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 core that repeats continuously (for example, 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.
From Kindergarten 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:
- Change the structure of the pattern: Explore more complex patterns by adding more elements or more attributes. Students will then face a cognitive challenge that will lead to new learning.
- Change the mode of representation: Present patterns with the same structure, but constructed in different modes of representation, and check whether students recognize that they have the same structure.
- Explore patterns with a missing element: Examining the pattern to determine the missing element at the beginning, middle, or end of the pattern core increases understanding of relationships. Many such explorations help students understand the pattern as a whole that contains several cores, rather than as a sequence of changing elements without any relationship.
- Spotting false leads: Recognizing that an attribute can be a false lead in a pattern helps develop algebraic reasoning.
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 pattern, we must look for the relationship between each term (recursive relationship), or the relationship between the position and the term (functional relationship). The number of elements that make up a term increases from one term to the next.
Source : En avant, les maths!, 3e année, CM, Algèbre, p. 3.
The study of relationships should first involve situations expressed using non-numerical patterns because these situations have a visual and kinesthetic dimension that makes the relationships they represent less abstract.
Example: Non-numeric growing pattern
Non-numeric growing patterns have the following characteristics:
- The elements that make up each term in the pattern are ordered and show a consistent change. For example, from one term to the next in the above pattern, a sun is added at the end of each branch.
- The pattern core can be found in each term, so that each term comes from the growth of the previous one. For example, in the following, the starting term is three suns placed in a "V" shape.
Note: Colour can be used to highlight the pattern core.
- The number and location of the elements that make up each term are predictable; for example, in the following, we can predict that the term in the 5th position will be composed of the pattern core, 3 suns placed in a V shape, and 4 more suns on each branch. It will thus be composed of 11 suns in all.
Familiarity with the characteristics of growing and shrinking non-numerical patterns enables students to recognize these types of patterns. For example, students can recognize that the following is not a non-numerical growing pattern, even though each term is composed of the same number of suns as in the previous pattern, that is, 3, 5, 7…, because the suns are not arranged with order and regularity that make it possible to predict their location 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 shrinking form, the student may also find that it is not considered a shrinking non-numerical pattern. Like the example above, the suns are not arranged with order and regularity that allow one to predict their location in the next term.
Skill: Determining Linear Growing Patterns
Linear growing patterns, like non-linear patterns, are arranged in an order and with a regularity that helps predict the location of the elements in the next term. However, linear growing patterns always increase in a constant way. Therefore, students can identify, examine, and use the constant rate that defines the linear pattern to distinguish it from other non-numerical patterns. In a coordinate system, a linear growing pattern is represented as a straight line.
Example: non-linear growing pattern
Linear increasing sequence.Rank one: one yellow square and one blue triangle.Rank 2: one yellow square and 2 blue triangles.Rank 3: one yellow square and 4 blue triangles.Rank 4: one yellow square and 5 blue triangles.Rank 5: one yellow square and 7 blue triangles.
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.
Example: linear growing pattern
This is a 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.
If we create a graphical representation of this pattern, it forms a straight line like this:
Source : En avant, les maths!, 6e année, CM, Algèbre, p. 3-5.
Knowledge: Non-Numeric Pattern
A series of shapes or objects arranged in a particular order and showing a regularity from one term to the next.
Knowledge: Repeating Pattern
A pattern in which a core repeats continuously (for example, AAB, AAB, AAB, …).
Repeating Pattern
Growing Pattern
Example 1 shows a growing pattern; the pattern core is a square to which 1 square is added to each subsequent term.
Example 2 shows a spiral shell, whose 1st term (pattern core) 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 sequence 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 regularity.
Example 3 shows a pyramid, whose 1st term is a rectangle-based prism at the top, to which additional rectangle-based prisms 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.
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.
Example
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 pattern that involves a regression (for example, a decrease in the number of elements) from term to term (for example, AAAA, AAA, AA, A).
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 decreasing numbers. The terms of a shrinking number 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: Pattern Core
The smallest part of a non-numeric pattern from which the pattern rule is created.
Repeating Pattern
The pattern core is:
.
Non-numeric Growing Pattern
The pattern core is:
.
By examining each of the terms in the 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 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 this pattern, the attributes that describe the 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.
Source: Adapted from the Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Knowledge: Term
Each element (for example, shape, object) that makes up a non-numeric pattern or each number in a number pattern.
Repeating Pattern
In the following, each of the two-dimensional shape is a term.
1st term:
2nd term:
3rd term:
Non-Numeric Growing Pattern
In the following, each set of elements is a term.
1st term:
2nd term:
3rd term:
Non-Numeric Shrinking Pattern
In the following, each structure 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: Pattern Rule
A description of the relationship between one term and the next, based on a generalization about the pattern structure.
Repeating Pattern
Pattern rule: repeat the blue rectangle, green trapezoid, orange triangle pattern core, always in the same order.
Non-Numeric Growing Pattern
Pattern rule: a cube is always added to the bottom row of the previous term.
Shrinking Non-Numeric Pattern
Pattern rule: a cube is always removed from the bottom row of the previous term.
Knowledge: Functional Relationship
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 term is 3 times its term number.
The number of triangles (t) is equal to three times the term number (n) : t = 3n.
In this next pattern, the green triangle is constant from one term to the next and the number of groups of 2 orange squares corresponds to the term number.
Nonnumeric sequence with increasing patterns. 1st row, there is 1 green triangle and 1 group of 2 orange squares. 2nd row, there is 1 green triangle and 2 groups of 2 orange squares.3rd row, there is 1 green triangle and 3 groups of 2 orange squares. 4th row, there is 1 green triangle and 4 groups of 2 orange squares.
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
Knowledge: Term Number
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.
Examples
In Pattern B and Pattern C, each term has its position (term number) : term 1 occupies the 1st position, term 2 occupies the 2nd position, etc.
Pattern B
Pattern C
Source : Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, p. 29.