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Grade Vs Difficulty:

“Your cousin has just gotten a new job as an announcer on the Golf Channel, and is so excited for you to watch every broadcast. While you may enjoy playing golf, you don’t really enjoy watching it. So you decide to place a small flatscreen TV between 2 windows above your kitchen sink. Now you can multitask. The space is 12 inches wide. TVs are sold by diagonal measurement. What size TV should you order?”
Pose this question to your students. It is intentionally phrased vaguely so that students must ask numerous clarifying questions. You’ll likely end up with two lists on the board: Questions, and Assumptions (and possibly a third, Conjectures).
This question can get students to need, use, and eventually prove the Pythagorean theorem. To get into proofs, once students need and use the theorem, ask, “How do you know that it works?”
Familiarize yourself with the following before class:
1. how TVs are sold (by the diagonal), and what sizes they typically come in (i.e. bestbuy.com)
2. aspect ratios, and their history/relation to TVs and the Golden Ratio (here’s a nice article about this: http://voices.yahoo.com/thegoldenratio2106072.html?cat=15)
Other questions may come up for exploration:
1. How do you calculate and compare ratios?
2. Which aspect ratio is closer to the Golden Ratio: the standard or the widescreen?
3. What are our preferences in TV screen shape, and what are the origins of our preferences?
4. What is an irrational number, and how can you prove their existence?
5. What other geometric considerations are there in TV selection and placement?
6. Why does the shortcut of moving the decimal point when dividing decimals work?
7. What is the history of the Pythagorean Theorem?
8. Who were the Pythagoreans?
9. What is the history of the irrationality of root 2 proof (Hippasus)?
10. What are different types of proofs?
11. What is a proof?
12. How can you evaluate the quality (or lack of quality) of a proof?
I wrote a blog report on what happened when I posed the above “TV Question” to my class: http://talkingsticklearningcenter.org/proofs1aspectratiosthegoldenratioandzstv/. I also blogged about where this question took our group in the following weeks: http://talkingsticklearningcenter.org/category/mathcircleblog/.