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.
Understanding Math: Number Patterns and Function Rules
A useful first step is to separate the position of a term from its value. In the list five, eight, eleven, fourteen, the fourth term has value fourteen. The position tells where the number sits, while the value is the number itself.
This matters when a rule must find a faraway term, such as the fiftieth term, without writing every earlier term. Start with the first value, then count how many equal steps are needed to reach the chosen position. There are one fewer steps than the term number because the first term is already present.
Checking the change between neighboring terms is a strong method, but it must be done carefully. Subtract each earlier value from the next value. If these first differences stay the same, one linear rule can describe the pattern.
If the differences change, inspect what happens next. A pattern such as two, four, eight, sixteen has growing differences, yet each value is made by multiplying the previous value by two.
Some patterns use square numbers, alternating operations, or groups of repeated values. A rule should explain every given term, not only the first few terms.
Function rules describe a different viewpoint. A sequence usually uses a term position as the input, while a function can accept any allowed input. For a rule that says output equals four times input minus two, an input of three produces ten.
Input zero produces negative two, which may be useful even if a sequence begins at position one. This difference prevents a common error.
Students sometimes use the first output as though it belongs to input zero when it actually belongs to input one. Labeling every row in a table makes the starting point clear.
Graphs reveal another part of the rule. A constant increase in output for every increase of one in input makes points that line up in a straight line. The steepness shows how quickly the outputs change.
A starting value shifts the line up or down. Multiplicative growth produces a curve rather than a straight line because the changes get larger or smaller over time. Students meet these ideas in saving money each week, taxi fares with a starting charge, temperature conversions, phone data costs, and populations that grow by a percentage.
When finding a rule, test it with several known values and use it to predict a new value. A correct rule should keep working beyond the examples shown.
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.