
Meaning
of operations Associative and Commutative
Laws Inverse operations  There are four basic operations in arithmetic: addition, subtraction, multiplication and division. It is important to understand the types of problems which each of these solves. Meaning is constructed through experience with materials and real life situations and through the language used to discuss what is happening. Only when an understanding of at least an initial meaning for an operation has been developed, and simple mental calculations can be carried out, should formal algorithms be introduced. A range of meanings for the operations is needed to illustrate arithmetic algorithms thoroughly
The table above describes the most common meanings for each of the operations. Note, however, that addition situations will give rise to problems to be solved by addition or by subtraction and that subtraction situations give rise to problems which are solved by addition. For example, consider the situation where I have some lollies and then I eat 3. This is a take away situation. However, if I know that there are 6 left, I can find out how many I had first by adding, not by subtracting. The same crossover holds for multiplication and division. Associative and Commutative Laws By appreciating the meaning of the operations, we can deduce their basic properties and then the basic facts. Knowledge of the basic facts of the four operations and place value is then the foundation for the algorithms. There are many basic properties which follow from two important principles: the commutative and associative laws for addition and multiplication. Commutative law for addition a + b = b + a for all real numbers a and b Reason: when two sets are combined, the total number of objects does not depend on which of the sets was placed down first. Associative law for addition (a + b) + c = a + (b + c) for all real numbers a, b and c Reason: when three sets are combined, the total number of objects does not depend on whether you combine the first two sets and then add the third, or start with the first and add the second and third combined. Commutative law for multiplication a x b = b x a for all real numbers a and b Reason: This can best be shown using arrays. It is explained in the Teaching Connections section. Associative law for multiplication (a x b) x c = a x (b x c) for all real numbers a, b and c Reason: This can best be shown using a three dimensional array. The number of objects in the array does not depend on whether the number in a horizontal crosssection is calculated first and multiplied by the number of horizontal layers, or if the number of vertical slices is multiplied by the number of objects in a vertical crosssection. Uses of the laws for additionThe commutative law means we only have to learn half the basic addition facts because we know that order does not matter for adding, e.g. 8 + 4 = 4 + 8. The associative law allows us to group numbers in convenient ways to add them, e.g. to add 8 and 5, we can split the 5 into 2 + 3 and use the known combination of 2 + 8 : 8 + 5 = 8 + (2 + 3) = (2 + 8) + 3 = 10 + 3 = 13. In the example below we have used these laws to reorder and regroup numbers to quickly add 38 and 14. The steps written out seem long, but they are carried out very fast. Example 1: 38 + 14 = 52
Uses of the laws for multiplication The commutative law means we only have to learn half the basic multiplication facts because we know that order does not matter , e.g. 8 x 4 = 4 x 8. The associative law allows us to group numbers in convenient ways to multiply them, e.g. to multiply 5 and 16, we can split the 16 into 8 x 2 and use the known combination of 5 x 8 and the easy calculation of 40 x 2: 5 x 16 = 5 x (8 x 2) = (5 x 8) x 2 = 40 x 2 = 80. In the example below we have used these laws to reorder and regroup numbers to quickly multiply 75 by 4. The steps written out seem long, but they are carried out very fast. Example 2: 75 x 4 = 300
Subtraction is the inverse operation of addition and division is the inverse operation of multiplication. Because it is the inverse operation, subtracting a number "undoes" adding it e.g. 8 + 27 27 = 8 and 8  27 + 27 = 8. Because it is the inverse operation, dividing by a number "undoes" multiplying by it e.g. 8 x 27 ÷ 27 = 8 and 8 ÷ 27 x 27 = 8. The properties of subtraction and division (which are NOT commutative nor associative) follow from their inverse status. Children learn basic subtraction facts alongside basic addition facts and basic division facts alongside multiplication facts. Also note that addition is the inverse operation of subtraction and multiplication is the inverse operation for division.
Distributive law Click here to read about the distributive law. The order of operations is fundamental to working with numbers. A comprehensive explanation of the correct order of operations, including common misconceptions and appropriate rules, can be found by clicking here, Order of Operations.
Estimation and mental computation The skills of estimation and mental computation are built on the understanding of the basic concepts for each operation and use knowledge of basic facts, place value and rounding. Estimation and mental computation are needed to provide both approximate answers in many real life situations and to check reasonableness of written and calculator answers. Rounding numbers so that they are easy to deal with is a key skill for quick estimates. Example: How can I round the following numbers to the nearest ten?
In calculations, it is important to appreciate the effect of rounding up or down. For example, when I estimate 29 x 37 by 30 x 40 = 1200, the answer is too big. If I estimate 29 divided by 37 by 30 divided by 40 = 0.75, I do not immediately know whether that answer is too big or too small.  Set A Set B  Set C  Set D 
To view the quiz answers, click here. 

© University of Melbourne 2003 