Objectives:
1. Students will learn to construct conics using the Geometer's Sketchpad (i.e. utilizing "technology").
2. Students will explore these conics by manipulating their constructions.
3. Students will further explore the relationships between the conics and "test" the conjectures they made in the previous lesson about these relationships.
Materials:
Computers equipped with GSP (2 or 3 students per computer)
Activity Sheet from Previous Lesson
Activity Sheet for this Lesson
Discussion of Lesson:
This lesson builds on the previous lesson which used the paper folding activity. In the same groups as for the paperfolding activity, students will explore conic sections and test their conjectures using GSP.
Activities:
Investigating Conic Sections Using GSP
Use GSP to test your conjectures from yesterday's lesson.
Activity One
Use GSP to draw a circle. Label the center of your circle C. Construct the "crease line" where an arbitrary point on the circumference is folded onto C. Trace the locus of this line as the circumference point moves around the circle.
1. Does this locus create the same pattern that the creases formed in yesterday's activity one? (Hint: it should)
2.Use GSP to test your conjecture about the point of tangency on each crease line.
3. If B is the point of tangency, make a conjecture about CB and BA. Use GSP to test your conjecture?
Activity Two
Rotate roles.
Use GSP to draw a circle. Label it's center C. Choose an arbitrary point within the circle and label it F. Construct the "crease line" where an arbitrary point A on the circumference of the circle is "folded" onto point F. Trace the locus of this line as A moves around the circle.
1. Does this locus create the same pattern that the creases formed in yesterday's activity two? (Hint: it should)
2. Use GSP to test your conjecture about the point of tangency on each crease line.
3. If B is the point of tangency, make a conjecture about CB + BF. Use GSP to test your conjecture?
Activity Three
Rotate roles.
Use GSP to draw a circle. Label the center of the circle C. Choose an arbitrary point outside the circle and label it F. Construct the "crease line" where the point F is "folded" onto an arbitrary point A on the circumference of the circle. Trace the locus of this line as A moves around the circle.
1. Does this locus create the same pattern that the creases formed in yesterday's activity three? (Hint: it should)
2. Use GSP to test your conjecture about the point of tangency on each crease line.
3. If B is the point of tangency, make a conjecture about FB - BC. Use GSP to test your conjecture?
Activity Four
Rotate roles.
Use GSP to construct a rectangle. Mark a point F roughly midway between it's two vertical sides. Choose an arbitrary point on the base of the rectangle. Label this point A. Construct the "crease line" where this point is folded onto point F. Trace the locus of this line as A moves along the base of the rectangle.
1. Does this locus create the same pattern that the creases formed in yesterday's activity three? (Hint: it should)
2. Use GSP to test your conjecture about the point of tangency on each crease line.
3. If B is the point of tangency, make a conjecture about the relationship between AB and BF. Use GSP to test your conjecture?
Here's an answer key to this activity:
Activity One:
Given a circle with center C, you can "fold" arbitrary points on the circumference of the circle to meet C and form a conic. It appears that the creases will form a circle with center C and radius one-half that of your original circle.
If A is a given point on the circumference of circle C, you can geometrically construct the crease line formed when point A is folded onto C. First construct the segment CA. Then construct the perpendicular bisector JK to CA. This will be the crease line. If you trace the locus of JK as A changes you see that it forms a circle with center C.
Any given crease line meeting the criteria of JK is tangent to this circle. Since the crease lines are tangent lines, you can use them to find the point of tangency. It turns out that for a given point A on the circumference of circle C, the point of tangency, N, to the inner circle is located at the intersection of AC with JK. If you trace the locus N as A changes, it forms a circle with center C.
If N is the point of tangency, it appears that the length of CN is equal to NA and will remain constant no matter where N is located on the inner circle. You can verify this by measuring the difference CN-NA and then varying A to see if the difference remains 0.
Activity Two:
Given a circle with center C and free point F inside the circle, you can "fold" arbitrary points on the circumference of the circle to meet F and form a conic. It appears that the creases will form an ellipse with foci C and F.
If A is a given point on the circumference of circle C, you can geometrically construct the crease line formed when point A is folded onto point F. First construct the segment FA. Then construct the perpendicular bisector JK to FA. This will be the crease line. If you trace the locus of JK as A changes you see that it forms an ellipse with foci C and F.
Any given crease line meeting the criteria of JK is tangent to the ellipse with foci C and F. Since the crease lines are tangent lines, you can use them to find the point of tangency. It turns out that for a given point A on the circumference of circle C, the point of tangency, N, to the ellipse is located at the intersection of AC with JK. If you trace the locus N as A changes, it forms an ellipse with foci C and F.
If N is the point of tangency, it appears that the value of CN+NF will remain constant no matter where N is located on the ellipse. you can verify this by measuring the sum CN+NF and then varying A to see if the sum changes.
Activity Three:
Given a circle with center C and a free point F outside the circle, you can "fold" the circle so that arbitrary points on the circumference meet F to form a conic. It appears that the creases will form a hyperbola with foci F and C.
If A is an arbitrary point on the circumference of circle C, you can geometrically construct the crease line formed when A is folded onto F. First construct the segment FA. Then construct the perpendicular bisector JK of FA. The line containing JK will be the crease line. If you trace the locus of JK as A changes you get a hyperbola with foci C and F.
The crease lines are tangent lines to the hyperbola. You can use these crease lines to find the point of tangency to the hyperbola. First construct segment AF and the line through JK as in figure 5. Then construct the line CA. Construct a point, M, at the intersection of CA and the line through JK. This should be the point of tangency. You can confirm this by tracing the locus of M as A changes.
If N is the point of tangency, it appears that FN-NC remains constant no matter the value of N. you can confirm this by calculating FN-NC and varying A. In doing so you find that the difference does not change.
Activity Four:
If you pick a free point F roughly midway between the two vertical sides inside a rectangle and "fold" arbitrary points on the bottom of the rectangle onto F, you can form a particular conic. It appears that the creases will form a parabola with F being it's focus and the line through RS, the bottom of the rectangle, it's directrix.
If A is an arbitrary point on the RS, you can construct the crease line that is formed when A is folded to F. First construct the segment FA. Then construct the line through JK that is the perpendicular bisector of FA. This line will be the crease line. If you trace the locus of the line through JK you get a parabola with focus F and directrix RS.
The crease lines are tangent lines to our parabola. you can use these crease lines to find the point of tangency. First construct segment FA and it's perpendicular bisector. Then construct the line AN parallel to the sides VR and TS of the rectangle. The intersection of AN and JK, N, is the point of tangency. You can confirm this by tracing the locus of N as A moves along RS. You get a parabola with focus F and directrix RS.
If point N is the point of tangency, it appears that AN=NF for any value of N on the parabola. You can confirm this by measuring the difference AN-NF and varying A to see that AN-NF is always equal 0.
Assessment:
Groups will complete a write-up of their findings and conjectures accompanied by their proofs and their GSP sketches.