Ideas from Classroom Teachers for
In general, vectors seem to be a counter-intuitive subject for students, possibly because separating a force (just the magnitude and direction part) from what it acts on is abstract. One possible tie-in would be anything involving wind, which is something familiar. Also, students know of translations and rotations from geometry, so invoking those transformations would help with the familiarity aspect.
Emphasize that a vector is defined by its direction and magnitude, not by its location.
To be considered: the zero vector; unit vector; vectors in terms of i and j; horizontal and vertical components of a vector; properties of vectors.
Students could use vectors to prove geometric statements, such as "the lines that join one vertex of a parallelogram to the midpoints of the opposite sides trisect the diagonal."
The angle between vectors can be found (proven) using the Law of Cosines and the dot product.
Wherever possible, tie vectors in with geometric transformations. For example, students already know what translations are; now they have the tools to formalize them.
Parametrics take a long time to introduce because students have been indoctrinated in x-y thinking since day one. Standard introductions to parametrics consist of some variant on the Ferris wheel problem, in which the x-coordinate and y-coordinate can both be visualized as directly dependent on time. One aspect of the Ferris wheel that needs to be brought out (even emphasized) is that by a set of parametric equations you can get a graph (in this case, a circle) which you cannot get by having a single y-in-terms-of-x function. Also (equally important) is that such a graph gets drawn in a way that represents the actual motion involved. In other words, it is usual to emphasize how to turn a pair of parametric equations into a single y = f(x), but that misses the point.
Technology note: Most graphing calculators have a "parametric mode" that allows students to investigate parametric equations graphically.
I would suggest teaching parametric equations as a separate mini-unit, perhaps just after polar coordinates (another alternative mode of graphing) and before vectors.
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