C1.3 Determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in patterns.
Skill: Determining and Using Rules to Extend Patterns
Understanding Change
Students live in a changing world. Understanding that change is a part of life and that most things change over time (for example, each year their height and weight increases and their feet grow in length) is the final component of developing algebraic thinking. Observed changes can be described qualitatively (for example, I am taller than last year; my hair is longer; the bucket filled with water quickly during the storm; it is colder now than this morning) and quantitatively (for example, I grew 2 cm this year; the bucket of water filled with 50 ml in 30 minutes; the temperature dropped 6°C in 3 hours). Students need to learn to observe and understand changes in patterns.
Change and regularities are two concepts that cannot be separated in the study of patterns. Students realize that a change in one position affects the next position. Subsequently, observing changes and relationships between positions in the pattern helps us to predict other positions in the pattern and, thus, generalize.
Source: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 21-22.
The main goal is to develop students' algebraic reasoning by teaching them how to justify the extension of a non-numeric or number pattern and by explaining the relationships that exist between the positions of the pattern.
Relationships Between Positions
Using a variety of representations and materials, students explore the concept of regularity in non-numeric and number patterns, and communicate, in their own words or through personal representations, their observations and perceptions of the relationships between the terms' positions in the pattern.
In the primary grades, students learn to recognize relationships that exist between terms positions in a pattern. By examining patterns, they identify how this information can be used to determine what needs to be added to a pattern to extend it longer. By discovering relationships, they realize that the next terms positions in the pattern are not chosen randomly. The search for patterns is, in itself, an important problem-solving strategy.
Students continually redefine their mental image of patterns. Their representation is often limited by the examples they are presented with or by their personal experiences. It is therefore important that teachers present a variety of representations and patterns during activities to facilitate the integration of the concept. The key is to develop students' algebraic reasoning by enabling them to justify the extension of a non-numeric or numeric pattern and by making explicit the relationships that exist between the terms of positions in the pattern.
Source: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 27.
Non-Numeric Patterns
In order to extend a pattern, students must recognize the elements of the pattern and how they repeat. determine their order. By extending a pattern while and justifying their choicethinking, students communicate their understanding of what the pattern rule is; for example, a student may say, "I'm going to be the sailboat because right before me is the balloonbeach ball, and the pattern is always sailboat, beach ball, sailboat, beach ball… repeating."
Using their bodies or manipulatives, students can explore the extension of a pattern and make changes more easily. They can also extend a pattern that others have constructed.
Source: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 38.
Number Pattern
Very early on, students become aware of patterns in their environment, in nature, in the objects around them. This is why it is possible to introduce number patterns as early as Grade 1. At the same time, students develop their sense of number, can count in intervals and backwards, and later acquire the concept of addition as a grouping of objects. All of these concepts have an important connection to learning number patterns.
As students begin to explore the base-10 number system, which is in fact the decimal system, they discover that the digits 0-9 are repeated when they count beyond 9 (10, 11, 12, 13, 14, 15…). Seeing and justifying this pattern in the decimal system enhances understanding of number sense and groupings (units, tens, hundreds, etc.). For example, when skip counting by 2, starting at 16, students observe a predictable regularity in the numbers 16, 18, 20, 22, 24, 26, 28, 30, 32.…This is a first step toward exploring multiples of 2. This understanding also leads to an ability to skip count from any number. Similarly, when students count by 5s, they quickly recognize a pattern, that is, that the units digit alternates between the digit 0 and the digit 5 (5, 10, 15, 20…). They can generalize this discovery informally by saying that any number that is a multiple of 5 will end in 5 or 0.
As is the case when learning concepts related to non-numeric patterns, it is by developing the ability to recognize, among other things, number patterns in addition and subtraction that students in the primary grades build their algebraic thinking. The approaches described above for developing this skill also apply to number patterns.
Source: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 49-51.
Skill: Making Predictions and Justifying Them
The use of concrete and semi-concrete materials, along with a variety of representations and rules, helps students learn to make and justify both near and far predictions.
Making a near prediction involves indicating or representing what the next terms in a given pattern will look like. This prediction can be verified simply by extending the pattern.
Making a far prediction involves indicating or representing what a pattern will look like well beyond the given section or the existing representation of the pattern. Calculations are often required to make a correct prediction or to check its plausibility.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Students can best describe a pattern when they understand the relationship between each term element in the pattern and the position it occupies in the pattern. They can do this by numbering each element in the pattern using a position number.
In this way, students can refer to specific terms' elements in the pattern; for example, in this pattern, the suns are in Positions 3, 6, and 9. By analyzing the relationship between the elements and their position in the pattern, it is easy to predict where any element will next appear in the pattern. This analysis helps students generalize; for example, a sun will be in the Position 12, since it is the third and final element in the pattern core of flower, tree, sun, and therefore it always appears in positions that are multiples of 3.
In the problem situation "How many suns are required to complete 10 repetitions in this pattern?", elementary students can informally discuss, model, represent, describe, and conclude how many suns will be needed, justifying their approach. Exploring this type of problem helps students develop their algebraic thinking and serves as a foundation for using a rule and variables in future grades.
Source: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 38-39.
In growing patterns, there is also a relationship between the positions in the pattern and the number of elements. This relationship is a very important mathematical concept, which leads to a more formal generalization, the pattern rule formulation.
For example, by carefully analyzing a pattern, students see that the representation in the Position 1 has one square, the representation in the Position 2 has two squares, the representation in the Position 3 has three squares, and so on. Students can recognize that the number of squares always corresponds to the position in a pattern. This observation, or rule, allows them to find any position in the pattern without having to build it.
Informal discussions dealing with the relationship between the position number and the number of elements in it can help in determining the pattern rule and extending the pattern.
Source: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 45-46.
Skill: Finding Missing Elements in Patterns
For a given non-numeric or numeric pattern, students should be able to determine what element came before, after, or is missing in the pattern.
For example, students can find missing numbers in a number grid or on a number line by using addition or subtraction. Students can use a calculator to solve this type of problem.
Examples
Missing Numbers in a Partial Number Grid
Missing Numbers on an Open Number Line
Source: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 67.
Knowledge: Regularity
A regularity is a uniform phenomenon that defines a pattern and helps to determine its elements.
Source: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 16.
Knowledge: Pattern Rules
A description of how a pattern repeats, grows, or shrinks, based on a generalization about the pattern structure.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Knowledge: Functional Relationship
The relationship between a position and the number of elements in a pattern.
Source: translated from Curriculum de l'Ontario, Programme de mathématiques de la 1re à la 8e année, 2020, Ministère de l'Éducation de l'Ontario.