C1.1 Identify and describe repeating, growing and shrinking patterns, including patterns found in real-life contexts
Skill: Recognizing and Describing Repeating Patterns
The repeated pattern is the simplest form of pattern. To recognize it, we must look for the core that repeats continuously (e.g., AAB, AAB, AAB, ...). Students must learn to identify the beginning and end of the core in the pattern.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 30.
In addition, it is important for students "read" the pattern by naming and touching each consecutive element of the pattern so that they are aware of the repetition.
For example, in the image below, the student has created a pattern where the beginning is a yellow hexagon with an orange square on top of it and the end is two green triangles, each having one vertex touch a vertex of the other triangle.
From Kindergarten through to the end of the Junior division, the patterns that students explore and create must be increasingly complex. From the primary division through to the end of the junior division, the patterns that students explore and create must be increasingly complex. The following are suggestions for increasing the complexity of repeating patterns:
- 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.
- Spot false leads: Recognizing that an attribute can be a false lead in a pattern helps develop algebraic reasoning.
Example
Pattern A
Sequence A: A red hexagon with an orange square on top and after two green triangles with vertices touching. A yellow hexagon with a beige square on top and after two green triangles with vertices touching. A blue hexagon with a green square on top and after two green triangles with vertices touching.
In this pattern, the colours used are not an attribute of the pattern rule. Therefore, students should eliminate the colour attribute and stick to the shape attribute (triangle, hexagon, square).
Pattern B
In this pattern, the different shapes and colours create false leads that must be eliminated in order to discover the attribute (figure on a vertex with a similar figure touching it on a flat side, figure placed on a flat side).
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 31-33.
Skill: Recognizing and Describing Growing and Shrinking Patterns
To recognize a growing geometric pattern, we must look for the relationship between each figure (recursive relationship), or the pattern between the term number and the term (functional relationship). The number of elements that make up a figure increases from one term to the next.
Source : En avant, les maths!, 3e année, CM, Algèbre, p. 3.
A relationship can be represented by a situation that is expressed using a sequence of concrete or semi-concrete figures or using words. The study of relationships should first involve situations expressed using non-numerical sequences because these situations have a visual and kinesthetic dimension that makes the relationships they represent less abstract.
Example of a non-numeric growing pattern
Growing non-numeric patterns have the following characteristics:
- The elements that make up each figure in the pattern are ordered and 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 pattern can be found in each figure, so that each figure comes from the growth of the figure in the previous row. For example, in the following, the base is three suns placed in a "V" shape.
Note: Colour can be used to highlight the basic pattern.
- The number and location of the elements that make up each figure are predictable. For example, in the following, the fifth row will have the basic pattern of three suns in a "V" shape and four more suns on each branch. It will thus be composed of 11 suns in all.
Familiarity with the characteristics of growing and shrinking non-numeric patterns enables students to recognize these types of patterns. For example, students can recognize that the following is not a non-numeric growing pattern, even though each figure is composed of the same number of suns as in the previous pattern, i.e., 3, 5, 7…, because the suns are not arranged with order and regularity that make it possible to predict the location of the suns that will make up the figure in the next term.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 42-43.
If the same illustrations are presented in descending form, the student may also find that it is not considered a shrinking non-numeric pattern. Like the example above, the figures are not arranged with order and regularity that allow one to predict the location of the suns that will make up the figure in the next term.
Knowledge: Non-Numeric Pattern
A series of figures or objects arranged in a particular order and showing a regularity from one term to the next.
Repeating Pattern
A pattern in which a core repeats continuously (e.g., AAB, AAB, AAB, …).
Example
Growing Pattern
A pattern that involves an increase from term to term (e.g., AB, AABB, AAABBB).
Example
Example 1 shows a growing pattern; the pattern is a square to which 1 square is added to each subsequent term.
Example 2 represents a spiral shell, whose 1st term (basic pattern) is the first spiral in the center of the shell. The other terms are obtained according to a certain regularity (often associated with the Fibonacci sequence and the golden ratio). The sequence formed by the different spirals is increasing because the size of the spirals increases according to a regularity.
Example 3 represents a pyramid, whose 1st term is a rectangular prism at the top, to which stones are added whose base area grows according to a certain regularity.
Source : Curriculum de l’Ontario. Programme-cadre de mathématiques de la 1re à la 8e année. 2020. Ministère de l’Éducation de l’Ontario.
Shrinking Pattern
A pattern that involves a regression (e.g., a decrease in the number of elements) from term to term (e.g., AAAA, AAA, AA, A).
Example
Knowledge: Numeric Pattern
Growing numeric pattern: A pattern where the terms are numbers that grow (increase). The terms of a growing 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 numeric pattern: A pattern where the terms are decreasing numbers. The terms of a shrinking pattern come from regularities involving subtraction and division.
Examples
- 14, 11, 8… (The recursive relationship for this pattern is to subtract 3 from the value of a term to get the next term)
- 144, 72, 36… (The recursive relationship for this pattern is to divide the value of a term by 2 to get the next term)
Knowledge: Core
The smallest part of a non-numeric pattern from which the pattern rule is created.
Repeating Pattern
The basic pattern is:
Growing Pattern
The basic pattern is:
By examining each of the terms in the pattern, we can recognize that they are created by building on the core, since we can see the core within each term.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 198.
Knowledge: Attribute
Repeating Pattern
An observable property of a person or object. Attributes may include colour, shape, texture, thickness, orientation, materials, motion, sound, objects, or letters. The attribute is reflected in an object by a characteristic. For example, if the attribute is colour, the characteristics may be red, blue, yellow.
In Pattern A, the attributes that describe the various terms of the pattern are shape and colour. The characteristics of the shape are rectangle, trapezoid and triangle. The characteristics of the colour are blue, green and orange.
Pattern A
Non-numeric growing or shrinking pattern
In a non-numeric growing or shrinking pattern, attribute analysis is no longer important since the focus is on pattern growth.
Knowledge: Term
Each element (e.g., figure, object) that makes up a non-numeric pattern or each of the quantities constituting a number in a sequence.
Repeating Pattern
In Pattern A, each of the figures is a term.
Pattern A
1st term:
2nd term:
3rd term:
Growing Pattern
In Pattern B, each of the figures is a term.
Pattern B
1st term:
2nd term:
3rd term: 
Shrinking Pattern
In Pattern C, each of the figures is a term.
Pattern C
1st term: 
2nd term: 
3rd term: 
Knowledge: Rule of Regularity
A description of the relationship between one term and the next, based on a generalization about the pattern structure.
Repeating Pattern
Pattern rule in Pattern A: repeat the blue rectangle, green trapezoid, orange triangle pattern, always in the same order.
Pattern A
Non-numeric growing pattern
Pattern rule: a cube is always added to the bottom row of the previous figure.
Pattern C
Non-numeric shrinking pattern
Pattern rule in Pattern D: a cube is always removed from the bottom row of the previous figure.
Pattern D
Knowledge: Matching Rule
A description of the relationship between two variables, the term number and the value of the corresponding term, based on a generalization about the pattern structure.
In the following, the functional relationship is that the number of triangles in the figure is 3 times its term number.
The number of triangles (t) is equal to three times the corresponding term number (n): t = 3n.
In this next pattern, the green triangle is constant from figure to figure and the number of groups of 2 orange squares corresponds to the term number of the figure.
|
In the 1st term, there is 1 green triangle and 1 group of 2 orange squares. In the 2nd term, there is 1 green triangle and 2 groups of 2 orange squares. In the 3rd term, there is 1 green triangle and 3 groups of 2 orange squares. In the 4th term, there is 1 green triangle and 4 groups of 2 orange squares. |
The number of plane figures(f) is equal to 2 times the term number (n) plus 1: f
=2n + 1
Knowledge: Structure
Representation with letters of the elements 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
Knowledge: Term
The position that each term occupies in a pattern, indicated by a number. It is used to help describe the functional relationships in a pattern and to predict the next terms in the pattern without having to extend it.
Repeating Pattern
In Pattern A, there is a blue rectangle in term number 1, term number 4 and term number 7, a green trapezoid in term number 2, term number 5 and term number 8, etc.
Pattern A
Growing non-numeric pattern
In the sequence B, each figure has its rank: figure 1 occupies the 1st rank, figure 2 occupies the 2nd rank, etc.
Pattern C
Non-numeric shrinking pattern
In the sequence C, each figure has its rank: figure 1 occupies the 1st rank, figure 2 occupies the 2nd rank, etc.
Pattern C
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 29.