Number patterns are lists of numbers that change in a predictable way. They help students notice structure, make predictions, and describe relationships using words, tables, graphs, and equations. A pattern like 3, 6, 9, 12, 15 shows a constant increase of 3 from one term to the next.
Learning to find the rule behind a pattern is an important step toward algebra.
A function rule works like a pattern machine: an input number goes in, a rule is applied, and an output number comes out. For example, the rule y = 2x + 1 turns inputs 1, 2, 3 into outputs 3, 5, 7. Input-output tables make it easier to test rules and extend patterns.
When students connect a sequence, a table, and a rule, they can describe patterns clearly and solve problems more efficiently.
Key Facts
- A sequence is an ordered list of numbers, such as 3, 6, 9, 12, 15.
- An arithmetic pattern adds or subtracts the same number each time, such as +3, +3, +3.
- A geometric pattern multiplies or divides by the same number each time, such as x2, x2, x2.
- A function rule can be written as y = expression in x, where x is the input and y is the output.
- For the rule y = 3x, inputs 1, 2, 3 give outputs 3, 6, 9.
- For an arithmetic sequence, nth term = first term + (n - 1)(common difference).
Vocabulary
- Sequence
- A sequence is an ordered list of numbers that often follows a pattern.
- Term
- A term is one number or item in a sequence.
- Rule
- A rule is a description, formula, or process that explains how a pattern changes.
- Input
- An input is the starting value placed into a function rule.
- Output
- An output is the result produced after a function rule is applied to an input.
Common Mistakes to Avoid
- Looking only at the first two terms, which can lead to a rule that fails later in the pattern. Always check several terms before deciding on the rule.
- Confusing the term number with the term value, which makes formulas hard to use correctly. The term number tells the position, while the term value is the number in that position.
- Adding when the pattern is multiplying, which gives correct results only by accident in some cases. Compare repeated differences and repeated ratios to identify the pattern type.
- Forgetting to apply the whole function rule, which gives the wrong output. In a rule like y = 2x + 5, multiply by 2 first and then add 5.
Practice Questions
- 1 Extend the pattern 4, 8, 12, 16, ... for the next three terms and write the rule in words.
- 2 Use the rule y = 5x - 2 to complete an input-output table for x = 1, 2, 3, 4.
- 3 A pattern begins 2, 4, 8, 16, 32. Explain why this pattern is not arithmetic and describe the rule that generates it.