Multiple Representations

As part of my education class, I conducted classroom observations at a precalculus class and a calculus class once a week. The following post is a reflection on one of my classroom observations. Note: The names used are pseudonyms.

The precalculus students had spent the previous few weeks studying sinusoidal functions, and last Friday they spent the class period deriving the tangent function graph. Mr. Brown started by having the students use their prior knowledge of sine and cosine functions to find the values of several expressions involving the tangent function.

For the first “derivation” of the tangent function graph, he had students recall tangent values for the most common angles on the unit circle. To fill in the gaps between those angles, he asked students to use their graphing calculators to find tangent values for those angles. Using the table of angles and corresponding tangent values, Mr. Brown asked the students to predict what the graph of the tangent function would look like.

The second derivation also involved the unit circle, but relied on software called The Geometer’s Sketchpad. Mr. Brown projected the Geometer’s Sketchpad document at the front of the classroom and drew a unit circle with the sine/cosine triangle and the tangent triangle.


Mr. Brown had the students recall their prior knowledge about similar triangles from their geometry class in order to convince them that the larger triangle in fact had a leg with length equal to the tangent of the opposite angle. Using Geometer’s Sketchpad, Mr. Brown animated the motion of a point following the circular path and simultaneously graphed the height of the tangent line. In the end, the students saw the graph of the tangent function.

Overall, I thought using the two derivations was a thoughtful way of introducing the students to the tangent graph. It gave the students better insight into why the tangent function looks the way it does. They would have missed that part had Mr. Brown simply told the students to graph the tangent function on their calculators.

That same day, the calculus students learned how to use their graphing calculators to find the indefinite integral of various functions. Previously they had learned how to find the area by hand. Mr. Brown had a program on his computer that projected the graphing calculator onto the screen, along with the most recently pressed sequence of buttons. As with many demos where students are expected to follow along in real time, many students fell behind, so Mr. Brown ended up helping quite a few students individually.

Eventually though, all the students figured out how to use their calculators to find the indefinite integral and were asked to demonstrate their knowledge and understanding by completing a worksheet. Mr. Brown stressed the importance of doing each problem by hand first and then using the graphing calculator to check their answers. Taking advantage of the immediate feedback from their calculators allowed students to practice metacognition and re-evaluate answers that they got incorrect by hand. Again, this was an effective way of using technology to advance student understanding by providing an additional mode of representation.

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