六又有九十一分之四十九-问约之得几何答曰-十三分

六又有九十一分之四十九-问约之得几何答曰-十三分 Some Operations on Fractions – An Historical Perspective and a Connection to the Present Cheng-Yao Lin Department of Curriciulum & Instruction , Southern Illinois University Jerry P. Becker Department of Curriciulum & Instruction , Southern Illinois University Abstract :This paper selected a few problems to examine that involve fractions and their applications from book of Jiuzhang Suanshu. In addition, this paper used these problems introducing how to learn and teach fractions through an open approach. Examples of several answers to each problem are provided. Keyword: fractions, fractions operation, open-ended approach, Jiuzhang Suanshu. Jiuzhang Suanshu , [Nine Chapters on the Mathematical Art], is the earliest Chinese textbook on practical mathematics. It consists of nine distinct chapters that include a total of 246 problems and their solutions that can be used to solve everyday problems in engineering, surveying, trade, and taxation. This book probably was composed in the 1st century AD, but perhaps as early as 200 BC. Liu Hui was a Chinese mathematician who lived in the 200s. In 263 he wrote a very detailed commentary on the book and provided solutions to the problems.In chapter 1 of this book - Fang tian/ Field measurement - two topics are addressed: (1) A method for calculating land area, and (2) addition, subtraction, multiplication and division of fractions and their applications (See Figure 1). Figure 1 Chapter One of Jiuzhang Suanshu. It is well-known that children of various ages experience considerable difficulty in learning the concepts of and operations on fractions (e.g.,Hart, 1981; Bezuk & Cramer, 1989). We think that providing a perspective from the history of mathematics may prove interesting and useful for both teachers and students. In this case, we go back two thousand years to see some interesting problems and operations on numbers. We have selected a few problems to examine that involve fractions and their applications. Two of them deal with practical problems in daily life. Reducing Fractions: Problem 6 in the ancient Chinese text is as follows: 〔 六 〕 又 有 九 十 一 分 之 四 十 九 。 問 約 之 得 幾 何 , 荅 曰 : 十 三 分 之 七 。 約 分 術 曰 : 可 半 者 半 之 , 不 可 半 者 , 副 置分 母 子 之 數 , 以 少 減 多 , 更 相 減 損 , 求 其 等 也 。 以 等 數約 之 。 Problem 6: Translation to English: [6] Reduce 49/91 to lowest terms. Method given in Jiuzhang Suanshu: If the denominator and numerator can be halved, halve them. If not, lay down the denominator and numerator, subtract the smaller from the greater. Repeat the process to obtain the greatest common divisor [GCD]. Reduce them by the GCD. That is, subtract 49 from 91 and get 42; Subtract 42 from 49 and get 7; Continually subtract 7, to finally get 7. Seven is the greatest common divisor because the subtrahend is equal to its counterpart in the minuend and the remainder is 0. In the problem, we find the GCD of 91 and 49 as follows: (91, 49) = (91-49, 49) = (42, 49) = (42, 49-42) = (42, 7) = ( 42-7-7-7-7-7, 7) = (7, 7) = 7. Divide the numerator and denominator by the common factor 7 and get 7/13 (Shen et. al, 1999). Fraction Computation: Problem 17 in the ancient Chinese text: 〔 一 七 〕 今 有 七 人 , 分 八 錢 三 分 錢 之 一 。 問 人 得 幾 何, 荅 曰 : 人 得 一 錢 、 二 十 一 分 錢 之 四 。 經 分 術 曰 : 以 人 數 為 法 , 錢 數 為 實 , 實如 法 而 一 。 有 分 者 通 之 , 重 有 分 者 同 而 通 之 。 Problem 17 translation to English: [17] 8 and 1/3 dollars equally shared among 7 people. How much does each person get? Method given in Jiuzhang Suanshu: Take the number of persons as the divisor and the amount of money as the dividend. Divide. If either the dividend or the divisor is a mixed fraction, convert it to an improper fraction. If both are mixed fractions, convert them to improper fractions with a common denominator. Multiply the numerator of the dividend by the denominator of the divisor, and multiply the divisor by the denominator of the dividend. That is to say, if both the dividend and divisor are mixed numbers, we first convert them to improper fractions, and then multiply the numerator of the dividend and by the divisor by the denominator (Shen et. al, 1999). Problem 22 in ancient Chinese text: 〔 二 二 〕 今 有 田 廣 三 步 、 三 分 步 之 一 , 從 五 步 、 五 分步 之 二 。 問 為 田 幾 何 , 荅 曰 : 十 八 步 。 大 廣 田 術 曰 : 分 母 各 乘 其 全 , 分 子 從 之, 相 乘 為 實 。 分 母 相 乘 為 法 。 實 如 法 而 一 。 Problem 22 translation to English: [19] A parcel of land is 3 and 1/3 bu wide and 5 and 2/5 bu long. What is the area? (Note: Unit of length: Bu is the basic unit of length - 1 bu is roughly equal to the length of a stride or 1 meter.) Method in Jiuzhang Suanshu: Multiply each denominator by its integral part; then add the corresponding numerator to get an improper fraction. In this manner the numerator and denominator are in the dividend. Multiply the sums as numerators; multiply the denominators. Divide. Here both the width and length are fractions, so first divide numerator and denominator by the product of the denominators, so the product of the denominators is regarded as the divisor. (Shen et. al, 1999). Connecting to teaching using the open approach The problems above offer some interesting insights about how fractions were used or operated on in ancient texts. They are relevant today. In fact, they can become part of a way of teaching mathematics that we can call an “open approach.” In the “open approach,” problems are selected and used in instruction so that they exemplify a diversity of approaches in getting answers or solutions. In this approach, students’ responses or answers are used by the teacher to provide experience to students in learning something new. Students combine what is newly learned with their previous knowledge, skills and mathematical ways of thinking. (Shimada, 1977; Becker and Shimada, 1997; Hashimoto and Becker, 1999) As a way of creating interest and stimulating creative mathematical activity in the classroom, the Japanese developed a tradition in teaching to focus on different ways to solve a problem when there is a unique answer, but many different ways to get it – i.e., the process is open. At the various grade levels it is possible to have students use their own mathematical thinking abilities in finding answers and solving problems. (Hashimoto and Becker, 1999, p. 102) The consequence is several or many ways to solve a problem that students can then share with the teacher and other students and which the teacher can have students discuss. In this approach students can have many opportunities to make comprehensive use of their mathematical knowledge and skills. Also, every student can respond to the problems in some significant way of his/her own. The problems above coming from the history of mathematics can fall in this category as will be seen below. The following fraction problems from Jiuzhang Suanshu will be presented using an “open approach.” Problem 6: Reduce 49/91 to lowest terms. In using the open approach, teachers first of all write down all the responses they anticipate their students will provide. Here are some examples for each of the problems from Jiuzhang Suanshu. Examples of expected responses from students: Expected response 1. Step 1, we list the prime factors of the numerator and denominator. 4977， ,91713， In step 2, we divide by, or cancel, the factor of seven that is common to both the numerator and denominator. We're left with a 7 in the numerator and 13 in the denominator. 49777， ,,9171313， Expected response 2. Let’s take a look the following algorithm. a=49; b=91 91=49×1 + 42 91×(1)+49×(-1)=42 49=42×1 + 7 91×(-1)+49×(2)=7 42=7×6 + 0 91×(7)+49×(-13)=0 So, gcd (49,91) =7 The greatest common divisor is 7. We divide by seven that is common to both the numerator and denominator. We got 7 in the numerator and 13 in the denominator. Expected response 3. If we know that if both numerator and denominator can be divided by the same number, then the result is an equivalent fraction; thus, divide both 49 and 91 by 7 and the result is 7/13. Problem 17: 8 and 1/3 dollars equally shared among 7 people. How much does each person get? Examples of expected responses: Expected response 1: 251，4125251254371, the answer is dollars. 8771,,,,,，,,712133372121，17 Expected response 2: 125725441, the answer is dollars. 871,,,,,213312121 Expected response 3: 1257257325212521254，,, 871,,,,,,,,,,33131333332121，, 4the answer is dollars. 121 Expected response 4: An Area model For an area model you can think of a rectangle whose are is 8 and 1/3 square units. Next, think of the length 7 units. Then find the width. That is 1425257x =, 7x= , x==. 81332121 Expected response 5: Fair-Sharing model For a fair-sharing model, first you have to define a 3 x 7 grid as equal to1 dollar. Then divide each dollar into 21 parts (pieces). Next, divvy up the pieces among the seven people. Then present the result for each person. Step 1: Define a 3 x 7 grid as equal to1 dollar. Divide each dollar into 21 equal parts (pieces). Step 2 : 8 and 1/3 dollars can be represented shown as follows: 8 and 1/3 dollars Step 3: Divvy up the pieces among the seven people: a,b,c,d,e,f, & g. 8 and 1/3 dollars shared by 7 a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g a b c d e f g Step 4: The result of the divvying up the pieces for each person. a a a a a a a a a a a a a a a a a a a a a a a a a =1 and 4/21 dollars Problem 22: A parcel of land is 3 and 1/3 bu wide and 5 and 2/5 bu long. What is the area? Examples of expected responses: Expected response 1: 12102710272539，，，， , the answer is 18 square bu. 352918，,，,,,，,35353535，， Expected response 2: 1212211265235(3)(5)353515，,，，，,，，，，，，，,，，，353553355315 635521825245，，,，，，,，，，,，,，,1515151531853351515151515，， 4So the answer is square bu. 9 Expected response 3: 1210271027270，4, the answer is square bu. 3518，,，,,,935353515， Expected response 4: Groups-of model For a groups-of-model (Ott,1990; Baroody, 1998), you could represent 1 unit as a line 15 grids long and 1 group of 5 and 2/5 as a line 5 X 15 grids or 75 grids long + another 6 grids long. This would be repeated two more times to represent a second and third group of 5 and 2/5. One third group of 5 would be represented by a fourth line which would be 1/3 as long any of the previous lines (i.e., 1/3 of a line 81 grids long or 27 grids long). To represent 1/3 of 2/5, you would need a fifth line that would be one third of 6 grids long. Figure 2 Example of groups-of model One group of 5 and 2/5 Two groups of 5 and 2/5 Three groups of 5 and 2/5 1/3 group of 5 and 2/5 Expected response 5: An Area model For an area model (Baroody, 1998), you would have to define a 15 x 15 grid as 1 square unit and a line 15 grids long as 1 linear unit (See Figure 3). Then construct a rectangle and determine the number of square units (15 x 15 grids). Figure 3 Area model Step 1. Define 15 x15 = 1 square unit and 15 grids long = 1 linear unit 12 rectangle. Step 2. Construct a 35，35 Concluding Remark Fractions are usually difficult for students to master in school. For example, most students have difficulty learning the various operations on fractions (Tourniaire & Pulos, 1985). In fact, operations on fractions such as addition, subtraction, multiplication, division, simplifying, etc. are more difficult than the concept of fraction for most of the students. The National Council of Teachers of Mathematics in its Principles and Standards for School Mathematics describes the need for students to develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use. Teaching students fraction operations using an open approach may provide a way to help them towards fluency in fractions operation. In addition, it may promote their higher order thinking and help them understand the meaning of fractions and the effects of arithmetic operations on them. References Baroody, S. J. (1998). Fostering children’s mathematical power: an investigative approach to K-8 mathematics instruction. Mahwah, NJ: Lawrence Erlbsum. Bezuk, N., & Cramer, K. (1989).Teaching About Fractions: What, When, and How? In P. Trafton (Ed.), National Council of Teachers of Mathematics 1989 Yearbook: New Directions For Elementary School Mathematics (pp. 156-167). Reston, VA: National Council of Teachers of Mathematics. Becker, J. & Shimada, S. (1997). The Open Ended Approach – A New Proposal for Teaching Mathematics. Reston, VA: National Council of Teachers of Mathematics. Hashimoto, Y. and Becker, J.P. (1999). The Open Approach to Teaching Mathematics - Creating a Culture of Mathematics in the Classroom: Japan. (In L. Sheffield (Ed.) Developing Mathematically Promising Students), Reston, VA: National Council of Teachers of Mathematics Hart, K. (1981). Children's Understanding of Mathematics, 11-16. London: Murray. Nunes, T. and Bryant, P. (1996). Children Learning Mathematics. Oxford, U.K. BlackwellOtt, J.M. (1990). A united approach to multiplying fractions. Arithmetic Teacher, 37(7), 47-49. Shen, K., Crossley, J. and Lun, A. (1999).The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford, U. K.: Oxford University Press. Shimada, S. (Ed.) (1977). The Open-ended Approach in Arithmetic and Mathematics - A New Proposal Toward Teaching Mathematics. Tokyo: Mizuumishobo, 1977. (In Japanese) Tourniaire, F.& Pulos, S. (1985). Proportional reasoning. Educational Studies in Mathematics, 16, 181-204.