Math Lesson - Working with Parabolas
The video will show the students real-world examples of how parabolas are used and where they can be found. I will explain to the students that I will ask them a few questions at the conclusion of the video. During the video, students are to take notes describing not only the different examples, but also the different shapes of the parabolas.
After students watch the video, ask them if they can think of any other real life parabolas. Explain that a parabola is a special arch-shaped curve. Each and every point of a parabola is at an equal distance from a fixed point (called the focus) and a fixed line (called the directrix).
Tape up two large pieces of paper (preferably on the board) that have these definitions written out in large lettering so the students can refer to it for the rest of the lesson. Tell the students that today they are going to learn how to graph the equations that make these parabolas. Pass out a sheet of paper. Ask the students to find a partner to work with for the rest of the lesson.
Ask them to cut the sheet of paper in half (width-wise) and each take one side. Pass out wax paper that has been cut ahead of time to fit those half sheets. Have them place the wax paper over the paper. Turn the papers so that they are “up and down.”
Model steps under document camera so that each student can follow along. Draw a straight horizontal line on the wax paper using a permanent marker at the bottom of the paper. Ask the students to do the same using their ruler to make the line straight and parallel from the bottom. Remind them that this will be their directrix. Ask for the definition of the directrix again and call on a student to remind the class.
Ask the students to find the center of the paper width-wise and place a dot with their marker. Let them know that they can place this dot as close to or as far away from the directrix as they choose. Ask the students what they think this dot is.
Tell the students that since every point on a parabola is at an equal distance from the focus and the directrix, we can find points of the graph by folding the wax paper so that the directrix and the focus intersect each other. Ask the students what the definition of intersect is in regards to parabolas.
Students will either use their Chromebooks or notebooks to take notes on the following information. Students are encouraged to offer up examples and provide sample points throughout the discussion.
- Have students graph the line y = 3x-2. Then, talk about the importance of the values (-2) and (3), and how they change the shape of the graph.
2. Change the equation to y =3x^2-2 and ask the class if they feel the graph will change. Have them build an x/y table and plot the points from it to see the graph of the new equation. They can choose any values for x and they plug them into the equation and see what happens to y.
- Negative two is the y intercept where the graph starts and 3 is the slope of the line rise over run.
2. Example if x = 0 then y = 3(0)^2-2 which is -2.
For the pairs of students that easily understand the concept, ask them to take their wax paper parabola and lay it on top of a piece of graph paper. Ask them to start plotting the points of their parabola. Ask them to discuss patterns in their results with their partners.
Ask the students to remember the video in the beginning of the lesson and some of the parabolas they saw. Ask them how they think what they learned about parabolas could be applied to those examples.
What do you know about the relationship between the focus, the directrix, and the points of a paraboloa?
What is the focal length of a parabola?
Architects or builders might be able to use parabolas to shape the bridges, roller coasters, and arches we saw. Encourage other answers as well.
Every point on the parabola is equidistant from the focus and the directrix.
It is the distance between the focus and the vertex.
Towards the end of class, students will be given five sample problems to evaluate their understanding of the material. For each of these problems, students will have to go through a variety of steps to get to the final answer. To show this, students will have to create a Google Slide depicting each of these steps. Once the students have completed the sample problems, randomly call on five different students, and have them explain how they got their answer.