After the discussion on first worksheet, students’ answer to the definition of tangent revised with several version. Some of these version are: “there are infinitely tangents at a point in a space curve”. The nest one is “there should be exactly one tangent, no matter whether a curve is plane or space”.
After some discussion, students identified the term “limit” to define tangent. Then the revised version of the definition was “ tangent is limiting position of secant line taking two points of the curve. In this exploration, the example and posing questions given by Martha and others were became very useful.
This dynamism of the tangent as limit of secant by dragging point B along the curve towards the point A
led question why what is the relation between secant and tangent line? This visualize the special case of the secant ‘transforming’ into the tangent.
I started to use GeoGebra to demonstrate concept image of “order of contact” between tangent line and given space curve. The warm up question in the beginning of the content was: How do you define
tangent line at a point in a curve? How many tangent line exists (if any) at a point in a curve? And what is number of contact points between curve and tangent?
The definition of tangent, as student say is "“a straight line that touched a given circle exactly once”. Another answer is "“a straight line that passes through exactly one point of given curve”. Along with definition, when discussion was brought into the case of space curve, there were several possibilities that a line can passes through exactly one point of given curve [move the line in the worksheet below].
Now, students were a little bit confused to identify whether there is possibility of several tangent line through a point or it is uniquely determined. If unique, which one is the correct?
In this case, tangent has two point contact with the curve at A, mean first order contact at A, like the root of a polynomial (x-2)^2, where the solution is 2 but has multiplicity 2. And this solution is different from the solution of a polynomial (x-2)=0 as discussed. In addition to this the Zeon’s paradox about Achilles and tortoise was also discussed to visualize the idea of limit. By this exploration and discussion, students are still struggling to accept that tangent has two point contact with the curve, this may be the reason that nowhere in the text book was written explicitly that “tangent has two point contact with the curve”.
Then, another content: the concept of osculating circle using GeoGebra was explored to connect the concept image of “order of contact” between osculating circle and given space curve. The warm up question in this content was: What should be the nature of three points to determine a triangle? Is it possible to circumscribe any kind of triangle by a circle? Is this circle be a plane figure? Even if it is drawn
in three dimensional space?
From the exploration, the dynamism of “osculating circle at P as limiting position of a circle passing through three points, say P,Q,R on a curve as Q, R approaches to P” was demonstrated and discussed. Here, ragging of the points Q, R along the curve towards the point P led question why does circle becomes small and large, and its dynamics?
With this exploration and connecting the concept definition from the textbook, students convinced that osculating circle at P has three point contact with the curve at P.
Here, the easy-to-use GeoGebra interface help students to understand the concept that, osculating circle at P has three point contact with the curve at P. And this understanding helped as metaphor to internalize that “tangent line at P has two point contact with the curve at P”.