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Proofs witho A Visual Application of Illustrating mathematical statements through the use of pictures—proofs without words—can help students develop their understanding of mathematical proof. carol J. Bell
Reasoning and Proof is one of the Process Standards set forth in NCTM’s Principles and Standards for School Mathematics (2000). Thus, it is important to give students opportunities to build their reasoning 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 students’ 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 statement 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 journals (see, e.g., Pinter  and Nelsen ). Some interactive proofs without words are available on the Internet. For instance, an animated version of “Proof without Words: Pythagorean Theorem” 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 Vijaganita (Root Calculation) by the Indian mathematician Bhaskara (1114–85 CE). An explanation of the diagram in figure 1 and a corresponding proof of the illustration are reproduced 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).
690 MatheMatics teacher | Vol. 104, No. 9 • May 2011
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hout Words Reasoning and Proof 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 composed of the same polygons, then they both have the same area. Thus, c2 = a2 + b2.
Source: http://illuminations.nctm.org/ActivityDetail.aspx?ID=30 Fig. 1 it is up to the observer to provide the reasoning that explains why the transformation of the ﬁgure represents a proof of the Pythagorean theorem.
CLASSROOM APPLICATION OF PROOFS WITHOUT WORDS Proofs without words cover a wide range of mathematical 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 interactive diagram, students are first shown a demonstration 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 mathematical 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 technology 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
Source: Burton (2007) Fig. 2 this diagram dates to circa 300 Bce.
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 questions 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. Vol. 104, No. 9 • May 2011 | MatheMatics teacher 691
Problem 1: Proof without Words: The Law of Cosines Explain 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 provide a revised construction.
Source: Kung (1990) Fig. 4 This proof of the law of cosines depends on the product of the segments of chords theorem.
Source: Kung (1990) Fig. 3 In preparation for the online class discussion of problem 1, students reviewed the law of cosines and discussed how it would generally be proved in a high school textbook.
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 communication of mathematical ideas. Written explanations 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: onstruct triangle ABC with longest side BC and C 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 692 Mathematics Teacher | Vol. 104, No. 9 • May 2011
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 indicated 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 concluded 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 diagrams created by four different students. Two examples show angle q entirely inside the circle, and two examples show angle q extending outside the circle.
ANOTHER EXAMPLE In 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. 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.
sent that statement. Problem 2 (see fig. 8) shows the problem given to the students. All students were able to answer the first question 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. Vol. 104, No. 9 • May 2011 | MatheMatics teacher 693
Fig. 7 several students were successful in constructing a proof without words for the law of cosines with q obtuse.
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. Looking 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.) 694 MatheMatics teacher | Vol. 104, No. 9 • May 2011
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 examples for one or two of the equations in the
Problem 2: Pythagorean Triples Consider 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 computed 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. 8 students with some experience with proofs without words may be able to tackle this more sophisticated problem.
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.
CONCLUSION I 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 explanation 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 mathematical statement, students are also improving their ability to reason through a problem.
Fig. 9 some students were able to devise an illustration of the ﬁrst equation in the pattern.
Fig. 10 the general pattern for problem 2 can be visually displayed.
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.
REFERENCES Burton, 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
CAROL J. BELL, [email protected]
, teaches mathematics education courses at Northern Michigan University in Marquette. She is interested in how future teachers communicate and make sense of the mathematics they will someday teach.
Vol. 104, No. 9 • May 2011 | MatheMatics teacher 695