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I have to deliver a lecture for secondary school, about one of these topics: integers, Euclid's Elements, polyhedra, prime numbers, non-Euclidean geometry, arithmetic functions or graphs.

It should be something interesting and easy to understand for high-school students, and 45 minutes long. My ideas are: platonic solids, something about prime numbers (but I don't know what exactly), golden ratio. They should be good but they seem to me not very original.

Do you have any suggestion? Maybe an interesting problem solved using one these mathematical tools? Or a particular aspect of one of these that could catch the eye?

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    $\begingroup$ As you already state here math.stackexchange.com/questions/2410881/…, this is an exam for you, to test your teaching capabilities. Thus, you can't expect someone here to do all the work for you. I would suggest to come up with some ideas of your own. Almost any topic can be either interesting or boring, depending on the one presenting it (you). So choose something you think you, personally, can present well, a topic that interests you and with which you can interest and fascinate others. No one here can tell you the "best" topic, as it depends on you. $\endgroup$ – Dirk Aug 30 '17 at 11:25
  • $\begingroup$ Good point, Dirk. That said, I'd go for Euler's formula, personally - it hits almost all of these things. After al, @xyzt still needs to actually write the talk ;) $\endgroup$ – kcrisman Aug 30 '17 at 11:39
  • $\begingroup$ @Dirk Liebhold I'm not expecting someone to do the work for me. I'm just asking if there's some of these topics you would have liked to know when you were a student. I'm not asking the "best" topic, I'm only looking for some ideas. For example now I came up with the number of partition of an integer. $\endgroup$ – xyzt Aug 30 '17 at 13:49
  • $\begingroup$ @kcrisman good advice, I will take into account. Thank you $\endgroup$ – xyzt Aug 30 '17 at 13:50
  • $\begingroup$ The Euclidean algorithm is a possible choice. You can make it seem impossible -- how do you find the GCD of two numbers that are too large to factor? Most students at that level think that the very definition of GCD would require factoring both numbers. Fibonacci numbers make an interesting appearance as well. You can also link it with RSA cryptography (finding modular inverses). $\endgroup$ – John Coleman Aug 30 '17 at 14:49
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One thing I like to do is to look at the Numberphile Youtube channel. They have good bits of deeper math but presented in a way that most high school students can understand. One idea that jumps out to me from your list (because I actually used it when I taught an Honors Sophomore class) was to play the game Brussels Sprouts. I won't tell you my exact layout and lesson plan, as it is up to you to come up with that. But keep in mind what others have said: Any topic can be fun or boring, it depends on your energy and presentation as well as how active you let your students be in the lesson. Are they sitting and listening for 45 minutes? Are they investigating some task on their own? Are they solitary or in groups? There are a ton of factors to consider when planning a lesson.

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  • $\begingroup$ Interesting channel! And nice game, didn't know it. I'm going to study it! Thank you $\endgroup$ – xyzt Aug 30 '17 at 13:56
  • $\begingroup$ @xyzt Just commenting to check: Did you deliver the lesson yet? I'm curious to know how it went. If you haven't delivered it yet, what are your plans for it? $\endgroup$ – ruferd Sep 5 '17 at 19:33
  • $\begingroup$ I haven't delivered it yet, still planning it. I'm thinking about graphs: Konigsberg's bridges, the four colour map theorem (there's an interesting video by Numberphile). Anyway, your advice has been very useful. I'll let you know! $\endgroup$ – xyzt Sep 8 '17 at 14:13
  • $\begingroup$ 4 Color Theorem also is an excellent option, plenty of opportunity to get the kids active and investigating the seemingly simple question on their own, and formulating their own ideas about the possible answer. Great choice! $\endgroup$ – ruferd Sep 8 '17 at 14:46
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I would personally choose Platonic Solids, every time.

In her book Geometry, Michèle Audin says,

In this section, I want to prove that there are five types of regular polyhedra in 3-dimensional space.

with the footnote

In my opinion, this statement is part of the cultural heritage of mankind, as are the Odyssey, the sonatas of Beethoven or the statues of Easter Island (without mentioning the Pyramids) and I cannot imagine how a citizen, a fortiori a maths teacher, could not know it.

I think she has a wonderful (if somewhat scathing!) viewpoint on this particular facet of geometry, and sadly, I don't know how often students meet polyhedra.

In 45 minutes, you should have time to

  • Introduce the Platonic Solids, although you really should have paper or wire-frame (e.g. Zometools) models for this.

  • Prove that at most five Platonic Solids exist (by examining how much of $360^\circ$ is taken up by various numbers of regular polygons, around any vertex). It's very elementary, although requires some explanation.

with time to spare. Good additional topics are to

  • Count faces of each dimension (probably not combinatorially, just by inspection. I've had students who have a hard time with the icosahedron, due to all the symmetry!) and verify Euler's famed $V - E + F = 2$ formula (which of course holds more generally than just for Platonic Solids).

  • Verify Descartes' Total Angular Defect Theorem (add up all the angles in a given vertex; that's the angular defect of that vertex. The sum of these defects, over each vertex of the polyhedron, is $720^\circ$), which also holds more generally than for just Platonic Solids.

I like these last topics because they show us that the world has some very cute, but not at all obvious, rules about what is physically possible; restrictions by the very nature of three-dimensional space!

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