Teaching Plan 3
Explore the Circumcenter of a Triangle

Title: Circumcenter
Topic: Geometry
Lesson Summary:
This lesson plan is to introduce the concepts of circumcenter by using computers with sketchpad software to explore. Students are able to observe and explore possible results (images) through computers by carrying out their ideas in front of screens.

IL Learning Standards
NCTM Standards
Lesson  Alignment

Goals
1. Understand the concepts of circumcenter of a triangle and other relative knowledge.
2. Be able to use computers with Geometer's Sketchpad to observe possible results and solve geometric problems.

Material
1. Computers and Geometer's Sketchpad software
2. Papers, pencils, and rulers

Lesson Plan

Day 1 - Introduction of basic definition, review of relative concepts, and class discussion
Day 2 - Group activity to answer questions by using computers with sketchpad
Day 3 - Group discussion, sharing results, and making conclusion
Day 1
1. The instructors introduces the basic definition of circumcenter and had better review similar concepts about centroid, incenter, and orthocenter of a triangle.
2. Discuss students' thought and other relative questions about circumcenter. Such as: How many circumcenters are there in a triangle? Is the point of circumcenter always on the inside of a triangle? If not, please describe the possible results and depend on what kind of triangle is.
3 Then, the instructor and students turn toward to play and test computers and discuss how to draw graphs and find their answers by using computers.

Day 2
The instructor has 2-3  students form a group team to work through computers  to collect data in order to decide the conclusion for questions. The instructor should turn around each group to observe students' learning and offer some help if students have problems on how to operate computers with sketchpad software.
Questions:
1. Is there only a point of circumcenter in a triangle? Explain your possible reasons.
2. Is the point of circumcenter always on the inside of a triangle? If not. Please describe the possible results and depend on what kind of triangle is. Worksheet#1 and GSP file.
3. What are the different properties among centroid, incenter, orthocenter, and circumcenter?
4. What kind of triangle will result in that centroid, incenter, orthocenter, or circumcenter in the same triangle will overlap? GSP file
5. Which three points among centroid, incenter, orthocenter, and circumcenter will be on a line? ( This line is called Euler line.) Describe your experimental result and explain it. GSP file.
6. In a triangle ABC, suppose that O is the point of circumcenter of triangle ABC. Observe the relation between angle ABC and angle AOC. Make a conclusion and explain it. Worksheet#2. and GSP file.
7. In a triangle ABC, suppose that O is the point of circumcenter of triangle ABC. Observe the length of OA, OB, and OC. Are they equal? Explain it.  Let O be the center, and the length of OA be the radius to draw a circle. Observe the situation of point B and C and explain it. GSP file. ( This circle is called circumscribed circle to the triangle ABC.)

Day 3
In this class, students offer their results to discuss and share among groups and make the final conclusion for the questions of Day 2 activity. Finally, if possible, the instructor should demand students to develop their geometric proof  for each of the above questions. And, let students know that lots of results from dynamic models do not represent and make a proof.
Assessment
Q1:
In a triangle ABC, AB= 3 cm, BC= 4 cm, CA= 5 cm.
1) What kind of triangle is it? Why?
2) Suppose that O is the point of circumcenter of triangle ABC, the sum of OA, OB, and OC is = ______.

Q2:
1) In a acute triangle ABC, suppose that O is the point of circumcenter of triangle ABC, and the angle BAC is 65 degrees, then the angle BOC is ________ degrees.
2) In a triangle DEF,  angle DEF is obtuse angle. Suppose O is the point of circumcenter of triangle DEF, and the angle DEF is 130 degrees, then the angle DOF is ________ degrees.

Q3:
In a triangle ABC, let A' be the midpoint of BC, B' be the midpoint of AC, and C' be the midpoint of AB. And let O is the circumcenter of triangle ABC. Please explain O is the orthocenter of triangle A'B'C'.  (Hint: perpendicular lines)

Q4:

There is an arc BCD which is a part of a circle. Could you find the center of this circle and draw the another part of this circle ? Explain your method. (Hint: Three points form a triangle and decide a circle.)

Summarizing statement:
Strength:
1. Replace traditional geometric teaching in which geometry is taught by a verbal description to dynatmic drawing.
2. Help teacher to teach and  replace traditional teaching which uses blackboards and chalks to draw graphs
3. Computers with sketchpad software not only allow students to manipulate geometric shapes to discover and explore the geometric relationships,  but also verify possible results,  provide a creative activity for students' ideas, and enhance students' geometric intuition.
4. Facilitate the creation of a rich mathematical learning environment to assist students' geometric proof and establish geometric concepts

Weakness:
1. It can not replace traditional logic geometric proof -lots of examples do not make a proof
2. Students can not get maximal and potential learning benefits from by using computers to learn if the instructor do not offer appropriate learning directions and guide. The instructor also should know what kind of learning environment with computers is most likely to encourage and stimulate students' learning.

Relative Article:
1. Szymanski, W. A., (1994). Geometric computerized proofs= drawing package + symbolic compution software. Journal of Computers in Mathematics and Science Teaching, 13, p433-444.

2. Silver, J. A. (1998). Can computers to teach proofs?  Mathematics Teacher, 91, 660-663

Any Comment: Yi-wen Chen  ychen17@uiuc.edu