proofs without words - carol j. bell.pdf

8/9/2019 Proofs without Words - Carol J. Bell.pdf

1/6

690 MatheMatics teacher |Vol. 104, No. 9 My 2011

RReasoning and Proof is one of the Process Stan-dards set forth in NCTMsPrinciples and Standards

for School Mathematics(2000). Thus, it is importantto give students opportunities to build their reason-

ing skills and aid their understanding of the proofprocess. Teaching students how to do proofs is adifficult task because students often will not know

how to begin a proof.The use of proofs without words is effective in

helping students understand the proof process, andhere I describe how I have used these proofs in myclassroom. Using proofs without words in teachingmathematical concepts can help students improvetheir ability to reason when asked to explain an

illustration, and this heightened reasoning canlead to understanding how to begin a formal proof.Understanding formal proofs not only deepens stu-dents understanding of mathematical concepts butalso prepares students for higher-level mathematics.

WHAT ARE PROOFS WITHOUT WORDS?A proof without words is a mathematical drawing

that illustrates the proof of a mathematical state-ment without a formal argument provided in words.Examples of proofs without words can be foundon various Web sites (e.g., illuminations.nctm.org,

www.cut-the-knot.org), in two books by Nelsen(1993, 2000), and in articles in mathematics jour-nals (see, e.g., Pinter [1998] and Nelsen [2001]).

Some interactive proofs without words are avail-able on the Internet. For instance, an animated ver-sion of Proof without Words: Pythagorean Theo-rem may be found on NCTMs Illuminations Website (http://illuminations.nctm.org/ActivityDetail.

aspx?ID=30). During the animation in this proofwithout words, the four triangles and the square on

the left side of figure 1are rearranged to form theright side of the figure. (Note that labels have beenadded to the figure to aid in understanding.)

The concept of a proof without words is notnew by any means. For instance, the proof ofthe Pythagorean theorem shown in figure 1wasinspired by the mathematical drawing shown in

figure 2. This drawing is found in one of the oldestsurviving Chinese texts,Arithmetic Classic of theGnomon and the Circular Paths of Heaven(ca. 300

BCE), and contains formal mathematical theories.

The proof eventually found its way into the Vijag-anita(Root Calculation) by the Indian mathemati-cian Bhaskara (111485 CE).

An explanation of the diagram in figure 1anda corresponding proof of the illustration are repro-duced here:

Draw a right triangle four times in the square of thehypotenuse, so that in the middle there remains

a square whose side equals the differencebetween the two sides of the right triangle.

Let cbe the side of the large square (hypotenuse).

Illustrating mathematical statements

through the use of picturesproofswithout wordscan help students

develop their understanding of

mathematical proof.

col J. Bll

A Visual Application o

Copyright 2011 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.


2/6

Vol. 104, No. 9 My 2011 |MatheMatics teacher 691

Reasoning and Proof

Label the legs of the right triangle as aandb, wherea b.

In the figure on the left, the area of the large squareis c2.

Rearrange the polygons of the figure on the left tocreate the figure on the right.

Now, the area of the figure on the right is composedof the area of two squares, the lengths of whosesides correspond to the legs of the right triangle,or a2+b2.

Since both the left figure and right figure are com-posed of the same polygons, then they both havethe same area.

Thus, c2= a2+b2.

CLASSROOM APPLICATION OFPROOFS WITHOUT WORDSProofs without words cover a wide range of mathe-matical conceptsincluding algebra, trigonometry,geometry, and calculusand can be used in suchcourses as the history of mathematics. I generallyintroduce students to proofs without words byusing those available on the Illuminations Website. Students are arranged into groups to discuss

the proof. If the proof without words is an interac-tive diagram, students are first shown a demon-stration of the diagram and then asked to discuss

in their groups a formal proof of what is depicted.This process allows students to work together tounderstand the diagram and prove the mathemati-cal result illustrated.

To further aid students in their understanding ofthe proof process, I also post a proof without wordson an online discussion board. Use of such technol-

ogy encourages class discussion about the diagramand why the diagram represents a proof of thestatement being illustrated. Students use the onlinediscussion board to post questions and any resultsthey have found. However, I ask students not to

post the entire solution because others should havethe opportunity to develop their own ideas about

how to prove the statement. Students who do notknow where to begin are encouraged to post ques-tions on the discussion board to get hints from me

or other studentsa process that promotes classdiscussion of the problem. Students are graded on

both their participation in the online discussionand their written work.

Fig. 1 i up o obv o povd onng xpln wy

nfomon of figu pn poof of Pygon om.

Source:http://illuminations.nctm.org/ActivityDetail.aspx?ID=30

Fig. 2 t dgm d o 300 Bce.

Source: Burton (2007)


3/6


An example of a proof without words that I haveused in my geometry course is provided in problem1 (see fig. 3). Through the online class discussion,students who understood the diagram provided hintsto those who did not see how to begin. The onlinediscussion allowed students to learn from their peersand also helped students improve their written com-munication of mathematical ideas. Written expla-

nations provided a means for students to organizetheir thinking about how they reasoned througha problem, and organizing their thinking, in turn,

helped them better understand the proof process. Inexplaining the diagram, most students found that itwas easier to add labels, as shown in figure 4.

The following student response to the diagramin figure 4is typical:

Construct triangleABCwith longest sideBCand

acute angle C, denoted by q. With centerBand

radiusBC, construct circleB. Extend BCto form

diameterDC. Extend ACso that it intersects circle

B. Label the intersection point W. Construct right

triangleDCW. This is a right triangle because any

triangle inscribed in a semicircle is a right triangle.

ExtendABso that it intersects circle Bat pointsP

and Qto form diameterPQ. Let the radius = a. Let

AB= c. LetAC=b. By right-triangle trigonometry,

WC= 2acosq, so WA= 2acosqb.

Some students used The Geometers Sketchpad(GSP) to construct the diagram and explain the

problem. A virtuostic example of how one studentused GSP to answer the first part of problem 1 isprovided in figure 5.

Because the law of cosines can also be applied toobtuse triangles, I asked students how the diagramwould change if angle qwere obtuse. In the onlineclass discussion, some students comments indi-cated that they believed that constructing a similardiagram having an obtuse angle was impossible.

Figure 6shows a students attempt to construct a

new diagram with angle qobtuse. The student con-cluded that the construction was impossible.

Other students indicated that if angle qwere

obtuse, the angle could not be drawn inside the circle.Some students were successful in constructing adiagram. Figure 7provides examples of correct dia-grams created by four different students. Two exam-ples show angle qentirely inside the circle, and twoexamples show angle qextending outside the circle.

ANOTHER EXAMPLEIn another course, after students had sufficientexperience with the concept of proof withoutwords, they were given a mathematical statementand asked to construct their own image to repre-

Fig. 4 t poof of lw of on dpnd on

podu of gmn of od om.

Fig. 3 in ppon fo onln l duon of pob-

lm 1, udn vwd lw of on nd dud

ow would gnlly b povd n g ool xbook.

Source: Kung (1990)

Problem 1: Proof without Words:The Law of CosinesExplain how each label in the figure is obtainedand then explain how the law of cosines can bededuced from the information in the diagram.That is, how can you use the diagram to provethe law of cosines? The figure shows qas an

acute angle. How would the figure change ifq

isobtuse? Use The Geometers Sketchpad to pro-vide a revised construction.

Source: Kung (1990)


4/6


sent that statement. Problem 2 (see fig. 8) showsthe problem given to the students.

All students were able to answer the first ques-tion and state a general rule for the pattern as(2n+ 1)2+ (2n2+ 2n)2= (2n2+ 2n+ 1)2for n= 1,

2, 3, . In response to the second question, somestudents used mathematical induction to prove thatthe statement was true for all integers n 1, andothers just used algebra to clear parentheses onone side of the equation, simplify, and obtain theother side of the equation. Most students were able

to construct an image to illustrate the pattern, andthey gave very detailed explanations of how thisimage can be used to generate the general equationin the pattern. In providing a visual statement interms of n, several students first provided imagesof one or two of the equations in the pattern. An

example of the first equation in the pattern createdby a student is shown in figure 9.

By looking at examples of images that represent

one or two of the equations, the student was ableto construct a diagram to represent the pattern in

terms of n. The students general representation isshown in figure 10. Although one student useddots in a manner similar to the polygons shown infigure 10, most students used an area model withsquares and rectangles for their illustration. In thefigure, notice that the large square on the right sideof the equation consists of the blue square of area(2n2+ 2n)2from the left side of the equation andthe yellow square of area (2n+ 1)2. Algebraically,(2n+ 1)2= 4n2+ 4n+ 1 = (2n2+ 2n) + (2n2+ 2n)+ 1, so the yellow square can be broken apart toform the yellow L-shaped region with a width of 1.

Fig. 5 On pv ud GsP o povd ooug nd xlln pon o poblm 1.

Fig. 6 a udn mp o vd dgm w qobu uld n n nompl dgm.


5/6


The student explained his illustration as follows:

The side length of the yellow square is 2n+ 1, and

the side length of the blue square is 2n2+ 2n. The

area of the yellow square is (2n+ 1)2or 4n2+ 4n+

1. The area of the blue square is (2n2+ 2n)2. Look-

ing at the final square: the side length of the square

to the far right will be 2n2+ 2nfor the blue square

and then an additional 1 for the width of the yellow

strips or 2n2+ 2n+ 1. Its total area is (2n2+ 2n+ 1)2.

We can see that this works for any n, n 1, by

looking at the area of that larger rectangle on the far

right. The side length of the blue square in the interior

of the larger square has already been established to

be 2n2+ 2n. The L-shaped yellow strip has a width of

1, as shown, and is broken into 3 sections. The long,

rectangular sections will have area (2n2+ 2n)(1). The

little square section will have an area of (1)(1). The

area of the entire yellow strip will be (2n2+ 2n)(1) +

(2n2+ 2n)(1) + 1. This can be simplified to be 4n2+ 4n

+ 1. (Note: This is also the area of the yellow square.)

The total area of the [largest] square equals the

area of the L-shaped yellow strip plus the area of

the blue square:

Total Area = Area of Yellow Strip + Area of

Blue Square

(2n2+ 2n+ 1)2= (4n2+ 4n+ 1) + (2n2+ 2n)2

(2n2+ 2n+ 1)2= (2n+ 1)2+ (2n2+ 2n)2

This can now be seen to be equivalent to the

original equation, and thus the original equation

holds true: (2n+ 1)2+ (2n2+ 2n)2= (2n2+ 2n+ 1)2.

Clearly, this student took time to think about theparts of the diagram and explain how they related to

the original equation. This level of thinking requiresreasoning through each part of the explanation,quite important in understanding the proof process.

A few students did not have a correct picture forthe general representation even though their exam-ples for one or two of the equations in the

Fig. 7 svl udn w uful n onung poof wou wod fo lw of on w qobu.

(a) (b)

(c) (d)


6/6


School Mathematics. Reston, VA: NCTM, 2000.

.Proof without Words: Pythagorean Theorem.

2008. http://illuminations.nctm.org.

Nelsen, Roger B.Proofs without Words: Exercises in

Visual Thinking.Washington, DC: Mathematical

Association of America, 1993.

.Proofs without Words II: More Exercises in

Visual Thinking.Washington, DC: Mathematical

Association of America, 2000.. Herons Formula via Proofs without Words.

The College Mathematics Journal32, no. 4 (2001):

29092.

Pinter, Klara. Proof without words: The Area of a

Right Triangle.Mathematics Magazine71, no. 4

(1998): 314.

CAROL J. BELL, [email protected],

teaches mathematics education cours-

es at Northern Michigan University in

Marquette. She is interested in how

future teachers communicate and make sense of

the mathematics they will someday teach.

pattern were correct. This was an indication thatone or two correct examples do not necessarilyimply that a general representation can be formed.Some students use examples as a way to try to provethe general result of a statement, but with morepractice they can overcome this misinterpretation ofproving a general result.

CONCLUSIONI have provided some ideas on how to use proofswithout words in the classroom, but no doubtthere are other ways of using them to help studentsimprove their understanding of mathematicalproof. When students write a formal proof of whatis being illustrated in a proof without words, theyare not just improving their proof-writing ability;they are also learning how to reason through amathematics problem better. Providing an explana-tion of the diagram is also a good way for students

to improve their ability to reason because they mustthink about the individual parts in the diagram. By

creating their own visual representation of a math-ematical statement, students are also improvingtheir ability to reason through a problem.

REFERENCESBurton, David M. The History of Mathematics: An

Introduction. New York: McGraw-Hill, 2007.

Kung, Sidney H. Proof without Words: The Law of

Cosines.Mathematics Magazine63, no. 5 (1990):

342.

National Council of Teachers of Mathematics

(NCTM). Principles and Standards for

Fig. 8 sudn w om xpn w poof wou

wod my b bl o kl mo opd poblm.

Problem 2: Pythagorean TriplesConsider the following pattern:

32+ 42= 52

52+ 122= 132

72+ 242= 252

92+ 402= 412

1. State a general rule suggested by the exampleabove that will hold for all integers n1where n= 1 corresponds to the pattern in

the first equation, n= 2 corresponds to thepattern in the second equation, and so on. Besure to include your work on how you com-puted the general rule.

2. Prove that your general statement is true forall integers n1.

3. Illustrate the pattern visually (e.g., with dots,lengths of segments, areas, or in some otherway).

Fig. 9 som udn w bl o dv n lluon of fi quon n

pn.

Fig. 10 t gnl pn fo poblm 2 n b vully dplyd.

proofs without words - carol j. bell.pdf

Documents