Eleven new Clickable-Calculus examples have been added to the Teaching Concepts with Maple section of the Maplesoft website. That means some 74 of the 154 solved problems in my data-base of syntax-free calculations are now available. Once again, these examples and associated videos illustrate point-and-click computations in support of the pedagogic message of resequencing skills and concepts.
This message has been articulated in earlier blog posts. Suffice it to say here that Maple's technologies allow concepts to be taught first, before emphasis is placed on manipulative skills. Moreover, Maple can be used to implement stepwise solutions so that the relevance of such skills becomes clear. In my 15 years' experience with Maple in the classroom, this proved to be an effective strategy, one that helped students master mathematics more efficiently than any other paradigm I had previously used.
The new additions to the site include two problems in algebra, and two in trig, three in differential calculus, two in multivariate calculus, and one each in integral calculus and linear algebra. In algebra, we obtain the coordinates of the intersection of two parabolas as a way of learning how to solve pairs of quadratic equations. A second algebraic problem is the analysis of a complete quadratic relation whose graph is an ellipse.
For trigonometry, we show how to invert graphically the inverse of a trig function, resulting in a graph of the principle branch of that function. Then, the Standard Functions tutor is used to illustrate the effects of varying a, b, h, k in the transformation a f(b (x - h)) + k for f, any one of the elementary trig functions.
In differential calculus, the Rational Function tutor illustrates graphing a rational function and finding its asymptotes. Taylor polynomials are accessed from the Context Menu and in the Taylor Approximation tutor. The optimization problem of finding a point on f(x) = sinh(x) - x e-3x closest to the point (1, 7) is solved. But linked to this blog is another post that explores that problem further, past just the "solution."
For integral calculus, we again tackle the question of "What does Maple do for methods of integration?" when we show how the Integration Methods tutor helps implement the method of trigonometric substitution. For the lines-and-planes section of multivariate calculus, one problem shows how to implement the equation of a line determined by a point and a given direction. Another problem comes to grips with skew lines.
Finally, in linear algebra, a curve describing a linear model is fit to data via a least-squares technique. (Shortly, the Maple Reporter will carry a Tips & Techniques article addressing the many ways that both linear and nonlinear least-squares fitting can be implemented in Maple.)
Once again, we hope that these examples illustrate the potential for changing the classroom dynamic, shifting math courses more towards the conceptual, by using Maple's syntax-free paradigm to support a resequencing of skills and concepts.