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I'm interested in learning two transitions:

(1) When can students reason (intuitively, but accurately) to conclude that two circles in the plane could intersect in $0$, $1$, or $2$ points, or are coincident so sharing $\infty$-many points?

(2) When can students prove this? E.g., distinguishing between circles and ellipses, which could intersect $4$ times?

I am hoping that in the US, (1): after a first course in geometry, and (2): after a first course in algebra. So sometime in "middle school" in the US. But I only teach college students, so I don't really understand when certain pre-college transitions are crossed. Maybe that algebra would need to await high school?

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    $\begingroup$ Do you want them to be able to describe where they intersect? Or just how many times? I'd expect a much younger student to be able to answer this, with a bit of guidance. I'm not sure what a proof would involve (because it's so obvious). $\endgroup$
    – Sue VanHattum ♦
    Commented Feb 28 at 1:38
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    $\begingroup$ Do you mean determining algebraically, or simply being able to draw sketches for all possibilities or something similar? The former would have to wait until quadratic equations are fairly thoroughly mastered (quadratic formula & discriminant, and more since we're talking about simultaneous quadratic equations), which would likely be 10th to 11th grade for students eventually making it to calculus before finishing high school. The latter is something I would expect young children could do, although you would have to give them some examples of ellipse shapes so they would know what to work with. $\endgroup$
    – Dave L Renfro
    Commented Feb 28 at 3:44
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    $\begingroup$ I don't understand why you drag ellipses in this question: in my youth, in Belgium in Europe, circles were taught at a very young age (I was ±10 years old) and their intersections were observed quite early too. Ellipses, however, or any kind of comic sections were taught much later (±16 years old). $\endgroup$
    – Dominique
    Commented Feb 28 at 7:53
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    $\begingroup$ "At how many points can to circles intersect? What about ellipses? Why?" given with some examples of both, and maybe formulae, seems to be a nice question suited for very many levels of education. The kinds of answers one expects would naturally vary, but it would be interesting to just give the question to pupils or students and see what they find out. $\endgroup$
    – Tommi
    Commented Feb 28 at 8:03
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    $\begingroup$ Your question would be improved with some focus. Maybe frame it in the context of Van Hiele levels? en.wikipedia.org/wiki/Van_Hiele_model Also, as I am sure you know, circles are not exceptions to the general expectation that two conics meet four times. Two circles always meet at the circular points at infinity $[1:\pm i:0]$ en.wikipedia.org/wiki/Circular_points_at_infinity Here we are talking about transition from Van Hiele Level 3 to Level 4 $\endgroup$
    – user52817
    Commented Feb 28 at 14:01

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