A rectangle is drawn so that its lower vertices are on the x-axis and its upper vertices are on the curve y = sin x, where 0 ≤ x ≤ π.

• Write down an expression for the area of the rectangle.

• Find the maximum area of the rectangle.

The first question one must ask oneself is: what is this problem assessing? The student must first translate the words into a mathematical model. The model is geometric, i.e., the graph of the sine function between 0 and π. The student must then sketch a rectangle with the given conditions and deduce the base and height both from the model and the given information.

(We need only ascertain that the area at the endpoints of the domain are less than this value, and we can easily see from the formula that when x=0 and x=π, A=0.)

Indeed there is no pedagogical value in the process involved in finding the maximum – the only skills being tested here are calculator ones. The student does not need to understand any calculus concepts to obtain the correct answer.

If we, however, would like to test some calculus concepts in obtaining the solution to this question, we could easily require an analytical solution to this question where the students would use a

CAS calculator.

The student must now interpret these results within the given domain and select x=2.431 or x=0.7105 (which is visible when the highlighted line is scrolled to the right).

To answer part (b) analytically, the student must know that the maximum of the function occurs when the first derivative is zero. Hence, using a CAS:

Teaching tip:

Throughout teaching with technology the teacher must be very clear as to the nature and amount of documentation the student is required to show in assignments and assessments. The student learns this best through the classwork the teacher does by way of examples and model questions.

The student must interpret what this means, i.e., a maximum occurs at a point when the 2nd derivative at this point is negative. Now, the student must evaluate the area function at this value, and obtains that the area of the rectangle is 1.12 square units!

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Once the problem has been correctly translated into a mathematical model, the student can easily now see that A = sin(x)(π-2x).

In order to answer (b) it is essential that the student arrive at the correct model. With a graphics calculator, there is no understanding of calculus necessary to successfully answer this question. The student need only enter the formula for the area obtained in (a) and using the calculate menu, read off the maximum value of the function within the given domain 0≤x≤π, A= 1.12 square units. > TOP

An example taken from the International Baccalaureate Higher Level May 2000 paper one final examination reveals that many of the concerns above can be allayed without sacrificing the pedagogical principles most mathematics educators hold dear. The IB requires a graphics display calculator for both papers one and two of the final examination in Higher Level and does not permit a calculator with symbolic capabilities.







						COMPUTER ALGEBRA SYSTEMS - TECHNOLOGY IN MATHEMATICS EDUCATION