# Fun with Proportions and Atoms

I came across this fun video on proportions, imagining how large some objects would be if atomic (and subatomic) length scales were magnified to the size of a tennis ball.

# 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 Jessica Martinez. Her topic, from Geometry: introducing proportions.

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

I would show a slideshow of classical artwork and architect such as the Mona Lisa, the Parthenon and the Pyramids in Egypt and then lead an open discussion of what qualities make these popular icons aesthetically appealing. Then I could lead into what makes a specific shape attractive, such as a rectangle, and have my students draw their ‘best looking’ rectangle, which would then lead into a discussion about the Golden Ratio. The Golden Ratio, also known as the “divine proportion”, appears in thousands of artworks and architecture pieces around the world. I would like to show a piece of artwork or architecture divided into the Golden Ratio rectangles and have my students calculate the Golden Ratio to the nearest hundredth -it’s about 1.618. Then I could go back to my original examples of classical artwork and show them with the Golden Ratio proportions drawn. I could also mention how famous painter Leonardo Da Vinci even illustrated a book about the Golden Ratio called De Divina Proportione, which talks about how mathematical and artistic proportions are used to create artwork and design architectural structures.

How could you as a teacher create an activity or project that involves your topic?

Continuing on the idea of C2 above, I would like to create a project that involves my students using the Golden ratio and proportions to create their own architecture. I could have them research some more famous examples to get ideas, and then have them design their project blueprints, explaining what measurements, ratios and proportions they used to develop it (I would also include a requirement of using the Golden Ratio somewhere in their design). Then in teams of 2-3 students, I would have them create a small scale model of their design using materials found at home or at a craft store. To make it more authentic to the students and possibly a longer, more intensive project, I could give them a scenario of designing their own dream homes; by adding a budget and some time to research, my students could use proportions to calculate the cost of materials needed to build their home (here I would most likely exclude materials not needed for the actual building; plumping, electric and air conditioning would make it more complicated but will most likely not fit in the time frame of my teaching).

How have different cultures throughout time used this topic in their society?

In American culture (and many other cultures), society considers beauty and attractiveness of high importance; its valued so much in some industries that people will go to the lengths of paying thousands of dollars altering their faces and body shapes to something more ‘aesthetically pleasing’ through plastic surgery. What is interesting to know is that plastic surgeons use proportions in order to create a more attractive look for their clients. Plastic surgeons will photograph a client’s face from the front and side views and divide their face into sections in the picture.  Then they will make corrections or marks on the photo (and on the client) of where the client wants surgery using specific proportions that create a look that is more symmetrical and ‘pleasing’ to look at. An example of this is shown below in the proportions of the nose to rest of the human face. (I would probably also remind my students that no one’s face is perfectly proportional and it’s a good thing because otherwise we would all look the same, and that’s boring.)

References

Obara, S. (n.d.). Golden Ratio in Art and Architecture. Retrieved October 13, 2016, from http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Obara/Emat6690/Golden Ratio/golden.html

Zimbler, M. S., & Ham, J. (n.d.). Aesthetic Facial Analysis. In Cummings Otolaryngology. Retrieved October 13, 2016, from http://www.marczimblermd.com/plasticsurgeonnyc/ResearchPublications/CummingsOtolaryngology.pdf

# 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

http://www.brightstorm.com

# 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 Michelle Nguyen. Her topic, from Geometry: introducing proportions.

A2. How could you as a teacher create an activity or project that involves your topic?

Using the video presented in E1, I would create a project that consists of the students making a poster of their own body with the proportion that they found within their body parts. For example, they would use the measurement of their foot and try to find out the amount of feet needed would create their height. Once they figure out all the proportion in their body, they would make a poster representing their finding. Throughout the project, the students will be able to write the proportion that compared the ratio of their feet to other part of their body. The outcome would similar to the pictures in the video that is shown in the engage. By doing this, the students can refer back to the engage to help them finish their project or use the engage to give them an example of what the project should look like. After the project, the students should be able to understand that proportion is the comparison of two ratios.

B2. How does this topic extend what your students should have learned in previous courses?

In previous courses, students should have covered ratios. Since proportion deals with fractions and ratios, students should be able to learn that proportion is the comparison of two ratios. This topic also extends the idea of comparing two different items to each others. With the ideas of ratios, the students should understand that units are important because they cannot compare two different ratios that are not related to each other. During algebra 1 the students should learn how to solve equations and when dealing with proportions the students may be required to solve for the missing variable in a proportion. With the knowledge of solving equations, the students will be able to cross multiply and solve for the missing variable. In conclusion, ratios, comparison of items, and solving equations should be learned before this topic is introduced. Proportion is the extended idea of ratio comparison.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

http://pbskids.org/cyberchase/videos/ecohaven-cse-ep-301

By showing this video in beginning of class, students are able to understand the basic meaning of proportion. This is a good video to engage students because the students are able to test out the real life situation. For example, in this video, the kids found out that the length of their foot is the same as the length of their face. Students can see that there is a proportional relationship with their own body part. With this whole episode of Cyberchase, students are able to see the different proportionality that is present with their own body. As the episode continues, the kids continue to measure different body parts to see how many foot spans would construct another body part. With the use of one type of measurement, the students will see the different proportionality that exists in the human body. During this episode, the kids measure that seven foot span is equal to the arm length and then they also discovered that the height is the same length as the arm length. Students will be able to make their own connection to proportion after seeing all the measurements mentioned in the episode.

# Trivia question for the day

Which is worth more: a pound of dimes, a pound of quarters, or a pound of dollars?

Rather than spoiling the fun for my readers, I’ll just leave this one unanswered and let you think about it. Feel free to Google for help.

# Engaging students: Solving 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 comes from my former student Kayla (Koenig) Lambert. Her topic, from Pre-Algebra: solving proportions.

C. Culture: How has this topic appeared in the news?

Solving proportions, or the idea of a proportion being solved, appears in the news more often than not. One specific example that can be used is the effect of the economy on real estate companies. Say we are given 25% of 16 real estate companies that have closed their businesses due to poor economy. We can use proportions to determine the number of real estate companies that closed. We know that the percent is 25 and that the whole is 16. Therefore $25/100 = x/16$ which gives us 4 real estate companies that closed (Review of Proportions). Proportions can also be used to determine how many miles we can drive on a certain amount of gas, and gas prices are constantly on the news. Also, this will be relevant to high school students who drive and need to find how much money they need to buy gas for the week, etc.

We can also use proportions to find the unit price of an item at a grocery store, or if an item costs a certain amount, you can find out how many of those items you can buy with a fixed amount of money you have. Buying items and saving money are also all over the news. If you find the unit price you can compare items therefore saving money by buying the item that you get the most out of your money.  Another way solving proportions can appear on the news is by the stock market. You can use proportions to find out how much the stock market will rise in a given amount of days given the current amount of points it has raised in a certain amount of days. Making a proportion problem for students to solve is relatively easy and can be related to anything that is on the news. We can use this to our advantage to get the students to be a little more interested in proportions (and mathematics) so they can see different ways it is related to real life.

The idea of proportions was adopted and used by many in the mathematical community. Proportions were used by Greek writers, including one named Nicomachus, who include proportions and ratios in arithmetic (Math Forum). Proportions were also adopted by Exodus who used them in geometry and by Theon of Smyrna who used proportions in music (Math Forum). In 2000 B.C., the Babylonians adopted proportions to represent place value notation (Pythagoras – Geometrical Algebra). Using proportions was accepted by mathematicians and was used to solve so many different equations used for so many different ideas, and is still used today. Early proportions were adopted by the Egyptians and were used to calculate fractions and measurement of farmland (Mathematics History). Later, proportions were adopted by so many more in the mathematical community like in Greece, China, India, and Babylonia in order to learn geometry. Greeks, like Plato, adopted proportions in order to study them with the Egyptians. I think that proportions were well liked by mathematicians and were adopted by many because you can use proportions to solve so many things.

D. History: How did people’s conception of proportions change over time?

From the beginning, people have used proportions. Early humans used proportions to see if one tribe was twice as large as another or if one leather strap is only half as long as another (Math Forum). It is obvious that the idea solving proportions hasn’t really changed that much, but what we can use proportions to solve has changed. In 2000 B.C. Babylonians used proportions to evolve place value notation by allowing arbitrarily large numbers and fractions to be represented (An Overview of Egyptian Mathematics). Around 1600 B.C. in Egypt, proportions were used to calculate the fraction and superficial measure of farmland (Mathematics History). Egyptians then used proportions to find volumes of cylinders and areas of triangles.

Vitruvius thought of proportions in terms of unit fractions for their architecture calculations (Proportion (architecture)). Also, scribes used “unit fractions” for their calculations in Egypt and Mesopotamia. Egyptians based proportions on parts of their body and their symmetrical relation to each other; like fingers, palms, hands, etc. Multiples of body proportions would be found in the arrangement of fields and buildings people lived in (Proportion (architecture)) and from here, proportions evolved. In 600 B.C., the idea of using proportions evolved and was then used for geometry (Mathematics History). Proportions are still used in geometry, like in architecture and land, like it was 3000 years ago. When you think about it, proportions have evolved, but the use of proportions has evolved even greater. There are so many topics we can now solve using proportions!

Works Cited

“Math Forum – Ask Dr. Math.” The Math Forum @ Drexel University. 7 Mar. 2012. <http://www.mathforum.org/library/drmath/view/64539.html&gt;.

“Mathematics History.” ThinkQuest : Library. 7 Mar. 2012. <http://library.thinkquest.org/22584/&gt;.

“Proportion (architecture).” Wikipedia, the free encyclopedia. 7 Mar. 2012. <http://en.wikipedia.org/wiki/Proportion_%28architecture%29&gt;.

“Review of Proportions.” Self Instructional Mathematics Tutorials. 7 Mar. 2012. <http://www.cstl.syr.edu/fipse/decunit/ratios/revprop.htm&gt;.

“An Overview of Egyptian Mathematics.”  7 Mar. 2012. < http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_mathematics.html >