Unit 6, Lesson 7
Distances and Parabolas
Geometry
7.1
Warm-up
Instructional Routines
Notice and WonderMaterials
To Copy (from Blackline Masters)
Blank Reference Chart
Activity Narrative
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.
Launch
Arrange students in groups of 2. Display the image. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things that they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
Student Response
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Activity Synthesis
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)
Lesson Synthesis
The goal is to make connections between coordinates of points and distances in the plane.
Tell students to imagine a parabola with focus and directrix . Ask them where the vertex of the parabola would be. They can make a quick drawing if it’s helpful. (The vertex would be at .) Now display this image.
Here are some questions for discussion:
- “How far is point from the parabola’s directrix?” (5 units)
- “How can you calculate the distance between any point on the parabola and the directrix?” (Subtract 1 from the -coordinate.)
- “How can you find the distance between any point and the focus?” (Subtract the coordinates to find the lengths of the legs of a right triangle for which the distance is measured by the triangle’s hypotenuse. Then use the Pythagorean Theorem.)
- “Compare and contrast what we’ve done so far with what you did when you wrote the equation for a circle.” (For both circles and parabolas, we are looking at distances between points on the figure and another point. For a circle, the “other point” was the center of the circle. Here, it’s the focus of the parabola. For parabolas, we also have a line to worry about. We didn’t have that for circles.)
Tell students that as we may have discussed in an earlier lesson, another way circles and parabolas are related is that they are both cross sections of a cone. This is why they are called conic sections. Another example of a conic section is an ellipse. Students will learn more about these figures in future courses. It's not important that students memorize the term “conic section,” but it’s helpful for them to know that the parabolas and circles are related in this way.
Student Lesson Summary
The diagram shows several points that are the same distance from the point as they are from the line . (Distance is measured from each point to the line along a segment perpendicular to the line.) The set of all points that are the same distance from a given point and a given line form a parabola. The given point is called the parabola’s focus and the line is called its directrix.
Focus and directrix on coordinate plane, no grid. X axis from negative 3 to 7. Y axis from negative 3 to 3. Parabola opens upward with vertex at 2 comma negative 1. Points plotted at negative 2 comma 1, 0 comma negative 0 point 5, 2 comma negative 1, 4 comma negative 0 point 5, and 6 comma 1. Focus plotted and labeled at 2 comma 1. Dotted lines drawn from each point to focus. Directrix, horizontal line, y equals negative 3. Vertical lines drawn from each point to directrix.
We can use this definition to test if points are on a parabola. The image shows the parabola with focus and directrix . The point appears to be on the parabola. Counting downward, the distance between and the directrix is 5 units.
Now use the Pythagorean Theorem to find the distance between and the focus, . Imagine drawing a right triangle whose hypotenuse is the segment connecting and . The lengths of the triangle’s legs can be found by subtracting the corresponding coordinates of the points.
Use those lengths in the Pythagorean Theorem to get . Evaluate the left side of the equation to find that . The distance, then, is 5 units because 5 is the positive number that squares to make 25. Now we know the point really is on the parabola, because it’s 5 units away from both the focus and the directrix.