C1.1 Identify and describe repeating elements and operations in a variety of patterns, including patterns found in real-life contexts.
Skill: Recognizing Elements and Operations that are Repeated in Various Patterns
Repeating Patterns
The non-numeric repeating pattern is 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 for students "read" the pattern by naming and touching each consecutive element of the pattern so that they are aware of the repetition.
From kindergarten through the end of primary school, 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 nesting 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 patternss by adding more elements 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 core (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.
image 6 children make a pattern 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 ‘’ pattern with a position of ‘’A’’ ‘’B’’. A drawing of a series of objects, 2 crossed hockey sticks and a puck, repeated 3 times. The legend illustrates ‘’pattern 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.
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.
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 5
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, i.e. 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.
Observing similarities and differences between certain patterns reinforces students' learning and develops their communication skills. Recognizing two patterns that are similar requires a higher level of reasoning, and helps students build on the relationships that exist between patterns. Understanding relationships is of primary importance, as recognizing relationships will later become a problem-solving strategy.
During mathematical exchanges, teachers should engage students in comparing the special characteristics of patterns such as:
- the attributes;
- the choice and quantity of elements in each pattern;
- the structure of each pattern;
- the pattern rule for each pattern.
Example 6
|
Pattern 1 |
Pattern 2 |
 |
 |
|
Attributes: Shape and Colour |
Attribute: Symbols |
|
Three-element pattern core: one yellow circle and two blue trapezoids |
Three-element pattern core: one x symbol and two o symbols |
|
Structure : ABB |
Structure : ABB |
|
Rule: a yellow circle followed by two blue trapezoids, always in the same order. |
Rule: the x symbol followed by two o symbols, always in the same order. |
The ability to compare patterns facilitates the acquisition of the ability to extend them and, subsequently, to create new ones.
To get students to create a mental image of a pattern, teachers can also have students compare patterns using examples and non-examples of patterns. This strategy allows students to recognize a pattern, find the rule, describe it and justify their reasoning while using appropriate mathematical vocabulary.
Example of a Pattern with a Repeating Pattern Core
Non-Example of a Pattern - No Pattern Core
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 33-34.
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 growing pattern in an array, as in Example C, they are practicing the concept of multiplication.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 41.
Number Patterns
Very early on, children become aware of the pattern rules in their environment, in nature, in the objects that surround them. This is why it is possible to introduce number patterns as early as Grade 3. At the same time, students develop a sense of number, can count in intervals and backwards, and eventually 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)
(Economopoulos, 1998)
As students begin to explore the base-10 numbering system, which is synonymous with 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, by counting in increments of 2, starting at 16, students observe a predictable pattern 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 count from any number by any leap. Similarly, when students count by 5s, they quickly recognize a pattern, i.e., that the units digit alternates between the digit 0 and the digit 5 (5, 10, 15, 20…). Students can generalize this discovery informally by saying that any number that is a multiple of 5 will end with the digit 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. 49-50.
Skill: Recognizing Elements and Operations that are Repeated in Various Patterns
Non-Numeric Repeating Patterns
During an activity, it is important for teachers to ask students relevant questions to get them to verbalize their observations, identify relationships and explain how they identified the rule.
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 movements snap, snap, raise your arm in front of you… , they ask them to say aloud, "snap, snap, raise…"; to describe a pattern of beads forming a necklace, they ask 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?
- Why is this a pattern?
As an example, they may describe the following pattern as follows: "The attributes are shape and position. The pattern has three elements: a triangle pointing up, followed by a triangle pointing down, 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:
Image A pattern of a repetitive patterns of 3 elements. The pattern 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 pattern, repeated 3 times.A tracing waves line is placed under each motif. The pattern rule: a triangle pointing towards the top, triangle pointing towards the bottom, and a sun is represented always in the same order.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 37.
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 consists in adding a star to the shape of the previous term.
Pattern B:
The pattern rule is that you always add a term of three squares below the squares in the previous term.
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 term, 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:
- Can you tell us how we managed to construct the 3rd term in pattern A? in pattern B?
- How will the 4th term of pattern B be constructed?
- What must be done to obtain the 5th term of pattern A? of pattern B?
- Where did you discover a pattern rule in pattern A? in pattern B?
Students should explore growing patterns using concrete materials. Then, students discuss ways to create them as well as the relationships they see. Students explain their understanding of the rule, "Each tree is built by adding a rectangle to the trunk of the previous tree."
To describe what happens next, students can also leave marks on their work, as in the examples below.
In growing patterns, there is also a relationship between the term number and the number of elements in each. This relationship is a very important mathematical concept, demonstrated with a table of values and leading to a more formal generalization in Grade 3, the formulation of the functional pattern rule.
Students learn to define the functional pattern rule. For example, by carefully analyzing the pattern of trees, students see that the shape in position 1 has five shapes, the shape in position 2 has six shapes, the shape in position 3 has seven shapes, and so on. Thus, students see that there are always four more geometric shapes than the position of the shape. This observation, the functional pattern 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 position will be composed of 14 geometric shapes.
The study of patterns, whether non-numeric or number patterns, is the cornerstone of understanding patternss. 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-numeric 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 non-numeric patterns.
Non-numeric pattern: set of shapes or objects arranged according to a pattern rule.
|
Non-Numeric Repeating Patterns |
Non-Numeric Growing Pattern |
|
Pattern A  |
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 shape, object, or motion that makes up a non-numeric 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 core in Pattern A: a blue rectangle followed by a green trapezoid and then an orange triangle. Each object that makes up the pattern is called an element. |
Element in Pattern B: a square (shape in term 1). Element in Pattern C: two superimposed cubes (shape in term 1). |
|
Pattern rule: uniform criterion that defines a pattern and helps to determine each of its terms. Pattern rule in Pattern A: repeat the blue rectangle, green trapezoid, orange triangle pattern, always in the same order. |
Pattern rule in Pattern B: a square is added to the previous shape. Pattern rule in Pattern C: a cube is always added to the bottom term of the shape in the previous term. |
|
Structure: letters representing the pattern rule of a repeating pattern. In Pattern A, each element of the pattern can be identified by a letter: blue rectangle (A), green trapezoid (B), orange triangle (C). The structure of Pattern A is therefore ABC. |
|
|
Position: Where each term is located 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 positions, a green trapezoid in the 2nd, 5th and 8th position, etc. |
In patterns B and C, each shape has its position: the first shape is in the 1st position, the second shape is in the 2nd position, etc. |
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 29.
Number Patterns
The terminology described above for describing non-numeric patterns also apply to number patterns.
Example 1
Pattern C:
2, 4, 8, 16, 32
The pattern rule is "double the previous number".
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 50.
Comparing patterns on the same number grid allows us to establish relationships between numbers.
Example 2
Introduce students to the following two number patterns:
Pattern A: 3, 6, 9, 12, 15…
Pattern B: 6, 12, 18, 24, 30…
Use transparent tokens to represent pattern A and shade the appropriate boxes to represent pattern B in a 100s grid.
Then, get students to think about and compare the patterns by asking questions such as:
- What do you observe once the two patterns are represented on the grid?
- Possible observations: some numbers are covered with a transparent token; some numbers covered with a transparent token are also shaded; the patterns form diagonal lines; there is always a difference of 3 between the numbers covered with a token; there is always a difference of 6 between the shaded numbers.
- Are there any numbers that are part of both patterns?
- What is the relationship between these numbers?
- What is the rule for each pattern?
- What similarity and difference is there between Pattern A: 3, 6, 9, 12, 15… and Pattern C: 5, 8, 11, 14, 17...?
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 59-60.
Vocabulary Related to 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.
|
Number Pattern: a Series of Numbers Based on a Pattern Rule |
|
|
Rule: a uniform phenomenon that defines a pattern and helps determine each of its terms. In the primary cycle, the pattern rule studied is one of addition or subtraction (from Grade 3) or multiplication or division (from Grade 3). |
|
|
Pattern rule in Pattern A: each term is always 5 more than the previous term. |
Pattern rule in Pattern B: each term is always 5 less than the pervious term. Pattern with a subtraction pattern rule |
|
Pattern with an addition pattern rule Pattern A 5, 10, 15, 20, 25, 30… Term: each number that makes up a number pattern. |
Pattern with a subtraction pattern rule Pattern B 30, 25, 20, 15, 10, 5, 0 Term: each number that makes up a number pattern. |
|
Position: Where each term is located 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, the number 5 is in the 1st position, the number 10 is in the 2nd position, etc. |
In Pattern B, the number 30 is in the 1st position, the number 25 the 2nd position, etc. |
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 50.
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 numerical. Patterns are found in many events, such as the succession of seasons and days of the week, the growth of living things, and daily activities; in rhymes and songs, such as rhythm, rhymes, and number of syllables; in stories with repeated structures, such as Goldilocks and the Three Bears; in music, such as the different sounds of instruments; in gestures, such as standing, crouching, or sitting; and in the world we have built, such as traffic lights, odd and even house numbers. Repeated patterns decorating objects, such as vases, clothing, and jewelry, from various cultures are also good examples of patterns.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 26.
Knowledge: Non-Numeric Pattern
There are two kinds of non-numeric patterns: non-numeric repeating patterns and non-numeric growing patterns.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 26.
Non-Numeric Repeating Patterns
The non-numeric repeating pattern is 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.
It is important to present the entire pattern at least three times before asking students to find the pattern core or extend the pattern. This makes it easier for them to identify the relationship between the elements of the pattern and between the different pattern rules for each attribute.
Students are able to generalize when they realize that a rule is created when the elements of the pattern are repeated in the same order.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 30.
Non-Numeric Growing Pattern
Non-numeric growing patterns are more complex than non-numeric repeating patterns since the number of elements that make up a shape increases from one term to the next in a predictable way.
Note: Initially, students should explore non-numeric growing patterns using manipulatives.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 40.
Non-numeric growing patterns have the following characteristics:
- The elements that make up each shape 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 can be found in each shape, so that each shape comes from the growth of the shape in the previous term. For example, in the following, the starting shape is three suns placed in a "V" shape.
Note: Colour can be used to highlight the basic pattern or starting shape.
- The number and location of the elements that make up each shape are predictable. For example, in the above pattern, it can be predicted that the 5th term 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.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 42.