C1.1 Identify and describe repeating, growing, and shrinking patterns, including patterns found in real-life contexts, and specify which growing patterns are linear.
Skill: Recognizing and Describing Repeating Patterns
The repeating pattern is the simplest form of pattern. To recognize it, we must look for the pattern core. It is created when the elements that make up the pattern repeat in the same order (for example, AAB, AAB, AAB, ...). Students must learn to identify the beginning and end of the pattern core.
From Kindergarten through to the end of the junior grades, the patterns that students learn to explore and create should be increasingly complex. When teaching repeating patterns, it is important to consider the following guidelines:
- Spot false leads: Recognizing that an attribute can be a false lead in a pattern. It is important for students to recognize upon which attribute(s) a pattern is based.
- Modify the structure of the pattern: Explore more complex patterns, adding more elements to the pattern or more attributes. Students will then face a cognitive challenge that will lead to new learning.
- Change the representation: Present patterns with different representations of the same structure and check if the students recognize that they have the same structure.
- Explore patterns with a missing element in the pattern: Examining the pattern to determine what element is missing from the beginning, middle, or end of a pattern increases student understanding of relationships. Many activities like this help students understand that the pattern, as a whole, may contain several smaller repeating elements, rather than seeing it as a collection of changing elements with no relationship to each other.
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. 30-33.
Skill: Recognizing and Describing Growing and Shrinking Patterns
To recognize a growing geometric pattern, we must look for the relationship between each figure (pattern rule), or the pattern between the term number and the term value (functional relationship). The number of elements that make up a figure increases from one term to the next.
Source: translated from En avant, les maths!, 3e année, CM, Algèbre, Habiletés liées aux relations dans les suites, p. 3.
A relationship can be represented by a pattern of concrete or semi-concrete figures or by using words. The study of relationships should first involve situations expressed using geometric representations because these representations have a visual and kinesthetic dimension that makes the relationships they represent feel less abstract.
Example of a Geometric Growing Pattern
Growing geometric patterns have the following characteristics:
- The elements that make up each term in the pattern are structured to show a regularity. For example, from one term to the next in the above pattern, a sun is added at the end of each branch.
- The elements that make up each figure in the pattern can show a consistent change (e.g., a sun is added at the end of each branch), as well as the constant (e.g., three suns placed in a "V" shape).
Note: Colour can be used to highlight the pattern core.
- The number of elements that make up each figure and their positions are predictable. In the following, for example, it can be predicted that the shape in the 5th row will be composed of the basic pattern, three suns placed in a V-shape, and four more suns on each branch. It will thus be composed of 11 suns in all.
When an increasing geometric pattern is not structured, as in the example below, it can be more difficult for students to recognize what elements are increasing. Both geometric patterns has an increase in suns from one figure to the next, i.e., 3, 5, 7...
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 42-43.
If the above pattern is represented in a descending order, then this pattern is considered shrinking pattern because the number of elements are decreasing from one term to the next.
Skill: Determining Linear Growing Patterns
The arrangement of elements in geometric growing patterns can help with predicting the location of the elements in the next term. If the number of elements increase at a constant rate then it is a linear growing pattern. Therefore, students can identify, examine, and use the constant rate in other representations to distinguish a linear growing pattern from other types of patterns. In a coordinate system, a linear growing pattern appears as a straight line.
Example: linear growing pattern
In the 1st term, there are 6 toothpicks to create the hexagon and 1 group of 3 toothpicks to create 1 square. In the 2nd term, there are 6 toothpicks to create the hexagon and 2 groups of 3 toothpicks to create the 2 squares. In the 3rd term, there are 6 toothpicks to create the hexagon and 3 groups of 3 toothpicks to create the 3 squares. From one term to the next term, 3 toothpicks are always added to create 1 more square. Since the growth from one term to the next is constant (+3), this pattern is a linear growing pattern.
If we create a graphical representation of this pattern, it forms a straight line like this:
Example: non-linear growing pattern
This pattern has 1 square and 1 triangle in the 1st term, 1 square and 2 triangles in the 2nd term, then 1 square and 4 triangles in the 3rd term. The number of triangles increases first by + 1 triangle compared to the previous term, then by + 2 triangles compared to the previous term. We alternate from + 1 to + 2 from one term to the next. Since the growth from one term to the next is not constant, we can say that it is not a linear growing pattern.
Source: translated from En avant, les maths!, 6e année, CM, Algèbre, Habiletés liées aux relations dans les suites, p. 3-5.
Knowledge: Geometric Pattern
Set of shapes or objects arranged according to a pattern rule.
Knowledge: Repeating, Growing and Shrinking Patterns
A series of figures or objects arranged in a particular order and showing a pattern rule from one term to the next.
Repeating Pattern
A set of figures or objects arranged in an orderly fashion, in which a repeated pattern (pattern core) can be found.
Growing Pattern
A set of figures or objects arranged in which the number of elements increases one term to the next.
Example 1 shows a growing pattern in which 1 square is added to each subsequent term.
Example 2 shows a spiral shell, whose 1st term (basic pattern) is the first spiral in the centre of the shell. The other terms are obtained according to a certain regularity (often associated with the Fibonacci pattern and the golden ratio). The pattern formed by the different spirals is a growing pattern because the size of the spirals increases according to a pattern rule.
Example 3 represents a pyramid, whose first term is a rectangular prism at the top, to which are added blocks whose base area grows according to a certain pattern rule.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Linear Growing Pattern
A pattern that increases (grows) by a value that remains constant. In a coordinate system, it is represented as a straight line.
Examples
Starts with three tiles in position 1 (constant) and, for each subsequent term, it increases by four tiles:
Starts with three tiles in position 1 (constant) and, for each subsequent term, it increases by three tiles:
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Shrinking Pattern
A geometric pattern that involves a decrease in the number of elements from term to term (e.g., 4 circles, 4 triangles, 3 circles, 3 triangles, ...).
Example
Knowledge: Number Pattern
Growing Number Pattern
A pattern where the terms are numbers that grow (increase). The terms of a growing number pattern come from regularities involving addition and multiplication.
Examples
- 4, 6, 8, 10... (The recursive relationship in this pattern is to add 2 to a term in order to obtain the next term)
- 1, 2, 4, 7, 11… (The recursive relationship in this pattern is to add 1 more than the previous time to get the next term)
- 3, 9, 27, 81… (The recursive relationship for this pattern is to multiply a term by 3 to get the next term)
Shrinking Number Pattern
A pattern where the terms are numbers that decrease by repeating subtraction of a positive or repeating multiplication by a factor between zero and one.
Examples
- 14, 11, 8… (The pattern decreases by 3 from one term to the next)
- 144, 72, 36… (The pattern decreases by one-half from one term to the next)
Knowledge: Pattern Core
Composition of the least number of elements in a pattern from which a pattern rule can be created.
Repeating Pattern
The pattern core is:
.
Growing Pattern
The pattern core is:
.
By examining each of the terms in a pattern, we can recognize that they are created by building on the pattern core, since we can see the core within each term.
Knowledge: Attribute
An attribute is a characteristic that describes an object that is observed or manipulated. Attributes can include colour, shape, texture, thickness, orientation, materials, motion, sounds, objects, or letters. The attributes that describe Pattern A below are shape and colour.
Source: Adapted from the Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Knowledge: Term
A term or a position number in a pattern is comprised of at least one value or element (e.g., shape, object, or motion).
Repeating Pattern
In Pattern A, each of the shapes is a term.
1st term:
2nd term:
3rd term:
Growing Pattern
In Pattern B, each of the shapes is a term.
1st term:
2nd term:
3rd term:
Shrinking Pattern
In Pattern C, each of the shapes is a term.
1st term:
2nd term:
3rd term:
Growing Number Pattern
2, 4, 6, 8…
4, 10, 16, 22, 28...
Shrinking Number Pattern
21, 14, 7...
72, 60, 48...
Knowledge: Recursive Relationship
Recursive pattern rule: a rule that is based on the relationship between one term and the next.
Repeating Pattern
Pattern rule in Pattern A: repeat the blue rectangle, green trapezoid, orange triangle pattern, always in the same order.
Note: Describing the pattern core is the pattern rule for a repeating pattern.
Growing Pattern
Recursive pattern rule in Pattern B: the number of squares increases by one from term to term.
Shrinking Non-Numeric Pattern
Recursive pattern rule in Pattern C: the number of cubes increases by one from term to term.
Knowledge: Functional Relationship
A rule that is based on a generalization of the relationship between any term number and its associated term value.
A) Geometric Pattern and Table of Values
Pattern rule: The number of triangles in any term is three times its term number.
The number of triangles (t) is equal to three times the term number (n): t = 3n.
B) Geometric Pattern and Words
Pattern rule: For any term there is one green triangle and the number of squares is twice the term number.
In the 1st position, there is 1 green triangle and 1 group of 2 orange squares.
In the 2nd position, there is 1 green triangle and 2 groups of 2 orange squares.
In the 3rd position, there is 1 green triangle and 3 groups of 2 orange squares.
In the 4th position, there is 1 green triangle and 4 groups of 2 orange squares.
The number of shapes (s) is equal to 2 times the term number (n) plus 1: s = 2n + 1.
Knowledge: Structure
Letters representing the pattern rule of a repeating pattern.
In Pattern A, each element of the pattern can be identified by a letter as follows: blue rectangle (A), green trapezoid (B), orange triangle (C). The structure of Pattern A is therefore ABC.
Pattern A
