C1.3 Determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating and gpositioning patterns.
Skill: Determining and Using Rules to Extend Patterns, Make and Justify Predictions, and Find Missing Terms
The study of relationships includes the representation of relationships using rules stated in everyday language.
Determining a functional relationship (between term number and term value) is more difficult than determining a recursive relationship. Determining the functional relationship in everyday language is an important step in the development of algebraic thinking, since it is a generalization of the relationship. The functional relationship allows students to explain the relationship between the two changing quantities as well as to determine any term (for example, the 25th term) without having to extend the pattern to the term being sought.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 52.
It is important to recognize that the path to the expression of a matching rule may differ from person to person, since reasoning develops from individual perceptions. The example below, inspired by a research by Radford ("Algebraic thinking and the generalization of patterns: a semiotic perspective", in J. L. C. S. Alatorre, M. Sáiz, A. Méndez (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, North American Chapter, vol. 1, Mérida, Mexico, 2006, pp. 2-21, as cited in Ontario Ministry of Education, A Guide to Effective Instruction in Mathematics, Grades 4-6 : Modeling and Algebra, 2008, p. 54), illustrates how students may perceive differently the relationship between the position of the term in a gpositioning pattern and the number of circles in it.
Example 1
Image Non-numeric gpositioning pattern of yellow circles. Term one, 5 yellow circles. Term 2, 7 yellow
circles. Term 3, 9 yellow circles.
Image “I see 2 circles in the upper position, there is always 1 circle
more (in blue) than the term position and in the lower position, there are always 2 circles more (in grey) Non-numeric
gpositioning pattern of yellow circles”. Term one, 5 circles made up of 2 yellow, 2 gray, and one blue circles. Term
2, 7 circles made up of 4 yellow, 2 gray, and one blue circles. Term 3, 9 circles made up of 6 yellow , 2 gray, and
one blue circles.
Continuing their analysis, the student determines that the term in the10th term will contain (10 + 1) + (10 + 2) circles, or 23 circles.
Image “To find the total number of circles, you add twice the term of the term and add 3. So, for term
1, the number sentence is one, plus, 3, equals, 5 for term 2, 2, plus, 2, plus, 3, equals, 7 and for term 3, 3, plus,
3, plus, 3, equals, 9”. Non-numeric gpositioning pattern of yellow circles. Term one, 5 circles made up of 2 yellow
and 3 blues circles. Term 2, 7 circles made up of 4 yellow and 3 blue circles. term 3, 9 circles made up of 6 yellow
and 3 blue circles.
Image “In position 1, I see 5 circles. In position 2, I see the 5
initial circles, to which we added 1 group of 2 circles. In position 3, I see the 5 initial circles, to which we added
2 groups of 2 circles”. Non-numeric gpositioning pattern of yellow circles. Term one, 5 yellow circles. Term 2, 7
circles made up of 5 yellow and 2 blue circles. Term 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. 53-55.
Note: Students may present their interpretation using a chart.
| Term number | Number of circles | Explanation using the Recursive Relationship | Relationship between the term number and the number of circles |
|---|---|---|---|
| 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 |
The pattern 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 pattern rule they formulate. It is important for teachers to encourage these different formulations of a pattern rule.
There may be students who move too quickly from the non-numeric pattern to the corresponding table of values. Starting with the previous non-numeric pattern, for example, they can immediately establish the following table.
| Term Number | 1 | 2 | 3 |
|---|---|---|---|
| Number of Circles | 5 | 7 | 9 |
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 Value tables. I compared the term to 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 double
the term plus 3”.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 56-57.
Teachers need to consider the different ways that students perceive the relationships between terms in a pattern and adapt their questioning accordingly to help each student accurately express the rule in words. The following situation highlights examples of adaptive questioning.
Example 2
Present the following situation.
Using the following questions, encourage students to analyze the pattern and relate the term number of the term to the number of squares it has:
- How many squares make up each of the terms in positions 1, 2, and 3?
- How many squares are needed to construct the term in position 4 ? Construct this term.
- How many squares are needed to construct the term in position 5 ? Construct this term.
Image
Non-numeric pattern with gpositioning patterns. term one, one square. term 2, 4 squares. term 3, 7 squares. term 4, 10
squares. term 5, 13 squares.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 58.
- What pattern rule do you see in the number of squares?
- Create a table of values that represents the relationship between the term number and the number of squares in each term. What patterns do you see in the table?
- Create a table of values that represents the relationship between the term number of the term and the number of squares in it. What pattern rule do you see in the table?
- How many squares will there be in the term in position 6 ? in the term in position 10 ? How do you know?
- What other ways are there to determine this?
Then facilitate a mathematical discussion that focuses on students' different perceptions of the relationship. Ask them questions, such as the ones below, to get them to explain and verbalize their strategy and rule.
- Have you found a quick way to count the number of squares in the 4th term? Can you explain it?
Student 1: In the term in position 4, I see that there are three squares on the left, three squares on the right, three squares at the top and one square in the middle, at the bottom. That's 3 + 3 + 3 + 1, or 10 squares.
Image term is made up of ten squares. One yellow square is found at the center of a 7 square
position. 7 squares are side by side horizontally, 3 squares are on top of another vertical square named ‘one’ at
the center.
- Who used the same method? Did anyone use a different method?
Student 2: I used a similar method. I see 1 square in the centre, and 3 branches of 3 squares. In total, that's 10 squares..
Image term is made up of ten squares. One yellow square is found at the center of a 7 square
position. 7 squares are side by side horizontally, 3 squares are on top of another vertical square named ‘one’ at
the center.
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..
Image term made up of ten squares. 7 squares are side by side horizontally, 3 yellow squares are
on top of another vertical square named ‘one’ of the center.
Student 4: I see the squares in a different way. If I look at the pattern, it starts with a square and in each term, a square is added in three places. In the fourth term, the three squares have been added three times. In all, we have 10 squares.
Image term made up of ten squares. 7 squares are side by side horizontally, 3 squares are on top
of another vertical square named ‘one’ in the center. Dotted lines make an arc with the number 3 on each position
of squares.
- Does your method work for counting the number of squares in the term in position 5 ?
Student 1: Yes. I see that there are four squares on the left, four squares on the right, four squares at the top, and one square in the middle, at the bottom. That's 4 + 4 + 4 + 1, or 13 squares.
Image term made up of ten squares. 7 squares are side by side horizontally, 3 squares are on top
of another vertical square named ‘one’ in the center. Dotted lines make an arc with the number 3 on each position
of squares.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 59.
Student 2: Yes. I see a square in the middle at the bottom and three branches of four squares. That's a total of 13 squares.
Image term is made up of ten squares. One yellow square is found at the center of a 7 square
position. 7 squares are side by side horizontally, 4 squares are on top of another vertical square named ‘one’ at
the center.
Student 3: Yes. I see a column of five squares, then four squares on the left and four squares on the right. In total, I count 5 + 4 + 4, which is 13 squares.
Image term made up of ten squares. 7 squares are side by side horizontally, 4 yellow squares are
on top of another vertical square named ‘one’ in the center.
Student 4: Yes. If I look at the pattern, it starts with a square, and in each term a square is added in three places. In the term at the 5th position, we added the three squares four times. In all, we have 13 squares.
Image term made up of ten squares. 7 squares are side by side horizontally, 4 squares are on top
of another vertical square named ‘one’ in the center. Dotted lines make an arc with the number 3 on each position
of 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 term in the 10th term?
Student 1: There will be 28 squares. There will be one square in the middle, at the bottom, then nine squares on the left, nine squares on the right, and nine squares at the top. In total, there will be 9 + 9 + 9 + 1, which is 28 squares.
- How do you know how many will be on the left, right and top?
According to the terms in the previous terms of the pattern, there is always one square less than the term of the term in 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)?
He (student 1) mentions that he adds 9 + 9 + 9. I multiply, that is, I do 3 × 9, because multiplying is like adding the same quantity several times.
Student 3: Me, I also got 28, since there will be a column of 10 squares, then nine squares on the left and nine squares on the right, so 10 + 9 + 9 = 28 squares.
- How do you know it's 10 squares, then nine squares twice?
From the first terms in the pattern, the number of squares in the column is equal to the term number of the term and the number of squares on the left and right is always 1 less than that term number.
Student 4: The way I see the terms in the terms, there is a square in the middle at the bottom. Then we add three squares to the term in the next term. We will have to do this for the terms in terms 2 through 10, so nine 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 term and the expression of this organization have generated different, but equivalent, pattern rules.
- Can your quick count method (your matching rule) be used to determine how many squares there will be in the term in the 25th term?
Students who explain in words how to determine the value of a term at a distant position (for example, the term at position 10 or position 25 ) in relation to the position of a term, use their matching rule to interpret the relationship.
Students may often have difficulty identifying the quantities involved. It is important to ask them questions such as the ones presented below to get them to express their pattern rule more clearly and to express the relationship using an equation.
- How can you determine the number of squares that make up a term given any term number?
Student 1: To determine the number of squares that make up a term at any position, I add a number three times, then add one.
Image term is made up of ten squares. One yellow square is found at the center of a 9 square
position. 9 squares are side by side horizontally, 4 squares are on top of another vertical square named ‘one’ at
the center.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 61.
- What is this number? How can you identify or name it?
It is always the previous term number.
- How can you then express your pattern rule (functional relationship) for determining the number of squares that make up a term in any term more accurately?
Student 1: To determine the number of squares that make up the term in any term number, I add the term number of the previous term 3 times, then add 1.
Student 2: To determine the value of any term, I multiply by three and add one.
Image term is made up of ten squares. One yellow square is found at the center of a 9 square
position. 9 squares are side by side horizontally, 4 squares are on top of another vertical square named ‘one’ at
the center.
- What do you multiply by three? What do you determine?
I multiply the term number of the previous term by 3 and add 1. This gives me the number of squares that make up the term in question.
Student 3: In the term, in each term, there are always three branches. One branch that has the same number of squares as the term of the term and two others that have one less square than the term of the term. To determine the number of squares, I add these three numbers together.
Image term made up of ten squares. 7 squares are side by side horizontally, 4 yellow squares are
on top of another vertical square named ‘one’ in the center.
- Can you explain this correspondence rule in a more concise way?
Student 3: To determine the number of squares that make up a term given any term number, I add 2 values, the term number of the term and two times the term number of the previous term.
Student 4: I do three times a number to determine the number of squares on the three branches and add one for the one in the center. Basically, I multiply the position of the previous term by three and add one.
Image term made up of ten squares. 7 squares are side by side horizontally, 4 squares are on top
of another vertical square named ‘one’ in the center. Dotted lines make an arc with the number 3 on each position
of squares.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 62.
Teachers then prompt students to check the validity of their pattern rule. To validate the rules, a table of values is used.
The previous example has some steps that can be summarized as follows: have students extend the pattern of terms, analyze the pattern rule, describe it, create a table of values, and then formulate a word matching rule. This procedure allows for the exploration of all kinds of relationships, even some that at first glance may seem to be beyond the students' reach.
Example 3
Here is a pattern. We will study the relationship between the of the term and the number of squares that compose it.
Image Non-numeric gpositioning pattern of the term of terms and squares. term one, 2 squares. term
2, 6 squares. term 3, 12 squares, term 4, 20 squares.
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 pattern corresponding to the number of squares that make up the term in each term (2, 6, 12, 20, …) does not present a simple addition regularity rule. Before putting such a situation aside, it is important to examine it more closely. First, we need to extend the pattern of terms, then analyze the regularity rule, and finally construct a table of values.
Extending a Non-Numeric Pattern
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 64.
Analysis of the Pattern Rule
If you look at the terms, you can see that there is a pattern rule, because a column and a term are always added.
Image Non-numeric gpositioning pattern of the term of terms and two different colour squares. term
one, 2 squares. term 2, 2 squares and 4 dark blue squares. term 3, 6 squares and 6 dark blue squares. term 4, 12
squares, 8 dark blue colour squares.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 65.
The number of squares increases by 4, by 6, by 8, and so on. These quantities can be observed in the non-numeric pattern.
To move from the term in the 1st term to the term in the 2nd term, four squares are added - a term of two squares and a column of two squares.
To move from the term in the 2nd term to the term in the 3rd term, six squares are added - a term of three squares and a column of three squares.
To move from the term in the 3rd term to the term in the 4th term, eight squares are added - a term of four squares and a column of four squares.
Table of Values
The table of values allows us to represent this pattern in a different way.
Image Value of table that represents the term of terms and the number of squares. term one, 2
sqaures. term 2, 6 squares. term 3, 12 squares. term 4, 20 squares. Rule of regularity of plus 4, plus 6, plus
8.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 65.
Determining a Pattern Rule Using Words
It is also possible to analyze each of the terms to recognize the relationship between the term number of the term and the number of squares in it. Students, for example, can explain their observation as follows.
- In the 2nd term, there are 3 columns of 2 squares.
- In the 3rd term, there are 4 columns of 3 squares.
- In the 4th term, there are 5 columns of 4 squares.
- So in the 10th term, there will be 11 columns of 10 squares.
- A term is always composed of columns of squares. The number of columns is one more than the term number of the term and the number of squares in each column is the term number of the term.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 65.
Knowledge: Pattern Rule
Recursive pattern rule : a description of the relationship between one term and the next, based on a generalization about the pattern structure..
Repeating Pattern
Image Non-numeric pattern with repetitive patterns: pattern A: term one to 9, rectangle, trapezoid,
triangle, repeated 3 times.
Pattern rule in Pattern A: repeat the blue rectangle, green trapezoid, orange triangle pattern, always in the same order.
Gpositioning Pattern
Pattern rule in Pattern B: a square is added to the previous term.
Image Non-numeric pattern with gpositioning patterns. pattern “B”. Term one: one square. Term 2: two
squares. Term 3: 3 squares.
Pattern rule in Pattern C: a cube is always added to the bottom term of the previous term.
Image Non-numeric pattern with gpositioning patterns. Term one, 2 cubes. Term 2, 3 cubes. Term
3, 4 cubes.
Image Value tables represents the term of a term and the number of cubes. Term one, 2 cubes.
Term 2, 3 cubes. Term 3, 4 cubes.
Knowledge: Functional Relationship
Functional Pattern Rule : a rule that extends a pattern by establishing the relationship between the term number and its term.
In the first pattern below, the pattern rule is that the number of triangles in the term is three times its term number. In the second pattern, the green triangle is constant from one term to the next and a group of two orange squares is always added to the next term.
Image Non-numeric pattern shows the correspondence rule by establishing the relation between
the term and its term. 1st term, 3 triangles. 2nd term, 6 triangles. 3rd
term, 9 triangles. 4th term, 12 triangles.
Image Number pattern with term of term and number of triangles. term one, three triangles.
term 2, 6 triangles. term 3, 9 triangles. term 4, 12 triangles. Correspondence rule times three.
Image Non-numeric pattern of the term and the shapes. term
one, one green triangle and one group of two orange squares. term 2, one green triangle and 2 groups of 2
orange squares. term 3, one green triangle and 3 groups of 2 orange squares. term 4, one green triangle and 4
groups of 2 orange squares.|
In the 1st term, there is 1 green triangle and 1 group of 2 orange squares. In the 2nd term, there is 1 green triangle and 2 groups of 2 orange squares. In the 3rd term, there is 1 green triangle and 3 groups of 2 orange squares. In the 4th term, there is 1 green triangle and 4 groups of 2 orange squares. |