C2.1 Identify and use symbols as variables in expressions and equations.
Skill: Determining and Using Symbols as Variables
Understanding the meaning of the symbol that represents a variable requires a high level of abstraction. Understanding the meaning of symbols and comfort in manipulating symbols requires a gradual process. Students first learn to describe relationships using a variety of representations. They should continue to 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) states 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, a 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 a 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 expression 3b +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.
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: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 85-87.
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: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 69.
The equation is a useful tool in algebra. Equations can be complex and are 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 variables (for example, a + 135 = 178 and m = 3 × b + 4). The ability to represent an equality relationship symbolically using an equation requires an understanding of the relationship and the use of symbols. Students' early experiences with equations involved representing and solving word problems.
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 the problem, 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 may need to use concrete materials before proposing the equation p = 4 + m, where p is the number of marbles Peter has and m is the number of marbles Misha has.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 83-84.
Knowledge: Variable
A variable can be a symbol or a letter used to represent an unknown value.
Example 1: Equation
n + 6 = 20 - 5
Variables can be used to generalize a relationship.
Example 2: Relationship Between Two Variables
n + p = 15
Formulas are generalizations of a relationship that usually involves multiple variables.
Example 3: Formula
Area of a rectangle or parallelogram: A = b × h
Variables are used to generalize number properties.
Example 4: Generalized Number Properties
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: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 85.
Knowledge: Numerical and Algebraic Expressions
Numeric expression: An expression containing only numbers, for example 5 – 2.
Algebraic expression: An expression with one or more terms containing numbers and symbols that are linked together by an operation.
- Example 1: 3ab is an algebraic expression with one term where the numbers and the variables are linked by the operation of multiplication.
- Example 2: a + 4 has two terms where the number 4 and the variable a is liked by the operation of addition. Terms are separated by the operations of addition and subtraction.
Expressions do not contain an equal sign; thus, they are not equations.
Knowledge: Equation
An equation is a symbolic way of representing an equality. In the junior grades, students encounter four types of equations.
- Equation with one variable: An equation such as 2 + n = 14 is solved to determine the value of the variable that makes the equation hold true.
- Equation with two variables: An equation such as c = 2n + 3 is used to express a relationship, such as the relationship between the term number (n) and the term value (c). The values for c are dependent on the values for n. This equation has multiple solutions.
- Equation that is a formula: A formula such as A = c × c can be used to represent the area (A) of any square, where c represents the side length. The area of a particular square can be determined by substituting a side length into the formula. The side length of a particular square can be determined by substituting the ara for that square into the formula.
- Equation that generalizes a property situation: The equality relationship between adding any two identical numbers and multiplying that number by two can be expressed using the equation n + n = 2 × n. This equation can be verified by testing that the equation holds true for various values of n, such as 4 + 4 = 2 × 4.
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: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 89.