# Patterns in Mathematics

## Overview

This chapter introduces students to the beauty and logic of patterns in mathematics. It covers number and shape sequences, their relationships, and how patterns can be visualized for deeper understanding. The chapter encourages creative thinking and reasoning through exploration of mathematical patterns seen in nature, art, and daily life.

## Key Topics Covered

### 1. What is Mathematics?

* Mathematics is about discovering and understanding **patterns** and the reasons they exist.
* Patterns are found everywhere — in nature, homes, schools, sports, technology, etc.
* Mathematics is described as both **an art and a science** because of the creativity involved in finding patterns.

### 2. Patterns in Numbers

* **Number Theory**: A branch that studies patterns in whole numbers.
* **Number Sequences**: Fundamental in mathematics. Some examples:

* All 1’s: `1, 1, 1, 1, ...`
* Counting numbers: `1, 2, 3, 4, ...`
* Odd numbers: `1, 3, 5, 7, ...`
* Even numbers: `2, 4, 6, 8, ...`
* Triangular numbers: `1, 3, 6, 10, ...`
* Square numbers: `1, 4, 9, 16, ...`
* Cube numbers: `1, 8, 27, 64, ...`
* Virahānka numbers: `1, 2, 3, 5, 8, ...`
* Powers of 2: `1, 2, 4, 8, ...`
* Powers of 3: `1, 3, 9, 27, ...`

### 3. Visualising Number Sequences

* Number sequences can be represented pictorially to better understand the pattern.
* **Visual Representations**:

* Triangular numbers form triangle shapes.
* Square numbers can be arranged in squares.
* Cube numbers create cubic arrangements.
* Some numbers like **36** can be both a square and a triangular number.

### 4. Relations Among Number Sequences

* Adding **odd numbers** sequentially results in **square numbers**:

```
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
...
```
* Adding **counting numbers up and down** also forms square numbers:

```
1
1 + 2 + 1 = 4
1 + 2 + 3 + 2 + 1 = 9
...
```
* These can be explained with visual aids like dot grids and layered illustrations.

### 5. Patterns in Shapes

* **Geometry** studies patterns in shapes of 1D, 2D, 3D or even more dimensions.
* **Shape Sequences**:

* Stacked triangles and squares.
* Complete graphs (`K2`, `K3`, `K4`, etc.)
* Regular polygons: Triangle, Square, Pentagon, Hexagon...
* Koch snowflake: Fractal pattern that evolves by modifying lines.

### 6. Relation to Number Sequences

* Shape sequences often relate to number sequences.
* **Examples**:

* Number of sides in regular polygons: `3, 4, 5, ...`
* Corners and sides follow the same number pattern.
* Stacked squares and triangles reflect square and triangular numbers respectively.
* Koch snowflake line segments: `3, 12, 48, ...` which follows `3 × powers of 4`.

---

**New Terms and Definitions**

| Term | Simple Definition |
| ----------------- | ----------------------------------------------------------- |
| Pattern | A repeated or regular way in which something happens |
| Sequence | A list of numbers or shapes arranged in order |
| Triangular Number | Numbers that can form a triangle when dots are arranged |
| Square Number | Numbers that make perfect squares (e.g., 1, 4, 9) |
| Cube Number | Numbers that are multiplied by themselves three times |
| Geometry | The study of shapes and their properties |
| Regular Polygon | A shape with all sides and angles equal |
| Complete Graph | A diagram where every dot is connected to every other dot |
| Koch Snowflake | A special shape created by repeating a pattern on each line |
| Visualise | To create a mental or drawn picture of something |

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**Practice Questions**

### Easy (3)

1. **What comes next in the sequence: 2, 4, 6, 8, ...?**
**Answer:** 10
**Explanation:** This is a sequence of even numbers, increasing by 2.

2. **What shape has 6 equal sides?**
**Answer:** Hexagon
**Explanation:** A regular polygon with 6 sides is called a hexagon.

3. **Which number is both a square and a triangular number?**
**Answer:** 36
**Explanation:** 36 = 6² and also the sum of the first 8 odd numbers.

### Medium (2)

4. **What is the 6th triangular number?**
**Answer:** 21
**Explanation:** Triangular numbers are formed by adding natural numbers: 1 + 2 + 3 + 4 + 5 + 6 = 21.

5. **Which number sequence do you get from the sides of regular polygons?**
**Answer:** 3, 4, 5, 6, ...
**Explanation:** Regular polygons start from triangle (3 sides) and increase by 1 side.

### Difficult (3)

6. **What is the sum of the first 10 odd numbers?**
**Answer:** 100
**Explanation:** Sum of first *n* odd numbers is *n²*. So, 10² = 100.

7. **Write the next three Virahānka numbers after 1, 2, 3, 5, 8.**
**Answer:** 13, 21, 34
**Explanation:** It’s a Fibonacci-like sequence: each number is the sum of the two before it.

8. **If you draw the next shape in the Koch Snowflake pattern, how many segments will there be? (Start from 3, 12, 48)**
**Answer:** 192
**Explanation:** Sequence is 3 × 4⁰, 3 × 4¹, 3 × 4², 3 × 4³ → next is 3 × 4⁴ = 192.

### Very Difficult (2)

9. **Find the total value of 1 + 2 + 3 + ... + 100 + 99 + ... + 2 + 1.**
**Answer:** 10,000
**Explanation:** This is two sequences of 1 to 100, so total = 2 × (100×101/2) = 10,100 – 100 = 10,000 (subtracting one duplicate 100).

10. **Multiply each triangular number by 6 and add 1. First three values?**
**Answer:** 7, 19, 37
**Explanation:** Triangular numbers: 1, 3, 6 → (6×1)+1 = 7, (6×3)+1 = 19, (6×6)+1 = 37.

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