Objectives:
1. Students will construct and explore conics using paperfolding.
2. Students will make conjectures about the relationships between these paper folding constructions and the geometric constructions of the conics.
3. Students will make conjectures about the relationships between the constructions of the different conics.
Previous Knowledge:
Before this lesson, students will have explored the terminology and standard equations of the conics.
Materials:
Wax Paper or Patty Paper (4 4"X4" sheets per student)
Compass (1 per student)
Activity Sheet (1 per student)
Discussion of Lesson:
In their groups, students will explore conics using the following paper folding activities. The teacher will provide guidance and encouragement throughout the activities by facilitating group discussion and keeping students on track but will allow students to make their own conjectures and discoveries.
Activities:
Conics Investigation With Paper Folding
Activity One
Using your compass, draw a circle with a radius of a least 6 centimeters. Cut out the circle and mark it's center C. Fold the circle so that any point on the circumference lands on point C. After creasing the fold sharply, unfold the circle. Choose another point on the circumference and make a second crease that lands that circumference point onto point C. Continue choosing circumference points and folding them to land on C. Once you have repeated this process several times (say 20 or more), answer the following questions as a group:
1. Predict what overall pattern the creases will form.
2. How would you construct geometrically the crease line formed when a point A on the circumference of the circle is folded onto point C? Justify your construction.
3. Locate the point of tangency on each crease line. Make a conjecture about the placement of this point.
4. If B is the point of tangency, make a conjecture about CB and BA. How would you prove your conjecture?
Activity Two
Rotate roles before beginning this activity.
Using your compass, draw a circle with a radius of at least 6 centimeters. Cut out the circle and mark it's center C. Choose an arbitrary point within the circle and label it F. Fold the circle so that any point on its circumference lands on point F. After creasing the fold sharply, unfold the circle. Choose another point on the circumference and make a second crease that lands that circumference point onto point F. Continue choosing circumference points and folding them to land on F. Once you have repeated this process several times (say 20 or more), answer the following questions:
1. Predict what overall pattern the creases will form.
2. How would you construct geometrically the crease line formed when a point A on the circumference of the circle is folded onto point F? Justify your construction.
3. Locate the point of tangency on each crease line. Make a conjecture about the placement of this point.
4. If B is the point of tangency, make a conjecture about CB + BF. How would you prove your conjecture?
Activity Three
Rotate roles before beginning this activity.
Using your compass, draw a circle with a radius of approximately 4 centimeters. (Do not cut this circle out). Mark the center of the circle C. Choose an arbitrary point outside the circle and label it F. Fold the circle so that point F lands on any point on the circumference of the circle. After creasing the fold sharply, unfold the circle and make a second crease so that point F lands on another circumference point. Continue the folding process. Answer and discuss the following questions as a group.
1. Predict what overall pattern the creases will form.
2. How would you construct geometrically the crease line formed when point F is folded onto point A on the circumference of the circle? Justify your construction.
3. Locate the point of tangency on each crease line. Make a conjecture about the placement of this point.
4. If B is the point of tangency, make a conjecture about FB - BC. How would you prove your conjecture?
Activity Four
Rotate roles before beginning this activity.
Take a sheet of patty paper and mark a point F roughly midway between it's two vertical sides (ie. mark a point F at the center of the paper). Choose a point on the bottom edge of the sheet. Fold the sheet so that this point lands on point F. Choose other points on the bottom edge and continue the folding process.
1. Predict what overall pattern the creases will form.
2. How would you construct geometrically the crease line formed when any point A on the bottom edge is folded onto point F? Justify your construction.
3. Locate the point of tangency on each crease line. Make a conjecture about the placement of this point.
4. If B is the point of tangency, make a conjecture about the relationship between AB and BF. How would you prove your conjecture?
Assessment:
Groups will complete a write-up of their findings and conjectures accompanied by their paper folding constructions.