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

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.