C1.3 Determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating and growing patterns.
Activity 1: Tree Growth
Ask students to represent the relationship between the age of the tree and its height using the following models:
- Table of values
- Graphic representation
- In words
Image A term that shows tree growth in meters over 4 years. Year one, 2 meters. Year 2, 4 meters. Year 3, 6 meters. Year 4, 8 meters.
Next, invite students to answer the following questions.
- What change is there from one box to the next in the table of values?
- What change is there in the graphic representation?
- By what number of hops does the age of the tree grow in the value table?
- By what number of hops does the height of the tree grow in the value table?
- What is the relationship between the height of the tree and its age?
- How tall will the tree be when it is 45 years old? Is this possible? Why or why not?
- Is there a pattern rule in this relationship?
- Which representation (table of values, graphical representation, words) allows the rule of regularity to be determined more quickly? The functional pattern rule more quickly? Why is this?
- Can all rules be represented in a table of values? How are they represented?
- Can all the rules be represented in a graphical representation? How are they represented?
- If we wanted to represent the height of the tree at 45 years, what would the curve look like?
Source : L’@telier - Ressources pédagogiques en ligne (atelier.on.ca).
Activity 2: A Square, Squares
Show students the following pattern and ask them to draw the fourth term.
Image Non-numeric growing pattern of squares. Term one, one light blue square. Term two, one light blue square and 3 medium blue square. Term 3, one light blue square and 3 medium blue square, and 4 dark blue squares.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 158.
Invite students to determine the pattern rule and use it to draw the fifth term.
Ask students to create a table of values that represents the relationship between the term number of the term and the number of squares in it.
| Term Number | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Number of Squares | 1 | 4 | 9 | 16 | 25 | ? |
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 158.
To get students to analyze the relationship between the term number and the number of squares within it, ask questions such as:
- What is the next number of square tiles to be entered in the table? How did you determine this?
- Will any terms following this pattern have 15 square tiles? 49 square tiles? 81 square tiles? 50 square tiles? Explain your answer.
- How can we determine the number of square tiles required to create any term in the pattern?
- How many squares will make up the paving in the 10th position? in the 12th position? in the 20th position?
- What is special about all the numbers of tiles?
- What is the relationship between the term number of the term and the number of squares?
To help students to recognize that the number of square tiles that make up a term always corresponds to the square of the term number (or the area of the term), read the values from the table aloud (6, 36; 5, 25; 4, 16…).
Encourage students to look at the term by the number of rows or columns (for example, the 3rd term has 3 rows of 3 squares or 3 columns of 3 squares) or to think about multiplication facts.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 158 et 159.