Finding Pattern Rules
A pattern rule is a description of how to get from one number (or shape) to the next in a pattern. Finding the rule is like cracking a code: once you know the rule, you can predict any term in the pattern.
In Class 5, you will work with input-output tables (function machines), growing shape patterns, and number rules. You will describe rules using words and simple expressions like “multiply by 3, then add 1”.
Pattern rules are the foundation of algebra. They teach you to see relationships between numbers and express those relationships clearly.
What is Finding Pattern Rules - Class 5 Maths (Patterns and Algebra)?
A pattern rule tells you how each term in a pattern is related to its position number or to the previous term.
- Input-output rule: A rule that connects an input number to an output number. Example: if the rule is “multiply by 2, add 3”, then input 4 gives output 4 × 2 + 3 = 11.
- Recursive rule: A rule that tells how to get the next term from the current term. Example: “add 5 each time.”
- Function machine: A visual way to show an input-output rule. A number goes in, the machine applies the rule, and a number comes out.
Types and Properties
Types of pattern rules:
- One-step rules: A single operation. Example: add 4, multiply by 2, subtract 3.
- Two-step rules: Two operations applied in order. Example: multiply by 3, then subtract 1.
- Growing shape patterns: Each figure in the pattern has more shapes than the previous one. The number of shapes follows a number rule.
- Input-output tables: A table of input values and their corresponding output values. Your job is to find the rule connecting them.
Solved Examples
Example 1: Example 1: One-Step Rule from a Table
Problem: Find the rule.
| Input | Output |
|---|---|
| 2 | 8 |
| 5 | 20 |
| 7 | 28 |
Solution:
8 ÷ 2 = 4, 20 ÷ 5 = 4, 28 ÷ 7 = 4.
Rule: Multiply by 4.
Answer: Output = Input × 4.
Example 2: Example 2: Two-Step Rule
Problem: Find the rule.
| Input | Output |
|---|---|
| 1 | 5 |
| 2 | 7 |
| 3 | 9 |
| 4 | 11 |
Solution:
Try “multiply by 2, add 3”: 1×2+3=5, 2×2+3=7, 3×2+3=9, 4×2+3=11. It works!
Answer: Rule = Multiply by 2, then add 3.
Example 3: Example 3: Finding Missing Outputs
Problem: Rule: multiply by 3, subtract 2. Find the output for inputs 5, 8, and 10.
Solution:
Input 5: 5 × 3 − 2 = 15 − 2 = 13
Input 8: 8 × 3 − 2 = 24 − 2 = 22
Input 10: 10 × 3 − 2 = 30 − 2 = 28
Answer: Outputs are 13, 22, 28.
Example 4: Example 4: Finding the Input
Problem: Rule: add 7. Output = 19. What is the input?
Solution:
Input + 7 = 19
Input = 19 − 7 = 12
Answer: Input = 12.
Example 5: Example 5: Growing Shape Pattern
Problem: A pattern uses matchsticks to make triangles. Figure 1 uses 3 matchsticks, Figure 2 uses 5, Figure 3 uses 7. How many matchsticks for Figure 6?
Solution:
Step 1: Sequence: 3, 5, 7, ... Common difference = 2.
Step 2: Rule: 2n + 1 (where n = figure number).
Step 3: Figure 6: 2(6) + 1 = 13.
Answer: Figure 6 uses 13 matchsticks.
Example 6: Example 6: Identifying the Rule from Words
Problem: Dev says, “I think of a number, multiply it by 5, and get 35.” What is the number?
Solution:
Rule: multiply by 5. Output = 35.
Input = 35 ÷ 5 = 7.
Answer: The number is 7.
Example 7: Example 7: Testing a Rule
Problem: Priya claims the rule for this table is “add 10”. Is she correct?
| Input | Output |
|---|---|
| 3 | 13 |
| 7 | 17 |
| 12 | 22 |
Solution:
3 + 10 = 13 ✓, 7 + 10 = 17 ✓, 12 + 10 = 22 ✓.
Answer: Yes, Priya is correct. The rule is “add 10”.
Example 8: Example 8: Pattern in Seating
Problem: Ria arranges chairs in rows. Row 1 has 4 chairs, Row 2 has 7 chairs, Row 3 has 10 chairs. How many chairs in Row 8?
Solution:
Step 1: Sequence: 4, 7, 10, ... Common difference = 3.
Step 2: Rule: Start at 4, add 3 each time. Or: 3n + 1.
Step 3: Row 8: 3(8) + 1 = 25.
Answer: Row 8 has 25 chairs.
Example 9: Example 9: Two-Step Rule (Divide then Add)
Problem: Find the rule.
| Input | Output |
|---|---|
| 10 | 7 |
| 20 | 12 |
| 30 | 17 |
Solution:
Try “divide by 2, add 2”: 10/2+2=7, 20/2+2=12, 30/2+2=17. It works!
Answer: Rule = Divide by 2, then add 2.
Example 10: Example 10: Creating Your Own Rule
Problem: Kavi creates a rule: “multiply by 4, then subtract 5”. Complete the table for inputs 2, 6, and 9.
Solution:
Input 2: 2 × 4 − 5 = 8 − 5 = 3
Input 6: 6 × 4 − 5 = 24 − 5 = 19
Input 9: 9 × 4 − 5 = 36 − 5 = 31
Answer: Outputs: 3, 19, 31.
Key Points to Remember
- A pattern rule describes the relationship between input and output (or between consecutive terms).
- To find the rule, study the differences (or ratios) between inputs and outputs.
- One-step rules use a single operation. Two-step rules use two operations in order.
- A function machine takes an input, applies a rule, and gives an output.
- Growing shape patterns can be described by number rules.
- To test a rule, apply it to every input and check if it matches all outputs.
- To find a missing input, work backwards using inverse operations.
Practice Problems
- Find the rule: Input 3 → Output 12, Input 5 → Output 20, Input 8 → Output 32.
- Rule: multiply by 2, add 5. Find outputs for inputs 3, 7, and 10.
- Find the rule: Input 4 → Output 9, Input 6 → Output 13, Input 10 → Output 21.
- A pattern uses squares. Figure 1 has 4 squares, Figure 2 has 7, Figure 3 has 10. How many squares in Figure 10?
- Rule: subtract 3. Output is 15. What was the input?
- Find the missing output: Rule is “multiply by 6, subtract 4”. Input = 5. Output = ?
- Is the rule “add 8” correct for this data? Input 2 → 10, Input 5 → 13, Input 9 → 17.
- Aditi’s pocket money rule: she gets ₹50 plus ₹10 for each chore she does. If she does 6 chores, how much does she get?
Frequently Asked Questions
Q1. What is a pattern rule?
A pattern rule is a description of how numbers in a pattern are related. It tells you what operation(s) to perform on an input to get the output.
Q2. What is a function machine?
A function machine is a visual tool that shows an input going in, a rule being applied, and an output coming out. It helps you understand input-output relationships.
Q3. How do I find a pattern rule from a table?
Look at each input-output pair. Try common operations (add, subtract, multiply, divide). If one operation does not work, try two operations in sequence.
Q4. What is the difference between one-step and two-step rules?
A one-step rule uses a single operation (like “multiply by 3”). A two-step rule uses two operations (like “multiply by 3, then add 1”).
Q5. How do I find the input if I know the output and the rule?
Work backwards using inverse operations. If the rule is “add 5” and output is 17, then input = 17 − 5 = 12.
Q6. Can patterns be made with shapes?
Yes. Growing shape patterns use shapes (like matchsticks, squares, or dots) that increase by a rule. The number of shapes in each figure follows a number pattern.
Q7. How do I check if my rule is correct?
Apply the rule to every input in the table. If it gives the correct output for all pairs, the rule is correct. If even one pair does not match, try a different rule.
Q8. What is the connection between pattern rules and algebra?
Pattern rules are early algebra. The rule “multiply by 2, add 3” can be written as 2n + 3, where n is the input. This is an algebraic expression.
Q9. Is this topic in the NCERT Class 5 syllabus?
Yes. Finding pattern rules and working with input-output tables are part of the Patterns and Algebra chapter in NCERT/CBSE Class 5 Maths.
