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, and use algebraic representations of the pattern rules to solve for unknown values in linear growing patterns.
Skill: Determining and Using Rules to Extend Patterns, Making and Justifying Predictions, and Finding Missing Elements
The study of relationships includes the representation of relationships using rules stated in everyday language.
Determining a functional relationship 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 describe the relationship between the two changing quantities as well as to determine the value of any term (for example, the value of the 25th term) without having to extend the pattern to the term being sought.
Students learn to use an equation to represent a functional relationship expressed in words. They must choose the variables and formulate an equation that represents the pattern rule, which helps develop a sense of symbols. It is in this context that the resulting equations and the relationships they represent become meaningful.
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:
Example 1
Dominic is saving up to buy a video game that costs $74.25 (including taxes). However, he only has $35 in her wallet. Every week, his parents give him $5, which he keeps in her wallet. The relationship between the number of weeks that pass and the amount of money in Dominic's wallet can be represented by a table of values.
| Week | 1 | 2 | 3 | |||
|---|---|---|---|---|---|---|
| Amount of Money in Dominic's Wallet | 40 | 45 | 50 |
How much money will Dominic have in his wallet after 5 weeks if he doesn't spend any?
Recursive Relationship (Addition)
The relationship between the number of dollars in Dominic's wallet from week to week can be represented using a table of values.
I extended the table of values and determined that he will have $60.
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 his wallet, which gives him $40.
- On the 2nd week, Dominique has $35 and 2 $5 bills in his wallet, which gives him $45.
- On the 3rd week, Dominique has $35 and 3 $5 bills in his wallet, which gives him $50.
- In week 4, Dominic has $35 and 4 $5 bills in his wallet, giving him $55.
- In week 5, Dominic has $35 and 5 $5 bills in his wallet, giving him $60.
b) After how many weeks will Dominic have enough money to buy the video game?
Recursive Relationship (Addition)
I extended the table of values until there is $75 in the wallet and determined that it will be after 8 weeks.
Value table that represents the number of weeks and the number of dollars in a wallet. Week one: 40 dollars. Week 2: 45 dollars.Week 3: 50 dollars.Week 4: 55 dollars.Week 5: 60 dollars. Week 6: 65 dollars.Week 7: 70 dollars.Week 8: 75 dollars. Arrows are representing the bonds of plus 5.
Functional Relationship
The number of $5 bills is always the same as the number of weeks that have passed, plus the $35 that was already in his wallet (the constant).
So, on the 8th week, Dominique has $35 and 8 $5 bills in his wallet, which gives him $75.
35 $ + (8 × 5 $) = 35 $ + 40 $ = 75 $
Note: In the past, such a situation would have been presented as part of problem-solving activities and only the second question would have been asked. The problem solving would have involved an arithmetic method, for example:
Dominic earns $5 a week. He is short $39.25. Since 39.25 ÷ 5 = 7.85, or about 8, it will take him 8 weeks.
However, the approach used here is more algebraic, since it relies on the comparison of two changing quantities.
Example 2
A delivery man has to deliver 45 newspapers in his neighbourhood. The table of values below 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 reasoning.
Recursive Relationship (Subtraction)
I determined that he has 3 fewer newspapers to deliver every minute. This means that he delivers 3 newspapers every minute. I extended the table of values maintaining this pattern rule, which allowed me to conclude that after 7 minutes, he has 24 newspapers left to deliver.
Value table that represents the number of minutes elapsed 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.
b) How long does the delivery man take to deliver all his newspapers?
I extended the table of values until there were 0 newspapers left and found that it took 15 minutes to deliver all of the newspapers.
| Number of Minutes Elapsed | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Number of Newspapers to be Delivered | 45 | 42 | 39 | 36 | 33 | 30 | 27 | 24 | 21 | 18 | 15 | 12 | 9 | 6 | 3 | 0 |
Functional Relationship
The number of newspapers delivered is equal to the initial number of newspapers (45) minus three times the number of minutes elapsed. The equation is therefore n = 45 - 3m, where n is the number of newspapers to be delivered and m, the number of minutes elapsed.
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 Number (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 Relationship (Multiplication)
I noticed that the salary doubles every day. I extended the table of values to 10 days. I then added up the amounts and found that Emily would earn a total of $1023 for 10 days of babysitting.
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.
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.
a) Construct a table of values that represents the relationship between the term number (day) and the number of teams playing a game on that day.
Recursive Relationship (Division)
b) On what day will the last game take place?
The last game will be played on 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 an equation may differ from student to student, as reasoning develops from individual perceptions. 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 and determine the corresponding equation.
The example below illustrates how differently students can perceive the relationship between the term number in a non-numeric growing pattern and the number of squares that make it up. The situation illustrates examples of appropriate questions.
Example
The teacher presents the following pattern.
Using the following questions, the teacher encourages students to analyze the pattern and relate the term number to the number of squares it has:
- How many squares are there in the 1st term, 2nd term, and 3rd term?
- How many squares are needed to construct the term in the 4th position? Construct this term.
- How many squares are needed to construct the 5th term of the pattern? Construct this term.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 58.
- What pattern rule (functional relationship) 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?
- How many squares will there be in term 6? In the10th term? How do you know?
- Are there other ways to determine this?
Teachers then facilitate a mathematical discussion that focuses on students' different perceptions of the relationship. The teacher asks questions such as the ones below to get students to explain their strategy and rule, and then to verbalize them.
- Have you found a quick way to count the number of squares in the 4th term? Can you explain it to us?
Student 1: In the 4th term, I see that there are 3 squares on the left, 3 squares on the right, 3 squares on the top and 1 in the middle, at the bottom. 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 centre and 3 branches of 3 squares. In total, that's 1 + (3 × 3), or 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 terms in a different way. If I look at the pattern, it starts with a square and in each subsequent term, a square is added in 3 places. In the 4th term, we have added the 3 squares 3 times. In all, we have 1 + (3 × 3), or 10 squares.
- Does your method work for counting the number of squares in term 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, at the bottom. That's 4 + 4 + 4 + 1, or 13 squares.
Student 2: Yes. I see 1 square in the centre, and 3 branches of 4 squares. In all, that's 1 + (3 × 4), or 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 pattern, it starts with a square and in each subsequent term, a square is added in 3 places. In the 5th term, we have added the 3 squares 4 times. In total, we have 1 + (4 × 3), which is 13 squares.
- According to your method, how many squares will there be in term 10?
Student 1: There will be 28 squares. There will be 1 square in the centre, then 9 squares on the left, 9 squares on the right and 9 squares above the centre. In total, there will be 9 + 9 + 9 + 1, or 28 squares.
- How do you know how many will be on the left, on the right, and above the centre?
Student 1: From the first few terms in the pattern, each always has 1 square less than the term number.
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?
From the first terms in the pattern, the number of squares in the column is equal to the term number 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, there will be 1 square in the centre. Then each time we add 3 squares to go to the next term. We need to do this for terms 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 term and the expression of this organization have generated different, but equivalent, pattern rules.
- How can your quick method (your pattern rule) be used to determine how many squares the term 25 will have?
By explaining in words how to make a far prediction (for example, the 10th or 25th term) in relation to the term number, students use their pattern rule 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 pattern rule more clearly and express the relationship using an equation.
- How can you determine the number of squares that make up any term?
Student 1: To determine the number of squares that make up any term, I add a number 3 times, then add 1.
- 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 any term more accurately?
Student 1: To determine the number of squares that make up any term, I add the previous term number 3 times, then add 1.
- How can you now express this pattern rule (functional relationship) with an equation?
The equation would be s =(n - 1) +(n - 1) +(n - 1) + 1, where n is the term number and s is the number of squares in it.
Student 2: To determine the value of any term, I multiply by 3 and add 1.
- What do you multiply by 3? What do you determine?
I multiply the previous term number by 3 and add 1. This gives me the number of squares that make up the term in question.
- If we represent the term number by n, how can we represent the previous term number?
This is n - 1. So the equation is s = 1 + 3 ×(n - 1), where n is the term number and s is the number of squares in that term.
Student 3: In each term, there are always 3 branches. One branch that has the same number of squares as the term number and two others that have 1 square less than the term number. To determine the number of squares, I add these 3 numbers.
- How would you explain this pattern rule to me in a more concise way?
To determine the number of squares that make up any term, I add 3 values, the term number and twice the previous term number. So, s = n +(n - 1) +(n - 1), where n is the term number and s is the number of squares that make it up.
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 centre. Basically, I multiply the previous term number by 3 and add 1. So, s =(n - 1) × 3 + 1, where n is the term number and s is the number of squares in it.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 59.
Teachers then prompt students to check the validity of their equation. To validate the equations, a table of values is used.
- According to your rule, if n takes a value of 4, what is the corresponding value of s?
Student 1: My equation becomes s = (4 - 1) + (4 - 1) + (4 - 1) + 1. Therefore, s = 10.
Student 2: My equation becomes s = 1 + 3 × (4 - 1). Therefore, s = 10.
Student 3: My equation becomes s = 4 + (4 - 1) + (4 - 1). Therefore, s = 10.
Student 4: My equation becomes s = (4 - 1) × 3 + 1. Therefore, s = 10.
- What does this value of s correspond to in your equation?
Student 1: The variable s represents the number of squares that make up the term. In the given pattern, the 4th term is composed of 10 squares.
- According to your equation, s = 10 when n = 4. What does this correspond to in the table of values?
Student 2: According to the table of values, the 4th term is composed of 10 squares. This matches what I get from the equation.
The previous example follows a certain approach that can be summarized as follows: have students extend the pattern, analyze the pattern rule and describe it, create a table of values, and then formulate a pattern rule in words and using an equation. This approach allows for the exploration of all kinds of relationships, even some that may, at first, seem out of reach for students. This approach is used for both growing and shrinking patterns.
Skill: Using Symbolic Representations of Rules to Find Unknown Values in Linear Growing Patterns
Among the symbolic representations of growing patterns, rules in the form of equations allow interpolations and extrapolations to be made more easily and precisely than graphical representations.
To establish these rules, it is important to help students generalize, that is:
- guide them in the observation and analysis of situations;
- help them to propose conjectures;
- ask them to support their conjectures with mathematical representations or arguments;
- invite them to verify their conjectures in other situations;
- accompany them in the formulation of a generalization, if necessary.
Source : Guide d’enseignement efficace des mathématiques de la 7e à la 10e année, p. 12.
Example of a Representation of the Relationship Between Two Changing Quantities
Write the equation s = g × 2 on the board and present its context:
Eric invites some friends to his house. When they enter, he asks them to take off their shoes and put them on a step of the stairs. We are interested in the relationship between the number of guests (g)and the number of shoes (s).
With students, create a table of values representing the relationship as well as a non-numeric pattern.
| Number of Guests (g) | 1 | 2 | 3 | ... |
|---|---|---|---|---|
| Number of Shoes (s) | 2 | 4 | 6 | ... |
Ask students to explain what variable g (the number of guests), variable s (the number of shoes), and "× 2" (the number of shoes per person) represent.
Relevant questions:
- How can you represent this situation using another model?
- What is the functional relationship (between the term number and the term value) in the growing pattern?
- How did you find the total number of shoes in the table of values?
- How many shoes will be on the stairs if Eric has 6 guests?
- How many shoes will there be when the 15th person takes off their shoes?
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 238.
Knowledge: Interpolation
The process of estimating values lying between elements of given data.
Source: The Ontario Curriculum. Mathematics, Grades 1-8, Ontario Ministry of Education, 2020.
Example
Determine the value of the cost when the quantity is 7.
| Quantity | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|
| Cost | 6 | 12 | 18 | 24 | 30 | 36 |
Note: In this case, the student can average the costs of quantities 6 and 8.
Knowledge: Extrapolation
The process of estimating values lying outside the range of given data.
Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Example
Determine the value of the cost when the quantity is 22.
| Quantity | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|
| Cost | 6 | 12 | 18 | 24 | 30 | 36 |
Note: In this case, the student can first determine that the equation is c =3q, and then calculate the cost by replacing q with 22, which results in $66.
Note: In order to avoid memorization of steps and to elicit the construction of meaning, vary the unknown value sought. This will give rise to cognitive flexibility and a questioning of the concept being explored.