Pre-Calculus and Arithmetic Series 03/18/12

This Week's Featured Activity
Pre-Calculus Class and Arithmetic Series
Related School-wide Learning Goals
or Community Beliefs and Commitments

This lesson is an example of Goal #3 – After completing a semester at Conserve School, a student Will have taken a strong step forward on their educational path equal to or greater than the expected progression in their sending school.

Why I Chose to Feature this Activity

This weekend was Family Weekend. It started on Friday with some of the families joining my classes for the morning. In Pre-Calculus, I was just introducing the topic of arithmetic series so they got to see a class where I developed a formula for the finite sum of an arithmetic series. I had forgotten that families might be joining us, but it was too late to change my plan, so went ahead with my original plan. I told the story of Friedrich, a young German boy, who was in a math class, many years ago, long before the creation of calculators. It goes something like this:

Friedrich was a young German boy who was very clever, so clever that he always finished his arithmetic way before anyone else in the class. Friedrich’s teacher wanted to keep him busy, so, the teacher devised a plan to slow Friedrich down, and challenge him as well. The next time that Friedrich finished his work early, the teacher gave him the assignment to add up all the numbers (integers) from 1 to 100. Now this was in a time long before the invention of calculators, so the teacher was quite confident that it would take Friedrich a long, long time to complete this assignment. Much to the teacher’s surprise, Friedrich was back in just a few moments. Pulling out the sheet upon which the teacher had carefully computed the sum in advance, the teacher checked Friedrich’s answer, 5050. No, it could not be! How in the world had Friedrich gotten the answer so quickly? The teacher asked Friedrich to explain how he had gotten his answer so quickly. Friedrich then explained that he started by attempting to add the numbers 1 through 10 first. He noticed that if he wrote 1 through 10 going down and then right next to it, the numbers 10 through 1 that the sum across the columns was always the same, 11! So he had ten 11’s, giving him 110 for his answer. However, he realized that he had actually added each number twice, so he cut his answer in half giving him the final answer of 55. Confident that his pattern was correct, Friedrich added 1 and 100, getting 101. This would mean that he would have a total of one hundred 101’s giving him 10100 as an answer, which he then cut in half, thus giving him the sum desired, 5050!

This is a somewhat true story, with a bit of literary license taken on my part. Friedrich was a real person, although Friedrich was the middle name not the first name of Carl Friedrich Gauss (originally known as Johann Friedrich Carl Gauss), one of the greatest mathematicians, and physicists, of all times.

This story was a wonderful jumping off point to create the formula for the sum of a finite arithmetic series, which we then tried out on some series that we could add up in our heads. This formula lead us to create a general formula for the nth term in a given arithmetic sequence, which enabled us to then use this formula for other arithmetic series.

After class, some parents came up to me and commented that they enjoyed the class and were surprised that they could follow what we were doing in class, and got actively involved in the lesson. The kids liked it too, especially since there was no homework because it was family weekend. However, we will apply the formula next week, when we do have homework on the topic.

 Citation: Gauss, Carl Friedrich. Photograph. Encyclopædia Britannica Online. Web. 18 Mar. 2012. <http://www.britannica.com/EBchecked/media/1685/Carl-Friedrich-Gauss-oil-painting-by-CA-Jensen-in-the>.

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