C2.4 Solve inequalities that include an operation and natural numbers up to 50, and verify and present solutions using models and graphical representations.
Skill: solve inequalities and verify and present solutions using models and graphical representations
To facilitate learning about the concept of inequality, teachers should provide students with activities that encourage them to analyze inequality situations and treat them algebraically. They then discuss with students the strategies used to analyze inequalities, emphasizing those that use concrete and semi-concrete representations, and that focus on the meaning of the inequality rather than on the mechanical application of a procedure or tedious calculations.
Representing solutions with a number line is a strategy that allows students to analyze an inequality using their sense of number, operations and symbol, and to find the range of valid values in an inequality situation.
Students need to consolidate these strategies as they form the basis for understanding the algebraic manipulations they will be exposed to in later grades. Students can also use these strategies to solve simple equations.
In what follows, each of these strategies is first presented in the context of developing a sense of inequality.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 200.
Represent solutions using a number line
This strategy involves carefully reading the given mathematical sentence and replacing the variable to find the range of valid values in the inequality situation. A table is used to find several values for the variable. Afterwards, the solution can be represented graphically on a numerical line.
Example
5 × n < 50
- The first column in the table represents the value by which the variable n will be replaced in the algebraic expression 5 × n.
- The second column in the table represents the solution of the algebraic expression when the variable n is replaced by the value in the first column.
Example
5 × n
5 × 6
5 × 6 = 30
- The third column in the table confirms or refutes the validity of the value of the variable n.
Example
Is 5 × 6 < 50? The answer is yes.
| n | 5 x n | < 50 |
|---|---|---|
| 0 | 0 | yes |
| 1 | 5 | yes |
| 2 | 10 | yes |
| 3 | 15 | yes |
| 4 | 20 | yes |
| 5 | 25 | yes |
| 6 | 30 | yes |
| 7 | 35 | yes |
| 8 | 40 | yes |
| 9 | 45 | yes |
| 10 | 50 | no |
| 11 | 55 | no |
| 12 | 60 | no |
The range of valid values can be represented with a numerical line:
Source: The Ontario Curriculum. Mathematics, Grades 1-8, Ontario Ministry of Education, 2020.
The solution is therefore n < 9.
Note: On a numerical line, an empty point indicates a strict inequality relationship ("is less than" or "is greater than"); a full point indicates a broad inequality relationship ("is less than or equal to" or "is greater than or equal to").
Knowledge: inequality
Relationship of order between two expressions or two quantities.
Inequality is represented by various signs including :
< (is less than, is smaller than);
> (is greater than, is greater than);
≤ (is less than or equal to);
≥ (is greater than or equal to).
Non-equality
Relationship between two expressions or two quantities that do not have the same value.
Non-equality is represented by the sign ≠ (does not equal, does not equal).
Example
5 ≠ 5 + 1
(3 × 5) + 4 ≠ 3 (5 + 4)
8a ≠ 25
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 70.