I'm going to participate in a course (as a teacher) where I'm suppose to teach high school math to high school students in about 15 days during the year. Each class has about 1 hour long. Now, I think it is quite impossible to really get deep into the concepts in so little time, and so I got the following idea: I could gather some texts/articles discussing advanced high school subjects and try to discuss them together with the class. My intention is to provide different experiences and to develope critical thinking. The problem is that I know very little material on this matter, so I'm looking for some references and tips on where to find appropriate material. Can someone help me, please?

EDIT: Let me add more information to my post. This course is not a regular one. It is actually a voluntary set of lectures taking place at the university and aimed to help high school students to pass the 'vestibular', which is a brazilian test (similar to the american SAT's) students have to do in order to get accepted in public universities. The (math) content of the test is basically all high school math taught in Brazil. This includes basic set theory, basic notions of functions, trigonometry, plane and analytic geometry, algebra of matrices and so on. Thus, our course is aimed to help these students with a set of prep lectures. Each teacher can choose the topics he wants to present. In my case, I'll probably teach about functions and first and second order degree equations but these topics are really demanding and 15 one-hour classes are not sufficient to cover all material, so I'm thinking about alternative ways to approach it. Because students often like class discussions, I thought about providing some discussions based on alternative material (not regular books, which are too long to get at any point in 15 days).

  • $\begingroup$ Thanks for the comments! I will add more information to my post! $\endgroup$
    – MathMath
    Commented Feb 9, 2020 at 16:37
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    $\begingroup$ "aimed to help high school students to pass the 'vestibular', which is a brazilian test (similar to the american SAT's) students have to do in order to get accepted in public universities" --- Given this, you will probably want to focus on "test prep" methods/topics, and not on supplementary and tangential topics, no matter how interesting they are. I think you will get more useful answers if you post a link to the specific topics the test covers and (especially) give some sample (or actual past) questions (perhaps a link to such questions). (continued) $\endgroup$ Commented Feb 9, 2020 at 19:59
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    $\begingroup$ Also, since there are presently 3 "close votes", you'll probably want to rewrite your question to be less specific to your situation (my guess for the close votes). For example, using the topics the test covers and giving some sample questions as an example, you could ask how one can simultaneously satisfy the specific task of helping students do well on a specific test without making major compromises in your desire to promote interest in mathematics and your desire to foster mathematical skills that you consider more worthwhile than just those needed to do well on the test. $\endgroup$ Commented Feb 9, 2020 at 20:08
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    $\begingroup$ Is this 'cursinho'? $\endgroup$
    – Sue VanHattum
    Commented Feb 10, 2020 at 2:57
  • $\begingroup$ Yes, it is! Do you know it? $\endgroup$
    – MathMath
    Commented Feb 10, 2020 at 13:00

2 Answers 2


You might engage them in billiards on an elliptical table (1st ref below), or on a circular table but aiming to hit a second ball (2nd ref below). Nice analytic geometry coupled with intuitive situations. See this Numberphile video.

          Fig.1 (detail) in Reznick et al.
          Fig.2 in Drexler-Gander.

Reznik, Dan, Ronaldo Garcia, and Jair Koiller. "Can the Elliptic Billiard Still Surprise Us?." The Mathematical Intelligencer (2019): 1-12. arXiv preprint abstract.

Michael Drexler and Martin Gander, "Circular Billiard." PDF download.


My advice: Look at the test itself. If possible get some statistics on common errors. If not, ask other teachers or at least use your intuition. I.e., make an intelligent guess, "Bayesian estimate".

Then design something (lecture and practice together) to hit the common mistakes. Don't reteach theory, concepts, etc. in some sort of organic manner. You don't have time for that and it will be a turnoff. But you will have their attention if you say (and mean it) that you will help them fix the most common mistakes and improve exam score.

Your approach should not be "you don't understand this, let me clarify theory to you". (The mistaken pedagogy of most of this site.) But more like..."hey watch out for this snake that wants to bite you." Maybe a little concept explanation, but not much. Be more practical instead. (Weaker students learn more from practice and correction than by explication.) Try to make the practice approximate the tests as much as possible. If old exams are public, mine those directly. If not, mimic.

P.s. I had a bunch of cool enrichment ideas for you, like JOR. But that is not what these kids need. See Dave Renfro's comments.


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