C1.1 Identify and describe repeating and growing patterns, including patterns found in real-life contexts.
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. Students must learn to identify the beginning and end of the pattern core. In the picture below, for example, the student who created this sequence shows, leaving a space between the patterns, that the orange bead indicates the beginning of the pattern and the red bead indicates the end. In addition, it is important to have students "read" the pattern by naming and touching each consecutive element of the pattern to identify the repetition.
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 grades, the patterns that students learn to explore and create should be increasingly complex. When teaching non-numerical repeating patterns, it is important to consider the following guidelines.
- 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.
Examples
Pattern A
In Pattern A, 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
Image A sequence of repeated forms with a purple pentagon, a red rectangle positioned vertical, a yellow rectangle position horizontal, one red square position vertical, a green trapezoid position horizontal, and a blue square position vertical.
In Pattern B, 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 (figure placed on a flat side or on a vertex).
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 31-33.
Skill: Recognizing and Describing Growing 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 pattern of concrete or semi-concrete figures or using words. 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 of an increasing pattern
Non-numerical sequences with increasing pattern have the following characteristics.
- The elements that make up each figure in the pattern are ordered and show a consistent change. For example, from one figure 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 term. For example, in the following, the starting figure is three suns placed in a "V" shape.
Note: Colour can be used to highlight the basic pattern or starting figure.
Image Non numerical sequence with increasing patterns of stars with different colors and underlined base. Rank one has 3 yellow stars. Rank two has 5 stars with 3 yellow and 2 white. Rank 3 has 4 stars.
- 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 figure 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.
Knowledge of the characteristics of an increasing non-numerical pattern helps students recognize this type of sequence. For example, students can recognize that the following pattern is not an increasing non-numerical sequence, even though each figure is composed of the same number of suns as in the previous sequence, i.e., 3, 5, 7, …, because the suns are not arranged according to an order and a rule that allows them to predict the position of the suns that will compose the next figure.
Image Non numerical sequence with increasing patterns of stars with different colors and underlined base. Rank one has 3 yellow stars. Rank two has 5 stars with 3 yellow and 2 white. Rank 3 has 7 stars with 3 yellow stars and 4 white stars.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 43.
Some pattern rules can be used to reinforce students' sense of number while developing their algebraic thinking. Contextual situations are best used because they are less abstract. Situations can feature a variety of pattern rules. Here are some examples.
Note: The situations where the focus is on rule analysis can be used in a problem-solving context or as an extension of learning. It is important to choose learning situations that are not necessarily oriented towards finding a rule since, in many cases, the rule is not within the reach of the students.
Example 1
Dominic is saving up to buy a video game that costs $74 (including taxes). He only has $35 in his wallet. Every week, his parents give him $5, which he keeps in his wallet.
| Number of weeks | 1 | 2 | 3 | … | … | … |
|---|---|---|---|---|---|---|
| Number of dollars in the wallet | 40 | 45 | 50 | … | … | … |
Describe the increasing sequence.
Recursive pattern rule (addition)
The relationship between the amount of money in Dominique's wallet from week to week can be represented by a table of values. Each term increases by a value of 5 from the value of the previous term.
Image A table with numeric sequence for money in wallet. Number of weeks from one to five with last numbers missing. Number of dolos in wallet 40, 45, 50, 55, 60 and the last numbers missing. The rule of regularity of plus 5.
Functional Relationship
The relationship between the number of weeks passed and the amount of money in the wallet can be represented using words.
On the 1st week, Dominique has $35 and 1 $5 bill in her wallet, which gives her $40.
In week 2, Dominique has $35 and 2 $5 bills in her wallet, giving her $45.
On week 3, Dominique has $35 and 3 $5 bills in her wallet, giving her $50.
In week 4, Dominique has $35 and 4 $5 bills in her wallet, giving her $55.
On week 5, Dominique has $35 and 5 $5 bills in her wallet, giving her $60.
Example 2
Emily babysits regularly. A couple asks her to look after their child for 6 hours a day for 10 days and offers to pay her according to the following table of values.
| Number of days | 1 | 2 | 3 | 4 | … |
|---|---|---|---|---|---|
| Salary for the day ($) | 1 | 2 | 4 | 8 | … |
Describe the increasing pattern.
Recursive pattern rule (multiplication)
I noticed that the salary doubles every day. So the pattern rule is ×2.
Image A table with a numeric sequence between 1 and 10 for salary in one day in the first row. The second row shows a numeric sequence between 1 and 512 representing daily salary. The sequence is as follows: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512. Below the table, arrows jumping from one number to the next indicate a pattern of multiplication by 2 for the daily salary.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 35-37.
Knowledge: Non-numeric Pattern
Non-numeric pattern: set of figures or objects arranged according to a pattern rule.
Repeating Pattern
A set of figures or objects arranged in an orderly and regular fashion, in which a repeated pattern (pattern core) is found.
Pattern A
Growing Pattern
A set of figures or objects arranged according to an order and a rule, in which a basic pattern is identified that grows from one term to the next.
Examples
Image Non numerical sequence with increasing patterns. Sequence ‘’B’’. Term one: one square. Term two: two squares. Term 3: 3 squares. A shell. A pyramid.
Example 1 shows a growing pattern; the pattern is a square to which 1 square is added to each subsequent term.
Example 2 shows a spiral shell, whose1st 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 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 pattern rule.
The third example represents a pyramid, whose first term is a rectangular prism at the top, to which are added stones whose base area grows according to a certain pattern rule.
Source : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
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 pattern rule is +2. It is possible to deduce the next term in the pattern by adding 2; for example, 10 + 2 = 12)
- 1, 3, 5, 7, 9, 11, … (The pattern rule is +2. It is possible to deduce that the next term will be 13, i.e. 11 + 2 = 13)
- 3, 9, 27, 81, … (The pattern rule is ×3. It is possible to deduce that the next term will be 243, i.e. 81 × 3 = 243.)
Shrinking numeric pattern
A pattern where the terms are decreasing numbers. The terms of a shrinking pattern come from rules involving subtraction and division.
Examples
- 14, 11, 8, … (Subtract three to get the next term.)
- 144, 72, 36, … (Divide by two to obtain the next term.)
Knowledge: Reason
Pattern core: smallest part of a non-numerical pattern from which the pattern rule is created.
Repeating Pattern
Pattern A
The reason is:
Growing Pattern
Pattern B
The reason 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.
Knowledge: Attribute
Attribute. A characteristic that describes an object that is observed or manipulated. Attributes can include color, shape, texture, thickness, orientation, materials, motion, sounds, objects, or letters. The attributes that describe Pattern A below are shape and colour.
Pattern A
Knowledge: Term
Term. Each element (figure, object, or motion) that makes up a non-numerical pattern or each number that makes up a numerical pattern.
Repeating Pattern
In Pattern A, each of the figures is a term.
Pattern A
1st term:
2nd term:
3rd term:
Growing non-numeric pattern
Pattern B
Shrinking non-numeric pattern
In Pattern C, each of the figures is a term.
Pattern C
1st term:
2nd term:
3rd term:
In Pattern B below, each of the figgmures is a term.
Pattern B
1st term:
2nd term:
3rd term:
Growing number pattern
2, 4, 6, 8, …
4, 10, 16, 22, 28, …
Knowledge: Pattern Rule
Recursive pattern rule : a description of the relationship between one term and the next, based on a generalization about the pattern structure..
Repeating Pattern
Pattern A
Pattern rule in Pattern A: repeat the blue rectangle, green trapezoid, orange triangle pattern, always in the same order.
Growing Pattern
Pattern rule in Pattern B: a square is added to the previous figure.
Pattern B
Pattern rule in Pattern C: a cube is always added to the bottom row of the previous figure.
Pattern C
Knowledge: Functional Relationship
Functional Pattern Rule : a rule that extends a pattern by establishing the relationship between the term number and
its term.
In the first pattern below, the pattern rule is that the number of triangles in the figure is three times its term number. In the second pattern, the green triangle is constant from one term to the next and a group of two orange squares is always added to the next term.
Image 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 row, there is 1 green triangle and 1 group of 2 orange squares. In the 2nd row, there is 1 green triangle and 2 groups of 2 orange squares. In the 3rd row, there is 1 green triangle and 3 groups of 2 orange squares. In the 4th row, there is 1 green triangle and 4 groups of 2 orange squares. |
Knowledge: Structure
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
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.
Repeating Pattern
In Pattern A below, there is a blue rectangle in the 1st, 4th and 7th rows, a green trapezoid in the 2nd, 5th and 8th rows, etc.
Pattern A
Growing Pattern
In Patterns B and C, each figure is a new term: figure 1 is the 1st term, figure 2 is the 2nd term, etc.
Pattern B
Pattern C