Sequences and functions
Generating sequences (2)
Sequences do not only come as numbers. Count the dots or matchsticks in a growing pattern and a sequence appears.
Patterns make sequences
Here is a sequence of patterns made from dots: pattern 1 — 3 dots pattern 2 — 5 dots pattern 3 — 7 dots The numbers of dots form the sequence 3, 5, 7, … Going from one pattern to the next, one extra dot is added at each end — so each pattern has TWO more dots than the one before. The term-to-term rule is 'add 2', and pattern 4 has 7 + 2 = 9 dots.
Matchstick triangles
A sequence of triangles made from matchsticks: pattern 1 — 3 matchsticks pattern 2 — 6 matchsticks pattern 3 — 9 matchsticks The sequence is 3, 6, 9, … and the term-to-term rule is 'add 3', because each new triangle needs 3 more matchsticks. So pattern 4 uses 12 matchsticks.
Explaining how it is formed
It is not enough to spot the rule — you should be able to say WHY it works. 'Add 3' because each extra triangle needs three more matchsticks. 'Add 2' because one extra dot goes on each end. Explaining the rule from the picture is what tells you the sequence will keep behaving that way.
Worked examples
Dot patterns of 3, 5, 7 — how many in pattern 4?
- Each pattern gains one dot at each end, so 2 more.
- 7 + 2 = 9.
Answer: 9 dots
Matchstick triangles: 3, 6, 9 — pattern 4?
- Each new triangle needs 3 more matchsticks.
- 9 + 3 = 12.
Answer: 12 matchsticks
Why is the dot rule 'add 2'?
- One extra dot goes on each end.
- Two ends means two extra dots.
Answer: Because of how the pattern grows
Why is the matchstick rule 'add 3'?
- Each extra triangle needs three more matchsticks.
Answer: One triangle, three sticks
Pattern 5 of the matchstick triangles
- Pattern 4 uses 12.
- 12 + 3 = 15.
Answer: 15 matchsticks
Spotting the rule is not enough
- You should be able to say WHY it works.
- Explaining it from the picture is what shows it will keep going.
Answer: Explain the rule, do not just state it
Where you see this in real life
- Tiling a path: each extra paving slab needs the same number of edging stones, giving a steady term-to-term rule.
- Seating at joined tables — each extra table adds a fixed number of new places, which is exactly a pattern sequence.
Key words
- Sequence of patterns
- a set of patterns that grow in a regular way
- Term-to-term rule
- how to get from one term to the next
- Pattern
- an arrangement that repeats or grows in a regular way
- Term
- one number in a sequence
- Generate
- produce the terms of a sequence from a rule