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Mysterious Circles
by Valorie Fayfich
Target audience
Subject: Geometry / grade level 9 - 12
Goals and objectives
Students will be able to make connections between geometry and a real-life
situation.
Archeologists need to determine the size of an object sometimes with only
fragments of the objects that are found. They may also want to determine
if the pieces belong to the same object or multiple objects.
Students will use the discovery method to make conjectures relating to circles:
a) The products of the lengths of two intersecting chords are equal.
b) The perpendicular bisector of a chord always passes through the center of the circle.
Students will be able to use proportions to determine the length of a chord and its segments.
Students will use the above conjectures to locate the center of a circle.
Students will use the center of the circle to determine the radius of the circle
sometimes only having an arc of the circle.
Academic content standards
The perpendicular bisector of a chord passes through the center of the circle.
If two chords intersect in the interior of a circle, then the product of the
lengths of the segments of one chord is equal to the product of the lengths of
the segments of the other chord.
The radius of a circle is a segment whose endpoints are the center of the
circle and a point on the circle; also the length of such a segment.
The circumference of a circle with radius r is 2
p r.
(C = 2 p r)
The area of a circle with radius r is
pr2.
(A = p r2)
A chord is a segment whose endpoints lie on a circle.
NCTM standards
Formulate mathematical definitions and express generalizations discovered through investigations
Make and test conjectures
Deduce properties of, and relationships between, figures from given assumptions
Reflect upon and clarify their thinking about mathematical ideas and relationships
Use and value the connections among mathematical topics
Apply and integrate mathematical problem solving strategies to solve problems from
within and outside mathematics
Recognize and formulate problems from situation within and outside mathemtics
Use and value the connections among mathematical topics
Use and value the connections between mathematics and other disciplines
Represent problem situations with geometric models and apply properties of figures
Reflect upon and classify their thinking about mathematical ideas and relationships
Formulate mathematical definitions and express generalizations discovered through investigations
Recognize equivalent representations of the same concept
Relate procedures in one representation to procedures in an equivalent representation
Use with increasing confidence, problem solving approaches to investigate and understand
mathematical content
Apply the process of mathematical modeling to real-world problem situations
Use and value the connections among mathematical topics
Reflect upon and clarify their thinking about mathematical ideas and relationships
Express
mathematical ideas orally and in writing
Needed resources and materials
Geometer Sketchpad
Students pages Part I, II, and III
Images to use with sketchpad.
Pieces of plates cut from different sizes of paper plates.
Activities
Part I - Students will draw circles with sketchpad and construct intersecting chords.
They will measure the segments and compare the products of the segment. Using patterns,
the students will make a conjecture about the products of the segments of the two
intersecting chords.
Part II - Students will draw circles with a chord. By locating the midpoint of the
chord and a perpendicular through the midpoint the students will observe that the center
of the circle always lies on the perpendicular bisector for the chord.
Part III - Using the conjecture in Part II, students will be asked to find the radius
of a circular object when given only pieces of the object. Finding the center will allow
the student to determine the radius, thus leading to other information such as the area
and circumference.
Evaluation
Teachers can use images that can be placed in sketchpad to have the students
find the center of the circle. Students might also be asked to find the
circumference and area.
Students might be given two pieces and asked if the two pieces could possible
come from the same plate. The students could also be given the circumference
or area and asked to verify the measurements by finding the radius by measuring
the object.
Paper plates of various sizes can be used to cut pieces of dishes. The students
will then be asked to find the center, radius, circumference and area of the
plates. Given two pieces determine if the two pieces come from the same plate
and justify their answers.
Teacher reflection
From my experience, minor errors in measuring sometimes makes it difficult for
students to make conjectures. I have asked students to measure the three angles
of a triangle and conjecture about the three angles adding up to 180°. Sometimes
the students don't see the idea because of the errors they make in measuring or
their inability to measure angles.
The same thing can happen when you ask the students to measure the segments and
look at the products. With the use of sketchpad, the measurements will be correct
giving the students a better chance to come up with a conjecture that will teach
the desired concepts.
Mysterious Circles
Student Pages
Scientists and historians sometimes need to find the size of
circular objects. The diameter, radius, and circumference are
some of the measurements that may be needed. Unfortunately, only
pieces of objects may be found.
How can a scientists or mathematician determine these measurement
from an arc of the circle or piece of the object? Through the
following investigations, we will determine a method to find the
center of a circle so that the radius, diameter and circumference
can be found.
1. Open Geometer Sketchpad.
2. Construct a circle with intersecting chords and label the points
as shown in the illustration.
3. Measure the lengths of the four segments. Multiply AE and EB. Also multiply CE and ED.
AE = __________ EB = __________
AE X EB = _______________
CE = __________ ED = __________
CE X ED = _______________
4. What do you notice about the two products?
5. Draw five circles of different sizes. Draw intersecting chords of
any length in each of the circles. Label the chords in each of the
circles with the same labels used in #2. Measure each segment in your
drawings, and complete the following chart.
| |
AE |
EB |
CE |
ED |
AE X EB |
CE X ED |
| Circle 1 |
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| Circle 2 |
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| Circle 3 |
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| Circle 4 |
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| Circle 5 |
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6. What pattern do you see?
7. Using the conjecture that you made in the previous investigation and
the given information, complete the following chart:
| AE |
EB |
CE |
ED |
| 7 |
6 |
2 |
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7 |
1.4 |
3.5 |
| 2.5 |
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12.25 |
4.3 |
| 5
Ö
2
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8 |
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2
Ö
2
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8.
Explain how you would find the measure of segment AE if segment CE is 15,
segment EB is 9, and segment ED is 12.
Part II
1. Draw a circle O with chord FG.
Find the midpoint of chord FG. Construct a line perpendicular to the chord
that passes through the midpoint of the chord. Repeat the process three
more times.
2. What do you notice about the perpendicular line in relation to the center
of the circle
Measure the radius and find the diameter of each of the circles constructed in #1.
Part III
1. Suppose you have a circle but do not know where the center of the circle
is located. From Part II, you know that the center of the circle is located
on a line perpendicular to a chord.
How might you use what you know to find the center of the circle?
2. Using sketchpad, draw a circle. Hide the center. Test the method that you
came up with in #1. To test your hypothesis, reveal the center to see if it
matches the one that you found.
3. Archeologists have found several pieces of pottery. They have come to you to
help them determine the original size of the pottery. They would like to know
the radius, diameter, and circumference of the pieces that they have found.
This will allow them to determine if the pieces possibly came from the same
piece or from a different piece.
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