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 get longer) 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 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 term affects the next term. Subsequently, observing changes and relationships between these changes helps predict other terms in the pattern and, thus, generalize.
By examining the change from one shape to the next, students observe a pattern rule that helps them predict the shapes in the next terms.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 21-22.
The main goal is to develop students' algebraic reasoning by making them capable of justifying the extension of a non-numeric or number pattern and by explaining the relationships that exist between the terms of the pattern.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 27.
Relationships Between Terms
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 among the terms in the pattern.
In the primary grades, students learn to recognize relationships that exist between terms 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 make it longer. By discovering relationships, they realize that the next terms 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 number pattern and by making explicit the relationships that exist between the terms of the pattern.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 27.
Non-Numeric Patterns
In order to extend a pattern, students must recognize the elements of the pattern and determine their order. By extending a pattern while justifying their choice, 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 balloon, and the pattern is always sailboat, balloon, sailboat, balloon… 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 : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, 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.
"Because our number system is built on a system of patterns and predictability, students must be able not only to identify the patterns that they see but also to give reasons and evidence for why the patterns exist." (Economopoulos, 1998)
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 : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 49-51.
Skill: Making Predictions and Justifying Them
The use of concrete and semi-concrete materials, and then a variety of representations and rules helps students make and justify near and far predictions.
Making a near prediction is to indicate or represent what the next terms in a given pattern will look like. The prediction can be verified simply by extending the pattern.
Making a far prediction is to indicate or represent what a pattern will look like far beyond a given section. Calculations are often required to make a correct prediction or to check its plausibility.
Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Students can best describe a pattern when they understand the relationship between each term in the pattern and the position each term occupies in the pattern. They can do this by numbering each term in the pattern in turn.
In this way, students can refer to specific terms in the pattern; for example, in this pattern, the suns are in the 3rd, 6th , and 9th rows from the left). By analyzing the relationship between position and term, it is easy to predict the next terms in the patternwithout having to extend it. This analysis helps students generalize; for example, a sun will be in the 12th position, since it is in the 3rd position of each pattern. The rank of the sun is always a multiple of 3.
In the problem situation "How many suns are required to complete 10 pattern cores in this pattern?", elementary students can informally discuss, model, create multiple representations, describe them, and conclude by finding the number of suns and 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 : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 38-39.
In growing patterns, there is also a relationship between the position of each shape and the number of elements in each shape. 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 shape in the 1st row has one geometric shape, the shape in the 2nd row has two, the shape in the 3rd row has three, etc. The students therefore see that there are always the same number of geometric shapes as the position of the shape. This observation, or rule, allows them to find any term in the pattern without having to extend it.
Informal discussions dealing with the relationship between position and the number of elements that make up the term can occur in determining the pattern rule and extending the pattern.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 45-46.
Skill: Finding Missing Terms in Patterns
In non-numeric and numeric patterns, students need to determine what is in a predetermined position (before, after, or within the pattern). Therefore, they need to determine the pattern rule and then identify the missing shape or number.
The study of patterns in a number pattern can continue with a number grid or number line with missing numbers. Students must first find the pattern rule in order to discover the missing numbers, and then explain the addition or subtraction pattern rule. Students can use a calculator to solve this type of problem.
Examples
Missing Numbers in a Partial Number Grid
A grid table which certain spaces are numbered. The model shows the numbers 11 and 12 and 2 followed by 2 shaded squares. The following row shows number 21 and 22 and 2 shaded squares. The third row has the first 2 squares shaded and number 33 and 34. In the last row, the first 2 numbers are 41 and 42 and the last 2 squares are shaded.
Missing Numbers on an Open Number Line
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 67.
Knowledge: Pattern Rules
A pattern rule is a uniform phenomenon that defines a pattern and helps to determine its terms.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 16.
Knowledge: Pattern Rules
Pattern Rule
Rule that helps to extend a pattern by respecting the difference between terms (also called constant jump).
Functional Relationship
Functional Pattern Rule : a rule that extends a pattern by establishing the relationship between the term number and its term.
Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.