Identifying and extending patterns, sequence basics
Elementary Foundations • K-5
Patterns are everywhere -- in wallpaper, music, seasons, and numbers. A pattern is something that repeats or follows a rule. Being able to spot patterns is one of the most important skills in all of math because patterns help you predict what comes next.
A repeating pattern has a core that repeats over and over:
Look at this pattern:
The core is: ▲ ● ● (triangle, circle, circle)
It repeats every 3 shapes. The next shape would be ▲ (triangle), because the core starts over.
A growing pattern (also called a sequence) gets bigger or smaller following a rule.
Look at this sequence:
Find the rule: What happens between each number?
The rule is "add 3" each time. The next number is 14 + 3 = 17.
Look at this sequence:
Each number is doubled (multiplied by 2):
The next number is 48 x 2 = 96.
You have already seen many patterns:
When a pattern is hard to see, organize the numbers in a table. Write the position (1st, 2nd, 3rd...) in one column and the value in another. This often reveals the rule.
| Position | Value | Change |
|---|---|---|
| 1st | 4 | -- |
| 2nd | 7 | +3 |
| 3rd | 10 | +3 |
| 4th | 13 | +3 |
| 5th | ? | +3 |
The 5th value is 13 + 3 = 16.
Students sometimes guess a rule from only two numbers. Always check your rule against ALL the numbers in the sequence. If the rule does not work for every step, it is not the right rule. For example, 2, 4, 8: the rule is NOT "add 2" (that would give 2, 4, 6), it IS "multiply by 2."
1. What are the next two numbers? 10, 20, 30, 40, ___, ___
The rule is "add 10." The next two numbers are 50, 60.
2. What is the rule? What comes next? 64, 32, 16, 8, ___
Each number is halved (divided by 2). The next number is 8 ÷ 2 = 4.
3. Complete the repeating pattern: A B C A B C A B ___
The core is A B C and it repeats. The next letter is C.
4. Find the rule and the next number: 1, 4, 9, 16, 25, ___
These are perfect squares: 1x1, 2x2, 3x3, 4x4, 5x5. The next is 6x6 = 36.
5. A pattern starts at 100 and subtracts 7 each time. Write the first five numbers.
100, 93, 86, 79, 72. (100-7=93, 93-7=86, 86-7=79, 79-7=72)
Summary: Patterns follow a rule that lets you predict what comes next. Repeating patterns have a core that cycles. Growing patterns increase or decrease by a consistent rule (adding, subtracting, multiplying, or dividing). To find a rule, look at what changes between each step and verify it works for every pair of consecutive terms.