Patterns

Overview

Patterns form one of the foundational concepts in primary mathematics, helping children develop logical thinking, prediction skills, and number sense. In the WB TET Paper I (Classes I–V), pattern-related questions test your understanding of how sequences progress and your ability to identify the rule governing them.

This topic bridges arithmetic and early algebra. Recognising patterns is essential for understanding multiplication tables, place value, and later algebraic thinking. For the exam, you must be able to identify, extend, and create number patterns, shape patterns, and growing patterns. Questions typically ask you to find the next term, identify the rule, or spot the odd element in a sequence.

Mastering patterns also supports pedagogical questions—you may be asked how to teach pattern recognition or why it matters in child development.

Key Concepts

• **Pattern**: A sequence that follows a definite rule or arrangement. The rule can involve numbers, shapes, colours, sizes, or positions.

• **Repeating Pattern**: A core unit repeats over and over (e.g., red-blue-red-blue or circle-square-circle-square). The smallest repeating unit is called the "core."

• **Growing Pattern**: Each term increases (or decreases) according to a rule. The change is not simple repetition but systematic growth (e.g., 2, 4, 6, 8 — adding 2 each time).

• **Number Pattern**: A sequence of numbers following a mathematical rule — addition, subtraction, multiplication, division, or a combination.

• **Shape Pattern**: A sequence of geometric shapes following a rule based on type, size, orientation, or colour.

• **Rule of the Pattern**: The instruction that tells how to get from one term to the next. Identifying the rule is the key skill tested.

• **Term and Position**: Each element in a pattern is a "term." The position tells where the term sits (1st, 2nd, 3rd, etc.). Sometimes the rule links position number to term value.

• **Odd One Out**: A term that breaks the pattern rule — often tested to check if students truly understand the underlying logic.

Formulas / Key Facts

| Pattern Type | Example | Rule | |--------------|---------|------| | Adding a constant | 3, 7, 11, 15, … | Add 4 | | Subtracting a constant | 50, 45, 40, 35, … | Subtract 5 | | Multiplying by a constant | 2, 6, 18, 54, … | Multiply by 3 | | Dividing by a constant | 64, 32, 16, 8, … | Divide by 2 | | Square numbers | 1, 4, 9, 16, 25, … | n² (position squared) | | Triangular numbers | 1, 3, 6, 10, 15, … | Add 1, then 2, then 3, … | | Fibonacci-type | 1, 1, 2, 3, 5, 8, … | Add previous two terms | | Repeating shapes | ○ □ △ ○ □ △ … | Core = ○ □ △ (3 elements) |

**Key facts to remember**: 1. To find the rule, calculate the difference (or ratio) between consecutive terms. 2. If differences are not constant, check second-level differences or look for multiplication/division. 3. In repeating patterns, divide the position by the core length; the remainder tells which element appears. 4. Growing patterns often model real-life situations (stacking blocks, saving money). 5. Patterns help introduce the concept of variables and functions at higher levels.

Worked Examples

### Example 1 — Number Pattern (Addition)

**Question**: Find the next two terms: 5, 9, 13, 17, ?, ?

**Solution**: Step 1: Find the difference between consecutive terms. 9 − 5 = 4, 13 − 9 = 4, 17 − 13 = 4. Step 2: The rule is "add 4." Step 3: 17 + 4 = 21, 21 + 4 = 25.

**Answer**: 21, 25

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### Example 2 — Shape Pattern (Repeating)

**Question**: A pattern is ○ ○ □ △ ○ ○ □ △ … What shape is in the 15th position?

**Solution**: Step 1: Identify the core: ○ ○ □ △ (4 shapes). Step 2: Divide position by core length: 15 ÷ 4 = 3 remainder 3. Step 3: Remainder 3 means the 3rd shape in the core. Core positions: 1st = ○, 2nd = ○, 3rd = □, 4th = △.

**Answer**: □ (Square)

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### Example 3 — Growing Pattern (Triangular Numbers)

**Question**: Rina arranges dots in triangles: Row 1 has 1 dot, Row 2 has 2 dots, Row 3 has 3 dots, and so on. How many dots in total up to Row 6?

**Solution**: Step 1: Total dots = 1 + 2 + 3 + 4 + 5 + 6. Step 2: Use sum formula n(n+1)/2 = 6 × 7 / 2 = 21.

**Answer**: 21 dots

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### Example 4 — Finding the Rule (Multiplication)

**Question**: 5, 10, 20, 40, ? — Find the rule and next term.

**Solution**: Step 1: Check differences: 10 − 5 = 5, 20 − 10 = 10, 40 − 20 = 20. Differences are not constant. Step 2: Check ratios: 10/5 = 2, 20/10 = 2, 40/20 = 2. Step 3: Rule is "multiply by 2." Step 4: 40 × 2 = 80.

**Answer**: 80

Common Mistakes

1. **Assuming the rule is always addition** → Students ignore multiplication or alternating rules. Always check both differences and ratios before deciding.

2. **Miscounting the core in repeating patterns** → Skipping or double-counting elements leads to wrong answers. Physically mark the core unit first.

3. **Using remainder 0 incorrectly** → When position divides exactly by core length, the answer is the last element of the core, not the first. (e.g., 12th position in a 4-element core → remainder 0 → 4th element.)

4. **Ignoring second-level differences** → In patterns like 1, 4, 9, 16, differences are 3, 5, 7 (not constant), but second differences are 2, 2 (constant). This signals a squared pattern.

5. **Mixing up "term" and "position"** → The 5th term is not necessarily the number 5. Keep position (where) separate from value (what).

Quick Reference

• **Rule first, answer second** — always identify the governing rule before extending the pattern.
• Repeating pattern: divide position by core length; use remainder.
• Growing pattern: look for constant addition, constant multiplication, or positional formulas (n², triangular).
• Second-level differences help detect square-number or quadratic patterns.
• Patterns build algebraic thinking — stress the "why" in pedagogy questions.
• Common exam traps: remainder = 0, multiplication disguised as addition, alternating rules.