# Engaging students: Deriving the Pythagorean theorem

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 Delaina Bazaldua. Her topic, from Geometry: introducing proportions.

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

How has this topic appeared in the news?

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

I found a really good blog from a teacher through Pinterest: http://mathequalslove.blogspot.com/2012/04/sugar-packets-and-proportions.html. This website is really great because it is posted from a teacher on a blog who actually tried the lesson. The lesson can be adjusted for a geometry class, but it is really remarkable the way it is without changing a thing especially as an introduction to proportions before going into deeper questions that involve geometry. Like the video above, it can be relatable by the audience of students because of how applicable it is to their life. Likewise, it could also help them eat/drink better! The goal of the lesson is to figure out how many packets of sugar are in a variety of food and drinks using proportions between packets of sugar and grams of sugar! The engage would include the video of someone eating packets of sugar, students brainstorming ideas of how many packets of sugar are in a drink, and then would escalate to students putting the drinks in order of most sugar to least sugar without looking at the nutritional label. After that, students would be given the fact that there are approximately 4 packets of sugar in a gram of sugar. They would also be given the nutritional labels to calculate how many packets are in the drinks using proportions. I think this is a good lesson because it engages the students by allowing them to relate to something that happens in everyday life when they drink/eat things. It is also a good way to introduce proportions with something concrete like bottles before introducing something that is somewhat abstract, such as shapes drawn on a paper which is how geometry is often seen.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Perhaps the most famous proportion in history is known as the “Divine Proportion.” Using the research found on the website: http://www.goldennumber.net/golden-ratio-history/, it can help students realize the history behind proportions because, despite popular belief, students need to learn the history of the concept they are being taught to fully grasp the concept of the topic. The website given is really great because it goes through the different names other than divine proportion, such as Golden Ratio and Fibonacci Sequence, and how it was discovered and rediscovered throughout time which is why there are so many unique names that exist now. I also found that the fact that the names that have the words ‘golden’ and ‘divine’ in the name are because of a spiritual background. Understanding divine proportion is important because it is around us every day and it is only a piece of the whole umbrella that engulfs all of probability. It is also applicable to students because it involves them and their physical body along with objects they interact with everyday. I found the topic of divine proportion very interesting and I would hope my students would as well which is why I think this is an extraordinary engage.

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