GEEM - Skills and Knowledge

B2.2 Recall and demonstrate multiplication facts from 0 × 0 to 12 × 12, and related division facts.

Skill: Demonstrating Understanding of Multiplication and Division Facts

Learning the basic number facts associated with multiplication and division is a must. They are the foundation for operations and an essential tool for working with larger numbers. It is important that students learn to master basic number facts.

However, there are teaching strategies that allow students to learn them. Facts do not have to be memorized by repeating them and using flashcards. This approach, which involves memorizing the facts as distinct pieces of knowledge, requires the memorization of nearly 400 isolated facts. It is very demanding on the memory, since learning the facts is not based on making connections.

Instead, learning number facts must build on students' ability to make connections and on their prior knowledge. The role of teachers is to present strategies and tools to students. For example, to learn multiplication facts by 9, students can use the "it's a group less than" strategy: multiply the number by 10 and subtract a group. Thus, a student might hold the following reasoning: $9\; \times \;8$, is like $10\; \times \;8\; = \;80$, minus a group of 8, so it's 72. This strategy helps the student answer correctly and quickly and understand the quantities involved in this multiplication.

With practice, students use the strategies less and less and develop automaticity in finding many basic facts. Students may not be able to spontaneously recall all the basic number facts, but they should be able to find them quickly using the learned strategies, as is the case with many adults who have excellent number sense.

Facts are learned up to $12\; \times \;12$ in Grade 5. However, it is important for students to know that if they learn the facts to $9\; \times \;9$ they can easily find the facts beyond. For example, if students need to multiply 8 by 12, they can use the x10 and x2 facts (for example, $12\; \times \;8\; = \;10\; \times \;8\; + \;2\; \times \;8$) or multiply 8 by 6 and double the answer.

The following list presents the most common strategies and properties for learning basic number facts. In general, not all students use the same strategies. They may not use all of the existing strategies, but choose one or more, depending on their comfort and the number facts being addressed.

Source: translated from Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 88-89.

Multiplication can be represented as repeated addition, arrays, a set of equal groups, or an area model.

Some properties and strategies that support conceptual understanding of multiplication include:

the identity principle of multiplication: ($a\; \times \;1$ always equals a);
the zero property of multiplication: ($0 \times n = 0 $);
the commutative property $\left( {2\; \times \;3\; = \;3 \times \;2} \right)$;
the distributive property $\left[ {5\; \times \;6\; = \;5\; \times \;\left( {4\; + \;2} \right)\; = \;\left( {5\; \times \;4} \right)\; + \;\left( {5\; \times \;2} \right)} \right]$;
the associative property $\left[ {5\; \times \;12\; = \;5\; \times \;\left( {2\; \times \;6} \right)\; = \;\left( {5\; \times \;2} \right)\; \times \;6} \right]$;
that it is the inverse operation of division.

Division can be represented as repeated subtractions, equal distribution or sharing.

Some properties and strategies that support conceptual understanding of division include:

the identity principle, which states that when dividing an amount by 1, the amount stays the same ($a\; \div \;1$ always equals a);
the relationship between division and fractions (12 candies divided into 3 groups represent both $12\; \div \;3$ and the whole divided into 3 equal parts);
that it is the inverse operation of multiplication.

Source: translated from Guide d'enseignement efficace des mathématiques de la maternelle à la 6^e année, Fascicule 5, p. 21-22.

Patterns in Numbers Sequences

Students develop a deeper understanding of the meaning of mathematical operations by exploring the patterns they generate in number sequences. They can use the number line or the hundreds grid to analyze the number sequences obtained when skip counting. For example, if one skip counts in intervals of 10 starting with any number from 0 to 9, all the results will end with that number, such as when one starts counting at 3, and gets the sequence 3, 13, 23, 33, 43…).

The study of patterns in number sequences also facilitates the learning of basic number facts (for example, all multiples of 5, that is 5, 10, 15, 20, 25…, end in 5 or 0) and the building of connections between numbers (for example, all multiples of 10[▲] are also multiples of 5[●] ).

… or, all multiples of 6 [▲] are also multiples of 3 [●]).

Students generally enjoy identifying patterns in the number sequences generated by mathematical operations, with each discovery usually evoking a sense of wonder. Activities related to finding patterns encourage students to juggle numbers and promote the development of flexible thinking and analytical skills, two important components of mathematical thinking. In addition, the ability to explore patterns is essential to the study of algebra and geometry.

Source: translated from Guide d'enseignement efficace des mathématiques de la 1^re à la 3^e année, Numération et sens du nombre, p. 30-31.

Strategies for Learning Multiplication and Division Facts

To learn the basic number facts about multiplication and division, it is not enough to memorize them. It is important to understand that these operations can be approached in different ways.

Basic Multiplication Facts

The basic multiplication facts are all the multiplication facts from $0\; \times \;0$ up to $9\; \times \;9$.
There are 100 basic multiplication facts.

Basic Division Facts

The multiplication grid can also be used for division.
The number facts of division are the inverse of the number facts of multiplication, from $81\; \div \;9$ to $1\; \div \;1$.
There are 90 basic facts for division.
There are no facts that include 0 because it is impossible to use 0 as a divisor.

Models for Representing Basic Multiplication and Division Facts

Several models can be used to help students understand multiplication and division. They enable students to:

grasp the meaning of operation symbols and make connections between drawings/pictures, words and symbols;
understand the meaning of operations and use different representations - concrete, visual (semi-concrete) and symbolic.

Many models can be developed using the materials below to help students understand multiplication and division:

concrete objects, such as counters, sticks, tiles, and various containers to place them in;
interlocking cubes;
visual material, such as illustrations;
arrays (row and column arrangement);
base ten blocks;
coins (1¢, 5¢, 10¢, 25¢, $1and $2);
grid paper;
number lines;
hundreds charts.

Models of all kinds can help students make connections and thus better understand what symbols represent in operations. Students need to work with representations that help them perceive relationships between multiplication and repeated addition, and between multiplication and division (students learning long division can also benefit from representations that help them to perceive long divisions as repeated subtractions or repeated additions). Understanding that arrays are a tool for multiplication or division develops with time and practice. Once students can represent facts such as $5\; \times \;6$ using arrays, they can use these to find the answer to $50 \; \times \;6$, and so on.

Representations of all kinds help students make connections between models, symbols and words. Below are different representations of the basic number fact $2\; \times \;6$ . Each of the seven representations is appropriate. Remember that students feel less need to resort to visual representations as they develop automaticity with basic number facts.

Source: translated from Guide d'enseignement efficace des mathématiques de la maternelle à la 6^eannée, Fascicule 5, p. 21-24.

Recall of Multiplication and Division Facts

Before they can learn multiplication and division number facts, students must understand how to make and count equal groups, and how to divide a whole into equal groups. They also need to know how to skip count in intervals of 2, 3, 4, 5, and 10. The strategies that cover basic multiplication and division number facts are intended to help students build on their knowledge and their previous experiences in order to learn how to use these new strategies (for example, dividing into equal groups, skip counting…).

The following strategies are not presented in any particular order. Some students may find some strategies more useful than others, or may ignore some strategies in favour of their own. Others may find it easier to memorize facts than to rely on a strategy. Whatever the case, the teacher's primary goal is to get all students to fully understand multiplication and division.

Commutative Property

Students benefit from experiences that help them recognize the commutative property of multiplication ($2\; \times \;4\; = \;4\; \times \;2$). Students who understand the commutative property can use half of the basic number facts to learn the other half.

Commutative Property Grid

Rules for 0 and 1

Students benefit from experiences that help them understand the Identity Principle which states that when multiplying or dividing an amount by 1, the amount stays the same (for example, $1\; \times \;3\; = \;3, \;6\; \div \;1\; = \;6$) and the Zero Property of Multiplication - the product of a number multiplied by 0 is 0 (for example, $100\; \times \;0\; = \;0$). To help students to understand these concepts, it is more useful to ask them problems such as "If I give you 3 packages of 0 cookies, how many cookies do you have?" than to have them memorize them.

Doubles

The multiplication facts of 2 should be related to the knowledge students have already learned about adding doubles. This strategy is particularly important because students who are familiar with the 2 times table can relate these number facts to the 3-times table. Indeed, if $2\; \times \;4\;$ is 8, it follows that $3\; \times \;4\;$ equals 8 plus another 4.

Double and Double Again

Once students know the double facts (the multiplication facts of 2), they can apply this knowledge to the multiplication facts of 4. Any number times 4 is the same as double the answer to two times that number (for example, $4\; \times \;6\;$ is the same as double $2 \times \;6\;$, with the answer 12 doubled to 24).

Double and One More Set

Again, once students know the double facts (the multiplication facts of 2), they can use this information to figure out the multiplication facts of 3. If $2\; \times \;4\;$ is 8, then $3\; \times \;4\;$ is 8 plus one more 4 which is 12.

(Van de Walle and Folk, 2005, p. 152)

Five Facts

Students who have experience counting by fives and identifying the five fact patterns on a hundreds chart usually do not have difficulty learning the five facts. A wide variety of songs, poems, and storybooks based on fives are available that will support learning this strategy.

Add One More Set

If the student knows $6\; \times \;7\; = \;42$, then they can figure out $6\; \times \;8$ by adding one more 6 to the 42 to make 48 . This strategy is especially useful for the multiplication facts of 6, which can be difficult to recall. If the multiplication facts of 5 are well retained, students figure out the multiplication facts of 6 by simply adding one more group of the factor that is not the 6. For example, if $5\; \times \;7\; = \;35$, then $6\; \times \;7$ is simply 35 plus one more 7, or 42. (Van de Walle and Folk, 2005, p. 152). Similarly, if the student needs to find the product of $8\; \times \;7$ and knows that $8\; \times \;8\; = \;64$, they can use the same strategy in reverse, that is, subtract 8 from 64 to get 56.

The Facts of 9

The 10 times facts are usually fairly easy to learn because students learn to count by 10s at an early age. Once they have mastered the 10 facts, students can make connections to the 9 facts by calculating the corresponding multiplication fact by 10 and subtracting one group of the factor that is not the 9. For example, if $10\; \times \;8$ yields 80, it follows that $9 \times \;8$ equals 80 minus 8, or 72.

Teachers can help students recognize some of the patterns in the facts of 9. For instance, all the digits in the products are equal to 9 if they are added $\left( {5\; \times \;9\; = \;45\;\text{ and\; then }\;4\; + \;5\; = \;9} \right)$, and the tens in the product are always one less than the multiplier. For instance, in $6\; \times \;9\; = \;54$, the tens digit is 1 less than the multiplier of 6 (so 5) and the sum of the two digits in the product (5 and 4) is 9. Students who conceptually understand the multiplication facts of 9 but who fail to retain them may benefit from learning the finger patterns, as shown below.

$4\; \times \;9$

A number of students find this kinesthetic method helpful in remembering the 9 times table. Students are asked to hold their hands out in front of them, fingers spread as shown. Their fingers (from the little finger of the left hand) are numbered from 1 to 10 (A). They must bend the finger which represents the multiplier of 9. Thus, in the multiplication $9\; \times \;4$, it is the finger number 4, that is the index finger of the left hand, which is bent (B). The number of fingers to the left of the bent finger represents the tens digit, the number of fingers to the right of the bent finger represents the ones digit. Thus, in this example the 3 fingers to the left of finger number 4 represent 3 tens, therefore 30, and the 6 fingers to the right of finger number 4 represent 6 ones, therefore 6, which gives 36 (C).

Half Then Double

Suppose a student cannot recall a fact and the fact involves even factors. One factor can be halved, and if that is known, the product can be found and then doubled. For instance, if $6\; \times \;8$ cannot be recalled, halve one factor $\left( {3\; \times \;8\; = \;24} \right)$, and then double the product of $\left( {2\; \times \;24} \right)$ for an answer of 48. Students should investigate this strategy by using arrays to provide a visual presentation. Sometimes, the strategies may seem cumbersome, but they provide students with a way to reason their way to a correct answer using known facts. (Van de Walle & Folk, 2005, p. 152)

Source: translated from Guide d'enseignement efficace des mathématiques de la maternelle à la 6^e année, Fascicule 5, p. 25-28.

Ten Facts

Students need to learn to master the effect of multiplication and division by 10 and multiples of 10. The explanation of these operations often reduces to the statement, "When you multiply by 10, you add a 0 and when you divide by 10, you take away a 0." This statement is not recommended because it does not take into account the understanding of operations and students are encouraged to apply a mechanical shortcut , without being able to make connections during multiplication or division by numbers such as 10, 100, or 1000.

It is helpful for students to recognize that operations like $5\; \times \;10$ or $60\; \div \;2$ can be thought of as $5\; \times \; 1$ tens and 6 tens ÷ 2. We can then better understand the appearance or disappearance of the digit “0” when performing the operation.

Operation	Interpretation	Result
$\;5 \times \;10$	$\;5 \times \;1$ ten	5 tens is 50

Source: translated from Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 91.

Multiplying by 11 and by 12

Facts are learned up to $12\; \times \;12$ in Grade 5. However, it is important for students to know that if they learn the facts to $9\; \times \;9$, they can easily find the facts beyond.

For example, if students need to multiply 7 by 11, they can use:

the facts of 10 and 1 $\left[ {11\; \times \;7\; = \;\left( {10\; \times \;7} \right) \; + \;\left( {1\; \times \;7} \right)\; = \;70\; + \;7\; = \;77} \right]$;
the observed pattern according to which the product corresponds to the factor that is multiplied by 11 written twice. It is important to let the students notice this pattern themselves.

$\begin{array}{l}1\; \times \;11\; = \;11\\2\; \times \;11\; = \;22\\3\; \times \;11 \; = \;33\\4\; \times \;11\; = \;44\\5\; \times \;11\; = \;55\\ \ldots \end{array}$

If students need to multiply 8 by 12, they can use:

the facts of 10 and 2 (for example $12\; \times \;8\; = \;10\; \times \;8\; + \;2 \times \;8\; = \;80\; + \;16\; = \;96$);
multiply 8 by 6 and double the answer $\left[ {\left( {8\; \times \;6} \right)\; \times \;2\; = \;48\; \times \;2 \;= \;96} \right]$.

Source: adapted and translated from Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 88.

Learning the Division Facts in Sequence

A suggested sequence for teaching division facts is to begin with facts for division by 2, then division by 1, by 5, by 3, by 4, by 6, by 7, by 8, and by 9.

Inverse Relationship of Division and Multiplication

Students who have mastered multiplication number facts should use this knowledge to find division number facts. It is recommended that division and multiplication be taught simultaneously to highlight the relationships between the two. When students create 4 groups of 5 items for a total of 20 items, teachers should encourage them to realize that when that total is divided among 4 groups, each group has 5 items.

Source: translated from Guide d'enseignement efficace des mathématiques de la maternelle à la 6^eannée, Fascicule 5, p. 29.