true of false that it converges to an actual value. So, students will have to ponder this idea for infinite arithmetic series in the future. Also, arithmetic series can be used to model certain situations in science within biology and physics. Thinking about arithmetic series provides information in tackling other types of series such as geometric in terms of behavior and solution. How does a geometric series behave? Well, each term increases with a common ratio instead of a common addition. Does the finite series converge? Yes, we know that every finite series does and this one basically behaves like the arithmetic in which we can easily find the total sum using a formula. Does the infinite series converge? Well, just like an arithmetic series it depends on the situation and the terms within the problem.

C1) How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

This topic has appeared in a particular movie called “All Quiet on the Western Front” which was released in 1930 and is an adaption of the novel that was published in 1929 by Erich Remarque. Within this movie, there is a scene in which a soldier states the formula for finding the sum of an arithmetic series. The soldier specifically states the formula S = A + N*(L / 2) and this corresponds to arithmetic series in accordance with the area of a rectangle and the area of a triangle. This is in a way a longer version of the short-hand formula we use today. One particular statement made from the soldier is that he mentions how beautiful the formula is. For some students, they can probably relate to the idea that something so complicated as adding 100000 terms that have a constant difference can be found using a short formula. Many problems in mathematics seem complicated at first in accordance with doing “grunt work” but many of them have beautiful solutions to them.

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