Three-Column ProofsAuthor(s): Michael ShieldsSource: The Mathematics Teacher, Vol. 71, No. 6 (SEPTEMBER 1978), pp. 515-516Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961329 .

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Three-Column Proofs

It seems that once mathematicians dis cover an idea they proceed to devise a con cise format for its presentation. Although this austere result may be good for using

mathematics, it is not always the best pro cess for teaching mathematics. For example, my high school geometry class was getting along fairly well until they began proofs.

Many had trouble following the reasoning in the concise two-column format. So many students were struggling with it that some

thing had to be done. I devised a three column proof format that seemed to help.

The columns of a three-column proof re flect the elements of the deductive reason

ing process: assumptions, implications, and conclusions (table 1). These topics are usu

ally taught early in the year.

An example of one step from a proof is seen in table 2 and a complete proof in table 3.

This format for deductive proof is easy to teach for several reasons. It is easy to fol low the reasoning in the proof and, there

fore, to show students errors in reasoning. In a proof it is clear which conditionals are

dependent on previous steps and which are

introducing independent ideas. Introducing a new conditional becomes a matter of ask

ing "What do we need to use this state ment?" (What goes in the first column?) and "What do we conclude from this state ment?" (What goes in the third column?)

This method of writing proofs requires more writing than two-column proofs, but it does allow students, especially the begin

TABLE l Format of a Three-Column Proof

IfP,then?

Statements already established true.

Implications that are known true and whose hypotheses are contained in the first column.

Conclusions of statements in the second column are asserted to be true.

TABLE 2

LA is supplementary to LB; LC is supplementary to ?B.

If two angles are supplementary to the same angle, they are congruent (^).

LA = LC.

TABLE 3

Given: AB\\VD; X midpoint CA Prove: XB s YB

AB\\CD X midpoint C?

angles 1,2 vertical

L3_= LA CX S XA } L? = LI >

AABXS ACDX

If the lines are ||, the alternate interior

angles are =. If X is the midpoint, it forms two = segments. If vertical, then they are =.

If ASA's are =, the triangles are =.

If triangles are =, corresponding parts are ?

Z3

CX

L?

AABX

Z4

XA

L2

ACDX

~AX = XC

September 1978 515

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ners, to recognize the deductive reasoning in the proofs.

The two-column proof may still be your goal. If so, the transition from three-col umn proofs to two-column proofs could go something like this: "In a two-column

proof you have statements and reasons.

Your statements are found in the third col umn and your reasons in the corresponding row of the second column. The first column is omitted."

Michael Shields Reorganized District ?3 Savannah, MO 64485

The Wrapping Function Kit A common way to introduce circular

functions makes use of the wrapping func tion p. Given a real number t,p(t) is defined as the point on the unit circle that is t units from the point (0, 1) as measured along the circumference in the counterclockwise di rection. The kit described below has been

very successful in acquainting students with the meaning and properties of the wrapping function.

Each kit consists of a can, a paper mea

suring tape, and a graph of the unit circle in the Cartesian plane (figure 1). The units on

Fig. 1. Units on the tape and the unit circle equal the

length of the radius of th? can.

the measuring tape and the graph paper are

equal in length to the radius of the can. The zero end of the measuring tape is attached to the side of the can at the bottom so that it is possible to measure around the base of the can (figure 2). By placing the can on the graph of the circle with the end of the mea

suring tape at the point (1,0), the students

may measure around the can with the tape measure to locate the points p(t) for various values of t. The following activity plan il lustrates some of the ways that the kit may be used.

Fig. 2. Using the wrapping function kit

1. Find a point on the circle to corre

spond to each distance on the tape measure and mark it on the graph paper:

a) 1 unit

b) 2 units c) 6.25 units

d) 7 units

516 Mathematics Teacher

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