B2.7 Multiply and divide integers, using appropriate strategies, in various contexts.
Skill : Multiplying and Dividing Integers, Using Appropriate Strategies, in Various Contexts
Eighth graders will multiply and divide integers using a variety of representations (two-coloured tokens, number line, patterns) and computational strategies that will allow them to generalize computational rules.
The use of two-coloured tokens and the number line is essential for illustrating integer multiplication and division. By using these models, the student creates visual representations of the operations to be performed and can better understand the rationale for the different rules that govern integer multiplication and division. Rather than teaching rules, it is better for the student to be able to generalize them as a result of some observations. The rules for integer computation are the end point, not the beginning point, of teaching and learning about integers.
Source: Les mathématiques... un peu, beaucoup, à la folie, Guide pédagogique, Numération et sens du nombre/Mesure, 8e année, Module 3, Série 1, Introduction, p. 15.
Multiplication
Some of the rules for multiplication of whole numbers, fractions, and decimals are appropriate for integers. However, the student will discover that the signs of integers influence the sign of the product.
Source: inspired by CFORP, Les mathématiques... un peu, beaucoup, à la folie, Guide pédagogique, Numération et sens du nombre/Mesure, 8e année, Module 3, Série 1, Activité 4 Multiplication de nombres entiers, p. 87.
Examples
Key:
White circle: positive
Red circle: negative
White and red circles : 0
\(3\; \times \;2\)
Representations using various conventions: \(({}^ + 3)({}^ + 2)\); \(({}^ + 3) \cdot ({}^ + 2)\)
Representation using words: 3 groups of 2
Representation using two-coloured tokens :
Representation using a number line:
Symbolic representation: \(3\; \times \;2\; = \;6\;{\rm{ou}}\;{\rm{2}}\;{\rm{ + }}\;{\ rm{2}}\;{\rm{ + }}\;{\rm{2}}\;{\rm{ = }}\;{\rm{6}}\)
\(3\; \times \;{}^ - 2\)
Representations using various conventions: \(({}^ + 3)({}^ - 2)\); \(({}^ + 3) \cdot ({}^ - 2)\)
Representation using words: 3 groups of \({}^ - 2\)
Representation using two-coloured tokens :
Representation using a number line:
Symbolic representation: \(3\; \times \;{}^ - 2\; = \;{}^ - 6\;{\rm{ou}}\;{}^ - 2\; + \;{} ^ - 2\; + {}^ - 2\; = \; {}^ - 6\)
\({}^ - 3\; \times \;2\)
The student can use the commutative property and perform \(3\; \times \;{}^ - 2\), which gives the same answer.
Representations using various conventions: \(({}^ - 3)({}^ + 2)\); \(({}^ - 3) \cdot ({}^ + 2)\)
Representation using words: the opposite of 3 groups of 2
Representation using two-coloured tokens :
Representation using a number line:
Symbolic representation: \({}^ - 3\; \times \;2\; = \;{}^ - 6\)
\({}^ - 3\; \times \;{}^ - 2\)
Representations using various conventions: \(({}^ - 3)({}^ - 2)\); \(({}^ - 3) \cdot ({}^ - 2)\)
Representation using words: the opposite of 3 groups of \({}^ - 2\)
Representation using two-coloured tokens :
Representation using a number line:
Representation using a pattern:
Source : Grandes idées pour l’enseignement des mathématiques – 9 à 14 ans, Marian Small, p. 102.
Source: Adapted from CFORP, Les mathématiques... un peu, beaucoup, à la folie, Guide pédagogique, Numération et sens du nombre/Mesure, 8e année, Module 3, Série 1, Activité 4 Multiplication de nombres entiers, p. 95
Some generalizations the student can make:
- There are conventions for expressing multiplication to make algebraic expressions clearer.
- A positive integer multiplied by a positive integer has a result that is positive.
- A positive integer multiplied by a negative integer has a result that is negative.
- A negative integer multiplied by a positive integer has the same result as a positive integer multiplied by a negative integer since the commutative property applies to the multiplication of integers.
- When the first factor is positive, it can represent the number of groups and the other positive or negative integer can represent the size of the group.
- Integer multiplication can be seen as repeated addition since it contains repeated equal groups.
- Patterns can be used to determine that a negative integer multiplied by a negative integer has a positive result. If we take the example of \({}^ - 3\; \times \;{}^ - 2\), when the first factor decreases by 1, the product increases by (-2) less which decreases the product. To extend the pattern,, the products must be positive.
Source: Adapted from Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
- When the integer is negative, we can say that it is the opposite of the positive integer (for example, -3is the opposite of 3).
- A summary table :
| x | Positive integer | Negative integer |
|---|---|---|
| Positive integer | Positive product | Negative product |
| Negative integer | Negative product | Positive product |
Source: inspired by CFORP, Les mathématiques... un peu, beaucoup, à la folie!, Guide pédagogique, Numération et sens du nombre/Mesure, 8e année, Module 3, Série 1, Activité 4 Multiplication de nombres entiers, p. 91
Division
Some of the rules for dividing whole numbers, fractions and decimals are appropriate for whole numbers. However, the student will discover that the signs of integers influence the sign of the quotient.
Source: Adapted from CFORP, Les mathématiques... un peu, beaucoup, à la folie, Guide pédagogique, Numération et sens du nombre/Mesure, 8e année, Module 3, Série 1, Activité 4 Multiplication de nombres entiers, p. 87.
Understanding division of integers requires a strong understanding of the operation and its relationship to multiplication.
Source: The Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Examples
Key:
White circle: positive
Red circle: negative
White and red circles : 0
\(6\; \div \;2\;\)
Representations using division sign or fractional notation: \(6\; \div \;({}^ + 2)\); \(\frac{6}{2}\;{\rm{ou}}\;\frac{6}{{{}^ + 2}}\)
Representation using words:
- Quotative: How many groups of 2 are in 6?
- Partitive: How much does each person get if 6 (cookies) are shared between 2 (people)?
Representation using two-coloured tokens (quotative) :
Representation using a number line (quotative):
Representation using a multiplication as an inverse operation: \(2\; \times \;?\; = \;6\)
Symbolic representation: \(6\; \div \;2\; = \;3\)
\(6\; \div \;{}^ - 2\;\)
Representations using division sign or fractional notation: \(6\; \div \;({}^ - 2)\); \(\frac{6}{{{}^ - 2}}\;\)
Use the inverse relationship between multiplication and division to rewrite this statement as \(\left( {^ - 2} \right)\; \times \;?\; = \;\left( {^ + 6} \right)\).
Representation using two-coloured tokens :
The opposite of two groups of 3 white tokens
Representation using a number line:
Representation using a multiplication as an inverse operation: \({}^ - 2\; \times \;?\; = \;6\;{\rm{ou}}\;{\rm{?} }\; \times \;{}^ - 2\; = \;6\)
Symbolic representation: \(6\; \div \;{}^ - 2\; = \;{}^ - 3\)
\({}^ - 6\; \div \;2\;\)
Representations using division sign or fractional notation: \(({}^ - 6)\; \div \;({}^ + 2)\); \(\frac{{{}^ - 6}}{2}\;{\rm{ou}}\;\frac{{{}^ - 6}}{{{}^ + 2}}\)
Representation using words:
- Partitive: How much does each person get if 6 red tokens(-6) are shared into 2 groups?
Representation using two-coloured tokens :
Representation using a number line :
Representation using multiplication as an inverse operation: \(2\; \times \;?\; = \;{}^ - 6\)
Symbolic representation: \({}^ - 6\; \div \;2\; = \;{}^ - 3\)
\({}^ - 6\; \div \;{}^ - 2\;\)
Representations using division sign or fractional notation: \(({}^ - 6)\; \div \;({}^ - 2)\); \(\frac{{{}^ - 6}}{{{}^ - 2}}\;\)
Representation using words:
- Partitive: How many groups of(-2) are in(-6)?
Representation using two-coloured tokens :
The opposite of two groups of 3 red tokens
Representation using a number line:
Representation using a multiplication as an inverse operation: \({}^ - 2\; \times \;?\; = \;{}^ - 6\;{\rm{ou}}\;{\ rm{?}}\; \times \;{}^ - 2\; = \;{}^ - 6\)
Symbolic representation: \({}^ - 6\; \div \;{}^ - 2\; = \;3\)
Some generalizations the student can make:
- A positive number divided by a positive number has a result that is positive.
- A positive number divided by a negative number has a result that is negative.
- A negative number divided by a positive number has a result that is negative.
- A negative number divided by a negative number has a result that is positive.
Source: The Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
- When the integer is negative, we can say that it is the opposite of the positive integer (for example, -3is the opposite of 3).
- A summary table :
| ÷ | Positive integer | Negative integer |
|---|---|---|
| Positive integer | Positive quotient | Negative quotient |
| Negative integer | Negative quotient | Positive quotient |
Source: Adapted from CFORP, Les mathématiques... un peu, beaucoup, à la folie, Guide pédagogique, Numération et sens du nombre/Mesure, 8e année, Module 3, Série 1, Activité 5 Divisons, p. 109.
Knowledge: Integers
The integers belong to the set \(\mathbb{I}\).
\(\mathbb{I}\)= {…, –3, –2, –1, 0, 1, 2, 3…}
Each integer is formed by a number and a sign (- or ).
Every integer has an opposite (for example, -3and 3). The opposite of a positive number is a negative number of the same quantity (absolute value), and vice versa.
The signs - and are associated with number. They are not operation signs. Thus, the number -4reads: the opposite of 4 or negative 4.
There is no space between the negative sign - and the number 4, since it is not the subtraction operation.
Zero is neither a positive nor a negative number, because it is the only number that is its own opposite.
On the number line, the numbers are in ascending order from left to right and descending order from right to left.
Source: adapted from CFORP, Les mathématiques... un peu, beaucoup, à la folie, Guide pédagogique, Numération et sens du nombre/Mesure, 7e année, Module 3, Série 1, Activité 1 Des nombres entiers sur la droite numérique, p. 35-37.