Generating and Analyzing Number Patterns
A number pattern is a list of numbers that follow a rule, and learning to find and use that rule helps you predict what comes next.
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A number pattern is a sequence of numbers where each number is connected to the next by the same rule. The rule tells you what to do to one number to get the next one, like always adding 5 or always multiplying by 2. When you generate a pattern, you create the list using the rule. When you analyze a pattern, you look at a list and figure out what rule was used.
Remember the rule
Find the rule: look at what you ADD, SUBTRACT, MULTIPLY, or DIVIDE to get from one term to the next, then check that same rule works for every step.
Key words
- Pattern
- A set of numbers that follow a repeating or growing rule.
- Rule
- The math operation you do to each number to get the next one, like add 3 or multiply by 2.
- Sequence
- Numbers listed in a specific order that follow a rule.
- Term
- Each individual number in a pattern is called a term. In 2, 4, 6, 8 the number 4 is the second term.
- Generate
- To create or build a pattern by starting with a number and applying the rule over and over.
- Analyze
- To look at a pattern that is already made and figure out its rule.
- Increasing pattern
- A pattern where the numbers get bigger, like 3, 6, 9, 12.
- Decreasing pattern
- A pattern where the numbers get smaller, like 20, 16, 12, 8.
Worked examples
Rule: Start at 4, add 6 each time. Write the first 5 terms.
→ 4, 10, 16, 22, 28 · 4+6=10, 10+6=16, 16+6=22, 22+6=28. Each jump is exactly 6.
Rule: Start at 80, subtract 10 each time. Write the first 5 terms.
→ 80, 70, 60, 50, 40 · This is a decreasing pattern. You can check by adding 10 back to each term to get the one before it.
What is the rule for this pattern? 3, 6, 12, 24, 48
→ Rule: multiply by 2 each time. · 3×2=6, 6×2=12, 12×2=24, 24×2=48. When numbers double, the rule is multiply by 2.
What comes next in this pattern? 5, 10, 15, 20, ___
→ 25 · The rule is add 5 each time. These are the multiples of 5.
Rule: Start at 1, multiply by 3 each time. Write the first 5 terms.
→ 1, 3, 9, 27, 81 · 1×3=3, 3×3=9, 9×3=27, 27×3=81. Numbers grow very fast when you multiply.
Find the rule and the missing number: 100, 90, 80, ___, 60
→ Missing number is 70. Rule: subtract 10 each time. · Check both sides: 80-10=70 and 70-10=60, so 70 fits perfectly.
Common mistakes
- Assuming the rule is always addition — some patterns use subtraction, multiplication, or division, so always check every step.
- Only checking the first two terms to find the rule, then not verifying the rule works for all the other steps too.
- Mixing up which number comes first — the starting number matters, so always begin with the correct first term.
- Confusing multiply-by-2 patterns with add-2 patterns. In 2, 4, 8, 16 the jumps get bigger, so it must be multiplication, not addition.
- Forgetting that a pattern can decrease. If numbers get smaller, the rule involves subtraction or division, not addition.
FAQs
How do I find the rule if I don't know it?
Look at the first two numbers and ask: what did I do to the first number to get the second? Then test that same operation on every other pair of numbers in the list. If it works every time, that is the rule.
Can a pattern have more than one rule?
In 4th grade, patterns usually have one rule. If you see the pattern 1, 2, 3, 1, 2, 3 that is a repeating pattern with a cycle, which is a different kind of pattern. Most number patterns you will see this year use just one operation.
What is the difference between a pattern rule and a feature of the pattern?
The rule is what you DO to get the next term, like add 4. A feature is something you notice ABOUT the pattern, like all the terms are even numbers or the terms end in 0 and 5. Both are useful, but they are different things.
My pattern uses multiplication. How do I check my work?
Divide each term by the number you multiplied by and you should get the term before it. For example if your rule is multiply by 3, then divide any term by 3 and you should land on the previous term.
Why do we learn about number patterns?
Patterns are the building blocks of algebra and multiplication. Recognizing patterns helps you solve problems faster, understand skip counting, learn multiplication tables, and later do algebra in middle school.
What if two different rules seem to work for the first few terms?
Keep testing further along the list. The correct rule will work for every single term without exception. If one rule breaks down at step 4 or 5, it is not the right rule.
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