C2.1 Describe how variables are used, and use them in various contexts as appropriate.
Skill: Using Variables Appropriately and Describing Them
Symbols in Everyday Life
At a very young age, children evolve in a world where omnipresent signs communicate, by their nature, a clear message or define a precise function.
Examples
- traffic lights dictate behaviour (red means stop or no go; green means go);
- the characters on the bathroom doors show the boy or girl which door to choose;
- at the daycare, several symbols indicate where to store toys, clothes, etc;
- a song or rhyme announces a specific activity (e.g., snack time, rest time, tidying up);
- WHMIS safety signs, often presented by parents, make children aware of the toxicity of cleaning products;
- road signs, logos and commercial signs quickly become meaningful to children.
These symbols, without being systematically taught, memorized or recited, are acquired through experience. Children quickly become familiar with them even though their nature is arbitrary. Thus, each symbol used in a given context is associated with a precise meaning and this association will no longer be contested. On the other hand, they can give rise to other interpretations if they are used in new contexts.
Symbols in Mathematics
Mathematical symbols represent fundamental concepts that underlie the learning of mathematics and algebra. Many of these concepts, which adults have known for a long time, are often introduced to students prematurely, in out-of-context situations, or without prior explanation. It is essential, therefore, to create situations in which students have time to become familiar with the symbols and their meanings in order to explain them clearly and accurately.
| Symbols in Mathematics | Examples |
|---|---|
| Symbols Designating Quantities - Numbers (Indo-Arabic Numerical or Graphic Symbols) |
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| Symbols representing fundamental operations Signs that designate the four operations in numeration |
– + × ÷ |
| Symbols that establish relationships The sign that indicates equality Signs that indicate inequality |
= ≠ > < |
Meaning of Symbol in Algebra
"As they have varied and meaningful experiences, students develop symbol sense. Symbol sense is a level of mathematical understanding that encompasses number sense." (Picciotto & Wah, 1993, p. 42)
Luis Radford (2001) sees learning algebra as the appropriation of a new mathematical way of thinking and acting that involves the production and use of symbols. These symbols carry meaning for students as they explore them in contextualized mathematical activities.
A key component of algebraic reasoning is the ability to recognize and explore relationships that link various quantities. These relationships are represented by various symbols that students must learn to use appropriately.
In the primary grades, a dash, geometric shape, or other symbol in an equation represents, to students, an unknown quantity. For example, in the equation 3 + Δ = 5, the triangle is a symbol that represents the quantity 2. Proper use of a box or dash is a first step toward the more formal symbolism of algebra.
Students with a fair understanding of this will be able to more effectively interpret or manipulate the literal symbols commonly used in algebra to represent the variable in the junior grades. Symbol meaning plays an important role in the development of algebraic thinking. Understanding of symbols is a function of students' maturity and diversity of experience.
Here are some points to consider to help students develop a sense of the symbol:
- Present symbols in challenging and realistic contexts so that students grasp their meaning and, ultimately, understand the concepts behind them.
- Encourage students to identify the elements to be represented in a problem situation, to choose an appropriate mode of representation, and to use personal symbols.
In the picture above, students use the first letter of each animal's name to formulate the math sentence; for example, C represents the toad, G represents the frog, and R represents the tree frog.
- Encourage students to identify when and how to use symbols (e.g., represent a missing quantity in an equation with a square).
- Provide students with activities that require them to make connections between various modes of representation, discuss these connections, and justify them using mathematical arguments.
To develop a sense of the symbol, students should:
- describe relationships using various modes of representation before exploring them symbolically;
- make connections between representations.
The effective use of different modes of representation is a progressive pathway integrated into the Number strand. Its purpose goes far beyond the simple mechanical use of symbols and memorization "tricks" in algebra. Rather, it is to develop the ability to perceive patterns, generalize relationships, and solve problems.
Here is an example of an authentic problem situation that illustrates a progressive path towards the development of the meaning of the symbol.
| Pathway to develop the meaning of the symbol | Examples |
|---|---|
| Present the situation and information. | A student could state the following: "There are 10 people on the bus after the first stop and 15 people after the second stop. How many people got on the bus at the 2nd stop? got off?" |
| Propose a conjecture. | "I think the number of people going down is always smaller than the number of people going up." |
| Represent the problem situation using concrete materials or drawings. | Students use cubes to represent people.
Students represent the people with drawings. 
|
| Gradually move away from semi-concrete representations and use personal symbols to communicate understanding of the situation. | In the photo, the student has chosen the symbols below as labels:
p for people; d for people getting off the bus; m for people boarding the bus. 
|
| Use conventional symbols to represent the problem situation with a number sentence. | In the picture below, the student makes sense of the number sentence 10 - 3 + 8 = 15.
The difference between the eight people going up and the three people going down is 5, which is the difference between the number of people after the 2nd stop (15) and the number of people after the 1st stop (10). |
| Generalize orally or with the help of symbols. | "The difference between the number of people getting on and the number of people getting off will always be the difference between the number of people after the second stop and the number of people after the first stop." |
To consolidate the meaning of the symbol, students should:
- recognize that the same symbol used more than once in a number sentence represents the same quantity, the same value.
Example
- use symbols to communicate a generalization.
Example
The equation symbolically represents the associativity property of addition.
- explore the quantities represented by the symbols by replacing the symbols with different numbers.
Example
If  represents |
3 x represents |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 3 | 9 |
Variables
In algebra, the use of symbols to represent variables requires a certain level of abstraction. In fact, being able to determine the value of an unknown quantity is an important step in the development of algebraic thinking. Students will grasp the meaning of the equation and perceive the relationship between the variable and the quantity it represents to the extent that the equation is representative of a meaningful problem situation.
Example
There are 15 sparrows perched on a telephone wire. Some of them fly away when a car passes. Then three sparrows come back to perch on the wire. I can now count 10. How many sparrows flew away when the car passed?
The equation below is the symbolic representation of this problem situation.
In mathematics, if the variable represents a quantity, a value, an entity in itself, it is however important to specify that the variable is also the name given to the value that makes the equation true. So let's specify that the variable can be used to :
- represent a specific quantity;
Example
- explore a situation of equality or inequality;
Example
- represent a mathematical property.
Example
Students in the primary grades must analyze variables in problem situations and gradually represent them in a more abstract way. At first, the variable is mostly represented by geometric shapes or dashes.
Examples
It can also be represented in a table.
Examples
| If represents |
7 - represents |
|---|---|
| 1 | 6 |
| 2 | 5 |
| 3 | 4 |
The variable can be represented by a letter (a literal symbol) that is directly related to the problem situation so that the letter has meaning for the students.
Example
15 - m + 3 = 10, the variable m represents the number of sparrows flown.
To develop a sense of variable, students need to explore a variety of authentic situations that engage meaningful data and promote the transition from concrete to abstract.
Source : Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, p. 51-59.
Representing problem situations is a fundamental component of algebraic thinking.
Real situations represented initially with concrete or semi-concrete material will gradually be represented by symbols.
In order to build a solid foundation for understanding algebraic concepts, it is important to move students gradually toward a more formal symbolic representation. The use of symbols facilitates a higher level of abstraction, especially when representing numbers in situations of equality.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 17.
Symbols can be used to represent and describe, in an abstract way, the pattern created in a concrete or semi-concrete representation.
For example, students use a semi-concrete representation to illustrate a pattern of bricks placed vertically and horizontally.
Example
Students quickly realize that it takes a long time to draw the bricks. As a result, they decide to use symbols such as D (standing) and C (lying) to represent the standing and lying brick elements in the pattern and describe its structure.
By using meaningful symbols, students deepen their understanding of a symbolic representation. The symbols then become an effective way to represent the situation and the observed pattern.
Example
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 17.
Here are other examples of variables used to represent quantities in a relationship.
- In the generalization of the properties of operations :
a + b = b + a (commutative property) - In the formulas :
Perimeter of a square = c + c + c + c
- In a table of values :
| Number of bicycles (b) | Number of wheels (wheels = 2 × b) |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Knowledge: Symbol
Symbols can be used to represent changing or unknown quantities.
Note: In an equation, if the same symbol appears more than once, its value is identical. For example, 
. Two different symbols in an equation can have different values. For example, 
, the two symbols can have values of 10 and 10, 19 and 1, etc.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Source : En avant les maths!, 2e année, CM, Algèbre.
Knowledge: Variable
A term in an equation (symbols or letters) that can be replaced by one or more values.
Example
In the equation 
or (10 = x + y), the shapes or letters are variables, because the quantities are unknown.
Source : En avant les maths!, 2e année, CM, Algèbre.
Knowledge: Constant
The quantities that remain the same are called constants.
Example
Looking at the pattern below, we notice that in each position, the constant is the group of eight purple squares. What changes are the number of groups of red squares.
