B2.1 use the properties of operations, and the relationships between operations, to solve problems involving whole numbers, decimal numbers, fractions, ratios, rates, and whole number percents, including those requiring multiple steps or multiple operations
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 relies on a good knowledge of the relationships between numbers and between
operations.
Understanding the properties of the operations allows one to develop efficient computational strategies; for example, since multiplication is distributive, one can compute \(5 \times 12\) by performing \((5 \times 10 ) + (5 \times 2) \).
In the primary grades, students were able to approach some of these properties intuitively. Students in the junior grades need to understand the properties of the operations presented below and learn to use them in problem solving situations.
Commutativite property
An operation is commutative if its result does not change when the order of its terms is reversed. 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 whether we do the opposite. Thus, \(44 + \;32\) is equal to \(32 + \;44\). We recognize that if the terms of an addition are switched, the result remains the same.
We can also demonstrate the commutative property of multiplication.
Examples
The two previous arrangements represent the same total quantity, organized in two different ways. Therefore, they
illustrate two different situations. For example, \(8 \times 3\) is 8 rows of 3 objects, while \(3 \times 8\) is 3
rows of 8 objects. It is important that students recognize the different representations.
You can also use an everyday example. For example, the teaching staff invites three students who have exactly one
brother or sister to represent the children of their family on the board. The total number of children is represented
by \(3\; \times \;2\) children, for a total of 6 children (Figure 1).
Then, teachers do the same with two children who have exactly two siblings. The total number of children is represented by 2 children, for a total of 6 children (Figure 2).
Figure 1
Figure 2
The two 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 pupils 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 using the commutative property of multiplication, we are more interested in the answer, regardless of the
situation. For example, even if we are looking for \(12 \times \;2\), we may choose to calculate \(2 \times 12\) if
the result is easier to obtain, even if the two 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 may use the commutative property of multiplication to develop a more efficient strategy for calculation. For example, students who use repeated addition would recognize that \(2 \times 12\;( 12 + 12)\) is simpler and less time-consuming to represent and compute 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.
An array can also be used to represent the commutative property of multiplying a whole number by fractions. For
example, a cook wants to cook a turkey that weighs 9 kg. Each kilogram takes \(\frac{1}{3}\) hours to cook. He must
calculate the total number of hours to cook the turkey.
It is possible to represent the number of hours by \(9 \times \;\frac{1}{3}\), for a total of 3 hours (Figure
1).
It is also possible to represent the total number of hours by \(\frac{1}{3} \times \;9\), for a total of 3 hours (Figure 2).
Figure 1
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(9 \times \frac{1}{3}\)
The turkey will cook for 3 hours.
Figure 2
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
1 hour
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
1 hour
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(\frac{1}{3}\)
1 hour
\(\frac{1}{3} \times 9\)
The turkey will cook for 3 hours.
Source: Adapted from A Guide to Effective Instruction in Mathematics, Grades 4-6, p 102-104.
Distributive Property
The distributive property means that you can perform an operation on a sum or difference of terms and obtain the same result as if the operation had been performed on each term. Multiplication is distributive over addition. For example, you can multiply 3, 5, 6 or 3.1 and get the same result as if you had performed the operation on each term \(3 \times (5 + 6)\) or \(3.1 \times (5.2 + 6) \) and arrive at the same result as if we had carried out \((3 \times 5) + (3 \times 6) \) or \((3.1 \times 5.2) + (3.1 \times 6) \) . Multiplication is also distributive over subtraction. For example, you can multiply \(3\; \times \;\left( {20\; - \;2} \right)\) or \(3,1\; \times \;\left( {20\; - \;2,5} \right)\) by making \(\left( {3\; \times \;20} \right)\; - \;\left( {3\; \times \;2} \right)\) or \(\left( {3,1\; \times \;20} \right)\; - \;\left( {3,1\; \times \;2,5} \right)\).
The following example illustrates how distributivie property can be used to calculate 6. In one case, we decompose
the factor 8 to get 5 + 3. Then we 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 \(3; + \;3\). One has then \(\left(
{3\; + \;3} \right)\; \times \;8\; = \;\left( {3\; \times \;8} \right)\; + \;\left( {3\; \times \;8} \right)\).
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 relationship between the distributive property and the usual multiplication algorithm. For example, to calculate \(3\; \times \;15\), the 15 is decomposed to get \(\left( {10\; + \;5} \right)\) :
\(3\; \times \;\left( {10\; + \;5} \right){\rm{,\;soit}}\;\left( {3\; \times \;10} \right)\; + \;\left( {3\; \times \;5} \right)\)
To calculate \(13 \times \;24\), the two factors are decomposed:
Only multiplication is distributive. One could recognize that division is partially distributive. For example, to compute 32;div 8, it is possible to decompose the dividend 32 to get 16; + 16. One has then \(\left( {16\; + \;16} \right)\; \div \;8\) and the division by 8 is distributed on the addition. We get \(\left( {16\; \div \;8} \right)\; + \;\left( {16\; \div \;8} \right)\; = \;2\; + \;2\), which is 4. However, if the divisor is decomposed, the distributive property does not work. Par exemple, \(32\; \div \;8\; \ne \;\left( {32\; \div \;4} \right)\; + \;\left( {32\; \div \;4} \right)\). This is the reason why distributive property is not a property of division.
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p 104-105.
Associative property
Associative property is a property of addition and multiplication. It allows you to combine the terms of an expression in different ways without changing the value. For example, in the expression \(15\; + \;13\; + \;17\), it is possible to combine 13 and 17 to obtain \(15\; + \;\left( {13\ ; + \;17} \right)\), which gives \(15\; + \;30\), i.e. 45. We can also associate 15 and 13 to obtain \(\left( {15\; + \ ;13} \right)\; + \;17\), which gives \(28\; +\;17\), or 45.
Associate 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, that is to say 6. We can also combine 5 and 0.75 to obtain \((5 + 0.75 ) + 0.25\), which gives 5.75; + 0.25, that is 6.
Addition and multiplication terms of fractions and percentages can also be combined, for example 5% + 10% + 20% could be associated as follows \(( 5\% + 10\% ) + 20\% \), which gives \(15\;\% \; + \;20\;\% \), or 35%.
The associative property of multiplication [for example, \(3\; \times \;2\; \times \;5\; = \;3\; \times \;\left( {2\; \times \; 5} \right)\) or \(3\; \times \;2\; \times \;5\; = \;\left( {3\; \times \;2} \right)\; \times \;5\)] is not easy to understand. Certainly, it can be seen by checking the results of the multiplications, but this does not constitute an understanding. To understand, one can use an extension of the model of a rectangular layout 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 floors
each containing \(2\; \times \;5\) cubes. The figure therefore represents \(3\; \times \;\left( {2\; \times \;5}
\right)\) cubes.
We can also consider figure 3, which illustrates 3 levels of 2 cubes, that is \(3\; \times \;2\) cubes. Figure 4 can
then represent \(\left( {3\; \times \;2} \right)\) cubes which appear 5 times, i.e. \(\left( {3\; \times \;2} \right
)\; \times \;5\).
So we see that \(3\; \times \;\left( {2\; \times \;5} \right)\) and \(\left( {3\; \times \;2} \right) \; \times \;5\) represents the same amount of cubes (even though each expression represents a different point of view) and each gives the same product as \(3\; \times \;2\; \times \ ;5\).
Devant une expression numérique telle que \(3\; \times \;\left( {2\; \times \;5} \right)\; = \;?\), some students sometimes attempt to apply distributive property so that they calculate \( (3 \times \;2) \times (3\; \times \;5) \), which results in \(2 \times \;5\) being multiplied by 9, rather than by 3. In such a case, recall that the distributive property of multiplication is performed only on 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 determine that of the expression \(\left( {2\; \times \;5} \right)\; \times \;3\). 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 \(\left( {19\; + \;11} \right )\; + \;27\), because the 9 and the 1 are complementary and give a ten, which makes it possible to obtain the answer mentally, i.e. \(30\; + \;27\; = \;57 \). It is by exposing students to a large number of activities that the teaching staff leads them to understand and use the different properties effectively.
Decomposing a number into a product of factors, along with 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:
\(\begin{align}24\; \times \;5\; &= \;\left( {12\; \times \;2} \right)\; \times \;5\\ &= \; 12\; \times \;2\; \times \;5\\ &= \;12\; \times \;10\\ &= \;120\end{align}\)
or
\(\begin{align}24\; \times \;5\; &= \;\left( {12\; \times \;2} \right)\; \times \;5\\ &= \; 12\; \times \;5\; \times \;2\\ &= \;60\; \times \;2\\ &= \;120\end{align}\)
Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 105-107.
Skill: Using the Order of Operations
The order of operations is on the curriculum in Grade 6, but students in other grades may address these rules informally. In the classroom, it is more important to focus on the meaning of expressions in a problem-solving context than on the ability to evaluate expressions with multiple operations.
The order of operations can occur in a problem-solving context or in operations presented as numerical expressions without context.
Operations in a problem-solving context
To solve a problem involving a series of operations, the order to be followed is dictated by the meaning of the problem.
Example 1
Simon has 3 envelopes containing 5 stamps each and his sister Annabelle has 7 envelopes containing 4 stamps each.
They decide to put all their stamps together to form a larger collection. How many stamps are in their joint
collection?
To solve this problem logically, students must first determine the number of stamps in Simon's collection (\(3\; \times \;5\; = \;15\)), then the number of stamps in Annabelle's collection (\(7\; \times \;4\; = \;28\)). Then they have to find the total number of stamps in the 2 collections (\(15\; + \;28\; = \;43\)). Here, multiplication takes precedence over addition.
To represent all these operations in a single numerical expression, we could write \(3; \times \;5; + \;7; \times
\;4\). However, this expression can be confusing. That's why it's better to use parentheses to specify the
order in which the operations must be performed, that is, \(3\; \times \;5)\; + \;(7\; \times \;4)\). The solution can
then be presented as follows:
\(\begin{align}\left( {3\; \times \;5} \right)\; + \;\left( {7\; \times \;4} \right)\; &= 15\; + \;28\ =
43\end{align}\)
So, the common collection contains 43 stamps.
In the following example, it is possible to solve the problem by giving priority to addition over multiplication.
Example 2:
Alphonse orders 12 comic books at a cost of $7 each. There is a $2 delivery charge per book. How much do the 12 books cost him?
Students can first determine the price of each book \(\left( {7\;\$ \; + \;2\;\$ \; = \;9\;\$ } \right)\), then the
total cost \(\left( {12\;\$ \; \times \;9\;\$ \; = \;108\;\$ } \right)\). As in the previous example, it is possible
to present both operations with brakets in the same numeric expression: \((7\; + \;2)\; \times \;12\) or \(12\; \times
\;(7\; + \;2)\).
Brackets can be used to combine certain elements of a numerical expression and to specify that these elements must be processed first.
Operations in the Form of Numerical Expressions
Ideally, numerical expressions should be presented in a context that allows for prioritization of operations. However, there are times when a numerical expression must be evaluated out of context. As presented earlier, brackets help prioritize the operations to be performed, as in the numerical expressions \((3 \times \;5) + (7 \times 4)\) and \(12 \times (7 + 2)\). However, a numerical expression presented without brackets and without context could generate a multitude of responses. For example, one might decide to treat the operations in the order in which they appear:
\(\begin{align}3\; \times \;5\; + \;7\; \times \;4\; &= \;15\; + \;7\; \times \;4\\ &= \;22\; \times \;4\\ &=
\;88\end{align}\)
We could also decide to give priority to addition over multiplication:
\(\begin{align}3\; \times \;5\; + \;7\; \times \;4\; &= \;3\; \times \;12\; \times \;4\\ &= \;36\; \times \;4\\ &=
\;144\end{align}\)
Faced with a numerical expression of this type, certain rules have been established in order to remove any ambiguity
and to standardize its treatment. One of these rules states that multiplications and divisions are done before
additions and subtractions. In this context, the conventional way to evaluate the preceding expression is:
\(\begin{align}3\; \times \;5\; + \;7\; \times \;4\; &= \;15\; + \;28\\ &= \;43\end{align}\)
The acronym PEDMAS is often presented to students to help them remember the set of rules that define the order of operations. The "P" stands for the parentheses that should be processed first. The "E" stands for exponents that are evaluated next. The "D" and "M" represent division and multiplication, operations to be performed in the order in which they appear. Finally, addition and subtraction correspond to the letters "A" and "S". These last two operations are performed last in the order in which they appear.
Source: A Guide
to Effective Instruction in Mathematics, Grades 4 to 6, p 108-110.
Some calculators respect the order of operations, while others do not (for example, if the keys \(3\; + \;4\; \times
\;5\; = \) are pressed, a calculator that respects the order of operations would display 23, while a calculator that
does not respect it would display 35). Students must then know the characteristics of their calculator, as well as the
order of operations, so as to change the order of operations if necessary (for example, the keys would have to be
pressed \(3\; + \;\left( {4\; \times \;5} \right)\; = \) or \(4\; \times \;5\; + \;3\; = \) to get the correct answer
on a calculator that does not does not respect the order of operations).
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 120.
Skill: Using Relationships Between Operations
Understanding the properties of operations and the relationships between them allows for more flexible use.
In the primary grades, students have made connections between operations through various activities. For example,
students know that addition and subtraction are inverse operations and that addition is commutative. Over time, they
develop their number sense and sense of operations 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.
- Addition and subtraction are inverse operations.
- Multiplication and division are inverse operations.
The fundamental operations of addition, subtraction, multiplication and division are closely related despite their apparent differences. The more opportunities students have to work with the operations, the more they can notice and understand the relationships between them.
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 97.
Addition and Subtraction
Addition and subtraction are simply operations that occur in problems. Therefore, it is important to avoid referring
to them as "subtraction problems" or "addition problems" because it is the understanding of the
situation, as well as the understanding of the operations, that leads to the choice of a problem-solving strategy, in
this case the choice of addition or subtraction. Thus, students must analyze the problem, choose a strategy and apply
it, just as adults do. In this context, the teacher's role is to assist students in their analysis and
understanding of operations.
It is important to note that the problems presented below appear similar because of their context. However, for students, each situation represents a unique problem. It is through mastery of these different types of problems that students gain mastery of addition and subtraction.
Types of Addition and Subtraction Problems
Adding and subtracting problems are seen by students as active situations, easier to model and “see” as the initial
quantity increases or decreases. Meeting problems, however, assume a static situation because no action or change
occurs, which makes them more abstract and harder to understand.
Comparing problems, on the other hand, deal with the relationship between two quantities by contrasting them: there
is no action, but a comparison of one quantity to another.
Since the students are regularly exposed to problems whose final quantity is sought, they solve them more easily. However, they have more difficulty solving problems in which the variable is the initial quantity, the quantity added or the quantity withdrawn. These problems help develop a more solid understanding of the operations of addition and subtraction and the connections between the operations. For example, in the case of addition problems where the variable is the initial quantity, students more easily see the advantages of addition (for example, \(?\; + \;12\; = \;37 \)) which makes it possible to respect the order in which the action takes place in the problem. This allows them to use a strategy (eg counting or counting down) to determine the initial quantity. These students demonstrate their understanding of the problem and their ability to use a strategy to solve it. However, they do not demonstrate an understanding of the meaning of difference (and subtraction). If they had used subtraction, that is \(37\; - \;12\; = \;?\), they would have demonstrated a broader understanding of the relationships between quantities in relation to this operation. But when students are learning, there is no need to impose a strategy on them. The obligation to subtract will not help students who do not see the relevance of this strategy. However, if they are regularly in contact with a variety of problems and participate in the mathematical exchanges that follow, they can see the links between various strategies and assimilate a variety of strategies. They then become more efficient.
Source: A Guide to Effective Instruction in Mathematics, Grades 4-6, p 81-84.
Multiplication and Division
To understand multiplication and division, one needs to recognize the three 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.
Types of Problems Related to Multiplication and Division
Source: A Guide
to Effective Instruction in Mathematics, Grades 4 to 6, p 84-85.
Effect of Operations
Each operation has an effect on the quantities involved. Depending on the operation, certain quantities increase or
decrease. They may 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 result 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 160 is multiplied by 8, you get 1280, whereas if you
add 8, you get 168. People with good operations sense recognize the effect of operations on whole numbers, but
learning students 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 may even be the opposite. For example, if you multiply one whole number by another
whole number, the product is larger than both factors (for example, if you multiply 3 by 6, the product 18 is larger
than 6 and 3), whereas if you multiply a proper fraction by a wholel number, the product is smaller than either factor
(for example, if you multiply \frac{1}{2}\ by 6, the product 3 is smaller than 6).
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p 90-91.
Skill: Solving Problems Requiring Multiple Operations
Whether it is a problem with one operation or multiple operations, students must make decisions and choices based on the context. A problem-solving approach will introduce them to this thinking. When faced with a problem, they must first analyze it to determine the data and understand that a calculation must be performed. Then, depending on the context, one must know whether one is looking for an approximate or an exact answer. In both cases, depending on the context and the numbers involved, it is then necessary to determine whether the calculation will be performed mentally, in writing or using a calculator. Finally, the desired calculation is performed.
Diagram of the Students' Thinking About a Problem
Source: A
Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 117.
Knowledge: Properties of Operations
A property of an operation is a characteristic that are specific to the operation, regardless of the numbers
involved.
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: \(13\; + \;0\; = \;13\), \(0\; + \;13\; = \;13\); \(25\; \times \;1\; = \;25\), \(1\; \times \;25\; = \;25\);
- Zero Property: \(8\; \times \;0\; = \;0\); \(0\; \times \;8\; = \;0\).
Source: A Guide to
Effective Instruction in Mathematics, Grades 4 to 6, p. 102.
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 \(8\; \times \;6\; =
\;6\; \times \;8\).
Generalization: \(a\; \times \;(b\; + \;c\;)\; = \;(a\; \times \;b\;)\; + \;(a\; \times \; c\;)\)
Knowledge: Associative Property
The associative property is a property of addition and multiplication. It allows the terms of an expression to be combined in different ways without changing their value.
Generalization : \(\left( {a\; + \;b} \right)\; + \;c\; = \;a\; + \;\left( {b\; + \;c} \right)\); \(\left( {a\; \times \;b} \right)\; \times \;c\; = \;a\; \times \;\left( {b\; \times \;c} \right)\)
Knowledge: Distributive Property
Multiplication is distributive over addition and subtraction.
The distributive property allows one 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 performing \(\left( {3\; \times \;20} \right)\
;-\;\left( {3\; \times \;2} \right)\).
Generalization: \(a\; \times \;\left( {b\; + \;c} \right)\; = \;\left( {a\; \times \;b} \right)\; + \; \left( {a\; \times \;c} \right)\)
Knowledge : Zero Property
In a multiplication, the 0 has the effect of “absorbing” the other factor. So no matter how many are multiplied by 0,
the product will always be 0 (for example, \(684\; \times \;0\; = \;0\); \(16.67\; \times \;0 \; = \;0\);
\(\frac{1}{4}\; \times \;0\; = \;0\)) and if 0 is multiplied by another number, the product will also be 0 (for
example, \(0\; \times \;684\; = \;0\); \(0\; \times \;16,67\; = \;0\); \(0\; \ times \;\frac{1}{4}\; = \;0\)). The
number zero is then qualified as an absorbing element for the multiplication.
Generalization: \(a\; \times \;0\; = \;0\), \(0\; \times \;a\; = 0\)
Knowledge: Identity Principle
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, \(133 \times 1 =133;\; 432.1\; \times \;1\; = 432.1\)). The Identity Principle does not apply to subtraction and division. In a subtraction, the number 0 has no effect when it is the second term (for example, \(3\; - \;0\; = \;0\)), but it does not if it appears as first term (for example, \(0\; - \;3\; \ne \;3\)). Similarly, in a division, the number 1 produces no effect when it is the divisor (for example, \(3\; \div \;1\; = \;3\)), but it is not the case if it appears as a dividend (for example, \(1\; \div \;3\; \ne \;3\)).
Generalization: \(a\; + \;0\; = \;a\), \(a\; - \;0\; = \;a\), \(a\; \times \;1\; = \;a\), \(a\; \div \;1\; = \;a\)
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p 102-107.
Knowledge: Relationships Between Addition, Subtraction, Multiplication and Division
Operational sense involves the ability to represent situations with symbols and numbers. Understanding the meaning of
the operations, and the relationships between them, allows one to choose the operation that best represents a
situation and allows one to solve the problem most effectively, given the tools available.
- 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.
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.