C2.1 Identify and use symbols as variables in expressions and equations.
Skill: Determining and Use Symbols as Variables
Grasping the meaning of the symbol that represents a variable requires a high level of abstraction. Acquiring the meaning of the symbol and comfort in manipulating symbols is a gradual process. Students must first learn to describe relationships using a variety of representations. They must use concrete and semi-concrete representations throughout their learning. Driscoll (Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10, 1999, p. 123, as cited in Ontario Ministry of Education, Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année – Modélisation et algèbre, 2008, p. 86) shows that to help students make the transition to symbolic representations using letters, it is beneficial to use drawings or shapes to represent quantities before using letters as variables.
In the primary grades, the variable is first represented by a box or line to be filled in (for example, 4 + _ = 6). Students then progress to using a drawing or geometric shape (for example, ♥ + 5 = 12) to represent a missing quantity in an equation. The use of the letter, as a variable, should occur in the junior grades because students need to be able to differentiate between its role in written language and its role in algebra. The transition is not easy, and students need to gradually become familiar with the use of literal symbols in an equation. How teachers present these symbols affects students' perceptions of them. According to Kieran and Chalouh ("Prealgebra: The transition from arithmetic to algebra," in D. Owens (Ed.), Research Ideas for the Classroom. Middle Grades Mathematics, 1999, p. 62, as cited in Ontario Ministry of Education, A Guide to Effective Instruction in Mathematics, Grades 4 to 6 : Modeling and Algebra, 2008, p. 86), the use of literal symbols to represent a quantity should be a topic of discussion in the classroom. Various research studies (Demonty and Vlassis, "Pre-Algebraic Representations of Students Leaving Elementary School: A Synthesis of Educational Research 231/97", Educational Information, No. 47, 1999, pp. 16-27, as cited in Ontario Ministry of Education, A Guide to Effective Instruction in Mathematics, Grades 4 to 6 : Modeling and Algebra, 2008, p. 86) have identified student misunderstandings regarding the use of these symbols in equations. Some examples include:
- Students think that a = 1, b = 2, c = 3, etc.
- Students think, for example, that the letter a must represent a lower value than the letter c, since the letter a precedes the letter c in the alphabet.
- Students think that if a letter takes any value in one equation, then it will take that same value in other equations.
- Students ignore the letter (for example, the algebraic expression 2p +3p is reduced to 5 by ignoring the p).
- Students think that the letter represents an abbreviation of a common noun or unit of measurement (for example, in the algebraic expression3b +6b, students think that the b may represent "cookies"). Note that in the primary grades, students may use a letter as a label in a number sentence (for example, 3b + 6b = 9b), but the letter does not represent a value.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 86.
Students may also confuse the letter x with the multiplication sign ×. In their mathematics journey, students learn that to represent multiplication, the sign × can be replaced by a dot (for example, 3 • 5 = 15, a •b = 32) or even omitted [for example, 3 × (2 + 4) = 3(2 + 4), 5 × a =5a]. Therefore, it is best not to use the letter x as a variable in the junior grades. Instead, letters that are contextually related are used (for example, the letter n to represent the term number or the number of objects). Students need to understand that the letter represents a quantity, not an object or person (for example, the letter m represents Mary's age, not Mary herself).
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 87.
In the junior grades, students continue to use symbols and to develop a sense of ownership of them. In particular, they gradually replace personal symbols with literal symbols to represent variables (for example, 13 + a = 19). They also use symbols to communicate algebraic reasoning.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 67-69.
The equation is probably the most useful tool in algebra. However, the equation is more complex than we think, so it is often misunderstood. Wagner ("Conservation of Equation and Function Under Transformation of Variable," Journal for Research in Mathematics Education, vol. 12, no. 2, 1982, pp. 107-118, cited in Ontario Ministry of Education, A Guide to Effective Instruction in Mathematics, Grades 4-6 : Modeling and Algebra, 2008, p. 83), for example, presented the equations 7 × W + 22 = 109 and 7 × N + 22 = 109 to students between the ages of 10 and 18 and asked them which of the variables, W or N, had the greater value. Less than half of the students answered correctly that both variables had the same value. Equations are equalities that have one or more indeterminate values, or variables (for example, a + 135 = 178 and m = 3 × b + 4). The ability to represent an equality relationship symbolically using an equation requires a good understanding of the relationship, a good sense of symbolism, and the use of algebraic thinking. Students' first experiences with equations come from problem situations. In fact, they are led to represent problem situations symbolically using an equation, as in the following example.
Example
Peter and Misha are preparing a bag of marbles to give to their friend. Peter puts 40 marbles in the bag. Misha puts some marbles in the bag. There are now 76 marbles in the bag. How many marbles did Misha put in the bag?
Once students understand this situation, they can represent it using the equation 40 + m = 76, where m is the number of marbles Misha put in the bag. It is important to recognize that it is not always easy for students to represent an equality relationship using an equation. For example, it is easy for them to understand the situation where Peter has four more marbles than Misha. However, students may have more difficulty representing this situation with an equation. They will need to use concrete materials before proposing the equation p = 4 + m, where p is the number of marbles Peter has and mis the number of marbles Misha has.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 83-84.
Knowledge: Variable
Variable: A term in an equation (symbols or letters) that can be replaced by one or more values.
A variable can be a symbol or a letter used to represent an unknown value.
Example 1
n + 6 = 20 - 5
There are variables in an equation that represent a relationship between two changing quantities.
Example 2
n + p = 15
There are variables in a formula.
Example 3
Area of a rectangle or parallelogram: A = b × h
There are variables in an equation that generalize an equality relation.
Example 4
a × b = b × a (commutativity of the multiplication)
Note: In an equation, it is possible for two different variables to take the same value at the same time. In the equation a + b = 6, for example, if a takes the value of 3, b will also have the value of 3.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 85.
Knowledge: Numerical Expression
Algebraic expression. An expression is a symbol or set of symbols that can be linked together using an operation sign. Specifically, an expression like 3 + n + 4 is an algebraic expression, since it contains numbers and symbols.
Notes: An expression containing only numbers is a numerical expression, for example 5 - 2.
Expressions do not contain an equal sign; they cannot be called equations.
Knowledge: Equation
An equation is a symbolic way of representing a relationship that can be difficult to understand, in part because there are various types of equations that vary in function depending on the situation. In the junior grades, students encounter four types of equations.
- Equation to solve: An equation such as 2 + n = 14 must be solved. It usually comes from a problem situation and describes an equality relationship. The letter n represents an unknown value that must be determined.
- An equation that represents a relationship between two changing quantities: An equation such as c =2n + 3 is used to express a relationship, such as the relationship between the term number (n) in a non-numeric pattern and the number of toothpicks (c) that make up the term. The letters n and c are variables, since they can take on various values.
- Equation that serves as a formula: To calculate the area (A) of a square, the equation A = c × c can be used. Such an equation is called a formula, since it is used to calculate the area of a square with sides of length c. Such an equation is not solvable.
- Equation that generalizes an equality situation: It is possible to generalize the equality relationship between adding any two identical numbers and multiplying that number by two (for example, 4 + 4 = 2 × 4) by the equation n + n = 2 × n. Such an equation is not solvable and is not a formula. Furthermore, it does not represent a relationship between two changing quantities.
Note: The details of the various types of equations are given in order to recognize that algebraic concepts are not encountered and treated exclusively in the context of modeling and algebra activities. For example, equations that are used as formulas in measurement are regularly found. However, in the junior grades, students do not necessarily have to distinguish between these types of equations.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 89.