Click on the following links to navigate this site.

 

Academic content standards

NCTM standards

Needed resources and materials

Activities

Evaluation

Teacher reflection

Student Pages Part I

Student Pages Part II

Student Pages Part III

Mysterious Circles

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 = 2p r)
  • The area of a circle with radius r is p r2. (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.

Students will need to know how to use sketchpad. You can download an instructor's evaluation eddtion of Geometer Sketchpad at http://www.keypress.com/sketchpad. Sketchpad tutorial web sites can be found on the "Resources and References" page.



Mysterious Circles

(Click here for a printable copy of these student pages.)

 Student Pages

Part I

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            
Circle 2            
Circle 3            
Circle 4            
Circle 5            

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  
  7 1.4 3.5
2.5   12.25 4.3
5 Ö 2 8   2 Ö 2


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. (Double click along the right edge of the page to select the images A-F. Right click on the images and copy. Open sketchpad and past the images into sketchpad by using Ctrl V. The images will be black. Click along the edge on the white to turn the images white. Use sketchpad and your knowledge of circles to find the radius, diameter, and circumference. Which pieces came from the same potery?)