C1.3 Determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating, growing, and shrinking patterns.
Skill: Identifying and using rules to extend sequences, making and justifying predictions, and finding missing terms
The study of relationships includes the representation of relationships by rules stated in everyday language.
Determining a correspondence rule is more difficult than determining a pattern rule. Determining the correspondence rule in everyday language is an important step in the development of algebraic thinking since it is a generalization of the relationship. The correspondence rule allows students to explain the relationship between the two quantities in change and to determine any term (e.g., the 25th term) without having to extend the sequence to the term being sought.
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 52.
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: These 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 $74.25 (including taxes). However, he only has $35 in his wallet. Every week, his parents give him $5, which he keeps in his wallet. The relationship between the number of weeks that pass and the number of dollars in Dominic's wallet can be represented by a table of values.
| Number of weeks | 1 | 2 | 3 | |||
|---|---|---|---|---|---|---|
| Number of dollars in the wallet | 40 | 45 | 50 |
How much money will Dominic have in his wallet after 5 weeks if he spends no dollars?
Recursive pattern rule (addition)
The relationship between the number of dollars in Dominique's portfolio from week to week can be represented by a table of values. Each term increases by a value of 5.
The pattern rule is +5.
I extended the value table and determined that he will have $60 after 5 weeks.
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 2nd , Dominique has $35 and 2 $5 bills in her wallet, giving her $45.
On week 3rd , Dominique has $35 and 3 $5 bills in her wallet, giving her $50.
In week 4th, Dominique has $35 and 4 $5 bills in her wallet, giving her $55.
On week 5th, Dominique has $35 and 5 $5 bills in her wallet, giving her $60.
Example 2
A peddler has to deliver 45 newspapers in his neighborhood. The following table of values represents the relationship between the number of minutes elapsed since the start of the delivery and the number of newspapers he has left to deliver.
| Number of minutes elapsed | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Number of newspapers to be delivered | 45 | 42 | 39 | 36 | 33 |
How many newspapers does he have left to deliver after 7 minutes? Explain your answer.
Rule of regularity (subtraction)
I found that he has 3 less newspapers to deliver every minute. This means that the vendor delivers 3 newspapers every minute. I extended the value table maintaining this pattern rule, which allowed me to conclude that after 7 minutes, he has 24 newspapers left to deliver.
image Value table that represents the number of minutes elasped and number of journals left to deliver.Week zero, 45 journals.Week one, 42 journals. Week 2, 39 journals.Week 3, 36 journals.Week 4, 33 journals.Week 5, 30 journals.Week 6, 27 journals. Week 7, 24 journals. Arrows represents the bonds of minus three.
Example 3
Emily babysits regularly. A couple asks her to look after their child for 6 hours a day for 10 days and offers to pay her according to the following table of values.
| Term (day) | 1 | 2 | 3 | 4 | … |
|---|---|---|---|---|---|
| Salary for the day ($) | 1 | 2 | 4 | 8 | … |
If she accepts the offer, how much money will Emily earn in total?
Recursive pattern rule (multiplication)
I noticed that the salary doubles every day. I extended the value table to 10 days. I then added up the wages and found that Emily would earn a total of $1,023 for 10 days of babysitting.
image Value table that represents the rank in days and one’s day salary.Rank one, one dollar.Rank 2, 2 dollars. Rank 3, 4 dollars. Rank 4, 8 dollars.Rank 5, 16 dollars.Rank 6, 32 dollars.Rank 7, 64 dollars.Rank 8, 128 dollars. Rank 9, 256 dollars. Rank ten, 512 dollars. Arrows representing the bonds of a multiplication of plus 2.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 35-37.
Example 4
In a soccer tournament, 64 teams are present. On the first day, all teams play a game and losing teams are eliminated. On the second day, the remaining 32 teams play a game and the losing teams are eliminated. The same process is followed on each subsequent day.
Construct a table of values that represents the relationship between the rank (day) and the number of teams playing a game on that day.
Pattern rule (division)
I noticed that the number of teams playing a game decreases by half each day. The pattern rule is therefore ÷ 2.
What day will the last game take place?
The last game will be played during the 6th day.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 35-37.
It is important to recognize that the path to expressing a rule 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 rank of the figure in a non-numerical sequence with an increasing pattern and the number of circles in it.
Example
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 54.
image Caption: ‘’ I see 2 rows of circles in the upper row, there is always one circle more (in blue) than the row of the figure and in the lower row, there are always 2 circles more (in grey) than the row of the figure’’.Nonnumeric increasing sequence of yellow circles.Rank one, 5 circles made up of 2 yellow, 2 gray, and one blue circles. Rank 2, 7 circles made up of 4 yellow, 2 gray, and one blue circles. Rank 3, 9 circles made up of 6 yellow , 2 gray, and one blue circles.
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 54.
Continuing the analysis, the student determines that the figure in the 10th row will contain (10 + 1) + (10 + 2) circles, or 23 circles.
image Caption: “To find the total number of circles, you add twice the rank of the figure and add 3. So, for rank one, the mathematical sentence is one, plus, 3, equals, 5 for rank 2, 2, plus, 2, plus, 3, equals, 7 and for rank 3, 3, plus, 3, plus, 3, equals, 9.”Nonnumeric increasing sequence of yellow circles.Rank one, 5 circles made up of 2 yellow and 3 blues circles.Rank 2, 7 circles made up of 4 yellow and 3 blue circles. Rank 3, 9 circles made up of 6 yellow and 3 blue circles.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 55.
image “In row one, I see 5 circles. In row 2, I see the 5 initial circles, to which we added one group of 2 circles. In row 3, I see the 5 initial circles, to which we added 2 groups of 2 circles”.Nonnumeric increasing sequence of yellow circles.Rank one, 5 yellow circles.Rank 2, 7 circles made up of 5 yellow and 2 blue circles. Rank 3, 9 circles made up of 5 yellow and 4 blue circles
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 55.
Note: Students may present their interpretation using a chart.
| Rank of the figure | Number of circles | Explanation using the Recursive Relationship | Relationship between the rank of the figure and the number of circles that compose it |
|---|---|---|---|
| 1 | 5 | 5 | 5 + 0 x 2 |
| 2 | 7 | 5 + 2 | 5 + 1 x 2 |
| 3 | 9 | 5 + 2 + 2 | 5 + 2 x 2 |
| 4 | 11 | 5 + 2 + 2 + 2 | 5 + 3 x 2 |
| 25 | ? | 5 + 2 + 2 + 2 + ... | 5 + 24 x 2 |
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 56.
The rules formulated by the three students came from their understanding of this relationship. Each student has perceived and generalized the situation in his or her own way, resulting in three rules expressed in different, but equivalent, words. None is better than the other. They do, however, demonstrate that students' interpretation of a relationship has an effect on the rule they formulate. It is important for teachers to encourage these different formulations of a rule.
A number of students move too quickly from the non-numerical sequence to the corresponding table of values. For example, from the previous non-numerical sequence, some students immediately establish the next table.
| Term position of the figure | 1 | 2 | 3 |
|---|---|---|---|
| Number of circles | 5 | 7 | 9 |
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 56.
Students then proceed by trial and error to determine the rule that defines the relationship, as evidenced by the student's explanations below. In each case, the depth of these students' understanding of the relationship it represents should be questioned.
image “I represented the relationship by a table of values. I compared the rank of each figure with the number of corresponding circles by making several trials. For example, I started with X 2, but it didn't work; then I tried X 3, but it didn't work either. I continued in this way and found that, in each case, it was "times two plus three" or twice the rank of the figure, plus 3”.
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 57.
Teachers need to consider the different ways that students perceive the relationships between terms in a sequence and adapt their questioning accordingly to help each student express the rule in words accurately. The following situation highlights examples of adapted questioning.
Example
The faculty presents the following sequence of figures.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 58.
Using the following questions, the teacher encourages students to analyze the sequence and relate the rank of the figure to the number of squares in it:
- How many squares make up the figures in rows 1, 2 and 3?
- How many squares are needed to create the figure in row 4? Create this figure.
- How many squares are needed to create the figure in row 5? Create this figure.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 58.
- What rule do you see in the number of squares?
- Create a table of values that represents the relationship between the rank of the figure and the number of squares in it. What rule do you see in the table?
- How many squares will there be in the figure in row 6? In the 10th row? How do you know?
- What other ways are there to determine this?
The teacher then facilitates a mathematical discussion that focuses on the students' different perceptions of the relationship. In order to get the students to explain and verbalize their strategy and rule, they ask questions such as :
- Have you found a quick way to count the number of squares in the figure in row 4? Can you explain it to us?
Student 1: In the figure in row 4, I see that there are 3 squares on the left, 3 squares on the right, 3 squares above the center and 1 in the center. In total, that's 3 + 3 + 3 + 1, or 10 squares.
- Who used the same method? Did anyone use a different method?
Student 2: I used a similar method. I see 1 square in the center, and 3 branches of 3 squares. That's a total of 10 squares.
Student 3: My method is different, but I got the same answer. I see a column of 4 squares, then 3 squares on the left and 3 squares on the right. In total, I counted 4 + 3 + 3, which is 10 squares.
Student 4: I see the figures in a different way. If I look at the sequence, it starts with a square and in each subsequent row, a square is added in 3 places. In figure 4, we added the 3 squares 3 times. In all, we have 10 squares.
- Does your method work for counting the number of squares in the figure in row 5?
Student 1: Yes. I see that there are 4 squares on the left, 4 squares on the right, 4 squares up and one square in the middle. That's 4 + 4 + 4 + 1, or 13 squares.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 59.
Student 2: Yes. I see 1 square in the center, and 3 branches of 4 squares. In all, that's 13 squares.
Student 3: Yes. I see a column of 5 squares, then 4 squares on the left and 4 squares on the right. In total, I count 5 + 4 + 4, which is 13 squares.
Student 4: Yes. If I look at the sequence, it starts with a square and in each subsequent figure, we add a square in 3 places. In the figure in row 5, we added the 3 squares 4 times. In all, we have 13 squares.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 60.
- According to your method, how many squares will there be in the figure in row 10?
Student 1: There will be 28 squares. There will be 1 square in the center, then 9 squares on the left, 9 squares on the right and 9 squares above the center. In total, there will be 9 + 9 + 9 + 1, or 28 squares.
- How do you know how many will be on the left, right and up?
Student 1: From the first few figures in the sequence, there is always 1 square less than the rank of the figure at each of these three places.
Student 2: It's almost the same thing.
- Why do you say this is the same as the other strategy (student 1)?
Student 2: He (student 1) says that he adds 9 + 9 + 9. I say that I multiply, that is to say that I do 3 × 9, because multiplying is like adding the same quantity several times.
Student 3: I also got 28, since there will be a column of 10 squares, then 9 squares on the left and 9 squares on the right, so 10 + 9 + 9 = 28 squares.
- How do you know it's 10 squares, then 9 squares twice?
Student 3: From the first few figures in the sequence, the number of squares in the column is equal to the rank of the figure, and the number of squares on the left and right is always 1 less than the rank.
Student 4: The way I see the figures, there will be 1 square in the center. Then each time we add 3 squares to go to the next row. We need to do this for rows 2 through 10, so 9 times. In all, we will have 1 + (9 × 3), or 28 squares.
Note: We notice that the calculations performed are similar. However, the different ways of seeing the organization of the squares in the figure and the expression of this organization have generated different, but equivalent rules.
- Can your quick counting method (your ruler) be used to determine how many squares there will be in the figure at row 25?
By explaining in words how to determine the value of a remote term (e.g., the figure at rank 10 or 25) in relation to the rank of a figure, students use their ruler to interpret the relationship.
Since students often have difficulty determining the quantities involved, teachers then ask questions such as the ones below to get them to communicate their rule more clearly and express the relationship using an equation.
- How can you determine the number of squares that make up a figure given any term number?
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 61.
Student 1: To determine the number of squares that make up a figure at any rank, I add a number 3 times, then 1.
- What is this number? How can you identify or name it?
Student 1: This is always the number of the previous row.
- Can you then express your rule for determining the number of squares that make up a figure at any rank more accurately?
Student 1: To determine the number of squares that make up a figure in any row, I add the number of the previous row 3 times, then 1.
Student 2: To determine the value of any term, I multiply by 3 and add 1.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 62.
- What do you multiply by 3? What do you determine?
Student 2: I multiply the rank of the previous figure by 3 and add 1. This gives me the number of squares that make up the figure in question.
Student 3: In each figure, there are always 3 branches. One branch that has the same number of squares as the rank of the figure and two others that have 1 square less than the rank of the figure. To determine the number of squares, I add these 3 numbers.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 62.
- Can you explain this rule to me in a more concise way?
Student 3: To determine the number of squares that make up a figure at any rank, I add 3 values, the rank of the figure and twice the rank of the previous figure.
Student 4: I do 3 times a number to determine the number of squares on the 3 branches and add 1 for the one in the center. Basically, I multiply the rank of the previous figure by 3 and add 1.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 63.
Teachers then prompt students to check the validity of their rule. To validate the rules, the table of values is used.
The previous example follows a certain approach that can be summarized as follows: have students extend the sequence of figures, analyze the pattern and describe it, create a table of values, and then formulate a rule in words. This approach allows for the exploration of all kinds of relationships, even some that may at first seem to be beyond the students' reach.
Example
Here is a decreasing sequence of figures. We study the relationship between the number of squares that make up each figure.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 64.
At first glance, this situation may seem very complex, since the sequence corresponding to the number of squares that make up each figure (42, 30, 20, 12…) is not a simple subtraction pattern. Before putting such a situation aside, we need to examine it more closely. Thus, we can first extend the sequence of figures, then analyze the pattern and create a table of values as follows.
Extending a non-numeric pattern
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 64.
Pattern analysis
If we examine the figures, we can see that there is a pattern, since we always remove a column and a row.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 65.
The number of squares decreases by 12, then by 10, then by 8, and so on. We can see these quantities in the non-numerical sequence. To go from the figure in row 1 to the figure in row 2, we remove 12 squares, that is 1 row of 6 squares and 1 column of 6 squares. To go from the figure in row 2 to the figure in row 3, we remove 10 squares, that is 1 row of 5 squares and 1 column of 5 squares. To pass from the figure in row 3 to the figure in row 4, we remove 8 squares, that is to say 1 row of 4 squares and 1 column of 4 squares.
Creation of a table of values
The table of values allows to represent this pattern.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 65.
Extend a decreasing sequence
Students can explain their observations as follows:
- The figure in row 1 is composed of 7 columns of 6 squares.
- The figure in row 2 is composed of 6 columns of 5 squares.
- The figure in row 3 is composed of 5 columns of 4 squares.
To extend the sequence, the student realizes that the figure in row 5 will be composed of 3 columns of 2 squares.
Knowledge: Pattern Rules
Pattern rule : a rule that extends a sequence by respecting the difference between terms (also called a constant jump).
Functional pattern rule : a rule that extends a pattern by establishing the relationship between the term number and its term.
Source : Curriculum de l’Ontario. Programme-cadre de mathématiques de la 1re à la 8e année. 2020. Ministère de l’Éducation de l’Ontario.
Non-numeric repeating pattern
Example of a functional relationship rule represented using words:
- If the term number is a multiple of 3, the term is a triangle.
- If the term number gives a remainder of 1 when divided by 3, the term is a rectangle.
- If the term number gives a remainder of 2 when divided by 3, the term is a trapezoid.
Non-numerical growing pattern
Pattern C
Example of a functional relationship rule represented using words:
- Each term is composed of a column of 2 cubes, to which 1 cube less than the term number of the figure is added.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, p. 29.
Non-numeric shrinking pattern
Pattern D
Example of a recursive pattern rule in words:
- Each term is composed of a column of 2 cubes. We remove 1 cube each new term.
Growing numeric pattern
4, 8, 16, 32, 64
Example of a recursive pattern rule in words and symbols:
- We double the value of the term to obtain the next term. The rule is : × 2.