C1.2 Create and translate growing and shrinking patterns using various representations, including tables of values and graphs.
Skill: Create Growing and Shrinking Patterns
Before creating patterns, students must first recognize, compare, describe, change the representation, and extend a wide variety of patterns. Initially, the use of manipulatives is essential for representing patterns. When creating patterns, students can easily change one element of the pattern and check for regularity. However, when drawing a shape on paper, students may focus on the shape to be reproduced rather than on the entire pattern or the nature of the pattern.
Here is an example in which students demonstrate their understanding of the concept of recursive relationships by creating a pattern and explaining it.
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 conversation, presentations of the patterns help to identify different representations of the same pattern and check how students communicate their understanding.
Teachers then ask students to create a sequence and exchange it with a partner. The teacher invites students to construct a different representation of the received sequence, 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, as this may make it difficult to identify the structure of the sequence.
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 (e.g., +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.
Skill: Representing Patterns and Making Connections Between Representations
The ability to communicate algebraic reasoning is developed when students express their understanding of a problem 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 4e à la 6e année, p. 18.
In order to develop a solid understanding, students need to have experiences in context by exploring problem situations. Contextualization allows students to make connections between various representations and develop an understanding of the algebraic concepts being explored. Teachers also use a variety of representations to help students take ownership of mathematical concepts and make connections between representations.
In the junior grades, students explore relationships and represent them in different ways. In grades 4 and 5, students learn that a relationship can be represented by a situation (non-numerical sequence), a table of values or a graphical representation. 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 4e à la 6e année, p. 41.
Using multiple representations of the same pattern to communicate understanding is an essential component of developing algebraic thinking. Non-numeric and numeric patterns with growing and shrinking patterns can be represented in a variety of ways, including oral description, a table of values, concrete materials, and graphs.
Oral Description
Represent a shrinking pattern using words
When students are presented with a shrinking numeric pattern, teachers should provide them with a variety of opportunities to represent it in words. This is an essential step in getting students to determine the algebraic expression of the pattern.
The non-numeric shrinking pattern can be described as follows:
- The figure in term 1 is composed of 3 blue squares and 8 red circles.
- The figure in term 2 is composed of 3 blue squares and 7 red circles.
- The figure in term 3 is composed of 3 blue squares and 6 red circles.
Students may also notice that the number of blue squares, 3, remains constant, while the number of red circles decreases by 1 each term.
Manipulatives
The manipulatives (e.g., algebraic tiles, nesting cubes, tokens, Cuisenaire rulers) 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 shrinking pattern using manipulatives
Have students represent the 14, 10, 6, 2 numeric pattern in a concrete way to make the pattern more visible by creating a shrinking pattern using manipulatives of their choice.
Relevant Questions
- How many cubes (tokens, blocks …) will you use to concretely represent the term of the sequence in the 1st row? In the 2nd row? In the 3rd row?…
- What change do you notice from one row to the next?
- What does not change from one row to the next (the constant)?
- What is the rule of regularity? How do you know?
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 212.
Illustration
Illustration 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 serve as justifications or explanations.
Many types of problems naturally prompt students to make a drawing to help them solve them. Students may also use manipulatives at the same time. As students become more abstract thinkers, they move away from manipulatives to drawings. The following are examples of how an illustration can be used.
To represent an increasing sequence by an illustration
Present students with the following situation:
| Number of the week | 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 her each week, and how much money has accumulated from week to week.
Relevant questions
- How does the illustration help determine the rule of regularity?
- 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 $5 after the first week?
- How can you determine the amount in the piggy bank after a large number of weeks, for example, week 10?
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 214-215.
Table of values
The table of values allows to semi-concretely represent the relationship between two changing quantities (variables), one of which depends on the other. The table of values is often created to represent the relationship between numerical values associated with the terms of an increasing non-numerical sequence and the position of these terms.
The table of values can also be used to present values of variables in an equation or in a problem solving context.
Represent a problem situation using a table of values
In a problem-solving context, students can also relate a situation expressed in words to a table of values.
Present the following problem situation:
Students can first use a table of values to represent the relationship between the number of movies viewed and the amount spent.
| Number of films viewed | Amount spent ($) |
|---|---|
| 1 | 9 |
| 2 | 14 |
| 3 | 19 |
| 4 | 24 |
| ... | ... |
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 51.
To better understand this relationship, students can model it using concrete or semi-concrete materials and words.
image Nonnumeric sequence with increasing patterns. Figure one, four red rounds and 5 blue rounds.Figure 2, 4 red rounds and 10 blue rounds.Figure 3, 4 red rounds and 15 blue rounds.Figure 4, 4 red rounds and 20 blue rounds.
|
When you rent 1 film, 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 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 $24. |
Afterwards, students can use a table of values to represent the relationship between the number of films rented and the amount of money spent.
| Number of movies viewed | Amount spent ($) |
|---|---|
| 1 | 9 |
| 2 | 14 |
| 3 | 19 |
| 4 | 24 |
| ... | ... |
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 51.
Graphical representation
A diagram is a schematic representation of a set of data. In the junior division, in the Data domain, students learn how to represent data using a variety of diagrams. Students can use this knowledge in the Algebra domain. The appearance of the data set in a graph (e.g., bands in ascending or descending order) allows for analysis of change and facilitates interpolation and extrapolation. The table of values is used to create the chart. The following are examples of how to use a chart.
Represent the relationship between two changing quantities using a graph
Present the following situation:
| Number of jugs sold | 1 | 2 | 3 | ||
|---|---|---|---|---|---|
| Amount collected ($) | 4 | 8 | 12 |
Then present the data in a diagram like this one:
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 233.
Relevant questions
- What changes are there from pitcher to pitcher in terms of sales? Are these changes always the same? Why or why not?
- Looking at the chart, how is it possible to determine the next cash inflow?
- What is the relationship between the number of pitchers sold and the amount collected?
- How can the first three values shown in the diagram help you determine the amount of money collected after eight pitchers are sold?
Represent the relationship between two changing quantities (with an illustration) using a bar graph
Present the following situation:
With the students, represent the gardens with an illustration.
For example:
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 234.
Using a table of values, represent the relationship between the perimeter of a garden and the number of square spaces it occupies.
| Number of square spaces | Perimeter (m) |
|---|---|
| 1 | 4 |
| 2 | 6 |
| 3 | 8 |
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 235.
Then, represent the data in a diagram like this one:
Relevant questions
- What change is there between the values of the number of square spaces? Between the values of the perimeters? Why do you ask?
- What would be the next entries in the value table?
- What is the relationship between the number of square spaces and the perimeter of a garden?
- How could you use the graph to determine the perimeter of an 8 square space garden?
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 235.
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 cycle, students have learned to create tables of values to represent relationships in problem situations.
The table of values is often created to represent the relationship between numerical values associated with the terms of a non-numerical sequence with an increasing pattern and the position of these terms.
Example
We can study the relationship between the position of a figure in a non-numerical sequence and the number of objects that make it up. The position of the figure is written in the first column (or row) and the number of objects that make up the position (term value) is written in the second column (or row). The regularity of the terms in the second column (or row) can be used to extend the value table.
Table of values in the vertical direction
| Position of the figure | Number of objects |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | … |
Horizontal table of values
| Position of the figure | 1 | 2 | 3 |
|---|---|---|---|
| Number of objects | 2 | 4 | 6 |
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 4e à la 6e année, p. 231.
Knowledge: graphic representation
Ascending pattern sequences can be used as an introduction to graphing. The position of the figures in the ascending pattern sequences 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 vertical axis also represents the number of elements that will exist at position 0 (the value of the constant), which is the ordinate of the graphical representation.
Source: A Guide to Effective Instruction in Mathematics, Grades 7-10, p. 63.
Example
Relationship between the position and the number of toothpicks
Source: En avant, les maths, Grade 6, Algebra, p. 5-6.