Folding Activity 3 Part 1 You re not done until everyone in your group is done! If you finish before someone else, help them finish before starting on the next part. You ll need: Patty paper Ruler Sharpie Compass Colored paper Tape or glue stick As you do each of the following, be careful not to smudge your work. For any step that includes the use of a Sharpie, wait about 30 seconds after marking before you do any folding. 1. Using a Sharpie, draw a point near the center of your wax paper. 2. Label this point O. 3. Using a pencil and a compass, draw a circle of radius 4 cm, using the point you drew in step 1 as the center. 4. Using a Sharpie, draw another point somewhere outside your circle. Where you put the point will affect the final result. Don t put your point too close to the edge of the paper or too close to the circle, or it will be difficult to do the rest of the activity. Try to arrange it so that everyone in your group has their point at different distances and in different positions from their circle. 5. Label this point F. 6. Fold the paper so that the point F lands on the circle (or the circle lands on F). 7. Crease the paper. 8. Slide the point along the circle just a little bit so that a different place on the circle is over the point F. 9. Crease again. 10. Keep sliding, folding and creasing the paper so that different places on the circle land on the point. The closer together your creases are, the more refined your shape will be. 11. Keep doing this until you have gone all the way around the circle. 12. Unfold your paper. Do you see a definitive shape? 13. Carefully darken the outline of your shape with a Sharpie. You ve just drawn a hyperbola! 14. Tape or glue the edges of your patty paper to a piece of colored paper. 15. Write The Hyperbola and your name at the top of the colored paper. If you don t know how to do this, ask one of your group members or your instructor.
Folding Activity 3 Part 2 You ll need: Patty paper Ruler Sharpie Compass Colored paper Tape or glue stick 1. With a pencil, trace the points you made on your hyperbola layer. 2. With the points still carefully aligned, draw a point on the top layer that lands somewhere on your hyperbola. Label this point T. 3. Using a pencil and a straightedge, draw a line segment that goes from O to T and extends beyond T to the circle. 4. Draw a point where this segment meets the circle and label this point D. 5. Draw a segment from T to F. 6. Carefully fold your paper so that OT lies on top of TF and crease. 7. Leave your paper folded, and draw a point on the back of the segment TF where O shows through from the other side. 8. Unfold your paper and label this point K. 9. With the points O and F carefully aligned, tape or glue this copy only at the top edge over the hyperbola on your colored paper.
Discovering the Hyperbola Parts 1 and 2 Name When you have finished making and labeling your hyperbola, answer the following questions. Each person should fill in their own answers, but work together to compare answers on your different ellipses before answering: a) Measure the distance (in cm) from O to D. How long is it? Is this surprising? Why or why not? b) Measure the distances (in cm) from F to T and T to O. Answer for your hyperbola: OD = cm Answer for your hyperbola: FT = cm TO = cm c) What do you notice about the measurements of FT and TO on your hyperbola when you compare them to those of your group members? d) How does this combination of measurements of FT and TO compare to the distance across your hyperbola (from vertex to vertex)? e) Compare your hyperbola to those of your group members, and compare your answers to the previous questions to your group members answers. Use what you learn to decide as a group what the definition of an ellipse is.
f) Since you know what makes an ellipse, and you know what the equation of an ellipse is, how do you think the similarity of a hyperbola to an ellipse and the answer to question c) affects the equation of an ellipse so that it becomes the equation of a hyperbola?
Folding Activity 3 Part 3 You ll need: Patty paper Ruler Sharpie Compass Colored paper Tape or glue stick 1. Using a pencil, trace the points you made on your hyperbola layer. You don t need the circle this time. 2. Using a pencil and a straightedge, draw a line segment that goes through, and extends beyond the points F and O. 3. Lay this copy on top of your finished hyperbola and carefully align the points, but don t tape it down yet. 4. Draw points where this new line segment crosses your hyperbola. 5. Label these points A 1 (near O), and A 2 (near F). 6. Using a pencil and a compass, draw the perpendicular bisector of this segment. 7. Draw a point where the two segments cross and label it C. 8. Measure the distance from A 1 to C and call it a. 9. Measure the distance from O to C and call it c. 10. Compute the value b = c 2 a 2. 11. Draw points b units along the bisector from the center. 12. Label these points B 1 and B 2. 13. Darken the segment A 1 A 2 with a Sharpie; this segment is called the transverse axis of a hyperbola, but you won t really have room to label it. 14. Darken the segment B 1 B 2 with a Sharpie; this segment is called the converse axis of a hyperbola, but you won t really have room to label it. 15. Using a protractor and a pencil, draw line segments through A 1 and A 2 that are at 90 to the segment A 1 A 2. 16. Using a protractor and a pencil, draw line segments through B 1 and B 2 that are at 90 to the segment B 1 B 2. 17. Draw points where these line segments intersect. 18. Using a pencil and a straightedge, draw long line segments that go through the opposite corners of the rectangle you just drew. 19. With the points carefully aligned, tape or glue this copy only at the bottom over the two previous layers on your colored paper. The perpendicular bisector of a line segment is the line that goes through your line segment at exactly 90, and at exactly the halfway point of the segment. We call the halfway point the Midpoint. If you don t remember how to do this, or if you never learned how, ask the instructor to show you. It s quick!
Discovering the Hyperbola Part 4 Part 4 You ll need: Scratch paper Graph paper (see back of packet) Pencil Now that you know something about hyperbolas whose center is the origin (the point (0,0)), discuss in your groups what you think the equation of an hyperbola would be if it were centered at a different point. g) In Part 3, you found the general form for the equation for a hyperbola. What would the equation be of the hyperbola where and a = 3 and b = 2? h) What would the equation be of the same hyperbola if it were shifted so that instead of having its center at the point (0,0), its center was at the point (1, 4)? i) Graph this hyperbola on one of the grids at the back of this packet. Label all of its features, including: i. Transverse axis ii. Coordinates of both endpoints of the transverse axis iii. Conjugate axis iv. Coordinates of both endpoints of the conjugate axis v. Coordinates of the center vi. Coordinates of both foci Your answer is your graph. Write the equation of the on the graph and add this graph to your portfolio. One focus, two or more foci (pronounce foh-sigh )
Folding Activity 3 j) How else can you write the equation you found in part l)? Using algebra, write the equation of this hyperbola in polynomial form, like this: ax 2 + bx + cy 2 + dy + k = 0. Hint: What this is asking you to do is expand each term so there are no parentheses, then combine like terms and make sure everything is on the left-hand-side of the resulting equation. k) Here is a new equation: 25y 2 100y 4x 2 16x 16 = 0 Rewrite this equation in the same form as you found in part l). l) In general, describe the difference between the equation of a hyperbola and the equation of an ellipse. m) In general, describe the difference between the graph of a hyperbola and the graph of an ellipse.