Chapter 4 DATA ANALYSIS AND RESULTS
5.5 Future research
As revealed in the literature review, a substantial amount research has investigated students general understanding of the concept of function, but very little is known about how stu- dents reason about parametric functions. This study aimed to fill this gap in the literature. However, further research on the topic of the teaching and learning of parametric functions is needed. This study investigated several different aspects involved in learning the concept of parametric function. It would be beneficial to expand on each of these.
1. How do students reason about the formal aspects of the concept of parametric function, such as domain, range, and uniqueness of function value?
2. How do students apply their general calculus graphing schema to the context of parametric functions in order to graph curves that cannot be expressed in standard form?
3. How do students reason about parametric functions in a multivariable context?
4. How does the study of non-planar curves contribute to students’ understanding of para- metric functions?
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Appendix A
INSTRUCTIONAL TASK
1. Imagine that the bottle1 above is being filled with water at a constant rate.
(a) Sketch a possible graph of the relationship between time and the volume of the water (with time on the horizontal axis and volume on the vertical axis).
(b) Sketch a possible graph of the relationship between time and the height of the water (with time on the horizontal axis and height on the vertical axis).
(c) Sketch a possible graph of the relationship between the volume of the water and the height of the water (with volume on the horizontal axis and height on the vertical axis).
(d) What would your graphs for (a)–(c) look like if the bottle was being filled faster (but still at a constant rate)?
(e) Specify the orientation of each of your graphs in (a)–(d). That is, indicate (with arrows) from where and in what direction the graphs are drawn with respect to time. 2. Let (x(t), y(t)) = 1 √ t,1 .
(a) Give an analytic rectangular representation of the above parametric representation. (b) Sketch the graph and specify the orientation.
3. Suppose a particle is traveling at a constant speed of 2 units per second along a linear course starting at the point (0,8) and ending at the point (4,0).
(a) What are the coordinates of the particle after 1.5 seconds? (b) What are the coordinates of the particle after t seconds?
Appendix B
FINAL EXAM QUESTIONS
1. Let x(t) = √1
t, y(t) = 1
t2, t >0.
(a) Express the above curve by an equation in x and y. Explain. (b) Sketch the curve. Explain.
2. (a) Is the relationship between x and y the same for both of the above graphs? Explain your reasoning.
(b) Do the curves above have the same analytic parametric representation? Explain your reasoning.
Appendix C
INTERVIEW QUESTIONS
1. State whether each of the following represents a function. (a) ρ(θ) = θ.
(b) g(x) =±√4−x2. (c) p(t) = (t2, t3).
2. Assume that Coolers 1 and 2 above are the same size. Imagine that they are full of water and being emptied at constant rates r1 and r2, respectively. Assume that |r1|<|r2|. (a) Sketch a possible graph of the relationship between time and the volume of the water
for each of the coolers. Sketch both of these graphs on the same coordinate plane with time on the horizontal axis and volume on the vertical axis.
(b) Sketch a possible graph of the relationship between time and the height of the water for each of the coolers. Sketch both of these graphs on the same coordinate plane with time on the horizontal axis and height on the vertical axis.
(c) Sketch a possible graph of the relationship between the volume of the water and the height of the water for each of the coolers. Sketch both of these graphs on the same