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Human Conics CA Standards: Algebra 1 - Needs to be identified. (The standards do not address a parabola in terms of a focus and directrix, nor vertex form.) I would add the Geometry standard for circles. Algebra 2 Learning Objectives Students will: Define conic sections as a locus of points Apply locus definitions to draw conic sections Collaborate with partners to solve a problem. Materials Sidewalk chalk Lightweight rope (about 10-12 feet per group) Right angle measures (e.g., 8.5 x 11 sheets of cardstock or right angle rulers) Compasses Markers or colored pencils or pens White boards for games Lesson Guide A. Opening: Vocabulary guessing game a. Using the CST released test questions, students will solve the problem and write their answers on the white board provided. The first group to provide the write (right) and complete solution has the chance to guess the scrambled word. b. After all the words are decoded the students will initially fill in the first and second column of the vocabulary worksheet. This activity will scaffold and build students’ understanding of these terms. (See attached worksheet for this activity. Words included on the worksheet are locus, parabola, ellipse, directrix, focal point/focus/foci, circle, radius, center, perpendicular, vertex, conics, and midpoint) B. Work time: Conic construction (In Door). a. Give each student a compass and the Human Conics Sheet, have the students illustrate the following (questions 1 & 2): i. Circle ii. Ellipse iii. Parabola b. Have them define the above-mentioned terms based from their illustration, and prior knowledge. Have them identify the other terminology that relates to it. i. Circle – radius, center ii. Ellipse – foci or focal points iii. Parabola – vertex, directrix Source: NCTM-Illuminations, Human Conics c. Provide time for students to discuss how they construct the locus of points. What are some of the limitations and conditions that should be followed to get the accurate path of points? C. Work time: Conic Construction (Out Door) Note: Before going outside, separate students into groups of three. Three students are needed for ellipse and parabola, two to represent foci or directrix and one to draw. For circle, only two students are required. Students will answer remaining questions on the worksheet. Task: They will be working in groups to draw a circle, an ellipse, and a parabola. Each group will have one piece of chalk and one piece of rope. Circle Students draw a perfect circle using the chalk and the rope. If students need a hint, suggest that they consider themselves to be a human compass. If they need further instruction: Fold the rope in half. One student puts the ends together, and holds them on the ground to be the center of the circle. The second student stretches the rope and puts the chalk in the bend at the midpoint. The second student then drags the chalk along the ground, while pulling the role taut. Note that students figuring out the activity independently may not fold the rope. This is not a problem. Questions: How many students are actually needed to draw the circle? What are their roles? What did the rope represent? As they finish have them complete the questions on the worksheet. Ellipse If students need hints, tell them that fact that there are three people in the group is significant and to consider what they did to draw the circle. They should also reference the significant parts of an ellipse, from the vocabulary sheet, as well as question 1 and 2 from the worksheet. If students need further instruction: Two students are human foci, holding the ends of the rope at fixed points on the ground. These students should not hold the rope taut. The third student uses the chalk to pull the rope taut and sweeps out the locus of points. Not sure if a picture here would help??? Questions As they finish, ask students to consider and discuss the questions on the activity sheet. Parabola Draw a focus approximately 4 feet from the directrix. This does not need to be precise. Assign roles to the three students in the group: F, D and A. Student D will be responsible for the directrix and will need a right angle measure, such as cardstock to approximate right angle measure. Student F will be responsible for the focus of the parabola. Student A will mark points on the parabola. Source: NCTM-Illuminations, Human Conics Assign each student a point on the rope. Student A is at the marked midpoint of the rope. At equal distance from her, measured by the folding the rope, are F and D. F should hold her point of the rope at the focus of the ground. D should place the right angle measured on the directrix and guide the rope along side of the measure. She should move the card and the rope along the directrix, A pulls the rope taut. When the rope is taut and perpendicular to the directrix, A should mark the point on the parabola. A picture here also might be beneficial.??? Students use the same rope length and repeat the procedure to draw a point on the other side of the parabola. Then, change the lengths and repeat for a total of at least ten points. Questions What is the shortest segment from the focus to the directrix? What is the midpoint of this segment? Why is it important to keep the rope perpendicular to the directrix? How can you find the vertex of the parabola using your rope right angle measure, and group members? D. Closing Questions for Students a. What effect does the length of the rope have on the shape of the conic? Is the rope ever too short? b. Why can you draw the circle with fewer people than the ellipse of the parabola? c. Which conics can you draw as a continuous line, without picking up your chalk? d. How could you use paper and pencil to draw or verify conics? e. Have the students complete the last column of the Vocabulary worksheet by writing the correct definition based from the activity and the discussion. E. Assessment options a. Give students a picture of an ellipse and a parabola with possible foci or directrix indicated. As them to sue a ruler and right angle measure to determine and explain whether or not the figure is actually the named conic. b. Give students two thumbtacks, string and a piece of cardboard to draw an ellipse. This is an individual reproduction of the chalk activity. Students may also be challenged to find a way to draw a parabola using these materials. LOVE THIS!! c. AS students to write a summary of either the ellipse or parabola constructions for the benefit of a classmate who has missed the lesson. The summary should include the definition and an explanation of how the drawing technique applies the definition. d. Students and teachers whoa re comfortable with the technology may construct these conics using geometry construction software. F. Extension a. Have the students construct a hyperbola. Source: NCTM-Illuminations, Human Conics Human Conics Vocabulary Worksheet 1. “soclu” What do you know? Illustration / Example Definition 2. “baaraplo” What do you know? Illustration / Example Definition 3. “pislele” What do you know? Illustration / Example Definition 4. “tceiixrid” What do you know? Illustration / Example Definition 5. “iocf” What do you know? Illustration / Example Definition 6. “lecric” What do you know? Illustration / Example Definition 7. “saruid” What do you know? Illustration / Example Definition Source: NCTM-Illuminations, Human Conics 8. “recnet” What do you know? Illustration / Example Definition 9. “treexv” What do you know? Illustration / Example Definition 10. “scocin” What do you know? Illustration / Example Definition 11. “dipenreplacur” What do you know? Illustration / Example Definition 12. “tonidimp” What do you know? Illustration / Example Definition 13. What do you know? Illustration / Example Definition 14. What do you know? Illustration / Example Definition Source: NCTM-Illuminations, Human Conics Human Circle Human Ellipse 1. What is the definition of a Circle? 1. What is the definition of an Ellipse? 2. In the space below, draw a circle using the 2. Draw a sketch to illustrate the definition. compass. 3. Working with a partner, sidewalk chalk, and 3. Working with 2 partners, sidewalk chalk and a rope, draw a perfect circle on the pavement. a rope, draw an ellipse on the pavement. Explain how you were able to be a human Explain the role that each person had in the compass and complete the drawing of your drawing. You may use a sketch to help circle. illustrate your explanation. ??? 4. What does the rope represent in your human 4. What would happen if the foci of the ellipse compass? moved closed together or further apart? 5. What does the length of the rope represent? Source: NCTM-Illuminations, Human Conics Human Parabola 1. What us the definition of a parabola? 2. Draw a sketch to illustrate the definition. 3. Working with 2 partners, sidewalk chalk, a rope and a right angle measure, draw a parabola on the pavement. Explain the role that each person had in the drawing. 4. How can you find the vertex of the parabola? 5. What can you say about the distance between the parabola and the focus or directrix at the vertex? 6. What would happen f the focus moved closer to the directrix? 7. What would happen if the focus moved further from the directrix? Source: NCTM-Illuminations, Human Conics