In Grade 4 you made patterns from a rule, and you can graph pairs of numbers. Now make two patterns side by side and compare them.
Key idea
Generate both patterns from their rules. Pair up the terms that are in the same position to make ordered pairs. Then look for a relationship: often each term of one pattern is a fixed number of times the matching term of the other.
Why it works
Pattern A adds 3 each time and pattern B adds 6. After any number of steps, B has added twice as much as A, so each term of B is twice the matching term of A. Graphing the pairs $(3, 6), (6, 12), (9, 18)$ shows the relationship as points on a straight line.
| Step | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Pattern A (add 3) | 0 | 3 | 6 | 9 | 12 |
| Pattern B (add 6) | 0 | 6 | 12 | 18 | 24 |
Worked examples
Worked example: Make the pairs
Pattern A: start at 0, add 3. Pattern B: start at 0, add 6. Write the first four ordered pairs.
- A: 0, 3, 6, 9. B: 0, 6, 12, 18.
- Pair by position: $(0, 0), (3, 6), (6, 12), (9, 18)$.
Answer: $(0, 0), (3, 6), (6, 12), (9, 18)$
Worked example: Use the relationship
When pattern A reaches 30, what is the matching term of pattern B?
- Each term of B is 2 times the term of A.
- $2 \times 30 = 60$.
Answer: 60
Common mistake
Pairing terms from different positions, such as the 3rd term of A with the 4th term of B, or putting B first in the pair. Check: count both patterns from the same start, and keep the order (term of A, term of B).