Support Materials --

Teaching Math: A Video Library, 9-12

Finding Proof

Process Standards
Reasoning, Communication, Connections
School: Boston University Academy; 75 students
Location: Boston, Massachusetts
Teacher: Larry Davidson
Years Teaching: 20
Students in Classroom: 11
Grade: 9 – 10
Finding Proof
Video Overview
Students are introduced to the historical context of proofs and to Greek mathematician Thales, who sought proof for mathematical conclusions. Then each student draws a circle; draws a diameter to create a semicircle; chooses and labels a random point on the circle; draws two line segments to create a triangle in the semicircle; then measures the three angles. Students discuss hypotheses based on the measurements and work in groups to develop a proof of the hypothesis that an angle inscribed in a semicircle must be a right angle.

The Class Challenge
Students perform the following work:

Create a triangle, ABC, in a circle, such that side AB is a diameter of the circle, and point C lies on the circle.
Develop a logical argument to prove that angle C must be a right angle regardless of location on a semicircle.

Course and Curriculum
At Boston University Academy, a school affiliated with Boston University, eleventh–grade students take half their courses at Boston University and twelfth–grade students take all their courses at the University.
This required Classical Geometry course focuses on the mathematics of the ancient world, primarily Euclidean geometry, but also Egyptian and Babylonian mathematics and number theory. Prior to this course, students took algebra. This lesson was the first section of a unit on angles, circles, proof, and deductive reasoning. In earlier lessons, students studied Egyptian mathematics; Egyptian unit fractions; Egyptian multiplication algorithms; and area and volume. They relied on basic geometry from prior courses, because they had not yet worked with formal axioms and theorems and were not given facts about circles and triangles.
The goal of this lesson was for students to discover Thales' Theorem, which states that any angle inscribed in a circle is a right angle, and to investigate its proof. During the lesson, Mr. Davidson explained that Greek mathematicians Thales and Pythagoras were the first to seek reasons for mathematical conclusions. Following this lesson, students did a parallel investigation using computer software, which confirmed their conjecture that the proof remains true however they position angle C and change the circle size. Students also considered the philosophies of Thales and Pythagoras and read from Journey through Genius: The Great Theorems of Mathematics, which provides historical contexts for theorems and proofs. They also investigated problems related to isosceles triangles, angle sums, and circles. The following unit introduced definitions and postulates of Euclidean geometry.

A Pre–Viewing Exploration for Teacher Workshops
Discovery with Thales' Theorem
Objective: To understand relationships in a triangle inscribed in a semicircle, where one side of the triangle is the diameter of the circle; to discover the triangle formed is a right triangle; and to develop a proof that shows the triangle formed is a right triangle.
Materials: Compass, protractor, straightedge, and paper for each small group.
Activity
Do this activity in small groups.
Draw a circle, O, with diameter AB (where the length of AB is at least seven inches).
Select any point on circle O (not A or B) and label it point C. Draw segments AC and BC. Use your protractor to find the measure of angles A, B, and C. Repeat the process five times, placing point C in different locations.
Compile your data with the other groups.
Exploring Content

What conclusions could students make based on the collected data?
What geometry must you know to prove this theorem?
How do you know that you need to draw the auxiliary segment OC to prove the conjecture?

Exploration Answers

Exploring Content

Angle C seems to be a right angle, the sum of the measures of the other two angles is 90°, the position of C on the semicircle does not matter.
Properties and theorems about angle addition, supplementary angles, isosceles triangles, and radii. Sum of measures of angles in a triangle equals 180°. To prove it, draw segment OC. OA, OC, and OB are radii; therefore OA = OC and OC = OB and triangles AOC and BOC are isosceles. The base angles of each triangle are equal. Assume the measure of the base angles in triangle AOC are x and the measures of the base angles in triangle BOC are y. In triangle ABC, the sum of the measure of angles A, B, and C equal 180°. Therefore, x + x + y + y = 180°, 2x + 2y = 180° and x + y = 90°. The measure of angle ACB = x + y = 90°. Therefore, angle ACB is a right angle. This is Thales' Theorem.
Answers will vary. Possible answers might be: Consider relationships of triangles formed by drawing segments of the triangle (median), or explore why the circle is necessary.

Topics for Discussion
How does this approach to find proof compare with traditional methods?

Why do you think Mr. Davidson interrupted the group discussions and offered two suggestions? How did his suggestions affect students' discussions?
What was the result of students sharing their reasoning as they worked together? How did Mr. Davidson guide students to use both inductive and deductive reasoning?
Do you think it is important for students to continue to use straightedges, compasses, and protractors in secondary–school mathematics? Why or why not?
How can technology enhance the teaching and learning of geometric concepts, specifically, concepts related to proof?
The NCTM Curriculum and Evaluation Standards state that "College–intending students should also see how their school mathematics fits into the larger picture of advanced mathematical studies" (p. 185). For example, these students could investigate properties of other geometry systems (i.e., spherical geometry) and compare them to Euclidean geometry properties. Discuss ways you could have your students conduct a similar investigation.
What is the value of extending a problem over several days?

Identify the lesson's mathematical components and the amount of time you would plan for each component.
How would you encourage students to feel ownership of the whole problem and to understand how the smaller parts connect over several days?
If you extend a problem over several days, how do you decide which mathematical topics will receive less class time?
The NCTM Curriculum and Evaluation Standards state that "Students gain a sense of the structure of mathematics over an extended time period through the general accumulation of experience, as well as through more focused activities" (p. 184). Describe how you could use an extended problem to help your students understand basic mathematical principles.
Consider the use of history in the study of mathematics.

How would you decide which classical mathematics topics are important today? How does technology influence these decisions?
What teaching issues would you have to consider if you wanted to provide a historical context for the mathematical topics in your classes?
Cite ideas on how to make history relevant and inspiring to mathematics students.