At a Glance...

C2.1 Describe how variables are used, and use them in various contexts as appropriate.

Skill: Using Variables Appropriately and Describing Them

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Symbols in Everyday Life

At a very young age, children evolve in a world where symbols are used to communicate a message or define a function.

Examples

• traffic lights dictate behaviour (red means stop or no go; green means go);
• the characters on the doors of the washrooms give information as to the purpose of the room;
• 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.

Meaning of Symbol in Algebra

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 an unknown quantity. For example, in the equation 3 + Δ = 5, the triangle is a symbol that represents a quantity when added to 3 has a result of 5. In this case the triangle is 2.

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 number sentence; for example, C represents the cat, G represents the goat, and R represents the rabbit.

• 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, including the use of symbols, not only exists within the algebra strand but should be nurtured throughout all of the strands. For example, students use symbols when working with measurement formulas.

Here is an example of an authentic problem situation that illustrates a progressive path towards the development of the meaning of the symbol.

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

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

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: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3^e année, Modélisation et algèbre, Fascicule 2, Situations d'égalité, 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.

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: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3^e année, Modélisation et algèbre, Fascicule 2, Situations d'égalité, 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:

Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.

Knowledge: Symbol

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Symbols can be used to represent changing or unknown quantities.

Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.

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: translated from En avant les maths!, 2^e année, CM, Algèbre, p. 2.

Knowledge: Variable

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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: translated from En avant les maths!, 2^e année, CM, Algèbre, p. 2.

Knowledge: Constant

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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.

Source: translated from En avant les maths!, 2^e année, CM, Algèbre, p. 2.