Geometry/History Lesson Plan
Rose Ann Solecki
Preparation and Transition to Two-Column Proofs
Objective: To acquaint the student with the use of Euclid’s methods of proof and
deductive reasoning. To create a smoother and more pleasant transition
from algebraic proofs to geometric two-column proofs.
According to Webster’s New World Dictionary of the American Language,
"inductive reasoning is a bringing forward of separate facts or instances esp.
so as to prove a general statement" and "deductive reasoning is going from a
known principle to an unknown, from the general to the specific or from
premise to logical conclusion."
Proclus, in his Commentary on Euclid, observes that geometry, literally,
means "measurement of land". This concept first arose in surveying practices of ancient Egyptians, for the flooding of the Nile compelled them each year to redefine the boundaries of land.
In Alexandria, Egypt, around 300 BC., the Elements, composed by
Euclid, was a systematic presentation of the properties of plane geometric
figures. His work became the pivotal contribution to theoretical geometry. The principle source for reconstructing pre-Euclidean mathematics is Euclid’s Elements. Euclid compiled the known fields of elementary geometry and arithmetic that had been
developed in the two centuries before him and called them the Elements. The compilation of the works was a total of thirteen books of which the first four books present constructions and proofs of plane geometric figures. Euclid divided mathematical propositions into "theorems" and "problems". "A theorem makes the claim that all terms of a certain description have a specified property; a problem seeks construction of a term that is to have a specified property."
In the Elements, all problems are constructible on the basis of three stated
postulates: that a line can be constructed by joining two given points, that
a given line segment can be extended in a line indefinitely, and that a circle can be constructed with a given point as centre and a given line segment
as radius. The postulates in effect restricted the constructions to the use of
the Euclidean tools- i.e.. a collapsible compass and a straightedge or unmarked ruler.
First, Euclid employs the deductive method of reasoning. He introduces fundamental concepts such as a point, line, plane and angle, and relates these to physical space by definitions. These constitute primitive or undefined terms.
Second, Euclid asserts certain primitive propositions or postulates about these primitive or undefined terms. These are propositions that the reader is asked to accept as true on the basis of the definitions of the primitive terms relative to the physical world.
Third, Euclid deduces from the postulates further propositions called
theorems. He shows that their truth follows logically from the truth of the
accepted primitive propositions. As each truth builds upon the one before, the argument for the two-column proof evolves.
G. Polya states in his book, How To Solve It, simplified steps to problem
First. You have to understand the problem.
Second. Devise a plan
a. Find the connection between the data and the unknown.
b. Consider auxiliary problems.
c. Use pictures, graphs, hands-on examples.
d. Obtain a plan of the solution.
Third. Carry out your plan.
Forth. Examine the solution obtained.
Prerequisites: Each student should have completed a course in Algebra I. Have
completed a section on basic constructions and have learned the basic
postulates and theorems on angles, parallel lines and similar triangles.
The target group should be basic high school geometry classes.
Materials needed to teach lesson: Each student should have a paper, pencil, ruler
a compass and blackboard and chalk (for teacher’s use for demonstration).
Explain the difference between inductive and deductive reasoning with some discussion of examples. (See dictionary definitions in history section.) Several minutes should be taken to ensure class members understand.
Explain the history of Euclid, his book the Elements and his method of construction and proof. (See history section.)
Explain simplified problem solving method by Polya and how it works. (See history section.)
On the board, construct triangle ABC, where segment AC is the base and having an angle bisector BD, where point D lies on base AC. Tell the students we need to prove
AD = BA
Instruct the students to consider the diagram and the given initial information and ask what they might do to start a plan of action in solving the proof.
If no ideas are forth coming, prompt the students by saying they may add auxiliary lines to the diagram to help in the solution.
Tell the students, they may extend line segment BC to line BC.
The students may consider Euclid’s Proposition 31 in Book I that states "you can draw a straight line through a given point parallel to a given line."
Demonstrate the construction now and label all points and angles.
Tell the students we know angle 1 = angle 2 because of the definition of an angle bisector.
Since BD is parallel to XA then angle 3 = angle 2 because they are corresponding angles.
Angle 4 = angle 1 by definition of alternate interior angles.
Angle 4 = angle 3 by substitution.
Because angle 4 = angle 3 then XA=BA
By theorem: If a line is parallel to one side of a triangle and intersects the other 2 sides it divides them proportionally, XA = AD.
By substitution, we can say BA = AD .
Outside of class, have the student put together a two-column proof showing the this lesson using the preceding diagram.
At the next class period have the students demonstrate how they ordered the proof. After this exercise, demonstrate to the students how the two column proof should look. (See proof below.)
1. Draw line CB through B and C 1.Two points determine a line
2. Through X draw AX parallel to BD 2. Through a point outside a line there is
exactly one parallel to that line
3. Line BC and line AX must meet at 3.They lie in a plane and cannot be parallel
point X since ray BD and line BC intersect
4. Angle 1 = angle 2 4. Definition of an angle bisector
5. AD = XB 5. A line parallel to one side of a triangle
DC BC and intersects the other 2 sides, it divides
6. Angle 1 = angle 4 6. If 2 parallel lines are cut by a transversal
then alternate interior angles are equal
7. Angle 2 = angle 3 7. If 2 parallel lines are cut by a transversal
then corresponding angles are equal
8. Angle 3 = angle 4 8. Substitution
9. XB = BA 9. If 2 angles of a triangle are equal then the
sides opposite those angles are equal
10. AD = BA 10. Substitution
To reinforce deductive reasoning, go to the article in the Mathematics Teacher, issue March 1989, Jigsaw Proofs, by Suzanne Goldstein. Suzanne had her students work with a partner and assemble a pre-designed proof with all the necessary information, statement and reason, given and prove. Each step of the proof is written on a separate strip of paper for the student to arrange in a logical order, so solving the proof. After the instructor checks for proper order, the groups will then trade with another to solve a different proof. (See attachment.)
"The New Encyclopaedia Britannica", 15th edition, copyright 1992, volume 23, pp. 565-570
"How to solve it. A new aspect of mathematical method", G. Polya, copyright 1945
"Webster’s New World Dictionary of the American Language", Second College Edition, copyright 1986
"Modern Geometry Structure and Method", copyright 1963, Houghton Mifflin Company
"Mathematics Teacher", March 1989, Jigsaw Proofs, Suzanne Goldstein