# Lesson 7

Distances and Parabolas

### Lesson Narrative

In this lesson, students learn that a parabola is the set of points equidistant from a given point, or focus, and a given line, or directrix. Students explore the relationship between the positioning of these elements and resulting shape of the parabola. Then, they practice using distance calculations to test if particular points lie on given parabolas. This work helps prepare students to write an equation for a parabola in an upcoming lesson.

Students have the opportunity to attend to precision in language (MP6) as they articulate what they notice and wonder about a diagram showing several points equidistant from a given point and line.

One of the activities in this lesson works best when each student has access to devices that can run the Desmos applet, because students will benefit from seeing the relationship in a dynamic way.

### Learning Goals

Teacher Facing

• Comprehend (in spoken and written language) that a parabola is the set of points equidistant from a given focus and directrix.

### Student Facing

• Let’s analyze the set of points that are the same distance from a given point and a given line.

### Required Preparation

Prepare additional copies of the Blank Reference Chart blackline master (double sided, 1 per student). Students can staple the new chart to their full ones as they will need to continue to refer to the whole packet.

Devices are required for the digital version of the activity Into Focus.

### Student Facing

• I know that a parabola is the set of points equidistant from a given point and line.

Building Towards

### Glossary Entries

• directrix

The line that, together with a point called the focus, defines a parabola, which is the set of points equidistant from the focus and directrix.

• focus

The point that, together with a line called the directrix, defines a parabola, which is the set of points equidistant from the focus and directrix.