Thursday, April 24, 2014
Parametric/Projectile Motion Simulated in Desmos - A Common Core Activity for Algebra/Precalculus
[Updated using folders to reduce amount of visible text. Click on the arrow next to the Folder icon to see the frames below. Thanks to Desmos team for this helpful hint!]
CLICK ON GRAPH TO ACTIVATE DESMOS...
The Desmos activity above is both an investigation of parametric representation and a tutorial for more advanced use of this remarkable WebApp. The The text in the side frames begins with a detailed background of the activity for the instructor and how Desmos can be used to demonstrate projectile motion using both parametric and rectangular coordinates. Some of the uses of slider 'variables' are demonstrated including animation, a powerful feature of Desmos.
In addition to showing how to use parameters in Desmos, the activity itself asks students to compare two different trajectories, representing an object dropped from some initial height, then a 2nd object two seconds later. The horizontal translation of the first graph is juxtaposed against the algebraic representations of these graphs using both system of coordinates.
The student activity starts about halfway down. There is a series of questions and actions the student needs to take in Desmos.
I'm hoping this will prove useful for both the instructor and the student. Desmos is powerful but, in my opinion, some of the illustrative examples provided by Desmos do not flesh out the ideas behind the various uses of slider 'variables'. I'm hoping this will fill in some of those gaps. I'm still a novice here so I'm sure more advanced users will be able to improve upon this...
Your comments and reactions are very helpful to me...
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Posted by Dave Marain at 7:01 PM 0 comments
Labels: activities, algebra 2, Common Core, Desmos, investigations, parametric, precalculus, projectile motion
Sunday, April 15, 2007
A Challenge Problem: Ellipses and Tangents and Normals
MAA members will likely recognize the following challenge that appeared on the outside of the envelope in the mailing to members or prospective members. I plan on giving this to my AP Calculus students as review for the exam or afterwards. As usual I will modify it for the student, place it in the context of an activity, broken into several parts with some hints. The original problem comes with a helpful diagram, however, unless I scan it, it would be difficult to reproduce. The problem involves a property of a point on an ellipse and requires basic understanding of the parametric form of this curve and some basic calculus and trig. The last part of the activity suggests a possible significance of this property but I'll leave the details to our astute readers.
Consider a standard ellipse, center at (0,0), with major axis of length 2a on the x-axis and minor axis of length 2b.
Let P(x,y) be a generic point on this ellipse with the restriction that P is not one of the endpoints of the major or minor axes. Consider the tangent and normal lines at P. Let P denote the point of intersection of the normal line with the x-axis and Q, the point of intersection of the tangent line with the x-axis.
Prove that (OP)(OQ) = a2-b2, where OP represents the distance between the origin and P and similarly for OQ.
Here is an outline with several parts for the student:
(a) Show that x = acos(t), y = bsin(t), 0<=t<2pi,>2-b2)/a)cos(t).
(f) Use (d) and (e) to derive the desired result: (OP)(OQ) = a2-b2
(g) Explain why we did not allow P to be an endpoint of the major or minor axes.
(h) What does the expression a2-b2 have to do with the foci of the ellipse? For EXTRA CREDIT, investigate this 'focal' property further.
Posted by Dave Marain at 8:40 AM 0 comments
Labels: calculus, ellipse, MAA, parametric, tangents, trigonometry