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How Many Circles Pass Through Two Given Points?

The impact of technology on education is often overstated.  However, some applications of technology are unequivocally transformative in mathematics teaching.

The question “How many circles pass through two given points?” is a wonderful prompt for a geometry class.  It’s simple, it provokes debate, it can be explored in a variety of ways, and it connects to many important geometric concepts.  And in the end, it requires some imagination on the part of the student to truly comprehend the answer.

And after all that classroom work, it is so powerful and satisfying to see something like this.

circles through two points

 

A simple demonstration that elegantly captures the essence of the problem, and leads to new compelling questions.  That shows students that mathematics is beautiful and inspiring.  And that takes just a few moments to put together in Geogebra.

And what’s truly transformative is how easy it is to get students using technology to create their own mathematics like this!  This is the real promise of technology.

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2 Comments

  1. DAVE MARAIN says:

    Dramatic powerful visual, Patrick, which invites students to explore and create their own art! Most students and adults for that matter would draw the “smallest” possible circle and just stop. After “seeing” the “family” of circles, what are some questions you would want teachers to ask to develop the underlying algebraic/geometric concepts linking the idea that 3 noncollinear pts determine a circle and the # of parameters in
    x²+y²+DX+Ey+F=0?

    • MrHonner says:

      I personally don’t link this to coordinate geometry, which I put off until the end of the course. And I’m not sure how I would; for me, the natural connection is through linear algebra. Thanks for the interesting suggestion.

      Instead, I leverage the uniqueness of circumcircles, a point of emphasis, to get at this notion. Does the addition of any third point determine a circle? If it does, how do we know the circle is unique?

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