In my numerous blog posts about Exploring Conic Sections with The Geometer’s Sketchpad, I’ve featured an assortment of hands-on and virtual models for drawing conic sections, ranging from pins and string to paper folding to 17th-century linkages. But in order for students to understand why any of these approaches work, they must first grapple with the distance definitions of an ellipse, hyperbola, and parabola. As a reminder:

• An ellipse is the set of point P such that PF₁ + PF₂ is constant for two fixed points F₁ and F₂ (the foci).
• A hyperbola is the set of point P such that the difference of the distances from P to two fixed points, F₁ and F₂, is constant (in absolute value).
• A parabola is the set of points P such that the distance from P to the focus (a point) is equal to the distance from P to the directrix (a line).

Digesting these definitions takes time—even my teachers at City College need opportunities to put these definitions into practice. As such, I’ve developed a sequence of tasks that focus on plotting points that satisfy the distance definitions of the ellipse, hyperbola, and parabola. The six-page Web Sketchpad model below (and here) collects these tasks in one place.

On page 1, students draw segments connecting point P to points F₁ and F₂ . They measure the lengths of the two segments and use the Calculate tool to compute the sum. The sum is automatically updated when point P is dragged to other locations.

Students pick a constant for the sum—for example, PF₁ + PF₂ = 12 cm. They then drag point P to pinpoint locations for which the sum is 12 cm (or nearly so, 11.9 or 12.1 cm is close enough.) For each location they find, they press Leave a Trace to place a mark on screen. Slowly, as the traces accumulate, students see the outline of an ellipse. With one ellipse in place, students pick a different constant sum and a different color for the traces (using the slider in the lower-left corner) and plot a new ellipse with the same foci. The short video here demonstrates the process.

Now, students are ready for a more systematic and precise way to identify points that sit on an ellipse. On page 2 of the websketch, points F₁ and F₂ serve as centers for two sets of concentric circles.  For each set, the radii of the circles increase by 1’s, from 1 unit all the way up to 9 units. Point A sits on an ellipse with foci at F₁ and F₂. The goal is to use the Point tool to mark other points that sit on the same ellipse. Students must think about how the circles help them with this task. (Again, see the video for a demonstration.)

From here, I will often introduce the Pins and String method of constructing an ellipse. The Web Sketchpad approach to modeling the pins and string can be strongly motivated by the concentric circles approach on page 2.

Pages 4-6 of the websketch follow the same approach for introducing the distance definitions of a hyperbola in parabola. Notice that on page 5, thee are no measurements. Students use a circle to identify points that are equidistant from the focus and directrix.