C1.1 Identify and compare a variety of repeating, growing, and shrinking patterns, including patterns found in real-life contexts, and compare linear growing patterns on the basis of their constant rates and initial values.
Skill: Identifying and Comparing a Variety of Patterns
When students justify their reasoning, it is an opportunity to step back and examine their approach logically.
Source : Guide d’enseignement efficace des mathématiques, de la 7e à la 10e année, p. 68.
That said, students' ability to justify their reasoning about the similarities and differences of certain patterns consolidates their learning and develops their communication skills. In addition, their ability to justify that two patterns 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 discussions, teachers should engage students in comparing the special characteristics of patterns such as:
For repeating patterns:
- the attributes;
- the choice and quantity of elements in the pattern core;
- the structure of each pattern;
- the recursive relationship for each pattern.
- the functional relationship in each pattern.
For growing and shrinking patterns:
- what changes from one term to the next;
- what is constant from one term to the next;
- the recursive relationship for each pattern.
- the functional relationship in each pattern.
The ability to compare patterns facilitates the development of the ability to extend patterns and, subsequently, to create new ones. To get students to create a mental image of a rule, teachers can also have students compare patterns using examples and counterexamples of patterns. This strategy allows students to recognize a pattern, find the recursive relationship or functional relationship, describe it and justify their reasoning while using appropriate mathematical vocabulary.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 33-34.
Strategic questions from teachers support and reinforce the ability to compare patterns.
Sample Questions
- How can you represent these patterns differently?
- What do you observe once the two patterns are represented in the table of values? as a graphical representation?
- Are there elements and/or numbers that are part of both patterns?
- What is the relationship between these elements, these numbers?
- What is the recursive relationship in each pattern?
- What is the functional relationship (between the term number and the term value) in each growing pattern?
- What similarities and differences are there between the patterns?
- Can you represent these patterns differently? Explain or demonstrate your answer.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 59-60.
Example of Comparison of Growing Patterns
First, it is important to note whether we are dealing with increasing or decreasing patterns. It is also very important to identify the pattern core and the recursive relationship in the growth of the terms.
The diagram below shows that the number of conifers (evergreens) and apple trees increases, the first in a linear way and the second in a non-linear way.
This can be illustrated with tables of values and graphical representations.
- Table of values
From a table of values, a growing pattern can be said to be linear or non-linear by determining the change in the term values ("+3, +5, +7" shown above). If the jumps are not constant, as it is the case here, the growing pattern is non-linear.
From a table of values, a growing pattern can be said to be linear or non-linear by determining the change in the term value ("+8, +8, +8" shown above). If the jumps are constant, as it is the case here, the growing pattern is linear.
- Graphic representation
In this case, the student should be able to note that the number of apple trees in each position represents a growing pattern that is non-linear and begins at 0 for term 0. This is non-linear, since the number of apple trees does not grow at a constant rate. In the table of values, as the term number increases by 1, the number of apple trees increases by an increasing constant rate (+3, +5, +7) . It produces a curved line in its graphical representation.
The number of conifers would also start at 0. In the table of values, each time the term number increases by 1, the number of conifers always increases by 8. This relationship is therefore a linear growing pattern, as it increases with a constant rate of 8 conifers per position. Its equation is represented by c =8n. Its graph represents a straight line in the Cartesian plane.
Source : Guide d’enseignement efficace des mathématiques de la 7e à la 10e année, p. 15-16.
Shrinking patterns, such as the number of treats left in a container, can be presented in the same way.
Skill: Comparing Linear Growing Patterns According to Their Constant Rates and Initial Values
When comparing linear growing patterns, we compare their constant rate as well as their initial value.
The equation of a linear sequence is y = mx +b, where m represents the constant rate of change and \(b\), the initial value .
If the ratio of the change in one variable to the change in another variable is equivalent between any two sets of data points, then there is a constant rate. An example of a real-life application of a constant rate is an hourly wage of $15.00 per hour.
Source : The Ontario Curriculum, Mathematics, Grades 1-8, Ontario Ministry of Education, 2020.
In a comparison of linear growing patterns, the pattern that has the greatest constant rate grows at a faster rate than the others and has a steeper incline as a line on a graph. For example, this is true for a constant rate of 3 compared to a constant rate of 2.
Constant rate of 3 :
Constant rate of 2 :
Source : En avant, les maths!, 7e année, CM, Algèbre, p. 5.
When representing a linear pattern graphically, the initial value corresponds to the value of y when x is 0.
Example
The initial value of the linear growing pattern y = 3x - 2 is -2, since when x = 0, y = -2.
Note that the graph of a linear growing pattern that has an initial value of zero passes through the origin at \((0, 0)\).
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.
Repeating Pattern
A pattern in which a core repeats continuously (for example, AAB, AAB, AAB, …).
Example
Growing Pattern
A pattern that involves an increase from one term to the next (for example, AB, AABB, AAABBB).
Examples
Example 1 represents a growing pattern; the pattern core is a square to which a square is added at 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.
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.
Examples
The pattern below starts with three squares in term 1 (pattern core and initial value) and each term increases by four squares (constant rate) in each subsequent term.
The pattern below starts with four squares in term 1 (pattern core and initial value) and each term increases by three squares (constant rate) in each subsequent term.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8 , 2020, Ontario Ministry of Education.
Shrinking Pattern
A set of shapes or objects, arranged in a pattern that involves a regression (for example, a decrease in the number of elements) from one term to the next (for example, AAAA, AAA, AA, A).
Example
The above pattern starts with 4 purple circles and 4 blue triangles in term 1, and each subsequent term decreases by one additional blue triangle (-1, -2).
Linear Shrinking Pattern
A pattern in which the value of the elements, which can be represented by y, decreases as the term number, represented by x, increases. The rate of change is constant, but it has a negative value.
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 core is:
Non-Numeric Growing Pattern
The core is:
By examining the above patterns, we can recognize that they are created according to a repeating pattern core.
Knowledge: Attribute
An observable property of a person or object. Attributes can include colour, shape, texture, thickness, orientation, materials, motion, sound, objects, or letters. The attribute of an object is represented by means of a characteristic. For example, if the attribute is colour, the characteristics might be red, blue, yellow.
In the following pattern, the attributes that describe 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 curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Knowledge: Term
Each element (for example, shape, object) that makes up a non-numeric pattern or each of the quantities constituting a number in a pattern.
Repeating Pattern
In the following, each shape is a term.
1st term:
2nd term:
3rd term:
Non-Numeric Growing Pattern
In the following, each set of shapes 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: Recursive Relationship
A description of the relationship between one term and the next, based on a generalization about the pattern structure.
Example
Recursive relationship: a cube is always added to 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 is equal to three times the term number: 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 one green triangle and one group of 2 orange squares. 2nd row, there is one green triangle and 2 groups of 2 orange squares.3rd row, there is one green triangle and 3 groups of 2 orange squares. 4th row, there is one green triangle and 4 groups of 2 orange squares.
|
In term 1, there is 1 green triangle and 1 group of 2 orange squares. In term 2, there is 1 green triangle and 2 groups of 2 orange squares. In term 3, there is 1 green triangle and 3 groups of 2 orange squares. In term 4, 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 this pattern, each element can be identified by a letter: blue rectangle (A), green trapezoid (B), orange triangle (C). The structure of this pattern is therefore ABC.
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 the patterns below, each term has its number: term 1 is in the 1st position, term 2 is in the 2nd position, etc.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 29.
Knowledge: Constant Rate
For any two sets of data, the equivalent ratio between the change in one variable and the change in another variable. In the equation y = mx + b, the constant rate is represented by m.
Example
Figure showing the relationships between the cost and the number of months. Number of months: 2, 4, 5.Costs: 22 dollars, 42 dollars, 52 dollars. The constant rate of change is 10 and it is illustrated with lines on top of each value.According to data reports 20 divided 2, 30 divided 3, and ten divided one, are equivalent to the constant amount of ten dollars per month.
Source : En avant, les maths!, 7e année, CM, Algèbre, p. 3.
Knowledge: Initial Value
The initial value of a linear growing pattern is the value of the term when the term number is zero. In the equation y = mx + b , the initial value is represented by b.
Example
In this situation, the initial value is $15, since this is the value of y when x is 0.
Graphic representation of a linear increasing sequence of the numbers of hours and the cost of location. Cost of location number zero to 70. Number of hours zero to 6. Linear line of dots drawn at values 15, 25, 35, 45, 55, and 65.
Source : En avant, les maths!, 7e année, CM, Algèbre, p. 4.