Chips, Chopsticks, and Polygonal Numbers to Foster Algebraic Thinking

Jeong Oak Yun – Seoul Global High School, [email protected] Alfinio Flores – University of Delaware, [email protected]

In these activities, students represent teacher explained how to form pentagonal triangular, square, pentagonal and hexagonal numbers with chips, she asked her students numbers with chips. Chopsticks are used to to find their own strategies for using break polygonal numbers into components. chopsticks to partition them and finding By representing each part with an algebraic numerical and algebraic representations of expression, students can find an algebraic the pentagonal numbers. Then she asked representation for the total, and establish students to do the same for central relations between different algebraic hexagonal numbers. The teacher prepared a expressions. In this article we focus on PowerPoint presentation and asked students breaking pentagonal and hexagonal numbers to draw imaginary chopsticks on the board into triangular numbers. Of course other where the slides were projected (Figure 1). decompositions of polygonal numbers are possible and are hinted at the end of the article. The goals of these activities are to help students find number patterns that have geometrical structure; develop their own strategies to count numbers that form a pattern; compare their strategies and learn from each other; and experience the beauty of mathematics. Figure 1. Student expressing a pentagonal nn1 number as nn1 The activities were conducted with 2 five groups in a public magnet school in Korea. The students are among the top 20% Students first wrote the numeric th of 11 grade students and have special talent representation of 2nd, 3rd, and in some cases for English and preference for social studies, 4th polygonal numbers, then wrote an but they are not especially gifted in algebraic representation. Students enjoyed mathematics. For the first activity the finding many different strategies for just one teacher illustrated how to represent the question and even became competitive. In triangular numbers algebraically after using one class students found eight different ways chopsticks to partition geometrically to represent pentagonal numbers. In another arranged chips and finding the numerical class students were eager to share their nd rd th representation of the 2 , 3 , and 4 strategies even after the bell had rung. Some triangular numbers. Then the teacher of the students who did not show much explained a strategy of representing square interest in the beginning, after watching numbers as the sum of two triangular their peers present their novel ideas, became numbers using chopsticks. Then, after the interested and tried to come up with their

Illinois Mathematics Teacher – Fall, 2009 ...... 6 own ideas. In some cases students were 12 T 1 amazed by the different strategies of their 1 2 peers. Although students were not 23 particularly fond of the chopsticks and T2 12 2 wanted to use tools that could be bent like 34 wire, thinking about where to put the T3 123 chopsticks seemed to help them to 2 45 concentrate on how to partition the array of T 1234 chips. Using tools to partition the arrays was 4 2 useful in finding patterns. nn1 Tn1234 n 2

Activity 1: Triangular Numbers Activity 2: Square Numbers 1. The sums of consecutive whole numbers like 1, 1 2, 1 2 3, , can be repre- The arrays in figure 5 are called square sented as triangular shapes (Figure 2). numbers. The number of chips in each row Represent the first four triangular is equal to the number of rows. So, if there numbers with poker chips. are n rows, the total amount of chips will be nn n2 . Use chips to build the first four square numbers.

Figure 2. The first four triangular numbers

Figure 5. Square numbers

Find triangular numbers within the square

Figure 3. The nth numbers and use a chopstick to separate them (Figure 6). 2. Make two copies of each triangular number with chips of two colors, and arrange them together to form rectangular arrays (see Figure 4). Describe the number in each rectangle as Figure 6. Triangular numbers and square numbers the product of the number of chips in each row by the number of rows. Express each algebraically as Express the number of chips for each the sum of two terms that represent triangular number as half the chips in the triangular numbers. For example, corresponding array. Generalize for the 23 34 th S 33 , n triangular number. 3 22 34 45 S 44 . 4 22 Generalize to the nth square number. Verify that the algebraic expressions on both

Figure 4. Rectangular arrays sides of the following identity are indeed

Illinois Mathematics Teacher – Fall, 2009 ...... 7 equivalent. Expand and simplify the right Generalize the pattern to obtain a formula side. for the number of chips in the nth pentagonal nnnn11 number. nn nnnnnn1131 22 PTTnnn221 Activity 3: Pentagonal Numbers 22 2

The arrays in figure 7 represent pentagonal Students can also break a numbers. Each new is into a triangular number and a square formed by adding a new layer consisting of number (Figure 10). In Figure 11, the thee sides at the bottom, thus extending the pentagonal numbers have been squeezed to two sides that meet at the upper vertex. This show more clearly how they are formed by a upper vertex has thus a different role than triangular number and a square number the other vertices in this kind of pentagonal (Meavilla Seguí 2005). array.

Figure 10. Pentagonal numbers as triangular numbers and square numbers Figure 7. Pentagonal numbers

Pentagonal numbers can be decomposed into triangular numbers. Figure 8 shows one kind of decomposition. Figure 11. Pentagonal numbers squeezed

Express the nth pentagonal number as the sum of the n 1 st triangular number and the nth square number. Verify that this algebraic expression is equivalent to the one obtained Figure 8. Pentagonal numbers and triangular above. numbers. Students found several partitions and the corresponding algebraic expressions. Express each of the pentagonal number as a Verify that indeed the total number of chips sum of triangular numbers. For example, is the same. For the partition in Figure 7 12 23 students found this expression P2 113 2 22 n 1322 332 342 3k 2 23 34 k1 P 336 2 3 22

th nn 121 n n Figure 8. The n pentagonal number Figure 12. 21 22

Illinois Mathematics Teacher – Fall, 2009 ...... 8 Generalize to the nth central hexagonal nn1 numbers, H 16 . n 2

Use the partition of H4 illustrated in Figure

17 to suggest another algebraic expression. nn1 Figure13. nn1 Express first this particular hexagonal 2 Students used “wire” for the following number as the sum of particular triangular partition. numbers and then generalize. Verify that the new general expression is equivalent to the one obtained above.

Figure 14. nn12 n 21 n

Activity 4: Hexagonal Numbers Figure 17. Another way to partition a hexagonal The arrays in Figure 15 are called the central number hexagonal numbers 45 34 23 H 4 4 222 nn 11 n n n 21 n H 4 n 222

Figure 15. Hexagonal numbers Below are other partitions found by students and the corresponding algebraic expressions. Use chopsticks to break each into triangular numbers. Several solutions are possible. Figure 16 shows one.

Figure 18. nn 34 nnnn 1 231 nn Figure 16. Hexagonal numbers and triangular 2222 numbers

Express the hexagonal numbers in terms of the triangular numbers. 23 H3 16 2 34 2 nn1 H 16 Figure 19. nn2 12 4 2 2 Write an expression for the number of chips in the fifth central hexagonal number.

Illinois Mathematics Teacher – Fall, 2009 ...... 9 represent them with letters. These activities can stimulate students to try to find various ways of solving a problem and appreciate the joy of finding various solutions. The activities also foster them to think how to nn1 2 find patterns, to express the patterns in Figure 20. 21n 1 2 2 numerical forms, and to generalize them into algebraic forms. A teacher can use Figure 21 illustrates the fourth central geometrical representations to help students hexagonal number as the sum of hexagonal as they learn to use algebra to generalize and shells. Students found an expression for the justify (Flores 2002). Polygonal numbers nth central hexagonal number as the sum of and other geometrical representations can n1 the shells 1616263 16 k . provide a more concrete step towards the k 1 more abstract use of letters as variables or Verify that this expression is equivalent to generalized numbers, which for beginners the expressions obtained above. may be a little complicated.

References Flores Peñafiel, Alfinio. “Geometric Representations in the Transition from Arithmetic to Algebra.” In Representations and Mathematics Visualization, edited by Fernando Figure 21. The fourth central hexagonal number Hitt, 9-29. México: Departamento

de Matemática Educativa del

CINVESTAV-IPN, 2002. Final Comments Meavilla Seguí, Vicente. La Historia de las

Matemáticas como Recurso Didác- Polygonal numbers are one kind of tico. Badajoz, Spain: Federación de geometrical representation of numbers and Sociedades de Profesores de relations among numbers. Students can use Matemáticas, 2005. such geometrical representations as a means Yun, Jeong Oak. Polygonal numbers. to explore algebraic ideas. With the help of Unpublished manuscript. Arizona these representations students can think State University, 2007. about the relations among the numbers, express them using their own words, and

2010 NCTM Annual Meeting and Exposition San Diego, California April 21-24, 2010

Illinois Mathematics Teacher – Fall, 2009 ...... 10