We also have Taylor Series, which have been regarded by some as the most interesting topic in calculus II. Power series are also useful in physics and chemistry. Power series are fundamental to the study of calculus because they provide a way to represent some of the most important functions in our field. One of these ideas is called a power series. In calculus II teachers teach students how the elementary ideas they learned in pre-calculus are now used in calculus applications. But from an educators’ standpoint, we understand how important sequences are. Sequences are often overlook by students in pre-calculus (high school) because it is different from what they have encountered in their math career thus far, but maybe if we show students how this topic evolves in calculus II then they will pay more attention to it (Or they will forget it more since many students will not take calculus II). Sequences and equations is a very important topic in mathematics, and unfortunately many students that take pre-calculus in high school will never get to experience how sequences evolve from simple arithmetic sequences to the more powerful ones in calculus II. “How can this topic be used in your students’ future courses in mathematics?” I would explain to the students that now that we have the formula we can easily find the nth term that contains our sum, and this parallels the same process as having an x value and finding a corresponding y value and by using this process I can assure the students that the methods they learned in algebra are still important in pre-calculus. Then just like the students did in algebra one, they can use the point slope formula to come up with an equation for the sequence. After writing out a few terms, I would expect the students to find the common difference between the terms and then compute the slope of the terms (I say slope because I hope they can see that this pattern is linear and therefore we can model the data using a linear equation and not just use the formula for arithmetic sequence but rather derive one ourselves). Q’s first bill happens to be $65, his total after the second bill is $130, after the third bill the running sum is $195, if this pattern continues, how many months will it take for the total to reach $780? To solve this problem we would write the terms in a sequence starting with the first term being $65 and up to three more terms. For this example, lets suppose that John Q, a pre-calculus student, has just bought a new phone from apple, but because of this new upgrade, Q’s parents are concern with the sum of money they will be paying for his monthly bill. There are many word problems we can do with arithmetic sequences but I am going to give one example that I believe students will understand. “What interesting word problems using this topic can your students do now?” His topic, from Precalculus: arithmetic sequences. This student submission comes from my former student Erick Cordero. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. 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.
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