C1.1 Identify and describe a variety of patterns involving geometric designs, including patterns found in real-life contexts.
Skill: Recognizing Non-Number Patterns
Repeating Patterns
Repeating patterns are the simplest form of pattern. To recognize it, you have to look for the pattern core. It is created when the elements that make up the pattern are repeated in the same order. Students must learn to identify the beginning and end of the pattern in the pattern. For example, in the picture below, the student who created this pattern shows by leaving a space between the patterns that the yellow bead indicates the beginning of the pattern core and the red bead, the end.
In addition, it is important to have students "read" the pattern by naming and touching each consecutive element of the pattern to become aware of the repetition.
From Kindergarten through the end of primary, the patterns that students learn to explore and create must be increasingly complex.
Here is a suggested approach to teaching repeating patterns:
- Observe a pattern with a single attribute and a two-element core
First explore a pattern that has only one attribute and whose core consists of only two elements. The attribute could be motion, position, sound, shape, or colour.
Example 1
In the pattern shown above, the attribute is position: one element on all fours and one element standing with arms outstretched constitute the pattern core.
Note: Colour change is easier to recognize and describe when a pattern is constructed with concrete materials, especially using objects that all have the same shape such as beads on a necklace, algebraic tiles, or interlocking cubes.
- Change the attribute
Then explore patterns with another attribute such as shape or size, while still having two elements in the pattern.
Example 2
In the pattern shown above, the attribute is the shape. The two elements, the sailboat and the balloon, constitute the pattern core.
- Modify the structure of the pattern
Explore more complex patterns by adding more elements to the pattern or more attributes. Students will then face a cognitive challenge that will lead to new learning.
Example 3
The following pattern consists of two attributes, shape and colour, and a three-element pattern core, blue rectangle, blue rectangle, orange triangle. Its structure is AAB.
Continue by introducing a third shape or colour. For example, the following pattern also has two attributes (shape and colour) and a three-element pattern (blue rectangle, red triangle, yellow oval). However, its structure is more complex, because there are three colours instead of two and three shapes instead of two. The structure of this pattern is ABC.
Note: At the beginning of the math period, introduce patterns that have cores already explored so that students can locate them more easily and effectively.
- Change the mode of representation
Present patterns that have the same structure but are constructed with different representations, and then test whether students recognize that they have the same structure. For example, present two patterns composed of different concrete materials, such as a pattern of beads and a pattern of interlocking cubes, or, as in the pictures below, two patterns, one of which is a concrete representation (a pattern of positions) and the other a semi-concrete representation (a pattern of drawn objects). When students can justify that the two patterns have the same structure, they are at a higher level of abstraction in their algebraic reasoning.
6 children make a sequence of repetitive pattern with their body. They alternate between a standing and kneeling position on the floor, repeated 3 times. ball position. ‘’The legend illustrates ‘’ sequence with a position of ‘’A’’ ‘’B’’.A drawing of a series of objects, two crossed hockey sticks and a puck, repeated 3 times. The legend illustrates ‘’sequence of objects drawn with a structure ‘’A’’ ‘’B’’.
- Explore patterns with a missing element in the pattern
An interesting challenge to present to students is the identification of a missing element in a pattern.
Examining the pattern to determine what element is missing from the beginning, middle, or end of a pattern increases their understanding of relationships. Many such explorations help students understand the pattern as a whole that contains several pattern cores, rather than as a pattern of changing elements without any relationship.
Example 4
The missing bead is a red bead.
- Identify false leads
Recognizing that an attribute can be a red herring in a pattern helps develop algebraic reasoning.
Examples
Pattern A:
In this pattern, the colours used are not an attribute of the pattern. Therefore, students should ignore the colour attribute and stick to the shape attribute (triangle or circle).
Pattern B:
In this pattern, the different shapes and colours create false leads that must be eliminated in order to discover the attribute, that is, the position of the base (shape placed on a flat side or on a vertex).
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 30-33.
Non-Numeric Growing Pattern
The simplest growing patterns are those where the term in position 1 is composed of one element and each of the subsequent terms increases by only one element (example A). Patterns where term 1 is composed of more than one element and where the subsequent term increases by one element (example B) or by more than one element (example C) are more complex.
Example: Non-Numeric Gowing Pattern
When students construct the shapes in a non-number pattern with an growing pattern in an array, as in Example C, they are simultaneously working on the concept of multiplication.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 40-41.
Skill: Describing Non-Number Patterns
Non-Numeric Repeating Patterns
During an activity, it is important to ask students relevant questions to get them to verbalize their observations, identify relationships, and explain how the rule was identified.
To help students establish an intuitive understanding of the structure of the pattern, teachers encourage them to verbalize the elements of the repeating pattern. For example, to describe the pattern of snapping, snapping, and raising one's arm in front of one's face…, teachers ask students to say aloud, "snap, snap, raise. To describe a string of beads that make up a necklace, he asks them to touch the beads while naming them aloud: "blue bead, purple bead, red bead. When students describe their string, it is important that they explain the relationship between the patterns (the elements of the pattern are represented in the same order) and use appropriate math vocabulary.
Sample Questions
- What are the attributes used to create the pattern?
- What are the elements of the pattern?
- Explain why it's a sequel.
As an example, students could describe the pattern shown below as follows: “Attributes are shape and term. The pattern is made up of three elements: an upward pointing triangle, followed by a downward pointing triangle, followed by a sun. The three elements of the pattern always repeat in this order."
Students can also record their understanding. Examples of factors to record include:
A sequence of a repetitive patterns of 3 elements. The sequence has a red triangle, purple triangle pointing downwards, and a sun, repeated 3 times. The 9 elements are numbered from one to 9. The elements are also represented by letters, ‘’A’’, ‘’B’’, ‘’C’’ the letters represent a structure of the sequence, repeated 3 times.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 37-38.
Non-Numeric Growing Pattern
Building on the change in shapes from one term to the next, students describe the pattern rule informally; for example, students might describe the rule in the patterns below as follows:
Pattern A:
"The pattern rule is that you always add a star to the shape in the previous term number."
Pattern B:
"The pattern rule is that you always add a row of three squares below the squares in the previous row."
Note: The pattern is described according to the number of elements and not according to the attributes which do not influence the growth of the pattern.
To describe the pattern specifically and see the relationships between each row, students must also be able to justify the arrangement of the elements that make up each term. Questioning helps students develop their ability to reason and recognize relationships. To help them, ask questions such as:
- How did we manage to construct the shape in the 3rd term of pattern A? that of the pattern B?
- How will the shape in the 4th term of pattern B be constructed?
- What must be done to obtain the shape at the 5th term of Pattern A? that of pattern B?
- Where did you discover a pattern rule in Pattern A? in Pattern B?
Students should explore non-number patterns with growing patterns using concrete materials. Then, students discuss ways to create them as well as relationships noticed or students explain their understanding of the rule, for example, "Each tree is built by adding a rectangle to the trunk of the tree in the previous row."
To describe what happens next, students can also make notes on their work, as in the following example.
The age of the tree and the number of shapes to represent it each year
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.
Students learn to define the matching rule of the pattern. For example, by carefully analyzing the pattern of trees, students see that the shape in term 1 has five shapes, the shape in term 2 has six shapes, the shape in tern 3 has seven shapes, and so on. Thus, students see that there are always four more pattern blocks than the term number of the shape. This observation, the matching rule, allows them to find any term in the pattern without having to extend it. Students can then predict that the shape in the 10th term will be composed of 14 pattern blocks.
The study of patterns, whether non-numeric or numeric, is the cornerstone of understanding patterns. Exploring patterns is a task that requires manipulation, intervention, and discussion that will allow each student to take a first step into the algebraic world.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 44-46.
Vocabulary Related to Non-Number Patterns
In order to fully grasp the concepts behind the big idea of pattern rules and relationships, here are some explanatory notes on the mathematical vocabulary related to number patterns.
| Non-Number Pattern: Set of Shapes or Objects Arranged According to a Pattern Rule | |
|---|---|
|
Non-Numeric Repeating Pattern Pattern A 
|
Non-Numeric Growing Pattern Pattern B
 Pattern C
 |
| Attribute: a characteristic that describes an object that we observe or manipulate. In Pattern A, the attributes that describe the Pattern are shape and colour. | In a growing pattern, attribute analysis is no longer important, since the focus is on pattern growth. |
| Term: Each element (shape, object, or motion) that makes up a non-number pattern or each number that makes up a number pattern. | |
| In Pattern A, each of the plane shapes is a term. | In Patterns B and C, each of the shapes is a term. |
| Pattern Core: the smallest part of a pattern from which the pattern rule is created. | |
| Pattern in Pattern A: a blue rectangle followed by a green trapezoid and an orange triangle. Each object that makes up the pattern is called a pattern element. | Pattern core in Pattern B: a square (shape 1). Pattern core in Pattern C: two cubes on top of each other (shape 1). |
| Rule: uniform phenomenon that defines a pattern and allows to determine each of its terms. | |
| Rule in Pattern A: repeat the blue rectangle, green trapezoid, orange triangle pattern core, always in the same order. | Rule in Pattern B: a square is added to the squares of the previous shape. Rule in Pattern C: a cube is always added to the bottom row of cubes in the previous shape. |
| Structure representation with letters of the regularity 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. | |
| 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. | |
| In Pattern A, there is a blue rectangle in the 1st, 4th and 7th rows, a green trapezoid in the 2nd, 5th and 8th rows, etc. | In the Pattern B and C, each shape has its position: shape 1 occupies the 1st position, shape 2 occupies the 2nd position, etc. |
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 29.
Skill: Recognizing and Describing Patterns Found in Everyday Life
Young children have a natural curiosity and interest in the patterns around them. Patterns can be literary, artistic, musical, scientific or numeric. Patterns are found in many events (for example, seasons and days of the week, growth of living things, daily activities), in rhymes and songs (for example, rhythm, rhymes, number of syllables), in stories with repeated structures (for example, the story Goldilocks and the Three Wise Men), and in stories with repeated structures (for example, the story Goldilocks and the Three Wise Men), (for example, Goldilocks and the Three Bears), in music (for example, different instrument sounds), in gestures (for example, standing, crouching, sitting), and in the world we have built (for example, traffic lights, odd and even house numbers). Repeating patterns decorating objects (for example, vases, clothing, jewelry) from various cultures provide good examples of patterns.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 26.