The 'Quilting' Lesson - Arithmetic Vs. Geometric Series

An example of arithmetic series and geometric series, with application for math classes, or leadership conferences and seminars

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The 'Quilting' Lesson - Arithmetic Vs. Geometric Series

(Published in Directory : Mathematics : Sequences And Series)

This is a lesson idea which, oddly enough, could be used either in a high school mathematics class, to explore the differences between arithmetic and geometric series, or as a lesson in a leadership/teaching seminar.

Begin with the premise that someone wants to make quilts, and wants to produce as many as possible. Since my mother enjoys making quilts, I use her as my "quilt-maker".

Imagine that she could make quilts at a rate of 1 quilt every month. Over the course of a year, she would be able to make twelve quilts. Just to make sure your class is with you, ask how many quilts she would have after two years, five years, ten years, etc.

This is an arithmetic progression, and if you are using this lesson in a math class, you should point that out.

Now suppose that, instead of just making quilts herself, she wants to teach someone else to do it. So she calls a friend and invites the friend to come work with her and learn to make quilts. Of course, since she is teaching, she cannot work as quickly. So let's say that over the course of the year, she can only make one quilt. Not very impressive. Clearly she's not going to make as many quilts this way.

And yet, when the next year rolls around, there are now two quilt makers, instead of just one. So the next year, they can each work at making quilts. So suppose each of those quilt-makers calls a friend, and teaches them to make quilts. How many quilts would be made in the second year?

Two. Because both quilt-makers would make one quilt while teaching their friends to do the work. So after two years there is a total of three quilts.

By the third year, there are now four quilt-makers. So the total number of quilts is 1 + 2 + 4 = 7.

"If this process continues", you can ask your students, "How long will it take before the second procedure for quilt-making 'catches up' with the first procedure?"

Students can spend some time doing the calculations, and discover that after seven years, the second procedure has produced a total of 127 quilts, while the first procedure has only produced 84 quilts.

Other questions can be asked, such as: "How many years will it take before the second method produces ten times as many quilts as the first method?" or "How many years would it take, using either method, to produce enough quilts to give one to every person in the world?"

This is a good exercise for a math class because, like the Chess Board Problem, it gives a simple but obvious demonstration of how a geometric series can "explode" in comparison to an arithmetic series.

But my preferred use for this illustration is to demonstrate the importance of education, and of taking the time for individuals. In our fast paced society, bigger is always better, and that which is more efficient immediately often takes precedence over that which is more efficient in the long run. A good leader is one who is willing to take things slowly in the present, teaching and training and preparing, in order to have far greater results in the future. In the present, it seems better to "do it all myself", but in the long run, taking the time to share the load with another produces greater results.

About the Author

Name: Douglas Twitchell

Website: http://www.jeorgethedodo.com/dougblog/

Bio: I am the designer, maintainer, and editor of this site (Articles For Educators).

I am a former educator in both elementary and secondary math and science. In addition to this site, I also built and maintain the following educational sites: The Problem Site, Tile Puzzler, and Quote Puzzler.

I'm also a ventriloquist, and you can find more about my puppets here: Jeorge The Dodo

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