B2.1 Use the properties of operations, and the relationships between operations, to solve problems involving whole numbers and decimal numbers, including those requiring more than one operation, and check calculations.
Skill: Using the Properties of Operations
Understanding the properties of operations and the relationships between them allows for more flexible use.
A good sense of operations is based on a good knowledge of the relationships between numbers and between operations. Properties of operations are characteristics that are specific to operations, regardless of the numbers involved; as an example, addition is commutative since \(3\; + \;5\; = \;5\; + \;3\); \(4\; + \;7\; = \;7\; + \;4\); \(1.22\; + \;3.51\; = \;3.51\; + \;1.22\).
Understanding the properties of the operations allows one to develop efficient computational strategies; for example, since multiplication is distributive over addition, one can compute \(5\; \times \;12\) by performing \(\left( {5\; \times \;10} \right)\; + \;\left( {5\; \times \;2} \right)\).
In the primary grades, students have been able to approach some of these properties intuitively. Junior students need to understand the properties of the operations presented below and learn to use them in problem-solving situations.
Commutative Property
Addition and multiplication are commutative. For example, the commutative property of addition can be demonstrated as follows: there are 44 apples in one basket and 32 in another. The total number of apples will be the same whether we add the apples in the first basket to the apples in the second basket or we do the opposite. Thus, \(44 + \;32 = \;76\) and \(32 + \;44 = \;76\). We recognize that if the order of the terms of an addition is switched, the result remains the same.
We can also demonstrate the commutative property of multiplication. For example, \(8\; \times \;3\;{\rm{et}}\;3\; \times\;8\).
And \(8\; \times \;3.5\;{\rm{and}}\;3.5\; \times \;8\)
The two previous arrays represent the same total quantity, organized in two different ways. As a result, they illustrate two different situations. Thus, \(8\; \times \;3\) represents 8 rows of 3 objects and \(8\; \times \;3.5\) represents 8 rows of 3.5 objects, while \(3\ ; \times \;8\) represents 3 rows of 8 objects and \(3.5\; \times \;8\) represents 3.5 rows of 8 objects. It is important that students recognize the different representations.
An example from everyday life can also be used. For example, the teacher invites 3 students who have exactly one sibling to come and represent the number of children in their family on the board. The total number of children, 6, is represented by children. (Figure 1).
Then, teachers do the same with two children who have exactly two siblings. The total number of children, 6, is represented by children. (Figure 2).
Figure 1
Figure 2
The 2 number sentences, \(3\; \times \;2\; = \;6\) and \(2\; \times \;3\; = \;6\), indicate the same result, even if the order of the factors is reversed. The students can then understand that 3 families of 2 children or 2 families of 3 children give a total of 6 children, without the situations being identical.
When we use the commutative property of multiplication, we are more interested in the answer than the situation.
For example, even if we are looking for the answer to \(12\; \times \;2\), we can choose to calculate \(2\; \times \;12\) if the result is easier to obtain, even if the 2 expressions do not represent the same situation.
When first learning multiplication, students often perceive multiplication as repeated addition. As they attempt to solve a variety of problems, they can use the commutative property of multiplication to develop a more efficient strategy for calculation.
For example, students using repeated addition would recognize that \(2\; \times \;12\;(12\; + \;12)\) is simpler and takes less time to represent and calculate than \( 12\; \times \;2\;(2\; + \;2\; + \;2\; + 2\; + \;2\; + \;2\; + 2\; + \;2 \; + \;2\; + 2\; + \;2\; + \;2)\).
An array is an excellent visual model for representing the commutative property of multiplication.
Associative Property
In the expression \(15\; + \;13\; + \;17\), it is possible to associate 13 and 17 to obtain \(15\; + \;(13\; + \;17) \), which gives \(15\; + \;30\), i.e. 45. We can also associate 15 and 13 to obtain \((15\; + \;13)\; + \;17\), which gives \(28\; + \;17\), or 45. The associative property also applies to decimal numbers, for example, in the expression \(5\; + \;0.75\; + \;0.25\), it is possible to associate 0.75 and 0.25 to obtain \(5\; + \;(0.75\; + \;0.25)\), which gives \(5\; + \;1\), i.e. 6. We can also associate 5 and 0.75 to obtain \((5\; + \;0.75)\; + \;0.25\), which gives \(5.75\; + \;0.25\), or 6.
The associative property of multiplication [for example, \(3\; \times \;2\; \times \;5\; = \;3\; \times \;(2\; \times \;5)\ ;{\rm{ou}}\;3\; \times \;2\; \times \;5\; = \;(3\; \times \;2)\; \times \;5)\) ] is not easy to understand. Certainly, it can be seen by checking the results of the multiplication, but this does not constitute an understanding. To understand, one can use an extension of the array model with the help of cubes.
In figure 1, we see that there are 2 rows of 5 cubes, that is, \(2\; \times \;5\) cubes. In figure 2, we see 3 tiers containing each one \(2\; \times \;5\) cubes. The figure thus represents\(3\; \times \;(2\; \times \;5)\) cubes.
One can also consider figure 3, which illustrates 3 tiers of 2 cubes, that is, \(3\; \times \;2\) cubes. Figure 4 can then represent these cubes that appear 5 times, \((3\; \times \;2)\) cubes 5 times, that is to say \((3\; \times \;2)\; \times \;5\).
So we see that \(3\; \times \;(2\; \times \;5)\) and \((3\; \times \;2)\; \times \;5\) represent the same amount of cubes (even though each expression represents a different point of view) and that each gives the same product as \(3\; \times \;2\; \times \;5\).
Faced with a numerical expression such as \(3\; \times \;(2\; \times \;5)\; = \;?\), some students sometimes try to apply the distributive property in order to calculate \(\left( {3\; \times \;2} \right)\; \times \;\left( {3\; \times \;5} \right)\), which has the effect that \(2\ ; \times \;5\) is multiplied by 9, rather than by 3. In such a case, remember that the distributive property says that multiplication is distributed over addition or subtraction and explain the situation using a concrete or semi-concrete model.
The associative property does not change the order of numbers in a numerical expression. However, the associative property and the commutative property can be combined to facilitate the evaluation of a numerical expression.
For example, to determine the value of the expression \(2\; \times \;3\; \times \;5\), one can first use the commutative property to obtain 2 x 5 x 3 which allows us to associate the 2 and the 5, then determine that of the expression \((2\; \times \;5 )\; \times \;3\). Indeed, it is usually easier to calculate \(10\; \times \;3\) than to calculate \(6\; \times \;5\). Similarly, to determine the value of the expression \(19\; + \;27\; + \;11\), one can determine that of \((19\; + \;11)\; + \; 27\), because the 9 and the 1 are complementary and give 1 ten, which makes it possible to obtain the answer mentally, that is \(30\; + \;27\; = \;57\). It is by exposing students to a large number of activities that teachings lead them to understand and use the different properties effectively.
Decomposing a number into a product of factors, along with the associative property, can also be useful. For example, the number 24 can be represented by \(24\; \times \;1,\;12\; \times \;2,\;8\; \times \;3,\;6\; \times \;4\) or even by \(2\; \times \;4\; \times \;3\) or \(2\; \times \;2\; \times \;2\; \times \ ;3\). To determine the value of a numeric expression such as \(24\; \times \;5\), students with good number and operation sense may choose to transform 24 into \(12\; \times \;2\) and use the operations properties as follows:
Distributive Property
Multiplication is distributive over addition. For example, we can multiply \(3\; \times \;\left( {5\; + \;6} \right)\) and arrive at the same result as if we had performed \(\left( {3\ ; \times \;5} \right)\; + \;\left( {3\; \times \;6} \right)\). Multiplication is also distributive over subtraction. For example, we can multiply \(3\; \times \;\left( {20\; - \;2} \right)\) by doing \(\left( {3\; \times \;20} \right)\; - \;\left( {3\; \times \;2} \right)\).
The following example illustrates how the distributive property can be used to calculate \(6\; \times \;8\). In one case, we decompose the factor 8 to obtain \(5\; + \;3\). We then have \(6\; \times \;\left( {5\; + \;3} \right)\; = \left( {6\; \times \;5} \right)\; + \ ;\left( {6\; \times \;3} \right)\). In the other case, we decompose the factor 6 to obtain \(3\; + \;3\). We then have \(\left( {3\; + \;3} \right)\; \times \;8\; = \left( {3\; \times \;8} \right)\; + \;\left ( {3\; \times \;8} \right)\).
Here is an example of using the distributive property when making a calculation involving decimal numbers, for example \(6.5\; \times \;4\).
In one case, we decompose the factor 4 to obtain \(2 + 2\). We then have \(6.5 \times (2 + 2) = (6.5 \times 2) + (6.5 \times 2)\). In the other case, we decompose the factor 6.5 to obtain \(6 + 0.5\). We then have \((6 + 0.5) \times 4 = (6 \times 4) + (0.5 \times 4)\).
There is an important connection between the distributive property and the standard multiplication algorithm. For example, to calculate \(3\; \times \;15\), the 15 is broken down to get \(\left( {10\; + \;5} \right)\):
To calculate \(13\; \times \;24\), both factors are decomposed:
Only multiplication is distributive over addition or subtraction. One could recognize that division is partially distributive. For example, to calculate \(32\; \div \;8\), it is possible to decompose the dividend 32 to obtain \(16\; + \;16\). We then have \((16\; + \;16)\; \div \;8\) and the division by 8 is distributed over the addition. We get \((16\; \div \;8)\; + \;(16\; \div \;8)\; = \;2\; + \;2\), that is, 4. However if the divisor is decomposed, the distributive property does not work. For example, \(32\; \div \;8\; \ne \;\left( {32\; \div \;4} \right)\; + \;\left( {32\; \div \;4} \right)\). This is the reason why distributivity is not a property of division.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 102-105.
Properties of operations can be used to verify an answer. For example, \(4\; \times \;9\) can first be determined using the distributive property, \(2\; \times \;9\; + \;2\; \times \;9 \) or by decomposing \(4\; \times \;9\) into \(2\; \times \;2\; \times \;9\) and using the associative property to get, \(2\; \times \;(2\; \times \;9)\).
source: adapted from Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Skill: Using the Relationships Between Addition, Subtraction, Multiplication and Division
Understanding the relationships between operations allows them to be used more flexibly.
The more opportunities students have to handle operations, the more they can notice and understand the relationships between them. Students may even use addition or subtraction strategies to solve multiplication and division.
In the primary division, students have made connections between operations through various activities. For example, they know that addition and subtraction are inverse operations and that addition is commutative. Over time, they develop their number sense and operations sense and gradually use them before performing operations. This practice, while often informal and mental, is still essential to understanding the relationships between numbers and between operations.
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 97.
Addition and Subtraction
Addition and subtraction are inverse operations. Any subtraction situation can be considered as an addition situation (for example, 154 - 48 = ? is equivalent to 48 + ? = 154) and vice versa.
However, students often have difficulty solving equations such as 17 + Δ= 31 when they are beginning learning. Many teachers then encourage their students to use the inverse operation, subtraction. However, this can be a learning trick unless students understand why subtraction is a possible strategy. They must first grasp the relationship of the whole to its parts and the meaning of "difference".
For example, a number can be represented as follows:
This way of representing the relationship between a number and its parts allows us to see that subtraction is the inverse operation of addition.
Since \(17\; + \;14\; = \;31\;{\rm{and}}\;14\; + \;17\; = \;31\), then \(31\; - \;17\; = \;14\;{\rm{and}}\;31\; - \;14\; = \;17\). Additionally, students can see why addition is commutative \((14\; + \;17\; = \;17\; + \;14\; = \;31)\) and why subtraction is not commutative \(\left( {31\; - \;17\; \ne \;17\; - \;31} \right)\). Those who have acquired good number sense and are able to break down and regroup numbers can use their knowledge to more effectively solve equations such as \(17\; + \;\Delta \; = \;31\) understanding that we are looking for the difference between 17 and 31.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 97-98.
It is also possible to represent decimal numbers as follows:
We can also use addition to perform a subtraction; this strategy is called "adding to subtract". For example, to solve \(31\; - \;17\; = \;?\), the student can add from 17 onward to 31, or \(17\; + \;?\; = \;31\).
This inverse relationship can be used to perform and verify calculations.
Multiplication and Division
Multiplication and division are also inverse operations. They can also be related to the concept of the whole and its parts. In multiplication, we group the equal groups, while in division, we partition a whole into equal groups. From this relationship between multiplication and division, students can use multiplication facts to perform division. Students often misunderstand the inverse operation relationship between multiplication and division (for example, recognizing that \(\rule{1em}{0.5pt}\; \times 6\; = \;234\) can be thought of as \(234\; \div \;6\; = \;\rule{1.5em}{0.5pt}\)) even after performing divisions and checking their calculations. It is therefore essential to regularly review the meaning of each of the operations in the context of problem solving.
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 98.
Multiplication represents the result of gathering equal groups of objects together while division represents the division of objects into equal groups. To understand multiplication and division, you need to recognize the 3 types of quantities involved: the total quantity (for example, 8 flowers), the number of equal groups (for example, 4 pots), and the size of each group (for example, 2 flowers per pot).
In the problems presented to students, division is too often associated with only one sense, the partitive sense of sharing. The grouping sense is usually neglected. Division has a partitive or sharing meaning when the total quantity and the number of groups are known (for example, 3 students want to share 15 apples equally and we look for the number of apples each will receive).
Division has a grouping meaning (quotative division or “measurement division”) when the total quantity and the number of items in each group (group size) are known (for example, we have 15 apples and we want to put them in bags, 3 apples per bag; we look for the number of bags we need).
It is essential that both types of situations be addressed, as they are the basis for the integration of other mathematical concepts. It is not necessary for students to know the names of the situation types, but it is essential that they have the opportunity to solve a variety of problem types, while using a variety of strategies.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 84-86.
Multiplication and Addition
The connection between multiplication and addition is often the starting point for introducing the concept of multiplication.
In the early stages of learning multiplication, students recognize when a situation presents the same quantity many times, and they use equal groups to represent the situation and repeated addition to get an answer. As students progress, it is important that they see the concept of multiplication rather than addition and learn other representations. The array, a row and column arrangement, is a powerful model for learning multiplication and provides a different perspective on multiplication.
Through solving a variety of problems and discussing strategies, students come to establish and grasp the connection between the word "times" and the "×" sign, a crucial step in developing an understanding of multiplication. Once their sense of multiplication is well established, they use the multiplication operation more regularly to obtain answers.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 85-86.
Division and Subtraction
Similarly, division can be associated with repeated subtraction, but not in the same way. The product of a multiplication is equal to the sum resulting from the repeated addition, while the quotient of a division is equal to the number of repeated subtractions (for example, to calculate \(20\; \div \;5 \), we do \(20\; - \;5\; = \;15,\;15\; - \;5\; = \;10,\;10\; - \;5\; = \; 5,\;5\; - \;5\; = \;0\); we subtracted 4 times; therefore \(20\; \div \;5\; = \;4\)).
It takes time for students to learn these relationships. To achieve this, teachers can use hands-on activities, problem solving, and mathematical discussions that focus on the connections between operations.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 98-99.
Effect of Operations
Each operation has an effect on the quantities involved. Depending on the operation, certain quantities increase or decrease. They can increase or decrease by a lot or a little.
Tracking the effect of operations on numbers allows students to make connections between operations and to anticipate the outcome of an operation. For example, if we subtract 8 from 160, we will notice little effect because the difference between 160 and 152 is relatively small. However, if we divide 160 by 8, the effect is large, because the resulting quotient, 20, is much smaller than 160. We can also compare the effect produced by addition to that produced by multiplication.
Compared to multiplication, addition increases a number by a small amount. For example, when the number 160 is multiplied by 8, you get 1280, whereas if you add 8 to it, you get only 168. People with good operations sense recognize the effect of operations on whole numbers, but students who are still learning are often impressed by the effect of, for example, multiplication.
One caveat is that care must be taken when generalizing, as operations on decimal numbers or fractions may have different effects than those on whole numbers.
In some cases, the effect can even be the opposite. Indeed, if we multiply a whole number by another whole number, the product is greater than the 2 factors (for example, if we multiply 3 by 6, the product 18 is greater than 6 and 3), whereas if we multiply a proper fraction by a whole number, the product is smaller than one of the 2 factors (for example, if we multiply \(\frac{1}{2}\) by 6, the product one third as large as 6).
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 90-91.
Explore these ideas with students, having them uncover the effect of the operations. In pairs, suggest some operations and then have them observe the effects of the operations on the numbers.
\(\begin{array}{l}160\; + \;8\\160\; - \;8\\160\; \times \;8\\160\; \div \;8\end{array}\)
Students can use models, such as the number line to show operations.
Facilitate a mathematical exchange to bring out the effects of the different operations.
Knowledge: Properties of Operations
A property of an operation is a characteristic that are specific to the operation, regardless of the numbers involved.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 102.
The properties of the operations are:
- Commutative Property: \(\left( {3\; + \;5\; = \;5\; + \;3} \right)\);
- Associative Property: \(\left( {2\; + \;9)\; + \;11\; = \;2\; + \;(9\; + \;11} \right)\);
- Distributive Property: \(8\; \times \;7\; = \;\left( {8\; \times \;5} \right)\; + \;\left( {8\; \times \;2} \right)\);
- Identity Principle of Multiplication: 25 x 1 = 25
- Zero Property of Addition and Subtraction: 13 + 0 = 13; 13 - 0 = 13; Zero Property of Multiplication: 8 x 0 = 0
Knowledge: Commutative Property
An operation is commutative if its result remains unchanged when the order of its terms or factors is reversed. Addition and multiplication are commutative. For example, \(27\; + \;63\; = \;63\; + \;27\) and \(8\; \times \;6\; = \;6\; \times \;8\).
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 102.
Knowledge: Associative Property
The associative property of addition and multiplication allows you to combine the terms or factors of an expression in different ways without changing the value of the expression. For example, \(3\; \times \;2\; \times \;5\; = \;3\; \times \;\left( {2\; \times \;5} \right)\).
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 105.
Knowledge: Distributive Property
Multiplication is distributive over addition and subtraction.
The distributive property allows you to perform an operation on a sum or a difference of terms and obtain the same result as if the operation had been performed on each term. For example,
- we can multiply \(3\; \times \;\left( {5\; + \;6} \right)\) and arrive at the same result as if we had performed \(\left( {3\; \times \;5} \right)\; + \;\left( {3\; \times \;6} \right)\);
- we can multiply \(3\; \times \;\left( {20\; - \;2} \right)\) by doing \(\left( {3\; \times \;20} \right)\ ;-\;\left( {3\; \times \;2} \right)\).
Source: based on Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 104.
Knowledge : Zero Property of Multiplication
When multiplying any number by 0, the result is always 0 (for example, \(684\; \times \;0\; = \;0;\;16.67\; \times \;0\; = \;0\)).
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 107.
Knowledge: Identity Principle of Addition and Identity Principle of Multiplication
The Identity Principle of Addition states that when adding 0 to any amount, the amount stays the same (for example, \(287\; + \;0\; = \;287\), \(4,5\; + \;0\; = \;4,5 \). The Identity Principle of Multiplication states that when multiplying an amount by 1 or dividing an amount by 1, the amount stays the same (for example, \(133 \times \;1 = 133 \); \(432.1\; \times 1 = 432.1 \).
There is no identity principle for either subtraction or division In subtraction, However, there is a Zero Principle for Subtraction which states that 0 subtracted from any number results in the same number, that is, n – 0 = n. Note: The number 0 has no effect when it is the 2nd term \(3\; - \;0\; = \;3\)), but this is not the case if it appears as the first term \(0\; - \;3\; \ne \;3\)). The specific wording of the Principle, along with the knowledge that subtraction is not a commutative operation, should be emphasized with students. Similarly, The Identity Principle states that when dividing an amount by 1, the amount stays the same (for example, 3 ÷ 1 = 3). The number 1 has no effect when it is the divisor. Again, we must make clear to students that since division is not a commutative operation, we cannot then assume that the number 1 has no effect when it is the dividend (for example, 1; div = 3; none = 3).
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 107.
Knowledge: Relationships Between Addition, Subtraction, Multiplication and Division
The fundamental operations of addition, subtraction, multiplication and division are closely related despite their apparent differences.
- Addition and subtraction are inverse operations.
- Multiplication and division are inverse operations.
- Multiplication can be associated with repeated addition.
- Division can be associated with repeated subtraction.