Virginia Mathematics Teacher Spring 2017

sequence with the first few iterations of the fraction by hand (e.g., 3, .75…). However, use of a calculator allows them to proceed in a more efficient way using the home screen of their calculator. Using this method, students can more easily generate a sequence and notice that it is “converging.” This can be done by entering a first iteration (e.g. 3) into their calculator, and then using the form of the continued fraction to generate subsequent iterations (see Figure 2 for screenshots of early and later iterations). By setting up the recursive process in this way, students can press ENTER for the next iteration, and keep pressing it until they notice a pattern or limit.

With this approach of course it is important for the teacher to ensure that students understand what they are doing, asking students to explain what each entry signifies and what is happening each time they ENTER. Recursive function strategy A second strategy involves generating and using a recursive function. With this method students first analyze the relationship between successive iterations, generated either by hand or with a calculator, and use this relationship to derive an appropriate recursive function and determine an appropriate starting point. For example, one preservice student teacher said, “I tried to take it one step at a time, starting with the first term of the sequence… it finally clicked that I was entering 3 divided by 1 plus the previous answer every time. It was at this point that I realized the pattern for the sequence.” From this realization he was able to create an equation that modeled the continued fraction (see Figure 3a). Students can enter their recursive function u (n) = 3/(1 + u(n-1)) , the beginning step number nMin=1, and the beginning value at that step u (nMin)={3} into the sequence mode editor of a TI- 84 graphing calculator as shown in Figure 3a, to generate a numerical sequence (Figure 3b) and generate a graphical representation (Figure 3c) of the sequence, both of which can help students visualize what is happening at each step of the process. Linking these representations can help students understand the mathematics more deeply. Quadratic equation strategy A third strategy for evaluating this continued fraction is to derive its value by connecting the recursive equation to a quadratic equation. Some students realize, often after working with one of the strategies shown above, that in the long run, the terms and approach a limit, and hence treat these two terms as equal, resulting in the revised equation: . (Note: we have seen few students use that limit, say L, in the revised equation). From this point students can use their algebraic manipulation skills to write this as the quadratic equation + – 3 = 0, and then use their knowledge of quadratic formula to

Figure 2 : Screenshots showing early and later iterations.

Figure 3 a, b, c : Using the sequence mode of a TI-84 graphing calculator to generate representations.

Virginia Mathematics Teacher vol. 43, no. 2

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