C2.3 Identify and use equivalent relationships for whole numbers up to 50, in various contexts.
Skill: Determining and Using Equivalence Relationships Involving Numbers Up To 50
In a mathematics class aimed at developing students' algebraic thinking, the traditional teaching objective of learning to calculate is not omitted; it is largely superseded.
Developing algebraic thinking is a complex process that relies on three fundamental processes: abstracting, generalizing, and operating on the unknown.
Note: Although properties of operations, such as the commutative property, are explored in the Number strand, the process of coming to understand and generalize them falls under algebraic thinking.
Generalize
It is drawing valid conclusions, true in all cases, from the observation and analysis of a few examples (adapted from Squalli, 2002, p. 9).
In situations of equality, students can more easily formulate a generalization when it follows a process of proposing and verifying a conjecture.
A conjecture is the expression of an idea that is perceived to be true in any similar situation.
When students see a recursive phenomenon while exploring various equality situations, they are able to propose a conjecture; for example, they might say that if you add the number 0 to any number, the initial quantity does not change.
Students should then check whether their conjecture is valid in other similar situations. For example, in the situation in the previous example, they could check it with various numbers as well as with concrete materials.
When a conjecture seems to apply to all similar situations, students formulate a generalization in words or using symbols.
Example 1
There are three important steps in the generalization process.
In the primary and junior divisions, conjectures are usually expressed in words. They may also be represented using concrete or semi-concrete materials to illustrate their mathematical reasoning as clearly as possible.
Teachers need to expose students to a variety of problem situations that challenge them to practice the skill of making and testing a conjecture; for example, they present the mathematical sentence (50 + 6 - 6 = 50) and ask them what they notice. Then, they offer them the following conjecture: "I wonder, when you add and subtract the same number in a math sentence, if it is the same as when you add or subtract zero." Teachers then invite students to discuss this conjecture with each other and determine if it is still true.
Students test this conjecture with other mathematical sentences. They may not be convinced that it applies to any mathematical sentence or to all numbers, especially large numbers. In the course of the discussion, they can propose their own conjectures as illustrated below.
2 students negotiate their mathematical reasoning. Student one says ‘’ The mathematical equation 100, plus, 5, minus, 5, equals, 100 is true because if we subtract a number from itself, it is like if we have never added it. Therefore, the phrase becomes 100, equals100. ‘'Student 2 says’’I believe that the mathematical equation is true because subtracting a number from itself is equivalent to adding a zero. The quantity does not change. Therefore, the equation becomes 100, plus, zero, equals, 100’’
After a check of various mathematical sentences, students can conclude that the conjecture is true and formulate a generalization.
Example 2
Since students' vocabulary in elementary school is not yet very developed and precise, initial conjectures usually need to be rephrased or clarified. Ideally, therefore, the formulation of a conjecture should be practiced in a class setting, as shown in the example below. During the discussion, students can point out the limitations of a peer's conjecture and contribute to the formulation of a clearer and more relevant common conjecture. However, it is important for teachers to establish a learning environment in which students perceive each other's questions as positive interactions that can fuel the exchange.
Example 3
Teachers present the mathematical sentence (34 + 0 = 34) and ask students if it is true or false.
- Student: "It is true."
- Teacher: "How can you tell?"
- Student: "When a zero is added to a number, it doesn't actually add anything. Therefore, we get the starting number."
The teacher presents other similar mathematical sentences. After several such exchanges, the teacher then asks students to formulate a conjecture.
- Student: "All numbers added with a zero remain the same."
- Another student presenting a counterexample: "No, because (10 + 30 = 40). The numbers 10 and 30 have zeros in them. When added together, they don't stay the same."
After further discussion, a student formulates another conjecture:
- Student: "When you add a zero to another number, you get the other number."
- Other student: "That's not true."
- Teacher: "So, are you referring to the number that is right next to the zero?"
- Student: "No, added to another number."
After much discussion, the following formulation is adopted: "Zero, when added to another number, is equal to that number." When students see that this conjecture applies to all numbers, they can generalize.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 8-11.
The change from a concrete or semi-concrete representation to a symbolic one, and vice-versa, helps to understand the relations of equality.
Note: The commutative property is an example of an equivalence relation.
Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Skills Related to Equivalence Situations
In the primary grades, students develop the ability to recognize, explain, create, restore, and maintain equivalence through the application of strategies and models such as 10 frames and the number line. These skills are to be developed each grade using progressively larger numbers, consistent with curriculum requirements.
Initially, situations of equality and inequality are explored primarily orally and with concrete materials, and then students are gradually exposed to symbolic representation. However, the use of concrete materials remains just as important and must be used in conjunction with the more abstract representations.
Ability to Recognize a Situation of Equivalence
"The effective use of concrete materials promotes the learning of algebraic concepts at all levels of students […]. Since this strategy involves the senses, including touch, sight, and hearing, it gives them the opportunity to make the transition from concrete to semi-concrete, semi-abstract, and abstract." (Conseil des écoles catholiques de langue française du Centre-Est, 2003, p. 6)
In a problem-solving approach, the use of concrete and semi-concrete materials, as well as models, helps students recognize and represent situations of equality and inequality.
Examples
Ten frame
Double open number line
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 40.
It is only after having manipulated various models several times with the same goal, that is, to recognize a situation of equality or inequality, that students will be able to approach the purely symbolic representation (the mathematical sentence) of this situation. Furthermore, to determine the nature of the relationship between quantities, they must understand that the elements on either side of the = sign are data to be analyzed and not just expressions to be calculated.
Ability to Explain a Situation of Equality
"Students need to discuss what is equal/unequal, the same/different, more than/less than, in balance/out of balance. It is through authentic dialogue that they construct meaning about equality." (Adapted from Taylor-Cox, 2003, p. 17)
To develop their ability to explain a situation of equality, students must go through different stages. The transfer from the concrete to the symbolic representation is easier when the equality relationship is constructed by following these different steps.
1. Explore using concrete or semi-concrete materials
Using toys, a student illustrates the following situation: adding 0 toys to 10 toys.
2. Describe with words and materials
If I add 0 toys to 10 toys, the quantity will not change, since I am not adding anything. It will still be equal to 10 toys.
3. To represent with symbols
The student can then express this equality symbolically by writing the mathematical sentence 10 + 0 = 10.
4. Propose a conjecture
By exploring several similar situations, the student can assume that adding 0 never changes the quantity.
5. Generalize for all numbers
Appropriate questions during other similar situations lead the student to generalize that adding 0 to any quantity does not change the quantity.
Students also develop their ability to explain a situation of equality by using models. Indeed, the use of models helps students communicate their reasoning effectively.
Example
To explain the expression 27 + 5 - 5 = 27, students can use an open number line to support their reasoning: "I take a leap of 27 and add a leap of 5. I take another leap of 5 in the other direction, so I am back to 27. It's like I never added a leap of 5."
Ability to Create a Situation of Equality
To get students to create an equality situation, it is important to start by presenting them with an equality situation represented with concrete materials. Ask them to represent it with a mathematical sentence, compare the different sentences proposed by the students and determine if they are all true.
Examples
2 images that represent a situation with equal parts. Image one shows an equal situation represented by concrete materials: 3 lizards, 4 loops, 2 balls, and one game. The second picture show an equal situation by making an equation: one, plus, 3, plus, 4, plus, 2, equals, ten. Ten, equals, 2, plus, 4, plus, 3, plus, one.
Afterwards, students can create their own equality situation. Teachers suggest that they write mathematical sentences and represent them using concrete materials. Students should have the opportunity to create equality situations represented by mathematical sentences with larger numbers so that they use the properties of operations or strategies instead of calculations.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 39-43.
Knowledge: Equivalence Relationship
A relationship that compares quantities to show that they have the same value.
Example
352 is equivalent to 350 and 2.
Knowledge: Equality
Relationship between two equal quantities.
Knowledge: Mathematical Sentence
A symbolic representation that represents a relationship. In a mathematical sentence, there is no unknown or variable.
Example
True (equal) sentences: \(75 + 5 = 5 + 75\) or \(50 = 20 + 20 + 10\)
False (unequal) sentences: \(100 = 95 - 5\) or \(45 + 10 = 15 + 45\)