C1.2 Create and translate repeating, growing, and shrinking patterns involving whole numbers and decimal numbers using various representations, including algebraic expressions and equations for linear growing patterns.
Skill: Creating Various Patterns
Before creating patterns, students must first recognize, compare, describe, translate and extend a wide variety of patterns. Students demonstrate their understanding of the concept of a relationship or rule by creating a pattern and explaining it.
Initially, the use of manipulatives is essential for representing patterns because, by creating them, students can easily change one element of the pattern and check the rule. When students draw it on paper, they focus on the element to be reproduced rather than looking at the whole pattern and thinking about what the rule is.
Teachers first ask students to 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.
Next, teachers may ask students to each create their own pattern and exchange it with a partner. Students can then translate the received pattern, describe its structure, extend it or create a completely different one for comparison. The number of elements in the pattern should be limited, however, as some students may use too many, making it difficult to identify the structure of the pattern.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e 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 maternelle à la 3e 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-solving situation 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", linked to a verbal or written explanation or description.
Source : Guide d’enseignement efficace des mathématiques, de la maternelle à la 6e 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.
Students explore relationships and represent them in different ways. Students learn that a relationship can be represented by a situation (non-numeric pattern), a table of values, an algebraic expression, an equation (rule) or a graph. The arrows in the diagram below indicate the connections between the various common representations of a relationship.
Source : Guide d’enseignement efficace des mathématiques, de la maternelle à la 6e 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, graphs, tables of values, equations, algebraic expressions, and oral description.
Oral Description
Represent a Repeating Pattern Using Words
The repeating pattern is composed of 4 red triangles, 2 blue squares, a pattern core that is always repeated in the same order.
Represent a Growing Pattern Using Words
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:
Term 1 is composed of 3 blue squares and 3 red circles.
Term 2 is composed of 3 blue squares and 4 red circles.
Term 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, algebraic tiles, interlocking cubes, tokens, Cuisenaire rods) are quite varied. Using them helps students explore, represent, and make changes easily during trials. The following are examples of how manipulatives can be used.
Represent a Numeric Pattern Using Manipulatives
Teachers can give students the equation below and have them represent it using interlocking cubes.
c = 1 + 2n
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:
In Term 1, there is 1 cube and 1 group of 2 cubes.
In Term 2, there is 1 cube and 2 groups of 2 cubes.
In Term 3, there is 1 cube and 3 groups of 2 cubes.
It is important to recognize that this pathway to expressing an equation may differ from student to student, as reasoning develops from individual perceptions.
Source: L'@telier - Online Educational Resources (atelier.on.ca).
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 solve them. They 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 an illustration can be used.
Represent a Growing Pattern Using an Illustration
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 deposits in his piggy bank. In order to keep track of the amount of money in his piggy bank, Simon has created the following table of values:
| Week (Term Number) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Amount in the Piggy Bank | 5 | 9 | 13 | 17 |
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 of money in the piggy bank after a large number of weeks, for example, after the 10th week?
Source: L'@telier - Online Educational Resources (atelier.on.ca).
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.
Represent a Situation Using a Table of Values
In a problem-solving context, students can also connect a situation expressed in words to a table of values.
Present the following problem-solving situation.
A new movie viewing website, Film Plus, is available. You have to pay a $4 subscription fee when you watch your first movie. So, if we include the cost of the subscription, after having viewed successively 1 movie, 2 movies, 3 movies and 4 movies, you will have paid $9, $14, $19 and $24.
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 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 $24. |
Afterwards, students can use a table of values to represent the relationship between the number of movies rented and the amount of money spent.
| Number of Movies Rented | Amount Spent ($) |
|---|---|
| 1 | 9 |
| 2 | 14 |
| 3 | 19 |
| 4 | 24 |
| ... | ... |
Source : Guide d’enseignement efficace des mathématiques, de la maternelle à la 6e année, p. 51.
Represent an Equation Using a Table of Values
Introduce the equation (g + k = 11) and ask students to determine the possible integer values of g and k.
| g | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| k | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
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 must 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: L'@telier - Online Educational Resources (atelier.on.ca).
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. The following are examples of how to use a graph.
Represent the Relationship Between Two Changing Quantities Using a Graph
Present the following situation.
Last Saturday, during the marathon, Louis and Gaëlle sold lemonade. After each pitcher they sold, they counted the money they'd received. The table of values below represents the relationship between the number of pitchers sold and the amount of money collected.
| Number of Pitchers Sold | 1 | 2 | 3 |
|---|---|---|---|
| Amount Collected ($) | 4 | 8 | 12 |
Then, represent the data in a graph like this one:
Source : Guide d’enseignement efficace des mathématiques, de la maternelle à la 6e 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?
- How can the first three values indicated on the graph help you determine the amount of money collected after eight pitchers have been sold?
- How many pitchers did they have to sell to get about $50?
Skill: Representing Linear Growing Patterns Using Equations or Algebraic Expressions
The learning progression described above and reflected in the mathematics curriculum content and research, including Lee (1996, p. 105), suggests three steps that lead to the representation of an 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-numeric pattern below, where the relationship between the term number and the number of cubes in it is explored (the functional relationship).
Possible reasoning of the student
- Visualize the relationship:
In the 1st term, there is 1 cube and 1 group of 2 cubes.
In the 2nd term, there is 1 cube and 2 groups of 2 cubes.
In the 3rd term, there is 1 cube and 3 groups of 2 cubes.
So, in the 10th term, there will be 1 cube and 10 groups of 2 cubes.
- Verbalize the relationship: To determine the number of cubes in a given term, I start with 1 and add the term number multiplied by 2.
- Represent the relationship using symbols: The relationship can be represented by the equation c = 1 + n × 2, where n is the term number and c is the number of cubes 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 + 2n |
Note: In grade 6, students have learned 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 example below, based on research by Radford (2006, pp. 2-21), illustrates how students may perceive differently the relationship between the term number in a non-numeric growing pattern and the number of circles in each 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 10th 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 determines the total number of circles by adding the term number twice and adding 3. So, for the total number of circles in the 1st term, the number sentence is 1 + 1 + 3 = 5; for the 2nd term, 2 + 2 + 3 = 7; and for the 3rd term, 3 + 3 + 3 = 9.
Continuing the analysis, the student concludes 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 that make up each term.
Student 3 sees:
- in the 1st term, 5 circles;
- in the 2nd term, the 5 initial circles, to which we added 1 group of 2 circles;
- in the 3rd term, the 5 initial circles, to which we added 2 groups of 2 circles.
Analyzing these observations, the student generalizes and finds 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 25th term, there will be \(5 + 24 \times 2\) circles, or 53 circles.
Note: Students may present their interpretation using a table of values.
| Term Number | Number of Circles | Explanation Using the Recursive Relationship | Relationship Between the Term Number and the Number of Circles that Make up Each Term (Functional Relationship) |
|---|---|---|---|
| 1 | 5 | 5 | 5+0x2 |
| 2 | 7 | 5+2 | 5+1x2 |
| 3 | 9 | 5+2+2 | 5+2x2 |
| 4 | 11 | 5+2+2+2 | 5+3x2 |
| 25 | ? | 5+2+2+2+... | 5+24x2 |
| n | 5+(n-1)x2 |
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 that make up each term.
Note: The student analyzed the recursive relationship (that is, adding 2 circles to the previous term), which leads them to the functional relationship using the expression n - 1 to represent the previous term number, the initial 5 circles, and the number of groups of 2 circles in term number n, that is, 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.
| Term Number (n) | 1 | 2 | 3 |
|---|---|---|---|
| Number of Circles (c) | 5 | 7 | 9 |
Students then proceed by trial and error to determine the equation that represents the relationship, as evidenced by the two students' explanations below. 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 \(c = n \times 2 + 3\), where n is the term number and c is the number of circles that make up each term.
Student 5 explains that in the second line of the table of values, there is a regularity: you add 2 to get from one term to the next. So in the rule, there is "× 2". They try the n × 2 rule and do not get the terms 5, 7, 9… They then try the \(n \times 2 + 1\) rule, then the \(n \times 2 + 2\) rule. When they try the rule \(n \times 2 + 3\), they get the terms 5, 7, 9…
The student concludes that the relationship can be represented by the equation \(c = n \times 2 + 3\) , where n is the term number and c is the number of circles that make up each term. In this case, this student appears to determine the equation by following a rote-learned procedure.
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: L'@telier - Online Educational Resources (atelier.on.ca).
Represent the Equation of a Linear Growing Pattern Using a Semi-Concrete Representation
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 : Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1er à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
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 two big ideas of algebraic reasoning.
Source : Guide d’enseignement efficace des mathématiques de la 7e à la 10e 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 term.
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 (term value) 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.
Vertical Table of Values
| Term Number | Number of Elements |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | … |
Horizontal Table of Values
| Term Number | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Number of Elements | 2 | 4 | 6 | … |
Source : Guide d’enseignement efficace des mathématiques, de la maternelle à la 6e année, p. 23.
Knowledge: Graphical Representation
Growing patterns can be used as an introduction to graphing. The term numbers in the growing pattern correspond 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 7e à la 10e année, p. 63
Example
Source : En avant, les maths!, 6e année, CM, Algèbre, p. 5.
Knowledge: Algebraic Expression
A collection of one or more terms involving variables, numbers, and operations.
Source: The Ontario Curriculum, Mathematics, Grades 1-8, Ontario Ministry of Education, 2020.
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 that can be difficult to understand because, among other things, there are various types of equations whose function varies depending on the situation. Four types of equations are presented to students:
- Equation to be solved: An equation like 2 + n = 14 must be solved. It usually comes from a problem 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 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 c. Such an equation is not solved.
- 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, \(4 + 4 = 2 \times 4\)) 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.
Source: L'@telier - Online Educational Resources (atelier.on.ca).