Experience First

Nothing evokes panic like seeing a spider out of the corner of your eye and wondering if it’s going to stay there! In this activity, students use parametric equations to track a spider’s position on the wall. Students see how horizontal and vertical distance are both functions of time that work in tandem.

In questions 6-8, students reason through how one can determine a spider’s vertical position when given its horizontal position. We expect students to first find the time at which a spider reaches that horizontal position and then evaluate y(t) at that specific value of t. This will set them up for the conceptualization of eliminating a parameter.

As you are monitoring groups, look for groups that are struggling to plot the ordered pairs. With three variables floating around, some students are unsure how to keep track of all the information. The labels of the x and y axis should help, but we encourage students to mark the ordered pairs with the t-value as well as indicate the direction/order of the points with an arrow. This is new for parametric equations, since Cartesian equations have no sequence to them.

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Formalize Later

In today’s lesson we focus on the parameterization of curves that do not involve trigonometry. Students should be comfortable making a table of values that includes t, x, and y and then plotting the ordered pairs generated. The focus in the conversions is on turning a parametric equation into a standard Cartesian equation, though occasionally students may be asked to introduce a parameter like in question 3 of the Check Your Understanding.

In question 5 of the Check Your Understanding, students reason through how to create parametric equations of line segments by thinking about the horizontal and vertical movement (components) separately. Restricting the domain of the parameter allows us to distinguish between a segment, ray, and line. Much of this work is related to fundamental ideas of linear functions, slope, and even some ideas about vectors.

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