C2.3 Identify and use equivalent relationships for whole numbers up to 100, in various contexts.
Skill: Determining and Using Equivalent Relationships Involving Whole Numbers up to 100
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students' algebraic thinking, the traditional teaching objective of learning to calculate is not omitted; it is largely superseded.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.
Important note: Although the properties of operations (for example, commutativity) are explored in the Numbers strand, the process of coming to understand and generalize them falls under algebraic thinking.
Generalize
To generalize is to draw valid conclusions, true in all cases, from the observation and analysis of a few examples
(H. Squalli, "The Development of Algebraic Thinking in the Elementary School: An Example of Reasoning with Mathematical Concepts", Mathematical Snapshots, Vol. XXXIX, Fall 2002, p. 9, adapted, as cited in Ontario Ministry of Education, Guide d’enseignement efficace des mathématiques, de la maternelle à la 34e5 année – Modélisation et algèbre6, fascicule 2, 2008, p. 16)).
In situations of equality, students can more easily formulate a generalization when it follows a process of proposition and verification of a conjecture.
A conjecture is the expression of an idea that is perceived to be true in all similar situations.
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, the conjecture could be tested using various numbers as well as 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 grades, conjectures are usually expressed in words by students. They may also be represented using concrete or semi-concrete materials to illustrate mathematical reasoning as clearly as possible.
It is important to expose students to a variety of problem situations that encourage them to practice the skill of making and testing a conjecture; for example, present them with the number sentence 50 + 6 - 6 = 50, and ask them what they notice. Then, propose the following conjecture to them: When you add and subtract the same number in a number sentence, is it the same to add or subtract zero. Then invite students to discuss this conjecture and determine if it is still true.
Students test this conjecture with other number sentences. They may not be convinced that it applies to any number sentence or to all numbers, especially large numbers. In the course of the discussion, they can propose their own conjectures as illustrated below.
Two 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 two 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 number sentences, students 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, the first conjectures usually need to be reformulated or clarified. It is therefore ideal to practice formulating a conjecture as a class, as shown in the example below. In the discussion, students can point out the limitations of someone else's conjecture and contribute to the formulation of a clearer and more relevant common conjecture. It is important, however, to establish a learning environment in which students perceive the questions of others as positive interactions that can fuel the exchange.
Example 3
Present students with the number sentence 54 + 0 = 54 and ask them if it is true or false.
- Student: The number sentence is true.
- Faculty : How can you tell?
- Student: When a zero is added to a number, it doesn't actually add anything, so we get the starting number.
Present the students with other similar number sentences. After several such exchanges, ask them to formulate a conjecture.
- Student: All numbers added with a zero remain the same.
- Another student presents a counterexample: No, 10 + 10 = 20. The numbers 10 and 10 have zeros in them. When added together, they do not remain 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 a zero?
- Student: No, added to another number.
After much discussion, the following formulation is adopted: Zero, 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.
When numbers are decomposed, the parts are equivalent to their whole.
Remark: Commutativity and associativity allow students to see the equality of a number sentence, demonstrate their understanding of these properties, and see the equality of a mathematical expression without performing calculations.
Students are not expected to be able to name or define these properties, but to use them appropriately.
Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Skills Related to Situations of Equivalence
In the primary grades, students develop the ability to recognize, explain, create, re-establish, and maintain equivalent situations through the application of strategies and models (for example, 10 frames, number line). These skills are to be developed in each grade using progressively larger numbers, consistent with curriculum requirements.
Initially, situations of equality and inequality are explored primarily orally and with concrete materials. Later, 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.
Skill: Recognizing 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 provides students with an opportunity to make the transition from concrete to semi-concrete, semi-abstract, and abstract
(Conseil des écoles catholiques de langue française du Centre-Est, Les mathématiques un peu, beaucoup, à la folie : Guide pédagogique - Modélisation et algèbre, 1 année, 2003, p. 6, as cited in Ontario Ministry of Education, Guide d’enseignement efficace des mathématiques, de la maternelle à la 36e7 année – Modélisation et algèbre8, fascicule 2, 2008, p. 51) .
In problem solving, the use of concrete and semi-concrete materials, as well as models, allows students to recognize and represent situations of equality and inequality.
Examples
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 are able to approach the purely symbolic representation (the mathematical expression) of this situation. Furthermore, in order to determine the nature of the relationship between the quantities, students need to understand that the items on either side of the equal sign are data to be analyzed, not just expressions to be calculated.
Example 1
Exploring the commutativity property allows students to see the equality of the number sentence 43 + 24 = 24 + 43. It is important, through interventions, to encourage students to observe that the same numbers are on both sides of the equality symbol and that the addition terms are simply switched.
Example 2
The strategy of canceling terms or equal expressions allows students to see the equality of a mathematical expression. Canceling terms involves crossing out the identical terms that appear on either side of the equal sign. Crossing out or canceling terms that are on either side of the equal sign makes it easier to establish relationships between the remaining terms and helps students develop their algebraic reasoning. For example:
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 39-41.
Skill: Explaining a Situation of Equality
Students need to discuss what is equal/unequal, same/different, more than/less than, balanced/unbalanced. It is through authentic dialogue that students construct the meaning of equality.
(J. Taylor-Cox, "Algebra in the Early years? Yes!" Young Children: Teaching and Learning about MATH, January 2003, p. 17, as cited in Ontario Ministry of Education, Guide d’enseignement efficace des mathématiques, de la maternelle à la 34e5 année – Modélisation et algèbre6, fascicule 2, 2008, p. 53).
In order to develop their ability to explain a situation of equality, students need to address different steps. The transfer from concrete to symbolic representation is easier when the equality relationship is constructed by following these different steps.
- Explore using concrete or semi-concrete materials.
The student demonstrates, using toys, the following situation: adding 0 toys to 10 toys. - Describe using words and materials.
Student says, "If I add 0 toys to 10 toys, the quantity will not change, since I am not adding anything. It will still equal 10 toys." - Represent using symbols.
The student expresses equality symbolically by writing the number sentence 10 + 0 = 10. - Generalize to all numbers
Appropriate questions in 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 allows students to 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 27 and add a leap of 5. I take another leap of 5 in the other direction, so I am back to 27. It is as if I never added a leap of 5."
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 41-42.
Skill: Creating a Situation of Equality
To get students to achieve equality, it is important to initially present them with a situation in which concrete materials are used. Ask them to represent it with a number sentence, compare the different expressions proposed by the other 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, two balls, and one game. The second picture show an equal situation by making an equation: one, plus, 3, plus, 4, plus, two, equals, ten. Ten, equals, two, plus, 4, plus, 3, plus, one.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 43.
Afterwards, students can create their own equality situation. They should be given the opportunity to write number sentences and represent them with concrete materials. Students should have the opportunity to create equality situations represented by numberl sentences made up of large numbers in order to be able to use the properties of operations or strategies instead of calculation.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 42-43.
Knowledge: Equivalent Relationships
A relationship that compares quantities to show that they have the same value.
Example
352 is equivalent to 350 and 2.
Source: En avant, les maths! grade3, CM, Algebra, p. 2.
Examples
- Represent a number in different ways using rods.
- Represent a number in different ways using base 10 materials.
- Represent a number in different ways using a Rekenrek.
- …
Knowledge: Equality
Relationship between two equal quantities.
Knowledge: Number Sentence
A symbolic representation that represents a relationship. In a number sentence, there is no unknown or variable.
Example
True (equal) sentences \(75 + 5 = 5 + 75\) or \(50 = 20 + 20 + 10\)
False sentence (inequal) \(100 = 95 - 5\) or \(45 + 10 = 15 + 45\)