 Proofs without Words in Geometry

• # Proofs without Words in Geometry

Pictures and diagrams help high school geometry students develop reasoning and proof-writing skills.

Since the 1970s, the Mathematical Association of America’s (MAA) journals Mathematics Magazine and College Mathematics Journal have published “Proofs without Words” (PWWs) (Nelsen 1993). “PWWs are pictures or diagrams that help the reader see why a particular mathematical statement may be true and how one might begin to go about proving it true” (Nelsen 2000, p. i). This article explains how I created and implemented a variation of using diagrams without words to facilitate the development of high school geometry students’ reasoning and proof skills.

The Common Core (CCSSI 2010) lists approximately twenty geometric theorems for students to prove (e.g., standards G.CO.9, G.CO.10, and G.CO.11) as well as the more general standard of using congruence and similarity criteria for triangles to prove relationships in geometric figures (i.e., standard G.SRT.5). Furthermore, Appendix A of the Common Core states that teachers should “encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words” (p. 29).

I used the following PWWs, adapted from Nelson (see fig. 1), when my Honors Geometry students studied right triangle trigonometry. Using the figure, students prove that sin(2 ) = 2cos sin and cos(2 ) = 2(cos )2 – 1, given that O is a semicircle with a radius of 1. (For further suggestions on how to use PWWs with students see Bell 2011).

Even though the Common Core’s Appendix A states that students should use diagrams without words, I was initially unsure of how I could use them for high school geometry theorems. As with figure 1, most PWWs that I was familiar with are a single diagram and seemed to fit the traditional norm of providing students with the diagram along with the “given” and the “prove” statements. After some thinking, I realized that a group of cards, each with the same base diagram but with different labels or markings on each card, could be arranged in a flowchart to “prove” a theorem. For example, consider how you would arrange the four cards in figure 2. Is there more than one way to arrange the cards? What theorems can you prove?

Four different ways to order the cards suggest meaningful reasoning. Figure 3 shows that when two parallel lines are cut by a transversal, the alternate interior angles are congruent. The diagrams lead an explanation of why this is true: Lines AC and BD are parallel and cut by a transversal (card d), so we have corresponding angles formed by parallel lines and ∠1 ≅ ∠2 (card b). Also, vertical angles are congruent, so ∠1 ≅ ∠3 (card a). Thus, by the transitive property, we can write ∠2 ≅ ∠3 (card c). On the other hand, figure 4 shows that a different arrangement of the cards can illustrate the converse. That is, if two lines intersect a third line and form congruent alternate interior angles (card c), then the two lines are parallel (card d). In yet other arrangements, the cards can demonstrate the corresponding angles theorem and its converse. After my students have arranged the cards, the diagram guides them to write the accompanying statements and reasons in paragraph, two-column, or flowchart form.

In my geometry class, I use PWWs as a bridge between informal reasoning and writing the complete proof. Preceding a five-part proof progression, students make many conjectures in small-group investigations both with and without dynamic geometry software. During the progression, students prove most of these conjectures (a few of them multiple times using different representations). In the first three stages of the progression, students reason informally with diagrams, put proofs in order when provided with the statements and reasons, and fill in missing details of given proofs. The fourth stage uses PWWs as scaffolding to help students write all the statements and reasons of a proof formally. In the final stage, students write complete proofs as well as critique and evaluate full proofs.

Depending upon the amount of scaffolding that students need and the complexity of the proof, I use PWWs in different ways within the fourth stage of the progression. Minimally, as with the activity associated with figure 2, I give students the set of cards and ask for arrangements that prove as many different results as possible. To add more support, I have sets of cards that also include a statement of what students are to prove. For example, figure 5 shows the nine cards, plus the card with the theorem, already arranged in a flowchart proof. This PWW illustrates that if a quadrilateral has a pair of parallel and congruent sides, then it is a parallelogram. Note that card a and card g are the same. Duplicate cards emphasize that the parallel sides are used at two different points in the proof: first, as part of the given information (card a), and then as a condition to apply the definition of a parallelogram (card g).

For some theorems listed in the Common Core that are more complicated for students to prove, I provide a flowchart arrangement of the cards, and students need only to write the statements and reasons by analyzing the diagrams. I also use bold print and color in the diagram, to help students focus on important parts. Figure 6 shows the PWW cards for the theorem “the medians in a triangle are concurrent.” Before students begin working, I explain that cards a and c show medians and intersecting at F. Then, based on card b, I explain that is extended to H, and that if we can conclude that intersection point I is the midpoint of then this would imply that is a median. Thus, all three medians in the triangle are concurrent at F. It would be nearly impossible for students (and most geometry teachers—including me) to come up with the proof themselves, but by using PWWs along with the short explanation I give before students begin, my students are able to write the statements and reasons needed to complete the proof. The proof in figure 6 uses two previously proven theorems (both also found in the Common Core): namely, the midsegment in a triangle is parallel to the third side (cards f and g), and the diagonals of a parallelogram bisect each other (card i).

Finally, to increase the complexity of the task, I include more cards than necessary to form a single proof. Figure 7 (p. 584) shows four of fifteen different cards that students can use to prove a geometric fact. (All fifteen cards are available as more4U content at www.nctm.org/mt.) Some cards in this set are extraneous: Card k, for example, shows AAA similarity, which would not be needed for proofs of congruence. I ask each student to glue cards onto paper in a flowchart that proves a result. Figure 8 (p. 584) shows how a student used eight cards to prove that if is parallel to and B is the midpoint of then B is also the midpoint of  Next, students exchange papers, and they write statements and reasons corresponding to the flowchart. I also instruct them to make corrections if they think anything is wrong with the flowchart they received in the exchange. Figure 9 shows a student’s statements and reasons for the PWW (fig. 8) created by a classmate. As an alternative activity, I give students multiple copies of the same PWW cards and ask them to prove as many different results as possible each time using a new set of cards. In the past, many of my students struggled to understand how the proofs for a theorem and its converse were different. However, students can use the same set of cards to create paired PWWs when I introduce a two-sided card, printed on a different sheet of colored paper to attract their attention. For example, in figures 10 and 11, different sides of card f are used to prove the two parts of this biconditional statement: A quadrilateral is a parallelogram if and only if the diagonals bisect each other. The forward implication (fig. 10) is read from the top down, whereas the converse is read from the bottom up (fig. 11).  Note three differences between the two flowcharts: Card f has been turned over, card e is moved, and the direction of the arrows is reversed. Also, note that in the forward direction, card c uses a previously proven theorem (the opposite sides of a parallelogram are congruent).

Proving this biconditional has benefited students who are unsure whether the congruent alternate interior angles or the parallel lines comes first in the proof. Some students have incorrectly attempted to make a generalization that one must always follow the other no matter what statement they are proving.

When I am implementing a set of cards that prove a biconditional, I do not tell the students what to prove, but instead I ask them to use the cards to prove a result. When they have something ready in flowchart form, I check it, offering feedback if necessary. When it is correct, students write the proof on paper, and then I check it. Once that flowchart and proof are correct, I ask students to try to reverse the proof. That is, they start with the conclusion they just proved as given information, and they try to prove that the originally given assumptions hold true. Because they created both proofs, they are able to see the two directions more clearly. Afterward, as part of a class discussion, I use a large bulletin board to display full-page versions of their PWWs. Student volunteers come up one at a time to begin arranging the cards. Once we have one direction of the proof on the bulletin board, we discuss reversing the proof. Student ownership of this PWW, and others like it that I have created, has helped students understand the differences between the proofs of a theorem and its converse.

The PWW approach to proof supports Sinclair, Pimm, and Skelin’s (2012) notion that working with diagrams is central to geometric thinking. According to Sinclair and colleagues, diagrams are artifacts that have a story to tell, but we must interpret the diagrams to learn the story. They suggest that although diagrams sit in silence, we need to improve upon hearing what they are telling us. PWWs put diagrams in the foreground instead of relegating them to the background where “marking the diagram” is only a part of the ritualized norm of “doing proof” (Herbst et al. 2009). PWWs emphasize that interpreting diagrams and reasoning are at the heart of geometric proof. The cards are a collection of individual still frames that students can organize to tell a coherent story of why a particular geometric invariance must be true.

Based upon the results from classwork, homework, and quizzes, my students have been very successful with proof when using PWWs. I believe this success is due to three reasons. First, arranging a PWW to prove a result is akin to solving a puzzle, and most people (including my students) enjoy working on puzzles and recreational mathematics with physical models. Second, focusing on diagrams initially frees students from getting bogged down with the formality of writing statements and reasons, even though they will have much of the same discussion in their groups as they complete a PWW. Third, working on PWWs is low-risk for students because, based upon discussion they have in groups and feedback they get from me, they can continue to revise the arrangement of the pieces of the PWWs before they begin the final process of writing down the statements and reasons. In summary, PWWs, such as the five examples described in this article, have made proof more accessible to more of my students while supporting them in developing their reasoning skills and in seeing the structure of how proofs are “assembled.”

ACKNOWLEDGMENTS

The author would like to thank his Troy High School colleagues, Brian Huelskamp and Samantha Potocek, for field-testing various versions of these PWWs with their students. The author also would like to thank participants in previous conference sessions and workshops for their feedback on preliminary versions of these PWWs.

REFERENCES

Bell, Carol J. 2011. “Proofs without Words: A Visual Application of Reasoning and Proof.” Mathematics Teacher 104 (9): 690–95.

Common Core State Standards Initiative (CCSSI). 2010a. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf

———. 2010b. Common Core State Standards for Mathematics Appendix A. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf

Herbst, Patricio, Chialing Chen, Richael Weiss, and Gloriana Gonzalez, with Talli Nachlieli, Maria Hamlin, and Catherine Brach. 2009. “Doing Proofs in Geometry Classrooms.” In Teaching and Learning Proof across the Grades: A K–16 Perspective, edited by Despina Stylianou, Maria L. Blanton, and Eric J. Knuth, pp. 250–68. Studies in Mathematical Thinking and Learning series. New York: Routledge.

Nelsen, Roger B. 1993. Proofs without Words: Exercises in Visual Thinking. Washington, DC: Mathematical Association of America.

———. 2000. Proofs without Words II: More Exercises in Visual Thinking. Washington, DC: Mathematical Association of America.

Sinclair, Nathalie, David Pimm, and Melanie Skelin. 2012. Developing Essential Understanding of Geometry for Teaching Mathematics in 9–12. Essential Understanding series. Reston, VA: National Council of Teachers of Mathematics.

WAYNE NIRODE, Nirode-w@troy.k12.oh.us, teaches mathematics at Troy High School in Troy, Ohio. His interests include geometry, statistics, and technology. 