One of my calculus students has difficulty relating spatial information with algebraic expressions. To address this challenge, I used the following approach when explaining the trapezoidal rule this afternoon after school. The entire explanation took about 30 minutes, after which the student left satisfied that he/she understood the big idea of the trapezoidal rule: using trapezoids to approximate the area under a curve.
Explaining the Trapezoidal Rule
The following graphic illustrates most of my explanation for this student related to the trapezoidal rule. Subsequent images focus on specific elements in this big picture view. The closed form depiction of the trapezoidal rule is missing from the rightmost side of this picture, but it is shown in a subsequent one.
My explanation purposefully links a graphical depiction of trapezoids under a curve to an algebraic representation. In doing so, I use different colors to enable rapid connection of specific regions in the diagram with their associated algebraic terms. The idea to use multiple colors for both the graph and the expression came to me as the student seemed to struggle with a black and white only approach.
Trapezoids Approximating Area Under Curve
Four trapezoids fill the region under the curve, f(x), from x = a to x = b below. Each trapezoid is further segmented into a rectangle and a right triangle to contrast a rectangular method with a trapezoidal method. It is clear that the trapezoidal method more accurately estimates the true area under f(x).
Isolating a Trapezoid
To further hone in on the essence of a trapezoid, especially how its area is computed, the following figure isolates one trapezoid, placing it into a more familiar orientation, as seen in most geometry courses. This depiction allows one to recognize the trapezoidal shape more quickly than in the context of the problem, and hence, recall the formula used to find the area of a trapezoid. In the figure below, note that the color of the top and bottom of the trapezoid map directly back to the left and right side of the vertically oriented trapezoid in the figure above, while the green side of the trapezoid below is the top of the vertically oriented trapezoid. Additionally, the same colors surround the specific terms in the formula for the area of a trapezoid.
Developing the Trapezoidal Rule
Once the student clearly understood how to compute the area of one trapezoid, finding the areas of subsequent trapezoids easily followed, as the multi-term expression at the bottom of the following photo shows. Note that color continues to highlight the connection between the graphical depiction and the algebraic representation of the trapezoidal rule. This particular use of color made the difference for this student between not having a sufficient handle on the method, and being able to follow the explanation to its logical conclusion in the second photo below.
Thirty minutes after the student arrived with little comprehension of how to handle trapezoidal approximation, he / she left confident they knew why it was called the trapezoidal method, and how to use the trapezoidal rule in a problem. I believe they could also derive the trapezoidal rule for use with any problem since my explanation emphasized conceptual understanding over procedural fluency. In fact, I did not work any problem with specific values. I simply explained the concept by linking a graphical depiction with its algebraic cousin.
I am proud of my student for coming in after school for help, and of my being able to explain and illustrate the concept for him / her. This reinforces why I teach.