Decartess multiplication and application in primary mathematics

In advanced mathematics, especially the set theory, decartess multiplication is an indispensable tool for solving some types of problems. In particular, Decartess multiplication will help students solve problems.

Scientific Journal − No27/2018 Part I SCIENTIFIC PAPERS DECARTESS MULTIPLICATION AND APPLICATION IN PRIMARY MATHEMATICS Nguyen Van Hao1, Dao Thi To Uyen2, Nguyen Thi Thanh Ha3 Department of Mathematics, Hanoi Pedagogical University Department of Primary Education, Hanoi Pedagogical University Faculty of Basic Science, Viet Tri Industrial University Abstract: In this article, we would like to present some applications of Decartess multiplication in teaching maths to teachers at primary schools Keywords: Decartess multiplication, application of Decartess multiplication Email: nguyenvanhaodhsphn2@gmail.com Received 11 October 2018 Accepted for publication 15 December 2018 PREAMBLE In advanced mathematics, especially the set theory, decartess multiplication is an indispensable tool for solving some types of problems In particular, Decartess multiplication will help students solve problems However, for students who are studying and with some students just graduated, Decartess not be application in mathematics especially for primary education In fact, nursery and primary school, the children were familiarize with the Decartess but with an approach through simple problems Teachers will help students understand application of Decartess multiplication through this problems Thus, in this article we will illustrate some of the problems of primary school in the language of Decartess multiplication CONTENT 2.1 Preparation To present some applications of Decartess to teach maths for primary teacher, we repeat some of the most basic knowledge about this concept Details on this, see the reference [1] Ha Noi Metroplolitan University 2.1.1 Concept of Decartess Decartess multiplication of two sets A and B denoted by A × B is a set containing all ordered elements (a, b ) with a a element of the set A and b an element of the set B , it mean that A × B = {(a, b ) : a ∈ A, b ∈ B } Example 1.1 Give two sets of elements as follows A = {1,5}; B = {2, 4, 7} In this case, it is easy to see that the decartess multiplication of these two sets is defined by the elements as follows A × B = {(1,2);(1, 4);(1, 7);(5, 2);(5, 4);(5, 7)} 2.1.2 Decartess multiplication of n the set Decartess multiplication of n the set A1, A2, , An is a set contains all ordered elements of the form (a1, a2, , an ) inside a1 ∈ A1, a2 ∈ A2, , an ∈ An Or write in the form of mathematical symbol { } A1 × A2 × × An = (a1, a2, , an ) : ∈ Ai , i = 1, n In particular, when the set A1 = A2 = = An = A is denoted by A1 × A2 × × An = An 2.1.3 Force of Decartess multiplication Force of Decartess multiplication is calculated by the force multiplication of each set A×B = A × B A1 × A2 × × An = A1 × A2 × × An Example 1.2 Give two sets A = {1, 2} ; with A = and B = {a, b, c}; with B = Scientific Journal − No27/2018 Decartess multiplication of these two sets is defined as follows A × B = {(1, a ),(1, b ),(1, c ),(2, a ),(2, b ),(2, c )} We see that the force of A × B is defined according to the above formula A×B = A × B = 2× = 2.2 Some applications of Decartess multiplication in primary mathematics For primary education, many problems are easily understood and taught to students when they know the concept of Decartess multiplication We illustrate this with some of the problems below Problem 2.1 Find and list all two-digit numbers and they divisible by When teacher guides student solve this problem, teacher knew in decimal system numbers are made up of digits is 0,1,2, 3, 4,5, 6,7, 8,9 Two-digit numbers have structure is XY with digits of tens different So, digits of tens are sets X = {1, 2, 3, 4, 5, 6, 7, 8, 9} ; with X = Based on the sign of the natural numbers divisible by have last digit or teachers can guide the students to find the units So, the digit of the units belong sets Y = {0, 5}; with Y = We can see that two-digit numbers divisible by are elements of Decartess multiplication of sets X and sets Y above So, teacher can know two-digit numbers divisible by are X × Y = × = 18 (numbers) Understanding the concept of Decartess multiplication, teachers can guide students knows two-digit numbers divisible by Beside, teachers can guide studens lists all this numbers So, we have numbers 10;15;20;25; 30; 35; 40,;45;50;55;60;65;70;75;80;85;90;95 Problem 2.2 Find two-digit numbers divisible by and (Pham Thanh Cong, Detailed explanation guide Violympic Math , General publishing house Ho Chi Minh City, 2013 , p 68 ) The same as proplem 1, Two-digit numbers have structure XY with digits of tens different So, digits of tens are sets X = {1, 2, 3, 4, 5, 6, 7, 8, 9} ; with X = Ha Noi Metroplolitan University 10 This numbers can divisible by , the last digit is one of the digits {0,2, 4, 6, 8} This numbers can divisible by , the last digit is one of the digits numbers divisible by {0, 5} Thus, two-digit and , the last digit is Y = {0} ; with Y = So, two-digit numbers divisible by and is force of Decartess multiplication X ×Y X × Y = × = (numbers) Problem 2.3 Give two triangles ABC and DEF , three free points in six points A, B,C , D, E , F not in line How many straight lines are connected from the vertices of the triangle ABC to the vertices of the triangle DEF ? Teacher paints two triangle satisfy problem, guides student connect vertices of the triangle ABC to the vertices of the triangle DEF and count those straight line But language of sets theory and knowledge of Decartess multiplication, we can see: { } { } Suppose X = A, B,C ;Y = D, E , F Thus, a straight line corresponding a element of Decartess multiplication X ×Y and all straight line is X ×Y = X × Y = × = (line segments) Problem 2.4 Give four digits from these four digits? 3, 4, 5, How many even numbers are three digits When students solves this problem, they will list all even numbers are three digits from these four digits and count However, they usually list coincide or lack For teachers, if they use Decartess multiplication, we see: three digit numbers have structure XYZ { } The digit of hundreds belonging X = 3, 4, 5, , the digit of tens belonging Y = {3, 4, 5, 6} and the digit of units is even numbers belonging Z = {4, 6} A even number are three digits from these four digits corresponding with a element of Decartess multiplication and these numbers are X ×Y × Z = X × Y × Z = × × = 32 (numbers) For this orientation, we can show some problems as follows Scientific Journal − No27/2018 11 Problem 2.5 Two teams A and B plays badminton, a team has three members Playing two pairs, a pair has team A ’s member and team B ’s member ( free choice) Team with at least two winners, that team will win How many assignment pairs? Problem 2.6 Class 5A have four groups, a group has seven students Pick out four students from four groups (a group has a student) to set up a group and cut grass How many set up groups? Problem 2.7 [ , Exercise 136 - p 24 ] With three digits 2; 0;5 : a ) Write three-digit numbers (three digits different) and divisible by b ) Write three-digit numbers (three digits different) and divisible by Problem 2.8 [ , Exercise 137 - p 24 ] With three digits number has three digits (three digits different) and divisible by 0;5;7 Write a odd Problem 2.9 [ , Exercise 138 - p 24 ] With four digits 0;1;4;5 Write three-digit numbers (three digits different) divisible by and divisible by CONCLUSION In this article, we present some applications of Decartess multiplication to find solutions to present the answer accordance with level of primary students Application of Decartess multiplication will help teacher guide students solve problems and help improve teaching effectiveness REFERENCES Nguyễn Đình Trí, Tạ Văn Đĩnh, Nguyễn Hồ Quỳnh (2009), Toán cao cấp tập 1, - Nxb Giáo dục Phạm Thành Công (2013), Violympic tốn 4, - Nxb Tổng hợp TP Hồ Chí Minh Đỗ Đình Hoan, Nguyễn Áng, Vũ Quốc Chung, Đỗ Tiến Đạt, Đỗ Trung Hiệu, Trần Diên Hiển, Đào Thái Lai, Phạm Thanh Tâm, Kiều Đức Thành, Lê Tiến Thành, Vũ Dương Thụy (2017), Toán 4, - Nxb Giáo dục Việt Nam Đỗ Đình Hoan, Nguyễn Áng, Đỗ Tiến Đạt, Đỗ Trung Hiệu, Phạm Thanh Tâm (2017), Bài tập Toán 4, - Nxb Giáo dục Việt Nam TÍCH DECARTESS VÀ ỨNG DỤNG TRONG DẠY TỐN TIỂU HỌC Tóm tắ tắt: Trong báo này, chúng tơi trình bày số ứng dụng tích Decartess dạy tốn giáo viên bậc Tiểu học Từ khóa: Tích Decartess, ứng dụng tích Decartess  ... defined according to the above formula A×B = A × B = 2× = 2.2 Some applications of Decartess multiplication in primary mathematics For primary education, many problems are easily understood and. .. by and divisible by CONCLUSION In this article, we present some applications of Decartess multiplication to find solutions to present the answer accordance with level of primary students Application. .. three free points in six points A, B,C , D, E , F not in line How many straight lines are connected from the vertices of the triangle ABC to the vertices of the triangle DEF ? Teacher paints two