C2.2 Solve equations that involve whole numbers up to 50 in various contexts, and verify solutions.
Skill: Solving Equations
In the primary grades, students usually solve equations in the Algebra and Number strands. These equations are usually derived from word problems and are often solved using concrete representations and illustrations. It is essential that they understand the concept of equality as an expression of equilibrium before they begin to solve equations.
In the junior grades, students are more likely to solve equations using a variety of strategies. These equations are usually derived from word problems dealing with a variety of contexts across all of the strands.
Solving such equations means determining the value of the unknown that maintains the equality. Solving equations must be done in a context of understanding and analyzing the equality. It is therefore important to regularly invite students to explain their approach to solving, to justify their actions, and to demonstrate their understanding of the concepts involved in order to prevent solving equations from becoming just a blind application of procedures.
Solving Equations by Systematic Testing (Trial and Error)
In this strategy, students systematically choose potential values of the unknown value and test until one of these values makes the equality true. To solve, for example, the equation 2 × p + 6 = 22, they successively choose p = 1, 2, 3… and find that the equality is true when p = 8.
To solve certain equations, such as 125 - b = 32, students can use number sense strategies to reduce the number of trials. For example, to solve this equation, it would not be wise to proceed using b = 1, b = 2, b = 3, and so on, because that would take too much time. They might think like this: "I know that 125 - 100 = 25, and 25 is close to 32. If I subtract 105, I get 20. I am getting away from the quantity I am looking for. So I'll subtract a little less than 100. I'll try b = 99, b = 98, and so on."
Advantages of solving equations by systematic testing:
- Students highlight what it means to solve an equation - to determine the value of the variable that maintains equality.
- Students work systematically, not randomly. They can also use their sense of numbers.
Solving Equations by Systematic Testing
Communication of the work done can be disorganized, as it can be difficult to keep track of the tests. It is possible to keep such records by creating a table of values. Here is an example of a table of values used to solve the equation 125 - b = 32:
| b | 100 | 105 | 99 | 98 | 95 | 93 |
|---|---|---|---|---|---|---|
| 125-b | 25 | 20 | 26 | 27 | 30 | 32 |
Note: Some notations should be avoided. To solve, for example, the equation 2 × p + 6 = 22, the student who tries p = 1 should not write "2 × 1 + 6 = 22," since this equality is false. Students can evaluate the left-hand side of the equation to get 2 × 1 + 6 = 8 or use the equation in interrogative form (for example, 2 × 1 + 6 22) or write 2 × 1 + 6 ≠ 22.
Solving Equations by Inspection
In this strategy, students recognize the equality relationship represented by the equation. Students compare the quantities involved and use their number sense to determine the value of the unknown. Here are three examples of solving the equation c + 45 = 98 by inspection.
Example 1
A student recognizes that he needs to find the number that, when added to 45, adds up to 98. To do this, he uses his number sense. Since he knows that 45 + 45 = 90, he concludes that the number he is looking for is 8 more than 45, or 53.
Example 2
A student recognizes that removing the same quantity from each side of the equality changes the equation, but the equality is maintained.
c + 45 = 98
c + 45 - 45 = 98 - 45
c = 98 - 45
c = 53
Note: It is important that students do this reasoning in steps, otherwise they may simply apply a misunderstood procedure mechanically. In addition, this reasoning can be used to better grasp the concept of inverse operation - subtraction is the inverse operation of addition.
Example 3
A student decomposes a number, then compares or cancels the numbers.
Image « c » plus 45 equals 98 « C » plus 45 equals 90 plus 8 « decomposed. » « c » plus 45 equals 45 plus 45 plus 8 « decomposed and compared» .« c » equals 53. A trait makes a link between « c » and 45 plus 8. « c » plus 45 equals 98. « C » plus 45 equals 90 plus 8 « decomposed. ». « c » plus 45 equals 45 plus 45 plus 8 « decomposed and compared. ». « c » equals 53. 45 is crossed out on each sign of the equality on the third line.
Advantages of solving equations by inspection:
- Students practice decoding the equation, that is, making sense of the symbolism of the equation. They develop their sense of symbol, equation and equality.
- Students think about operations and numbers instead of trying to use a meaningless procedure.
In the examples above, we see that students can solve the same equation by inspection using various strategies. The strategies comparing terms, decomposing terms , and modifying the equation that were explored in the analysis of an equality apply very well in the situations of solving equations by inspection, since the equation represents an equality. It is important to help students make this connection by pointing out the similarity between an equality and an equation to solve.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 90-92.
Knowledge: Equation
Equations are number statements such that the expressions on both sides of an equal sign are equivalent.
Example
20 + 44 = ____ + 20
a + b = 10
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Knowledge: Variable
A symbol or letter in an equation can represent an unknown value that makes the equality hold true.
Example
In the equation, 10 = Δ + 9, the triangle is a variable, because the value is unknown. It is possible to replace the unknown with a single value, namely 1, to make the equation true.
In the equation 10 = Δ + * or 10 = x + y, the symbols or letters are variables, as they can be replaced by different values.
Source: translated from En avant, les maths!, 3e année, CM, Algèbre, p. 2.