Catanduanes State UniversityCOLLEGE OF EDUCATION

Virac, Catanduanes

MATHEMATICAL INVESTIGATION

Submitted by:

MARK RAYMOND T. DOMINGOBSE-MATH 3D

Submitted to:

RUDY P. TABLIZOAssistant Professor II

MATHEMATICAL INVESTIGATION

DETERMINING THE RULE THAT RELATES ARITHMETIC

OPERATIONS ON FOUR CONSECUTIVE EVEN AND

ODD INTEGERS

INTRODUCTIONMathematics is a way of describing

relationships between numbers and other measurable quantities. Mathematics can express simple equations as well as interactions among the smallest particles and the farthest objects in the known universe. Mathematics allows scientists to communicate ideas using universally accepted terminology. Making it truly been the language of science.

We benefit from the results of mathematical research every day. The fiber-optic network carrying our telephone conversations was designed with the help of mathematics. Our computers are the result of millions of hours of mathematical analysis. Weather prediction, the design of fuel-efficient automobiles and airplanes, traffic control, and medical imaging all depend upon mathematical analysis.

For the most part, mathematics remains behind the scenes. We use the end results without really thinking about the complexity underlying the technology in our lives. But the phenomenal advances in technology over the last 100 years parallel the rise of mathematics as an independent scientific discipline.

Thus, without mathematics to describe physical phenomena, we might be living in a world with beautiful art, literature, and philosophy, but no technology. Even the medical advances of the last 50 years might not have occurred. Science and technology, in their turn, have provided many of the problems that motivated progress in mathematics. Such problems include the behavior of weather systems, the motion of subatomic particles, and the creation of speedier and smaller computers that can perform multiple tasks simultaneously.

And to understand mathematics and make use of numbers wisely we make use of the two of its branches: arithmetic and algebra. 

Arithmetic, one of the oldest branches of mathematics, arises from the most fundamental of mathematical operations: counting. The arithmetic operations—addition, subtraction, multiplication, division, and placeholding—form the basis of the mathematics that we use regularly. In many countries arithmetic is the primary area of mathematical study during the first six years of school.

Although arithmetic itself is not an area of mathematics research, research on how best to teach arithmetic is crucial to the field of mathematics education. Models of learning and mastering the basics of arithmetic are often used in cognitive science—the study of the processes of acquiring, storing, and using knowledge. Cognitive sciences encompass a range of activities, including the design of computer-aided instructional systems and the study of artificial intelligence. Arithmetic and logic also form the basis for all computer software—the instructions that tell computers what to do.

Algebra is the branch of mathematics that uses symbols to represent arithmetic operations. One of the earliest mathematical concepts was to represent a number by a symbol and to represent rules for manipulating numbers in symbolic form as equations. 

Making use of arithmetic and algebra I was able to make my investigation with regards to determining the rules that relates arithmetic operation into four consecutive even and odd numbers.

DISCUSSION

The following are the general observations about the rules of arithmetic operations regarding even and odd numbers.

Even and Odd NumbersEven numbers are numbers that can be divided evenly by 2. Even numbers can be shown as a set like this:

{ … -4, -2, 0, 2, 4, … }

Odd numbers are numbers that cannot be divided evenly by 2. Odd numbers can be shown as a set like this:

{ … -5, -3, -1, 1, 3, 5, … }

Zero is considered an even number.

Is It Even or Odd?To tell whether a number is even or odd, look at the number in the ones place. That single number will tell you whether the entire number is odd or even.

An even number ends in 0, 2, 4, 6, or 8.An odd number ends in 1, 3, 5, 7, or 9.

Consider the number 3,842,917. It ends in 7, an odd number. Therefore, 3,842,917 is an odd number. Likewise, 8,322 is an even number because it ends in 2.

Adding Even and Odd Numbers

even + even = even4 + 2 = 6

even + odd = odd4 + 3 = 7

odd + odd = even5 + 3 = 8

Subtracting Even and Odd Numbers

even - even = even4 - 2 = 2

even - odd = odd4 - 3 = 1

odd - odd = even5 - 3 = 2

Multiplying Even and Odd Numbers

even x even = even4 x 2 = 8

even x odd = even4 x 3 = 12

odd x odd = odd5 x 3 = 15

Division or the Fraction ProblemAs you can see, there are rules that tell

what happen when you add, subtract, or multiply even and odd numbers. In any of these operations, you will always get a particular kind of whole number.

But when you divide numbers, something tricky can happen—you might be left with a fraction. Fractions are not even numbers or odd numbers, because they are not whole numbers. They are only parts of numbers, and can be written in different ways.

For example, you can't say that the fraction 1/3 is odd because the denominator is an odd number. You could just as well write that same fraction as 2/6, in which the denominator is an even number.

The terms “even number” and “odd number” are only used for whole numbers and their opposites (additive inverses).

And now here is my observation about the rules of arithmetic operations regarding the four consecutive even and odd numbers.

FOUR CONSECUTIVE EVEN AND ODD NUMBERS

(a,b,c,d) (a,b,c,d) (a,b,c,d) (a,b,c,d)

(2,4,6,8) (10,12,14,16) (1,3,5,7) (9,11,13,15)

2+8=10 10+16=26 1+7=8 9+15=24

4+6=10 12+14=26 3+5=8 11+13=24

8-2=6 16-10=6 7-1=6 15-9=6

6-4=2 14-12=2 5-3=2 13-11=2

2x8=16 10x16=160 1x7=7 9x15=135

4x6=24 12x14=168 3x5=15 11x13=143

In addition of consecutive odd and even numbers, a + d is always equal to b + c which means that the sum of the means is equal to the sum of the extremes.

In subtraction of consecutive odd and even numbers, d - a is thrice to c-b which means that the difference of the extremes is thrice the difference of the extremes.

In multiplication of consecutive odd and even numbers, b x c minus a x d is equal to 8 which means that the products of the means minus the products of the extremes is always 8.

In division of consecutive odd and even numbers, it is not possible since the quotient is a fraction. Thus fraction is not in the concept of odd and even numbers since they are not whole numbers.

CONCLUSIONGenerally speaking, even and odd

numbers are numbers which are whole numbers. Which make all the basic fundamental operations in real numbers possible except for division that makes the quotient a fraction and seems only part of the whole. Thus, making division serves as a limitation regarding the concepts of rules with even and odd numbers.

With the investigation, it came out that the rules that relates arithmetic operations on four consecutive even and odd numbers includes addition, subtraction and multiplication with regards to the relationships of its extremes and means.

THANK YOU