Lesson 7

Distances and Parabolas

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7.1: Notice and Wonder: Distances (10 minutes)

CCSS Standards


Building Towards

Routines and Materials


Instructional Routines

Warm-up

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, 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

Display the image for all to see. 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 ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions.

Student Facing

Point F with 10 dashed lines extending from point F to other points. These points are then extended down to a solid line.

What 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 for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

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

After the discussion, display this image for all to see.

Graph of parabola with focus and directrix.  

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.

Ask students to add this definition to their reference charts as you add it to the class reference chart:

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)

Graph of parabola with focus and directrix.  

7.2: Into Focus (15 minutes)

CCSS Standards


Building Towards

Routines and Materials


Instructional Routines

Activity

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.

Launch

Provide students access to internet-enabled devices.

Conversing: MLR2 Collect and Display. As students work on this activity, listen for and collect the language students use to describe how the positioning of the focus and directrix affects the shape of the parabola. Write the students’ words and phrases on a visual display. As students review the visual display, create bridges between current student language and new terminology. For example, the phrase, “the point must be underneath the line” can be rephrased as “the focus must be underneath the directrix.” This will help students use the mathematical language necessary to precisely describe the relationship between the focus and directrix and the shape of the parabola.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness

Student Facing

The applet shows a parabola. In the applet, move point \(F\) (the focus) and the line (the directrix) and observe how the shape of the parabola changes.

  1. What happens as the focus and directrix move farther apart?
  2. Try to make the parabola open downward (that is, to look like a hill instead of a valley). What needs to be true for this to happen?
  3. The vertex of the parabola is the lowest point on the curve if it opens upward, or the highest point if it opens downward. Where is the vertex located in relationship to the focus and the directrix?
  4. Move the directrix to lie on the \(x\)-axis and move the focus to be on the point \((2,2)\). Plot a point \(P\), with coordinates \((6,5)\). It should lie on the parabola.
    1. What is the distance between point \(P\) and the directrix?
    2. What does this tell you about the distance between \(P\) and \(F\)?

Student Response

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Launch

Conversing: MLR2 Collect and Display. As students work on this activity, listen for and collect the language students use to describe how the positioning of the focus and directrix affects the shape of the parabola. Write the students’ words and phrases on a visual display. As students review the visual display, create bridges between current student language and new terminology. For example, the phrase, “the point must be underneath the line” can be rephrased as “the focus must be underneath the directrix.” This will help students use the mathematical language necessary to precisely describe the relationship between the focus and directrix and the shape of the parabola.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness

Student Facing

Here are several images of parabolas.

Coordinate plane, x, negative 2 to 5 by 1, y, negative 2 to 5 by 1. Parabola opens up, through approximately 0 comma 2 point 5, 2 comma point 5, 4 comma 2 point 5. Point F, 2 comma 1. Line y = 0.
Coordinate plane, x, y. Parabola opens up, drawn through approximately 0 comma 2 point 1, 2 comma 1 point 5, 4 comma 2 point 1. Point F at 2 comma 3.  Line y = 0.
Coordinate plane, x, y. Parabola opens up, drawn through approximately 0 comma 2 point 9, 2 comma 2 point 5, 4 comma 2 point 9. Point F at 2 comma 5.  Line y = 0.
Coordinate plane, x, y. Parabola opens down, through negative 3 comma negative 5, 1 comma negative 1, 5 comma negative 3. Point F at 1 comma negative 2. Line y = 0.
Coordinate plane, x, y. Parabola opens down, negative 4 comma negative 5, 1 comma negative 2 point 5, 5 comma negative 4. Point F at 1 comma negative 5. Line y = 0.
Coordinate plane, x, negative 3 to 6 by 1, y, negative 2 to 7 by 1. Parabola opens up, through 0 comma 2, 2 comma 1, 4 comma 2. Point F at 2 comma 2. Line y = 0.

Look at the focus and directrix of each parabola. In each case, the directrix is the \(x\)-axis.

  1. How does the distance between the focus and the directrix affect the shape of the parabola?
  2. What seems to need to be true in order for the parabola to open downward (that is, to be shaped like a hill instead of a valley)?
  3. The vertex of the parabola is the lowest point on the curve if it opens upward, or the highest if it opens downward. Where is the vertex located in relationship to the focus and the directrix?
  4. In the final image, the directrix is on the \(x\)-axis and the focus is the point \((2,2)\). Point \(P\) on the parabola is plotted.
    1. What is the distance between point \(P\) and the directrix?
    2. What does this tell you about the distance between \(P\) and \(F\)?

Student Response

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Anticipated Misconceptions

Students may be unsure how to find the distance between point \(P\) and the directrix. Remind them that the distance from a point to a line is the length of the segment between the point and the line and perpendicular to the line.

Activity Synthesis

The goal is to deepen students’ understanding of the role distances play in the definition of a parabola. Display this image for all to see. Ask students to identify the location of the focus (point \(F=(4,5)\)) and the directrix (the horizontal line \(y=1\)).

Graph of parabola with focus and directrix.  

Here are some questions for discussion:

  • “The vertex is a special point on a parabola. How can we use distances to describe how it’s unique?” (The vertex appears to be the point that is closest to both the focus and the directrix. We measure distance from a point to a line by drawing a segment from the point perpendicular to that line. So, the vertex is the midpoint of the segment drawn from the focus perpendicular to the directrix.)
  • “What is the distance from point \(B\) to the directrix? What does that tell us about the distance from \(B\) to \(F\)?” (\(B\) is 6.5 units from the directrix. That means \(B\) must also be 6.5 units from \(F\).)
  • “How could we verify that \(B\) really is 6.5 units from \(F\)?” (Draw in a right triangle where segment \(BF\) is the hypotenuse. Find the lengths of the legs by subtracting coordinates. Substitute into the Pythagorean Theorem and check that the result is 6.5.)
Engagement: Develop Effort and Persistence. Break the class into small discussion groups. Invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention

7.3: On Point (10 minutes)

CCSS Standards


Addressing

Building Towards

Routines and Materials


Instructional Routines

Activity

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.

Launch

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, students can label the directrix and the focus, and draw the legs of the triangles used to figure out the distance.
Supports accessibility for: Visual-spatial processing

Student Facing

The image shows a parabola with focus \((6,4)\) and directrix \(y=0\) (the \(x\)-axis).

Parabola on the coordinate grid. The focus is at point 6 comma 4. The directrix is a y equals 0, the x axis.
  1. The point \((11, 5)\) looks like it might be on the parabola. Determine if it really is on the parabola. Explain or show your reasoning.
  2. The point \((14,10)\) looks like it might be on the parabola. Determine if it really is on the parabola. Explain or show your reasoning.
  3. In general, how can you determine if a particular point \((x,y)\) is on the parabola?

Student Response

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Student Facing

Are you ready for more?

The image shows a parabola with directrix \(y=0\) and focus at \(F=(2,5)\).

parabola with directrix y=0 and focus at F = 2 comma 5. points F prime = 2 comma 2 and F prime prime = 2 comma 0 plotted.

Imagine you moved the focus from \(F\) to \(F’=(2,2)\).

  1. Sketch the new parabola.
  2. How does decreasing the distance between the focus and the directrix change the shape of the parabola?
  3. Suppose the focus were at \(F''\), on the directrix. What would happen?

Student Response

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Anticipated Misconceptions

If students are stuck, suggest they refer to the definition of a parabola on their reference chart. Ask them what must be true about the points in order for them to be on the parabola.

Activity Synthesis

Invite a previously selected student to share their picture of a right triangle. Ask them to describe how they found the lengths of the triangle’s legs.

Then, invite students to share their strategies for testing a general point \((x,y)\). If possible, ask some students who have used words and others who have used equations for variables to share, and draw connections between these two strategies. For example, if students discuss “the distance between the point and the directrix,” ask them if there is an expression that gives this distance (\(y\)).

Speaking: MLR8 Discussion Supports. As students share how they would verify whether a point \((x,y)\) is on the parabola with focus \((6,4)\) and directrix \(y=0\), press for details by asking students how they know that the distance between the point \((x,y)\) and the directrix \(y=0\) is \(y\). Show concepts multi-modally by drawing and labeling the distance between the point \((x,y)\) and the directrix \(y=0\) and the distance between the point \((x,y)\) and the focus \((6,4)\). This will help students explain how to check if a particular point \((x,y)\) is on the parabola.
Design Principle(s): Support sense-making; Optimize output (for justification)

Lesson Synthesis

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 \((\text-1, 3)\) and directrix \(y=1\). 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 \((\text-1,2)\).) Now display this image for all to see.

Graph of parabola with focus and directrix.  

Here are some questions for discussion:

  • “How far is point \((3,6)\) from the parabola’s directrix?” (5 units)
  • “How can you calculate the distance between any point \((x,y)\) and the directrix?” (Subtract 1 from the \(y\)-coordinate.)
  • “How can you find the distance between any point \((x,y)\) 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.

Cross section of a cone.
Cross section of a cone.
Cross section of a cone.

7.4: Cool-down - A Point and a Line (5 minutes)

CCSS Standards


Building Towards

Cool-Down

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Student Lesson Summary

Student Facing

The diagram shows several points that are the same distance from the point \((2,1)\) as they are from the line \(y=\text-3\)​​​. (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.

Parabola, focus and directrix on coordinate plane.

We can use this definition to test if points are on a parabola. The image shows the parabola with focus \((2,3)\) and directrix \(y=1\). The point \((\text-2,6)\) appears to be on the parabola. Counting downwards, the distance between \((\text-2,6)\) and the directrix is 5 units.

Now use the Pythagorean Theorem to find the distance \(d\) between \((\text-2,6)\) and the focus, \((2,3)\). Imagine drawing a right triangle whose hypotenuse is the segment connecting \((\text-2,6)\) and \((2,3)\). The lengths of the triangle’s legs can be found by subtracting the corresponding coordinates of the points.

Parabola with the focus at point 2 comma 3. Point negative 2 comma 6 is on the parabola. Directrix is at y equals 1.

Use those lengths in the Pythagorean Theorem to get \((\text-2-2)^2+(6-3)^2=d^2\). Evaluate the left side of the equation to find that \(25=d^2\). The distance, then, is 5 units because 5 is the positive number that squares to make 25. Now we know the point \((\text-2,6)\) really is on the parabola, because it’s 5 units away from both the focus and the directrix.