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690 MatheMatics teacher | Vol. 104, No. 9 • May 2011
RReasoning and Proof is one of the Process Stan-dards set forth
in NCTM’s Principles and Standards for School Mathematics (2000).
Thus, it is important to give students opportunities to build their
reason-ing skills and aid their understanding of the proof process.
Teaching students how to do proofs is a difficult 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, and here I describe how I have used
these proofs in my classroom. Using proofs without words in
teaching mathematical concepts can help students improve their
ability to reason when asked to explain an illustration, and this
heightened reasoning can lead to understanding how to begin a
formal proof. Understanding formal proofs not only deepens
stu-dents’ understanding of mathematical concepts but also 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 found on 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 NCTM’s Illuminations
Web site (http://illuminations.nctm.org/ActivityDetail.aspx?ID=30).
During the animation in this proof without words, the four
triangles and the square on the left side of figure 1 are
rearranged to form the right side of the figure. (Note that labels
have been added to the figure to aid in understanding.)
The concept of a proof without words is not new by any means.
For instance, the proof of the Pythagorean theorem shown in figure
1 was inspired by the mathematical drawing shown in figure 2. This
drawing is found in one of the oldest surviving Chinese texts,
Arithmetic Classic of the Gnomon 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 (1114–85 CE).
An explanation of the diagram in figure 1 and a corresponding
proof of the illustration are repro-duced here:
Draw a right triangle four times in the square of the
hypotenuse, so that in the middle there remains a square whose side
equals the difference between the two sides of the right
triangle.
Let c be the side of the large square (hypotenuse).
Illustrating mathematical statements through the use of
pictures—proofs without words—can help students develop their
understanding of mathematical proof.
carol J. Bell
A Visual Application of Reasoning and ProofProofs without
WordsProofs without WordsProofs without Words
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.
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Vol. 104, No. 9 • May 2011 | MatheMatics teacher 691
A Visual Application of Reasoning and ProofProofs without
WordsProofs without WordsProofs without Words
Label the legs of the right triangle as a and b, where a ≥
b.
In the figure on the left, the area of the large square is
c2.
Rearrange the polygons of the figure on the left to create the
figure on the right.
Now, the area of the figure on the right is composed of the area
of two squares, the lengths of whose sides 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 have the same area.
Thus, c2 = a2 + b2.
CLASSROOM APPLICATION OF PROOFS WITHOUT WORDSProofs without
words cover a wide range of mathe-matical concepts—including
algebra, trigonometry, geometry, and calculus—and can be used in
such courses as the history of mathematics. I generally introduce
students to proofs without words by using those available on the
Illuminations Web site. 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 to
understand the diagram and prove the mathemati-cal result
illustrated.
To further aid students in their understanding of the proof
process, I also post a proof without words on an online discussion
board. Use of such technol-ogy encourages class discussion about
the diagram and why the diagram represents a proof of the statement
being illustrated. Students use the online discussion board to post
questions and any results they have found. However, I ask students
not to
post the entire solution because others should have the
opportunity to develop their own ideas about how to prove the
statement. Students who do not know where to begin are encouraged
to post ques-tions on the discussion board to get hints from me or
other students—a process that promotes class discussion of the
problem. Students are graded on both their participation in the
online discussion and their written work.
Fig. 1 it is up to the observer to provide the reasoning that
explains why the
transformation of the fi gure represents a proof of the
Pythagorean theorem.
Source:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=30
Fig. 2 this diagram dates to circa 300 Bce.
Source: Burton (2007)
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692 MatheMatics teacher | Vol. 104, No. 9 • May 2011
An example of a proof without words that I have used in my
geometry course is provided in problem 1 (see fig. 3). Through the
online class discussion, students who understood the diagram
provided hints to those who did not see how to begin. The online
discussion allowed students to learn from their peers and also
helped students improve their written com-munication of
mathematical ideas. Written expla-nations provided a means for
students to organize their thinking about how they reasoned through
a problem, and organizing their thinking, in turn, helped them
better understand the proof process. In explaining the diagram,
most students found that it was easier to add labels, as shown in
figure 4.
The following student response to the diagram in figure 4 is
typical:
Construct triangle ABC with longest side BC and acute angle C,
denoted by q. With center B and radius BC, construct circle B.
Extend BC to form diameter DC. Extend AC so that it intersects
circle B. Label the intersection point W. Construct right triangle
DCW. This is a right triangle because any
triangle inscribed in a semicircle is a right triangle. Extend
AB so that it intersects circle B at points P and Q to form
diameter PQ. Let the radius = a. Let AB = c. Let AC = b. By
right-triangle trigonometry, WC = 2a∙cos q, so WA = 2a∙cos q –
b.
Some students used The Geometer’s Sketchpad (GSP) to construct
the diagram and explain the problem. A virtuostic example of how
one student used GSP to answer the first part of problem 1 is
provided in figure 5.
Because the law of cosines can also be applied to obtuse
triangles, I asked students how the diagram would change if angle q
were obtuse. In the online class discussion, some students’
comments indi-cated that they believed that constructing a similar
diagram having an obtuse angle was impossible. Figure 6 shows a
student’s attempt to construct a new diagram with angle q obtuse.
The student con-cluded that the construction was impossible.
Other students indicated that if angle q were obtuse, the angle
could not be drawn inside the circle. Some students were successful
in constructing a diagram. Figure 7 provides examples of correct
dia-grams created by four different students. Two exam-ples show
angle q entirely inside the circle, and two examples show angle q
extending outside the circle.
ANOTHER EXAMPLEIn another course, after students had sufficient
experience with the concept of proof without words, they were given
a mathematical statement and asked to construct their own image to
repre-
Fig. 4 this proof of the law of cosines depends on the
product of the segments of chords theorem.
Fig. 3 in preparation for the online class discussion of
prob-
lem 1, students reviewed the law of cosines and discussed
how it would generally be proved in a high school textbook.
Source: Kung (1990)
Problem 1: Proof without Words: The Law of CosinesExplain how
each label in the figure is obtained and then explain how the law
of cosines can be deduced from the information in the diagram. That
is, how can you use the diagram to prove the law of cosines? The
figure shows q as an acute angle. How would the figure change if q
is obtuse? Use The Geometer’s Sketchpad to pro-vide a revised
construction.
Source: Kung (1990)
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Vol. 104, No. 9 • May 2011 | MatheMatics teacher 693
sent that statement. Problem 2 (see fig. 8) shows the 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)2 for n = 1, 2, 3, …. In response to the second question, some
students used mathematical induction to prove that the statement
was true for all integers n ≥ 1, and others just used algebra to
clear parentheses on one side of the equation, simplify, and obtain
the other side of the equation. Most students were able to
construct an image to illustrate the pattern, and they gave very
detailed explanations of how this image can be used to generate the
general equation in the pattern. In providing a visual statement in
terms of n, several students first provided images of one or two of
the equations in the pattern. An
example of the first equation in the pattern created by a
student is shown in figure 9.
By looking at examples of images that represent one or two of
the equations, the student was able to construct a diagram to
represent the pattern in terms of n. The student’s general
representation is shown in figure 10. Although one student used
dots in a manner similar to the polygons shown in figure 10, most
students used an area model with squares and rectangles for their
illustration. In the figure, notice that the large square on the
right side of the equation consists of the blue square of area (2n2
+ 2n)2 from the left side of the equation and the 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 to
form the yellow L-shaped region with a width of 1.
Fig. 5 One preservice teacher used GsP to provide a thorough and
excellent response to problem 1.
Fig. 6 a student’s attempt to create a revised diagram with q
obtuse resulted in an incomplete diagram.
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694 MatheMatics teacher | Vol. 104, No. 9 • May 2011
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)2 or 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 + 2n for 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 the parts of the
diagram and explain how they related to the original equation. This
level of thinking requires reasoning through each part of the
explanation, quite important in understanding the proof
process.
A few students did not have a correct picture for the general
representation even though their exam-ples for one or two of the
equations in the
Fig. 7 several students were successful in constructing a proof
without words for the law of cosines with q obtuse.
(a) (b)
(c) (d)
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Vol. 104, No. 9 • May 2011 | MatheMatics teacher 695
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.
———. “Heron’s Formula via Proofs without Words.” The College
Mathematics Journal 32, no. 4 (2001): 290–92.
Pinter, Klara. “Proof without words: The Area of a Right
Triangle.” Mathematics Magazine 71, 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 that one or two
correct examples do not necessarily imply that a general
representation can be formed. Some students use examples as a way
to try to prove the general result of a statement, but with more
practice they can overcome this misinterpretation of proving a
general result.
CONCLUSIONI have provided some ideas on how to use proofs
without words in the classroom, but no doubt there are other ways
of using them to help students improve their understanding of
mathematical proof. When students write a formal proof of what is
being illustrated in a proof without words, they are not just
improving their proof-writing ability; they are also learning how
to reason through a mathematics problem better. Providing an
explana-tion of the diagram is also a good way for students to
improve their ability to reason because they must think about the
individual parts in the diagram. By creating their own visual
representation of a math-ematical statement, students are also
improving their 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 Magazine 63, no. 5 (1990): 342.
National Council of Teachers of Mathematics (NCTM). Principles
and Standards for
Fig. 8 students with some experience with proofs without
words may be able to tackle this more sophisticated problem.
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 example above that will
hold for all integers n ≥ 1 where n = 1 corresponds to the pattern
in the first equation, n = 2 corresponds to the pattern in the
second equation, and so on. Be sure to include your work on how you
com-puted the general rule.
2. Prove that your general statement is true for all integers n
≥ 1.
3. Illustrate the pattern visually (e.g., with dots, lengths of
segments, areas, or in some other way).
Fig. 9 some students were able to devise an illustration of the
fi rst equation in the
pattern.
Fig. 10 the general pattern for problem 2 can be visually
displayed.