B2.4 Use objects, diagrams, and equations to represent, describe, and solve situations involving addition and subtraction of integers.
Skill: Representing, Describing and Solving Addition and Subtraction of Integers
In grade 7 , students will add and subtract integers using a variety of representations (two different colours of tokens, number line) and computational strategies that will allow them to generalize computational rules. The use of two colours of tokens and the number line is essential to illustrate addition and subtraction of integers. By using these models, students create visual representations of the operations to be performed and can better understand the rationale for the different rules that govern addition and subtraction of integers. 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… , Guide pédagogique, Numération et Sens du nombre/Mesure,7e année, Module 3, Série 1, p. 15.
As with whole numbers, the addition of integers is commutative. So the order in which the integers are written in an addition is not important.
However, as with whole numbers, the commutative property does not apply to subtraction. The order in which the integers are written in a subtraction is important.
When writing an equality, integers are often enclosed in parentheses, for example, \(( \;{}^ + 3)\; - \;(\;{}^ - 2)\; = \;(\;{}^ + 5)\).
- If a sign is not included, the number is considered positive.
- These conventions help reduce confusion between number and operation.
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Coloured tokens model (2 different colours)
One colour represents positive numbers and another represents negative numbers.
In this model, 2 tokens of opposite colours cancel each other out or give 0. These are “Zero pairs” where the sum of a positive and a negative number results in zero.
Examples
\({}^ - 5\; + \;{}^ + 3\)
I represent each term with coloured tokens, that is 5 red and 3 white tokens.
I combine 1 red token with 1 white token to get zero pairs since their value is 0. In all, there are 3 zeros. This set of tokens represents the integer -2. So, \({}^ - 5\; + \;{}^ + 3\; = \;{}^ - 2\).
\({}^ - 5\; + \;{}^ - 3\)
I represent each term with coloured tokens, that is 5 red and 3 red tokens.
In total, there are 8 red tokens. This set of tokens represents the integer -8. Then, \({}^ - 5; + \;{}^ - 3; = \;{}^ - 8\).
\(5\; - \;{}^ + 3\)
I represent each term with coloured tokens. I have 5 white tokens and 3 red tokens.
I combine a red token with a white token to get zero pairs. In all, there are 3 zeros. This set of tokens represents the integer 2. So, \(5\; + \;{}^ - 3\; = \;{}^ + 2\;{\rm{or \;2}}\).
\(5\; - \;({}^ - 3)\)
I represent 5 with 5 white tokens. I have to remove -3. If I remove -3, it is like adding its opposite. So I add 3 white tokens.
In total, there are 8 white tokens. This set of tokens represents the integer 8. Then, \((- 5)\; + \;(- 3)\; = \;(- 8)\).
Number Line Model
In this model, integers are represented by arrows. The direction of the arrow determines whether the value represents a positive or negative quantity. The length (always a positive value) of the arrow represents the quantity, and its direction, the sign.
Example
\({}^ + 8\; + \;{}^ - 5 = \;{}^ + 3 \)
The black arrow that is positive (points to the right) represents 8.
The red arrow that is negative (points to the left) represents -5.
Then, \( 8\; + \;(- 5) = \; 3 \).
When 2 positive integers are added, the result is positive. This can be visualized on a number line as :
- 2 arrows moving in a positive direction (to the right or up);
- 1 arrow moving in a positive direction from a positive starting number.
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Example
\(20\; + \;60 = \;80\)
From 0, the student jumps 20 and then adds a jump of 60 so the final position is the sum that equals 80. So, \(60\; + \;20\; = \;80\).
When 2 negative integers are added, the result is negative. This can be visualized on a number line as :
- 2 arrows moving in a negative direction (left or down);
- 1 arrow moving in a negative direction from a negative starting position.
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Example
\(({}^ - 2)\; + \;({}^ - 5)\)
From 0, the student draws 1 red arrow 2 units to the left. From -2, they move another 5 units to the left. The student stops at -7. Then, \((- 2)\; + \;(- 5)\; = \;(- 7)\)
When positive and negative integers are added together, the result is negative if the absolute value of the negative integer is greater than the absolute value of the positive number. This can be visualized on a number line as :
- 1 arrow moving in a positive direction and the other arrow of greater amplitude moving in a negative direction;
- 1 arrow moving in a negative direction from a positive starting position (tip is to the left or below 0);
- 1 arrow moving in a positive direction from a negative starting position (arrowhead is to the left or below 0).
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Example
\(\ ({}^ - 6) + (4) \)
From 0, the student draws 1 red arrow 6 units to the left. From -6, they move 4 units to the right by drawing 1 black arrow. The student stops at -2.
Then, \(({}^ - 6)\; + \;(4)\; = \;{}^ - 2\)
When positive and negative integers are added, the result is positive if the absolute value of the positive integer is greater than the absolute value of the negative integer. This can be visualized on a number line as :
- 1 arrow moving in a negative direction and the other arrow of greater amplitude moving in a positive direction;
- 1 arrow moving in a positive direction from a negative starting position (the arrowhead is to the right or above the 0);
- 1 arrow moving in a negative direction from a positive starting position (arrowhead is to the right or above 0).
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Example
\(\ ({}^ - 3) + (7)\)
From 0, the student draws a red arrow 3 units to the left. From -3, they move 7 units to the right by drawing a black arrow. The student stops at 4. Then, \(({}^ - 3)\; + \;(7)\; = \;4\).
The subtraction on the number line can also be seen as the difference between the 2 numbers.
Example
\((5)\; - \;({}^ - 3)\; = \;8\)
\(5; - \;({}^ - 3)\) means the difference between -3and 5
Everyday situations that include positive and negative integers provide a starting point for understanding how they describe change (for example, temperature, elevator travel, sea level, golf score, acquiring and losing money, steps forward and backward).
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Example 1
According to the weather report, it is currently -4°Cin Iqaluit. The temperature will rise by 9 degrees during the day. What is the maximum temperature expected?
Here is an example of a possible solution:
I know that 5°Cis 9 degrees warmer than -4°C.
Therefore, the maximum temperature expected in Iqaluit is 5°C.
Source: Les mathématiques… , Guide pédagogique, Numération et Sens du nombre/Mesure,7e année, Module 3, Série 1, p. 81.
Example 2
In the spring, Patrick borrows $75 from his parents to buy a bike. Over the summer, he saves up to pay his parents back. In September, he gives his parents $25. In October, he gives them
$30. What is Patrick's financial situation as a result of his loan?
Examples of possible solutions include:
Debt: $75 or -75
Money saved: $55((25; + \;30; = \;55)\) or $55
Patrick paid back $55 of the $75 he owed his parents.
\(\begin{align}(- 75)\; + \; 25\; + \; 30\; &= \;(- 75)\; + \; 55\\ &= ((- 20)\; + \;(- 55))\; + \; 55\\ &= (- 20)\; + \;0\\ &= (- 20)\end{align}\)
Patrick still owes his parents $20.
Source: Les mathématiques… , Guide pédagogique, Numération et Sens du nombre/Mesure,7e année, Module 3, Série 1, p. 78.
Knowledge: Integers
The integers belong to the set ℤ.
ℤ = {…, -3, -2, -1, 0, 1, 2, 3,…}
Each integer consists of 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.
0 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: Les mathématiques… , Guide pédagogique, Numération et Sens du nombre/Mesure,7e année, Module 3, Série 1, p. 35-37.