Introduction: The Fascinating Journey of Adding and Subtracting

Relevance of the Topic

Basic operations in Mathematics, addition and subtraction form the basis for numerous more complex mathematical journeys. Addition is the most basic of all addition concepts and is present in many fundamental aspects of everyday life, from counting items in the supermarket to time spent on activities. Subtraction, on the other hand, is an extension of addition, applied to effectively determine remaining quantities or which quantities have been removed from a total sum.

Contextualization

In the vast universe of Mathematics, addition and subtraction are the first challenges we encounter after learning basic numbers. They are vital parts for understanding numerical structures. In fact, they are so intrinsically linked to the fabric of numerical reasoning that we unconsciously use them in many daily life situations. In terms of curriculum, addition and subtraction operations are the gateway to more advanced concepts such as multiplication, division, and algebra. Without a solid understanding of these operations, the comprehension of future topics is compromised. Therefore, now is the perfect time to embark on this journey through addition and subtraction, establishing a solid foundation for your future mastery in mathematics!

Theoretical Development

Components

• Relevant Terms

  • Operand: The terms in a mathematical expression to which an operation is applied. In addition, they are the numbers we are actually adding. In subtraction, the first number is called the 'minuend', the second is called the 'subtrahend', and the result is called the 'difference'.

  • Sum (Addition): Basic mathematical operation of combining two or more quantities to form a total quantity. Represented by the symbol '+'.

  • Subtraction: Mathematical operation of taking one quantity away from another. Represented by the symbol '-'.

  • Integer: Numbers that do not have fractions or decimal parts. It can be positive, negative, or zero. At this stage of mathematics, we will mainly focus on integers in our examples of addition and subtraction.

  • Result (Sum/Subtraction): The answer to an addition or a subtraction.

• Key Processes

  • Commutativity of Addition: The property that allows us to change the order of the numbers we are adding without changing the sum result. For example, 2+3 is equal to 3+2.

  • Regrouping (Exchange): This is a technique used in subtraction when the subtrahend is larger than the minuend. The number being subtracted is broken down into its place values to be adjusted in the minuend. For example, in 23-18, we can take a '1' from the tens (2) and move it to the ones (3), making the ones 13. Then we subtract 8 from 13 to get 5.

Key Terms

• Operand: Number used in an addition or subtraction operation. In addition, all numbers are operands; in subtraction, minuend and subtrahend are operands.

• Sum: Result of an addition. Use the symbol '+' to express addition.

• Subtrahend: The number to be subtracted from the minuend.

• Minuend: The number from which another number (the subtrahend) will be subtracted. The result of a subtraction is called the difference.

• Difference: Result of a subtraction. Use the symbol '-' to express subtraction.

• Exchange: In subtractions, it is the process of moving a value from one place to the place to the left. This is necessary when the subtrahend is larger than the minuend.

Examples and Cases

• Addition Example: Consider the expression '5 + 3'. Here, 5 and 3 are the operands, and the operation is expressed by addition. The result, or sum, is 8.

• Subtraction Example: In the expression '9 - 4', we have 9 as the minuend and 4 as the subtrahend. The operation being performed here is subtraction, and the result, or difference, is 5.

• Commutativity of Addition Example: If we have the expression '3 + 2', we can commute the numbers and still get the same result: '2 + 3', which is also equal to 5.

• Regrouping (Exchange) Example: Consider the subtraction '43 - 27'. Here, we need to exchange a unit from the tens to the ones, making the minuend '33'. Now, subtracting 27 from 33, the difference is '6'.

Detailed Summary

Key Points

• Operand: Term used in arithmetic operations. In additions and subtractions, all involved numbers are operands.

• Sum (Addition): Fundamental operation to combine quantities and obtain the total. Expressed by the symbol '+'.

• Subtraction: Operation to subtract a quantity from another. Represented by '-'. Unlike addition, subtraction is not commutative, meaning the order of terms alters the result.

• Integer: Without fractions or decimal parts. Can be positive, negative, or zero.

• Result (Sum/Subtraction): The answer to an addition is the 'sum', in subtraction is the 'difference'.

• Commutativity of Addition: Characteristic of addition that allows the reordering of terms without altering the total sum.

• Regrouping (or Exchange): Technique used in subtraction when the subtrahend is larger than the minuend. Allows for exchanges in the units to facilitate subtraction.

Conclusions

• Mastery of Fundamental Operations: Addition and subtraction are the cornerstones of mathematical reasoning. Mastering these operations is crucial for understanding more advanced concepts such as multiplication, division, and algebra.

• Manipulation of Integers: Through addition and subtraction operations, we strengthen our understanding and skills with integers, recognizing their properties and applying techniques for effective manipulation.

• Development of Strategies: Practice in addition and subtraction helps develop mathematical strategies, such as the use of exchange in more complex subtractions.

Suggested Exercises

1. What is the difference between 17 and 8?
2. If João had 12 apples and ate 7, how many does he still have?
3. The sum of the numbers 9 and 3 is equal to which number?
4. If Maria had 14 chocolates and gave 5 to her brother, how many chocolates does she have now?