The purpose of this Warm-up is to elicit the idea that the set of points equidistant from a given point and line form a parabola. While students may notice and wonder many things about this image, the important discussion points are the equal distances and the smooth curve.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Monitor for the use of vocabulary such as equidistant, congruent segments, and parabola.

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If equal distances do not come up during the conversation, ask students to discuss this idea.

After the discussion, display this image.

Ask students if they have seen a curve like this before, and if so, what characteristics about it they recall. Students may remember the term parabola from earlier courses. The words quadratic, vertex, and axis of symmetry may come up. If not, there is no need to bring them up.

Add the following definition to the class reference chart, and ask students to add it to their reference charts:

A parabola is the set of points that are equidistant from a given point, called the focus, and a given line, called the directrix. (Definition)

In this task, students look at the relationship between the positioning of the focus and directrix and the size and shape of the resulting parabola. This visual understanding of parabolas will be helpful when students write an equation for a parabola, in an upcoming activity.

As students consider the positioning of the vertex, encourage them to use precise vocabulary. Monitor for responses such as, “The vertex is halfway between the focus and the directrix.” Point out that if they mean the vertex is an equal distance from the focus and directrix, this is true for all points on the parabola. Ask these students how the vertex differs from other points on the parabola. As students carefully describe the relationship between points on the parabola, the focus, and the directrix, they are increasing their precision of language (MP6).

In the digital version of the activity, students use an applet to change the parameters of a parabolic equation. The applet allows students to see how changing the focus, directrix, and vertex of a parabola affects the shape of its graph. The digital version may be helpful for connecting, in a dynamic way, the parts of an equation to the graph of a parabola.

Students use distance calculations to test whether points are on a parabola. In an upcoming activity, students will generalize this process when they write an equation for a parabola given a focus and directrix.

Monitor for students who draw right triangles on the image to help calculate distances.