C1.4 Create and describe patterns to illustrate relationships among rational numbers.
Activity 1: Talking Numbers (Analysis of Regularities to Discover the Properties of Numbers)
Project a table of numbers that represents a regularity that is incomplete. The student individually analyzes the table and completes it. The student must be able to explain their reasoning.
Examples of number tables
a) What's going on?
| Input | Output |
|---|---|
| 104 | 10 000 |
| 103 | 1 000 |
| 102 | 100 |
| 101 | 10 |
| 100 | ? |
| 10-1 | ? |
| 10-2 | ? |
- Can you deduce the next values?
- What would be the value of 10-4? How do you know this?
- What relationships exist between the numbers? Explain your reasoning.
b) What do you notice in this picture?
| \(2 \times 4 = 8\) |
| \(2 \times 3 = 6 \) |
| \(2 \times 2 = 4\) |
| \(2 \times 1 = 2\) |
| \(2 \times 0 = 0\) |
| \(2 \times (-1) \ = \ ?\) |
| \(2 \times (-2) \ = \ ?\) |
| \(2 \times (-3) \ = \ ?\) |
| \(2 \times (-4) \ = \ ?\) |
- Can you predict the next values in this pattern? Explain.
- What would be the value of (2 times (-100)) Why?
- How would you show that (-2) times (-4) = 8 from a table of numbers like the one above?
Note: Functional thinking necessarily involves identifying two sets of numbers and the relationship between them. It is sometimes difficult for students to identify these two sets of numbers in order to put them in a table of values. This is a skill that should be practiced by students in this type of exercise.
Activity 2: The Chessboard (Conjecture and Generalization From a Number Pattern)
Materials
- Chessboard reproduced on a grid sheet or on a paper
- Non-permanent felt-tip pen
- Calculator
Divide the students into teams. Present the scenario to them. Each team must answer the questions. Discuss the proposed strategies in a debriefing session.
Situation
Anastasia is fascinated by the regularities and squares of a chessboard. Using her chess board and a non-permanent felt-tip pen, she creates a sequence of numbers.
She writes numbers on the boxes. In square A1 she writes the number 100 000 000, in square B1 the number 10 000 000, in square C1 the number 1 000 000, in square D1 the number 100 000 and so on.
She would like to identify the 64 squares of the chessboard with a number.
- What number will be written in box H1? How do you know? (last square of the first row)
- Can you explain the rule for finding the number assigned to each box? Is there a more efficient way to write your number?
- What number will be written on the 64th square of the chessboard? Describe the number pattern you would use to find this answer.