At a Glance...

C1.2 Create and translate repeating, growing, and shrinking patterns using various representations, including tables of values, graphs, and, for linear growing patterns, algebraic expressions and equations.

Skill: Creating Various Patterns

Before creating patterns, students must first recognize, compare, describe, change the representation, and extend a wide variety of patterns.

Students demonstrate their understanding of the concept of a rule by creating a pattern and explaining it.

Initially, the use of manipulatives is essential for representing patterns because, in constructing patterns, students can easily change one element of the pattern and check the rule, whereas in drawing it on paper, they may focus on the drawing to be reproduced rather than examining the pattern and thinking about the nature of the rule.

Initially, teachers have students work in small groups to encourage the exchange of ideas, and can provide the attributes and structure of the pattern as well as the materials needed to create it.

During a mathematical discussion (consolidation), presentations of the patterns help to identify different representations of the same rule and to check how students communicate their understanding.

Teachers then ask students to create a pattern and exchange it with a partner. The teacher invites students to construct a different representation of the received pattern, describe its structure, extend it or produce a completely different one for comparison. However, students should be instructed to limit the number of elements in the pattern since too many elements may make it difficult to identify the structure of the pattern.

Source : Guide d’enseignement efficace des mathématiques de la 1^re à la 3^e année, p. 39-40.

Example

Provide a recursive relationship of addition or subtraction (for example, +23 or -23). Have students create different patterns with this relationship and compare them.

Source : Guide d’enseignement efficace des mathématiques de la 1^re à la 3^e année, p. 63-64.

Skill: Translating Patterns and Making Connections Between Different Representations

The ability to communicate algebraic reasoning is developed when students express their understanding of a problem or concept and defend their ideas using different representations:

• concrete, related to exploration, manipulation and creation using concrete materials;
• semi-concrete, related to an illustration, drawing or other representation on paper;
• symbolic, related to any representation made from numbers or symbols;
• “in words”, related to a verbal or written explanation or description.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 18.

In order to develop a solid understanding, students need to have experiences in context by exploring problem-solving situations. Contextualization allows students to make connections between various representations and to develop an understanding of the algebraic concepts being explored. Teachers also use a variety of representations to help students take ownership of mathematical concepts, make connections between representations, facilitate transfer, and elicit cognitive flexibility in relation to the concepts.

In the junior grades, students explore relationships and represent them in different ways. In Grade 6, they learn that a relationship can be represented by a situation (non-numeric pattern), a table of values, an algebraic expression, an equation (functional relationship rule) or a graph. The arrows in the diagram below indicate the links between the various common representations of a relationship.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 41.

Using multiple representations of the same pattern to communicate understanding is an essential component of developing algebraic thinking. Repeating patterns, growing patterns, and shrinking patterns can be represented in a variety of ways, including concrete materials, graphical representations, tables of values, equations, algebraic expressions, and oral description.

Oral Description

The repeating pattern is composed of 4 red triangles, 2 blue squares, 4 red triangles, 2 blue squares, 4 red triangles, 2 blue squares.

When students are presented with a growing pattern, teachers should provide them with a variety of opportunities to describe it in words. This is an essential step in getting students to determine the algebraic expression of the pattern.

The growing pattern can be described as follows:

• The term in position 1 is composed of 3 blue squares and 3 red circles.
• The term in position 2 is composed of 3 blue squares and 4 red circles.
• The term in position 3 is composed of 3 blue squares and 5 red circles.

Students may also notice that the number of blue squares, 3, remains constant, while the number of red circles increases by 1 each term.

Manipulatives

The manipulatives (for example, algebra tiles, interlocking cubes, counters, Cuisenaire rods) are quite varied. Using them helps students explore, represent, and make changes easily during explorations.

Examples of the use of manipulatives include:

Have students represent the 2, 6, 10, 14… number pattern in a concrete way to make the pattern rule more visible, by creating a growing pattern using manipulatives of their choice. Encourage them to add materials (cubes, tokens, etc.) in the same way from one term to the next and to rearrange them as needed.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 212.

Relevant Questions:

• How many cubes (tokens, blocks ) will you use to concretely represent the 1^st term of the pattern? The 2^nd term? The 3^rd term?
• What change do you notice from one term to the next?
• What does not change from one term to the next (the constant)?
• Are your cubes added the same way from one term to the next? Explain.
• How would you describe the next term in this growing pattern?
• How many cubes will the tenth term contain? How can you determine this?
• What is the pattern rule? How do you know?

Teachers can give students the equation below and have them represent it using interlocking cubes.

c = 1 + 2n

Two steps can be illustrated that lead to the representation of an equation using manipulatives, during which the emphasis is placed on the relationship between the term number and the number of cubes that compose the term.

Example of reasoning:

• Verbalize the relationship: To determine the number of cubes in a term, I start with 1 and add the term number multiplied by 2.
• Visualize the relationship:

Term 1 has 1 cube and 1 group of 2 cubes.

Term 2 has 1 cube and 2 groups of 2 cubes.

Term 3 has 1 cube and 3 groups of 2 cubes.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 53-54.

Illustration

Drawing allows students to create a semi-concrete personal representation of their observations and understanding, which helps them clarify their thinking. It is particularly beneficial for students who have difficulty writing or using symbols as a means of representation, as the drawings can serve as justifications or explanations.

Many types of problems naturally prompt students to make a drawing to help them with the solution. Students may also use manipulatives at the same time. As students become more abstract thinkers, they move away from manipulatives to drawings.

Here are some examples of how a drawing can be used:

Simon wants to buy a toy, but he doesn't have enough money in his piggy bank. His parents decide to give him the same amount of money every week, which he puts into his piggy bank. In order to keep track of the amount of money he has accumulated, Simon has created the following table of values:

Ask students to illustrate the situation and determine how much money the parents give him each week, and how much money has been accumulated from week to week.

Relevant questions:

• How does the illustration help determine the pattern rule?
• How much money will Simon have in the piggy bank in week 5? How do you know?
• If Simon receives the same amount of money each week, why does he have $5 after the first week?
• How can you determine the amount in the piggy bank after a large number of weeks have passed, for example, in week 10?

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 214-215.

It is also possible, for example, to represent the equation \(y = 2x + 4\) using the non-numeric growing pattern below with the first three terms:

Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.

Table of Values

The table of values allows to semi-concretely represent the functional relationship between two changing quantities (variables), one of which depends on the other. The table of values is often created to represent the functional relationship between two numeric values associated with a non-numeric growing pattern: the term number and the value of the corresponding term.

The table of values can also be used to present values of variables in an equation or in a problem-solving context.

In a problem-solving context, students can connect a situation expressed in words to a table of values.

Present the following situation:

To better understand this relationship, students can model it using concrete or semi-concrete materials and words.

When you rent 1 movie, there is a $4 subscription fee and rental fee for 1 movie, 1 x $5, so you pay $9.

When you rent 2 movies, there is a $4 subscription fee and rental fee for 2 movies, 2 x $5, so you pay $14.

When you rent 3 movies, there is a $4 subscription fee and 3 x $5, so you pay $19.

When you rent 4 movies, there is a $4 subscription fee and 4 x $5, so you pay $24.

Afterwards, students can use a table of values to represent the relationship between the number of movies viewed and the amount of money spent.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 51.

Present the following pattern:

Ask students to create a table of values that represents the relationship between the term number and the number of squares in the term.

Relevant questions:

• Describe the relationships in the table of values.
• What is the value of the next terms?
• Describe the relationship between the non-numerical pattern and the table of values.

Present the equation g + k = 11 and ask students to determine the possible integer values of g and k.

Relevant Questions:

• If g takes the value of 10, what value must the variable k take?
• If k takes the value of 8, what value should the variable g take?
• Are there other possible values? How can you verify this?
• How can you organize the different possibilities in a table of values?

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 231-233.

Graphical Representation

A graph is a schematic representation of a set of data. In the Data strand for the junior division, students learn how to represent data using a variety of graphs. Students can use this knowledge in the Algebra strand. The appearance of the data set in a graph (for example, bars in ascending or descending order) allows for analysis of change and facilitates interpolation and extrapolation. The table of values is used to create the graph.

Here are some examples of how to use a graph:

Present the following situation:

Then graph the data as shown below.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 233.

Relevant Questions:

• What changes in sales happen as more pitchers are sold? Are these changes always the same? Why or why not?
• Looking at the graph, how is it possible to determine how much money they will have collected after selling an additional pitcher of lemonade?
• What is the relationship between the number of pitchers sold and the amount collected?

Skill: Representing Linear Growing Patterns Using Equations or Algebraic Expressions

The described learning progression reflected in the mathematics curriculum content and research, including Lee (1996, p. 105), suggests three steps that lead to representing a growing non-numerical pattern using a rule:

• Visualize the relationship: perceive the terms in a certain way and recognize a connection between them.
• Verbalize the relationship: use what has been perceived to state a pattern rule in everyday language.
• Represent the relationship using symbols: express the pattern rule using an equation made up of variables and constants.

These three steps to developing an equation can be illustrated using the non-numerical pattern below, where the relationship between the term number and the number of cubes in it is explored (the functional relationship).

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 53.

Possible reasoning of the student

1. Visualize the relationship:
• In the1^st term, there is 1 cube and 1 group of 2 cubes.
• In the 2^nd term, there is 1 cube and 2 groups of 2 cubes.
• In the 3^rd term, there is 1 cube and 3 groups of 2 cubes.
• So, in the 10^th term, there will be 1 cube and 10 groups of 2 cubes.
2. Verbalize the relationship:

   To determine the number of cubes in a term, I start with 1 and add the term number multiplied by 2.

3. Represent the relationship using symbols:

   The student concludes that the relationship can be represented by the equation c = 5 +(n - 1) × 2, where n is the term number and c is the number of circles in it.

This relationship can be confirmed by creating a table of values and a graphical representation of the linear growing pattern.

Term Number (n)	Number of Cubes (c)
1	3
2	5
3	7
n	\(\ 1 + 2c \)

Note: In Grade 6, students learn to perform arithmetic operations using order of operations. The equations presented in this document to express a rule reflect this knowledge.

It is important to recognize that this path to expressing an equation may differ from student to student as reasoning develops from individual perceptions. The following example, based on research by Radford (2006, pp. 2-21), illustrates how students may perceive differently the relationship between the term number in a growing non-numerical pattern and the number of circles in the term.

Example

Student 1 sees 2 rows of circles. In the top row, there is always 1 more circle (in blue) than the term number, and in the bottom row, there are always 2 more circles (in gray) than the term number.

Continuing the analysis, the student determines that the 10^th term will contain (10 + 1) + (10 + 2) circles, or 23 circles. The student then concludes that the relationship can be represented by the equation c =(n + 1) +(n + 2), where n is the term number and c is the number of circles in it.

Student 2 finds the total number of circles by adding the term number twice and adding 3. So, for the total number of circles in the 1^st term, the number sentence is 1 + 1 + 3 = 5; for the 2^nd term, 2 + 2 + 3 = 7; and for the 3^rd term, 3 + 3 + 3 = 9.

As the student continues their analysis, they conclude that the relationship can be represented by the equation c = n + n + 3, where n is the term number and c is the number of circles in it.

Student 3 sees:

• in the 1^st term, 5 circles;
• in the 2^nd term, the 5 initial circles, to which we added 1 group of 2 circles,
• in the 3^rd term, the 5 initial circles, to which we added 2 groups of 2 circles.

Analyzing these observations, the student generalizes and determines that in each term there are 5 circles and a number of groups of 2 circles, this number corresponding to the previous term number. Thus, in the 25^th term, there will be 5 + 24 × 2 circles, or 53 circles.

Note: Students may present their interpretation using a table of values.

The student concludes that the relationship can be represented by the equation c = 5 +(n - 1) × 2, where n is the term number and c is the number of circles in it.

Note: The student analyzed the recursive relationship (that is, adding 2 circles to the previous term), which leads her or him to the functional relationship using the expression n - 1 to represent the previous term number, the initial 5 circles, and a number of groups of 2 circles, namely, 5 + (n - 1) × 2.

The pattern rules formulated by the three students came from their understanding of this relationship. Each student perceived and generalized the situation in their own way, resulting in three different but equivalent equations. None is better than the other. They do, however, demonstrate that students' interpretation of a relationship affects the rule they formulate. It is important that teachers encourage these different formulations of a pattern rule (functional relationship). In doing so, they help students develop a better understanding of equations and variables.

Sometimes students move too quickly from the non-numeric pattern to the corresponding table of values. For example, from the non-numeric pattern in the example below, students immediately establish the following table.

Students then proceed by trial and error to determine the equation that defines the relationship, as evidenced by the two students' explanations. In each case, the depth of these students' understanding of the resulting equation and the relationship it represents should be questioned.

Student 4 explains that she represented the relationship using a table of values. She compared the term number to the corresponding number of circles by making several attempts. For example, she started with "× 2," but that didn't work; then she tried "× 3," but that didn't work either. She continued in this way and found that in each case it was "times two plus three" or double the term number, plus 3.

The student concludes that the relationship can be represented by the equation c = n × 2 + 3, where n is the term number and c is the number of circles that make up each term.

Student 5 explains that in the 2^nd row of the table, there is a regularity, because from one term to the next, 2 is added. So in the rule, there is "x 2". She or he tries the n x 2 rule and does not get the terms 5, 7, 9… She or he then tries the n x 2 + 1 rule, then the n x 2 + 2 rule. When he or she tries the n x 2 + 3 rule, they get the terms 5, 7, 9…

The student concludes that the relationship can be represented by the equation c = n × 2 + 3, where n is the term number and c is the number of circles that make up each term. In this case, the student appears to determine the equation by following a memorized approach.

Teachers should consider the different ways that students perceive the relationships between terms in a pattern, and adapt their questions accordingly to help each student accurately express the rule in words and determine the corresponding equation.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 53-57.

Technology

Technology is a powerful tool for observing and analyzing relationships between variables. The graphing calculator and some web-based applications are useful for representing, creating and analyzing data. The graph that results from entering data helps students observe if the data was entered correctly, determine the relationship represented, and change some of the data to check the effect of those changes on the graph. Using the graphing calculator or an app will require learning how to operate the tool. Time spent learning the technology is reinvested in analyzing the relationships represented by a graph or table of values. Fundamental processes and representations of relationships are essential components in the development of the the big idea of algebraic reasoning.

Source : Guide d’enseignement efficace des mathématiques de la 7^e à la 10^e année, p. 15-16.

Knowledge: Table of Values

The table of values is a semi-concrete representation of the relationship between two changing quantities (variables), one of which depends on the other. By the end of the primary division, students have learned to create tables of values to represent relationships in problem-solving situations.

The table of values is often created to represent the functional relationship between two numeric values associated with a non-numeric growing pattern: the term number and the corresponding value of the terms.

Example

We can study the relationship between the term number in a non-numeric pattern and the number of elements that make up that term. The term number is written in the first column (or row) of a table of values and the number of elements that make up that term (value of the term) is written in the second column (or row). The recursive pattern of the terms in the second column (or row) can be used to extend the table of values.

The table of values can be arranged vertically or horizontally and is separated into columns or rows. It is a good idea to vary the layout of the table of values so that students get used to both layouts.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 231.

Knowledge: Graphical Representation

Growing patterns can be used as an introduction to graphing. The term numbers in the growing pattern corresponds to the values on the horizontal axis. The total number of elements in each pattern is represented by the values on the vertical axis. The number of elements at term number 0 (the constant value) will be indicated by a point on the vertical axis.

Source : Guide d’enseignement efficace des mathématiques de la 7^e à la 10^e année, p. 63.

Example

Source : En avant les maths!, 6^e année, CM, Algèbre, p. 5.

Knowledge: Algebraic Expression

An expression is a numeric or algebraic representation of a quantity. An expression may include numbers, variables, and operations. Specifically, an expression such as 3 + n + 4 is an algebraic expression, since it contains numbers and symbols.

Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8 , 2020, Ontario Ministry of Education.

Note: An expression containing only numbers is a numeric expression, for example 5 - 2.

Knowledge: Equation

An equation is a symbolic way of representing a relationship. It can be difficult to understand, in part because there are several types of equations that vary in function depending on the situation. In the junior grades, students encounter the following four types of equations:

• Equation to solve: An equation such as 2 + n = 14 must be solved. It usually comes from a problem-solving situation and describes an equality relationship. The letter n represents an unknown value that must be determined.
• Equation that represents a relationship between two changing quantities: An equation such as t =2n + 3 is used to express a relationship, such as the relationship between the term number (n) in a non-numeric pattern and the number of toothpicks (t) that make up the term. The letters n and t are variables, since they can take on various values.
• Equation that serves as a formula: To calculate the area (A) of a square, we can use the equation A = b × h. Such an equation is called a formula, since it is used to calculate the area of a square with sides of length b and h. We do not solve such an equation.
• Equation that generalizes a situation of equality: This type of equation is called an identity. For example. we can generalize the relationship of equality between the addition of any two identical numbers and the multiplication of that number by 2 (for example, 4 + 4 = 2 × 4) by the equation n + n = 2 × n. Such an equation is not solved and is not a formula. Moreover, it does not represent a relationship between two changing quantities.

Note : The clarification of the various types of equations is provided to recognize that algebraic concepts are not dealt with exclusively in modelling and algebraic activities; for example, equations that serve as formulas in measurement are regularly found elsewhere. However, in the junior grades, students do not necessarily have to distinguish between these types of equations.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 89.