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.

Part I SCIENTIFIC PAPERS

DECARTESS MULTIPLICATION AND APPLICATION

DECARTESS MULTIPLICATION AND APPLICATION

IN PRIMARY MATHEMATICS

IN PRIMARY MATHEMATICS

Nguyen Van Hao 1 , Dao Thi To Uyen 2 , Nguyen Thi Thanh Ha 3

1

Department of Mathematics, Hanoi Pedagogical University 2

2

Department of Primary Education, Hanoi Pedagogical University 2

3

Faculty of Basic Science, Viet Tri Industrial University

Abstract:

Abstract: In this article, we would like to present some applications of Decartess

multiplication in teaching maths to teachers at primary schools

Keywords:

Keywords: Decartess multiplication, application of Decartess multiplication

Email: nguyenvanhaodhsphn2@gmail.com

Received 11 October 2018

Accepted for publication 15 December 2018

1 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 do 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

2 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].

2.1.1 Concept of Decartess

Decartess multiplication of two sets A and Bdenoted 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

{1,5}; {2, 4, 7}

A = B =

In this case, it is easy to see that the decartess multiplication of these two sets is defined by the elements as follows

{ (1,2);(1, 4);(1, 7);(5, 2);(5, 4);(5, 7) }

A × B =

2.1.2 Decartess multiplication of nthe set

Decartess multiplication of nthe set A A1, 2, , An is a set contains all ordered elements of the form ( , , , a a1 2 an) inside

1 1, 2 2, , n n.

a ∈ A a ∈ A a ∈ A

Or write in the form of mathematical symbol

1 2 n ( , , ,1 2 n) : i i, 1,

A × A × × A = a a a a ∈ A i = n

In particular, when the set A1 = A2 = = An = A is denoted by

A × A × × A = A

2.1.3 Force of Decartess multiplication

Force of Decartess multiplication is calculated by the force multiplication of each set

A × B = A × B

A × A × × A = A × A × × A

Example 1.2. Give two sets

{ } 1, 2 ;

A = with A = 2 andB = { , , }; a b c with B = 3.

Decartess multiplication of these two sets is defined as follows

{ (1, ),(1, ),(1, ),(2, ),(2, ),(2, ) }

A × B = a b c a b c

We see that the force of A × Bis defined according to the above formula

2 3 6

A × B = A × B = × =

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 5

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 0 So, digits of tens are sets

{ 1, 2, 3, 4, 5, 6, 7, 8, 9 ; }

Based on the sign of the natural numbers divisible by 5 have last digit 0 or 5

teachers can guide the students to find the units So, the digit of the units belong sets

{0, 5};

We can see that two-digit numbers divisible by 5 are elements of Decartess multiplication of sets X and sets Y above So, teacher can know two-digit numbers divisible by 5 are

9 2 18

X × Y = × = (numbers) Understanding the concept of Decartess multiplication, teachers can guide students knows two-digit numbers divisible by5 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 2 and 5 (Pham Thanh Cong,

Detailed explanation guide Violympic Math 4, 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 0 So, digits of tens are sets

{ 1, 2, 3, 4, 5, 6, 7, 8, 9 ; }

This numbers can divisible by 2, the last digit is one of the digits { 0,2, 4, 6, 8 } This numbers can divisible by 5, the last digit is one of the digits { } 0, 5 Thus, two-digit numbers divisible by 2 and 5, the last digit is 0

{ } 0 ;

So, two-digit numbers divisible by 2 and 5 is force of Decartess multiplication

X × Y

9 1 9

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 DEFand 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 × Yand all straight line is

3 3 9

X × Y = X × Y = × = (line segments)

Problem 2.4. Give four digits 3, 4, 5, 6 How many even numbers are three digits from these four 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, 6 }, the digit of tens belonging

{ 3, 4, 5, 6 }

Y = 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

4 4 2 32

X × × Y Z = X × Y × Z = × × = (numbers) For this orientation, we can show some problems as follows

Problem 2.5. Two teamsA 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. [4, Exercise 136- p 24 ] With three digits 2; 0;5:

)

a Write three-digit numbers (three digits different) and divisible by 2

)

b Write three-digit numbers (three digits different) and divisible by 5

Problem 2.8. [4, Exercise 137- p 24] With three digits 0;5;7 Write a odd number has three digits (three digits different) and divisible by 5

Problem 2.9. [4, Exercise 138- p.24] With four digits 0;1;4;5 Write three-digit numbers (three digits different) divisible by 5 and divisible by 9

3 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

1 Nguyễn Đình Trí, Tạ Văn Đĩnh, Nguyễn Hồ Quỳnh (2009), Toán cao cấp tập 1, - Nxb Giáo dục

2 Phạm Thành Công (2013), Violympic toán 4, - Nxb Tổng hợp TP Hồ Chí Minh

3 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

4 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 TOÁN TIỂU HỌC

Tóm t ắắắắt: t: t: Trong bài báo này, chúng tôi trình bày m ột số ứng dụng của tích Decartess trong dạy toán đối với giáo viên bậc Tiểu học

T ừ ừừ ừ khóa: khóa: khóa: Tích Decartess, ứng dụng của tích Decartess