What are patterns and sequences?
A pattern is a set of numbers, shapes, or objects that follow a rule. A sequence is a list of numbers in a pattern. Understanding patterns helps you predict what comes next and solve problems in math and real life.
- 2, 4, 6, 8, __ → This pattern adds 2 each time. The next number is 10.
- 15, 12, 9, 6, __ → This pattern subtracts 3 each time. The next number is 3.
- 1, 2, 4, 8, __ → This pattern multiplies by 2 each time. The next number is 16.
- 32, 16, 8, 4, __ → This pattern divides by 2 each time. The next number is 2.
Every pattern has a rule. The rule tells you how the sequence changes from one term to the next. Look for the operation—addition, subtraction, multiplication, or division—and the number being used.
Identifying the rule in a pattern
The rule is the operation or set of operations that change one number into the next. To find the rule, ask yourself: “What is happening to each number to get the next number?” Then test your rule on at least two pairs of numbers.
- 5, 10, 15, 20, __ → Rule: Add 5. The next number is 25.
- 50, 45, 40, 35, __ → Rule: Subtract 5. The next number is 30.
- 3, 6, 12, 24, __ → Rule: Multiply by 2. The next number is 48.
- 81, 27, 9, 3, __ → Rule: Divide by 3. The next number is 1.
Sometimes the rule uses more than one operation. For example, in the pattern 2, 5, 11, 23, __ the rule is “multiply by 2 and add 1.” The next number is 47. Always check your rule with at least the first three terms to make sure it works every time.
Increasing patterns (addition and multiplication)
An increasing pattern is a sequence where each term is larger than the term before it. Increasing patterns often use addition or multiplication. When the rule adds a constant number, it is called an arithmetic sequence. When the rule multiplies by a constant number, it is called a geometric sequence.
- Addition: 7, 14, 21, 28, __ → Rule: Add 7. The next number is 35.
- Addition: 100, 120, 140, 160, __ → Rule: Add 20. The next number is 180.
- Multiplication: 4, 12, 36, 108, __ → Rule: Multiply by 3. The next number is 324.
- Multiplication: 2, 10, 50, 250, __ → Rule: Multiply by 5. The next number is 1,250.
With multiplication patterns, numbers can grow very quickly. Pay close attention to the factor you multiply by. If the numbers double, the factor is 2. If they triple, the factor is 3.
Decreasing patterns (subtraction and division)
A decreasing pattern is a sequence where each term is smaller than the term before it. Decreasing patterns often use subtraction or division. When the rule subtracts a constant number, the numbers get smaller in equal steps. When the rule divides by a constant number, the numbers shrink by a repeated factor.
- Subtraction: 99, 88, 77, 66, __ → Rule: Subtract 11. The next number is 55.
- Subtraction: 250, 200, 150, 100, __ → Rule: Subtract 50. The next number is 50.
- Division: 64, 32, 16, 8, __ → Rule: Divide by 2. The next number is 4.
- Division: 625, 125, 25, 5, __ → Rule: Divide by 5. The next number is 1.
When a pattern uses division, the numbers become fractions or whole numbers that are smaller. Always check if the division leaves a remainder. If it does, the pattern might use subtraction instead, or the rule may be different.
Patterns with mixed operations
Some patterns use mixed operations, meaning the rule changes between steps or uses a combination of operations. For example, a pattern might add 3, then multiply by 2, then add 3 again. These patterns require you to find a two-step rule or a repeating operation cycle.
- 2, 5, 10, 13, 26, __ → Rule: Add 3, then multiply by 2. After 26, add 3 to get 29.
- 1, 3, 6, 10, 15, __ → Rule: Add 2, then add 3, then add 4, then add 5. Each time the added number increases by 1. The next number is 21 (add 6).
- 48, 24, 26, 13, 15, __ → Rule: Divide by 2, then add 2. After 15, divide by 2 to get 7.5 (or 7 ½ in fraction form).
Mixed operation patterns can be tricky. Look for a repeating cycle or a pattern within the differences. Write the differences between terms in a separate row to see a secondary pattern.
Finding missing terms in a sequence
A sequence may have a missing term. To find it, first determine the rule that connects the numbers. Then apply the rule to find the missing number. You may need to work forward from a known term or backward from a later term.
- 3, 7, 11, __, 19 → Rule: Add 4. Missing term: 11 + 4 = 15.
- 100, __, 80, 70, 60 → Rule: Subtract 10 each time. Missing term: 100 − 10 = 90.
- 2, 6, 18, __, 162 → Rule: Multiply by 3. Missing term: 18 × 3 = 54.
- 512, __, 32, 8, 2 → Rule: Divide by 4. Check: 512 ÷ 4 = 128, 128 ÷ 4 = 32. Missing term: 128.
When a term is missing in the middle, use the rule to go forward from the term before the blank and backward from the term after the blank. Both methods should give the same number.
Using patterns to solve real-world problems
Patterns are not just in math textbooks—they appear in real life. You can use patterns to predict events, calculate costs, schedule activities, or understand growth. Recognizing a pattern helps you make logical conclusions without calculating every single step.
- A gardener plants 3 trees on day 1, 6 trees on day 2, and 12 trees on day 3. If the pattern continues, how many trees will be planted on day 5? Rule: Multiply by 2 each day. Day 4: 24 trees. Day 5: 48 trees.
- A library fines $0.25 for the first overdue day, $0.50 for the second day, $0.75 for the third day. If the pattern continues, what is the fine for the 7th day? Rule: Add $0.25 each day. Day 7 fine: 7 × $0.25 = $1.75.
- A race car’s speed: 50 mph in lap 1, 60 mph in lap 2, 70 mph in lap 3. If the pattern continues, what is the speed in lap 8? Rule: Add 10 mph each lap. Lap 8 speed: 50 + (7 × 10) = 120 mph.
In real-world patterns, always identify the starting value and the rule. Then use the rule to extend the pattern as far as needed. Write down each step to avoid errors.
Creating your own patterns
You can create your own patterns by choosing a starting number and a rule. The rule can involve addition, subtraction, multiplication, division, or a combination. Creating patterns helps you understand how sequences work and strengthens your number sense.
- Start with 12. Rule: Subtract 4. Sequence: 12, 8, 4, 0, −4.
- Start with 1. Rule: Multiply by 10. Sequence: 1, 10, 100, 1,000, 10,000.
- Start with 3. Rule: Add 5, then subtract 2. Sequence: 3, 8, 6, 11, 9, 14.
- Start with 256. Rule: Divide by 2, then add 8. Sequence: 256, 128, 136, 68, 76, 38.
When you create a pattern, write the rule clearly. Then ask a classmate or family member to find the next three terms. This helps you see if your rule is easy to follow and correct.
Common mistakes with patterns and sequences
Even strong mathematicians make mistakes with patterns. Recognizing common errors helps you avoid them. The most frequent mistakes include misidentifying the operation, skipping a step in mixed-operation patterns, and forgetting to check the rule on all given terms.
- Mistake: Pattern 3, 6, 9, 12. Saying “multiply by 2” is wrong because 3 × 2 = 6, but 6 × 2 = 12, not 9. Correct rule: Add 3.
- Mistake: Pattern 100, 90, 81, 73. Assuming subtraction of 10 each time fails after 90 − 10 = 80, but the third term is 81. The actual rule: subtract 10, then subtract 9, then subtract 8.
- Mistake: Pattern 2, 4, 8, 16. Saying “add 2” works for the first step but fails for 4 to 8. Correct rule: Multiply by 2.
Always test your rule on at least three steps. If it works for the first two steps but fails on the third, your rule is incorrect. Look for a new rule that fits all given terms.
Extending patterns beyond given terms
Once you know the rule of a pattern, you can extend it to find any term, no matter how far ahead. This skill is useful for predicting future events, understanding growth, and solving advanced math problems.
- Pattern: 5, 9, 13, 17, … Rule: Add 4. The 10th term: Start at 5 and add 4 nine times: 5 + (9 × 4) = 5 + 36 = 41.
- Pattern: 3, 6, 12, 24, … Rule: Multiply by 2. The 8th term: 3 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 3 × 2⁷ = 3 × 128 = 384.
- Pattern: 200, 190, 180, 170, … Rule: Subtract 10. The 15th term: 200 − (14 × 10) = 200 − 140 = 60.
To find a term far in the sequence, find the number of steps from the first term. Multiply the step size by the number of steps. For subtraction, subtract the total; for addition, add the total. For multiplication, use exponents or repeated multiplication.
Common Core alignment: CCSS.MATH.CONTENT.4.OA.C.5 – Generate and analyze patterns. Identify apparent features of the pattern that were not explicit in the rule itself.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.OA.C.5. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework.