Engaging students: Introducing proportions

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 Deborah Duddy. Her topic, from Pre-Algebra: introducing proportions.

How can this topic be used in your students’ future courses in mathematics or science?

Proportions, in the form a/b = c/d, is a middle school math topic. The introduction of proportions builds upon the students’ understanding of fractions and ability to solve simple equations. This topic is used in the students’ future Geometry and Statistics courses. The use of proportions is used in Geometry to identify similar polygons which are defined as having congruent corresponding angles and proportional corresponding sides. The use of similar triangles and proportions are used to perform indirect measurements. In Statistics, proportions are used throughout measures of central tendency. Additionally, statistics uses sampling proportions including the proportion of successes.

The ability to use proportions for indirect measurements is also included in the study of Physics, Chemistry and Biology.  Chemistry uses proportions to determine based upon the chemical structure of a compound, the number of atoms pertaining to each element of the compound.  The study of Anatomy also uses many proportions including leg length/stature or the sitting height ratio (sitting heigh/stature x 100).

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

In art, proportions are expressed in terms of scale and proportion.  Scale is the proportion of 2 different size objects and proportion is the relative size of parts within the whole.  An example of proportion is Michelangelo’s David.  The proportions within the body are based on an ancient Greek mathematical system which is meant to define perfection in the human body.  Da Vinci’s  Vitruvian Man is also an example of art based upon proportions or constant rates of fractal expansion.  The music of Debussy has been studied to show that several piano pieces are built precisely and intricately around proportions and the two ratios of Golden Section and bisection so that the music is organized in various geometrical patterns which contribute substantially to its expansive and dramatic impact.

The use of proportions is also a constant within Greek and Roman classical architecture.  Many classical architecture buildings such as the Parthenon illustrate the use of proportions through the building.  Additionally, classical architecture uses specific proportions to determine roof height and length plus the placement of columns.

How has this topic appeared in the news?

Proportions are constantly in the news even though they may not be presented in a/b=c/d format.  However, the concept of proportion is used throughout news reporting and even advertising.  The current news topic is the upcoming Presidential election.  Daily, we are provided with new and different poll results.  These results are derived via a proportion.  For example, 100 people are polled, these results are then derived via proportional concepts to provide a percentage voting for each candidate.  Percentage is a specific type of the  a/b = c/d proportion.  Daily news uses proportions when reporting growth trends for national debt, crime and even new housing starts in DFW.   Today, proportions were used when discussing the new Samsung Note7 and its ability to explode.  During the winter, proportions are used to tell us how many inches of rain would result from 2 inches of snow. Sports broadcasters also use proportions when discussing the potential of athletes.  If the athlete can hit 10 homeruns in 20 games, then he will potentially hit 50 homeruns in 100 games.  Proportions even appear in advertising for new medicines detailing the data associated with the medicine trial.

References:

Debussy in Proportion: A Musical Analysis, Dr Roy Howat

Michelangelo’s David

https://abcnews.go.com/images/PollingUnit/MOEFranklin.pdf

http://www.brightstorm.com