# Engaging students: Geometric mean

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 again comes from my former student Matthew Garza. His topic, from Geometry: the geometric mean.

How has this topic appeared in high culture?
Crockett Johnson was an artist, writer, and mathematician who worked as an art editor for McGraw-Hill in the 1920s.  By the 1930s, he was making cartoons; in the 40s he was known for his “Barnaby” comic strips, which appeared in several American Newspapers.  He wrote “Harold and the Purple Crayon” in 1955, which may be one his most famous works.  In the 1960s he created a series of more than 100 paintings to honor of geometry and geometric mathematicians.  Among them was this painting of a construction of the geometric mean of two numbers – line up the lengths and use that as the diameter of a circle, and draw a line from where the two lengths meet up to the circle.  If the students know the Pythagorean theorem, they could try to prove that. Crockett Johnson also created a new construction of a regular septagon, using a compass and marked ruler (and trigonometric identities).  I found another one of his mathematical paintings on the Smithsonian’s website, of a golden rectangle, and laid it over the geometric mean painting. It seems he included the golden ratio in his work, although I could not find anything verifying this.  In general, Crockett Johnson is an interesting person, and that should help engage students.

Painting at Smithsonian: http://americanhistory.si.edu/collections/search/object/nmah_694664
Another Bio: https://divisbyzero.com/2016/03/23/a-geometry-theorem-looking-for-a-geometric-proof
Regular septagon proof: http://www.jstor.org/stable/3616804?seq=1#page_scan_tab_contents

How have different cultures throughout time used this topic in their society?
Finding much information on the history of the geometric mean is pretty difficult. Pythagoras seems to be generally credited for “discovering” the geometric mean, and the Greek mathematicians are famous for the three means – arithmetic, geometric, and harmonic.  Although one not-necessarily credible source explained the word “geometry” comes from words meaning “land measurement.”  From this, we can easily consider the task of land management – to find a square plot of land of equal area to a rectangular one, the side length of the square should be the geometric mean of the two sides of the rectangle.  For this reason, I believe the geometric mean of at least two numbers must have been used as far back as math has been used for commerce; so pretty close to as far back as math has been used (I wouldn’t be surprised if Egyptians, or even Babylonians, were at least aware of such a relationship, whether or not a constructive proof existed).

http://hsm.stackexchange.com/questions/3057/what-is-the-history-of-the-meanings-behind-the-word-geometric

How has this topic appeared in the news?
Geometric mean is extremely useful for rates and values on varying scales.  Rates are used as products – consider something like an investment with a varying return rate for each year.  The regular arithmetic mean of the different rates would not give correct results – after one year at rate a, a quantity k becomes ak; after a second year at rate b, the original k is now bak.  The yearly average, if taken arithmetically, gives [(a+b)2/4]k after 2 years; if the geometric mean is used, it gives (√ab)2k = abk, so it’s more appropriate. With regard to values on varying scales, it prevents a top-heavy average.  Clearly, geometric mean is very useful, which is why finding news will work in a pinch, like if you forgot to plan.  Just do a google news search for geometric mean and you find several articles.  It’s mostly economic news.  The following were not.  Alternatively, a search in a scholarly database gives plenty of examples of geometric mean in action, although the technical writing may be difficult for students to get into for an engage.

Geometric mean to measure water quality: http://www.lajollalight.com/sd-beach-water-advisories-20161004-story.html
To measure general wellness of a nation: http://247wallst.com/healthcare-economy/2016/09/22/obesity-violence-helps-push-us-to-no-28-in-global-health-ratings
College sports stats: http://www.usatoday.com/story/sports/ncaaf/2016/10/11/sec-dominates-college-football-computer-composite-rankings/91910190/