Symmetry and Patterns
1. Line Symmetry
A shape has LINE SYMMETRY (also called REFLECTION SYMMETRY) if it can be folded into TWO IDENTICAL halves that match exactly.
'The fold line is called the LINE OF SYMMETRY or the MIRROR LINE. The two halves are MIRROR IMAGES of each other.'
Examples of Symmetry
| Shape | Number of Lines of Symmetry | Description |
|---|---|---|
| Square | 4 | Through midpoints of opposite sides (2) and through opposite vertices (2) |
| Rectangle | 2 | Through midpoints of length and breadth |
| Circle | INFINITE | Any line through the centre |
| Equilateral triangle | 3 | Through each vertex to the midpoint of the opposite side |
| Isosceles triangle | 1 | Through the vertex between the equal sides |
| Scalene triangle | 0 | No lines of symmetry |
| Regular pentagon | 5 | 5 lines through vertices and midpoints |
Symmetry in Letters
| Letter | Lines of Symmetry | Letter | Lines of Symmetry |
|---|---|---|---|
| A | 1 | H | 2 |
| B | 0 | I | 2 |
| C | 1 | M | 1 |
| D | 1 | O | INFINITE (in circle form) |
| E | 0 | T | 1 |
| F | 0 | X | 2 |
'Not all letters are symmetrical. The letters A, H, I, M, O, T, U, V, W, X, Y have at least one line of symmetry.'
Symmetry in Nature
- Butterfly wings are symmetrical.
- A human face (roughly) has one line of symmetry down the centre.
- Leaves often have one line of symmetry along the midrib.
- Starfish have five lines of symmetry.
- Snowflakes have six lines of symmetry.
'Nature LOVES symmetry. Most animals and plants show some form of symmetry. But PERFECT symmetry is rare in nature — look closely and you will find small differences.'
2. Mirror Halves
When you place a MIRROR on the line of symmetry, the reflection shows the COMPLETE shape.
Completing a Symmetrical Figure
If half of a symmetrical shape is given, you can complete it by reflecting each point across the line of symmetry.
| Left Half | Line | Right Half (Mirror Image) | Complete Shape |
|---|---|---|---|
| / | | | \ | V |
| (_ | | | _) | (_) |
| > | | | < | >< |
'When drawing the mirror half, imagine the line is a MIRROR. Every point on the left side has a corresponding point on the right side at the SAME distance from the line.'
Checking Mirror Halves
To check if two halves are mirror images:
- Count the squares from the line of symmetry for each point.
- The distance on BOTH sides must be EQUAL.
- The shapes must be EXACT reversals of each other.
3. Patterns in Numbers
Number Patterns
| Pattern | Rule | Next Three Terms |
|---|---|---|
| 2, 4, 6, 8, 10, ... | Add 2 | 12, 14, 16 |
| 3, 6, 9, 12, 15, ... | Add 3 | 18, 21, 24 |
| 1, 4, 9, 16, 25, ... | Square numbers (1², 2², 3²...) | 36, 49, 64 |
| 1, 3, 5, 7, 9, ... | Odd numbers (add 2) | 11, 13, 15 |
| 2, 5, 10, 17, 26, ... | Add odd numbers (+3, +5, +7, +9...) | 37, 50, 65 |
| 100, 90, 80, 70, ... | Subtract 10 | 60, 50, 40 |
'Look for the RULE. Does the pattern add a constant number? Multiply? Follow a sequence of squares? Once you find the rule, you can PREDICT any term.'
Triangular Numbers
1, 3, 6, 10, 15, 21, ...
- 1 = 1
- 3 = 1 + 2
- 6 = 1 + 2 + 3
- 10 = 1 + 2 + 3 + 4
'TRIANGULAR numbers are formed by adding consecutive natural numbers. They are called triangular because they can be arranged to form a TRIANGLE of dots.'
Fibonacci Pattern
1, 1, 2, 3, 5, 8, 13, 21, ...
Each term is the SUM of the two preceding terms.
1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8
'This pattern appears in nature — in the petals of flowers, the spirals of shells, and the branching of trees.'
4. Patterns in Shapes
Repeating Patterns
A repeating pattern uses a CORE that repeats.
| Core | Pattern |
|---|---|
| ▲ ● | ▲ ● ▲ ● ▲ ● ▲ ● |
| ○ ○ ● | ○ ○ ● ○ ○ ● ○ ○ ● |
| ▲ ▲ ● ○ | ▲ ▲ ● ○ ▲ ▲ ● ○ |
'The CORE of a repeating pattern is the smallest part that repeats. Find the core and you can EXTEND the pattern infinitely.'
Growing Patterns
Growing patterns INCREASE in size or number each time.
Stage 1: #
Stage 2: # #
Stage 3: # # #
Stage 4: # # # #
The number of '#' increases by 1 each stage.
Stage 1: □
Stage 2: □□
□□
Stage 3: □□□
□□□
□□□
The side length increases by 1 each stage. The number of squares = side².
5. Tessellations
A TESSELLATION is a pattern of shapes that fit together WITHOUT GAPS or OVERLAPS.
'Think of a tiled floor — the tiles cover the ENTIRE surface with no spaces between them. That is a tessellation.'
Shapes That Tessellate
| Shape | Does It Tessellate? | Reason |
|---|---|---|
| Square | YES | Four 90° corners meet at 360° |
| Equilateral triangle | YES | Six 60° corners meet at 360° |
| Regular hexagon | YES | Three 120° corners meet at 360° |
| Regular pentagon | NO | 108° × 3 = 324°, 108° × 4 = 432° — neither sums to 360° |
| Circle | NO | Leaves gaps |
'For a shape to tessellate, the angles meeting at a point must SUM to EXACTLY 360°. This is why squares, triangles, and hexagons work, but pentagons and circles do not.'
Tessellations in Real Life
- Floor and wall tiles
- Honeycomb (hexagonal pattern by bees)
- Brick walls
- Islamic art and architecture
- Checkerboard patterns
Key Facts to Remember
- A shape with line symmetry has TWO identical halves that are mirror images.
- A shape can have MORE than one line of symmetry (circle has infinite).
- A tessellation covers a surface with NO gaps and NO overlaps.
- 'Number patterns follow a RULE. Find the rule by looking at the DIFFERENCE between consecutive terms.'
- Only three REGULAR shapes tessellate on their own: triangle, square, hexagon.
Common Mistakes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| Saying a rectangle has 4 lines of symmetry | A rectangle only has 2 — through the midpoints of sides | Diagonal lines on a rectangle do NOT create mirror halves |
| Thinking all triangles are symmetrical | Only ISOSCELES and EQUILATERAL have symmetry | Scalene triangles have NO lines of symmetry |
| Confusing repeating and growing patterns | Repeating patterns have a fixed core; growing patterns keep increasing | Identify whether the core repeats or the pattern grows |
| Drawing incomplete tessellations | Shapes must meet EXACTLY — no gaps allowed | Check that each shape edge aligns perfectly with its neighbour |
Exam Focus (ICSE Class 5)
| Topic | Marks (Typical) | Question Type |
|---|---|---|
| Lines of symmetry | 3-4 marks | Draw lines of symmetry for given shapes |
| Completing symmetrical figures | 3-4 marks | Given half, draw the complete figure |
| Number patterns | 3-4 marks | Find the next terms and state the rule |
| Tessellations | 2-3 marks | Identify which shapes can tessellate |
| Mirror halves | 2 marks | Identify if two halves are mirror images |
Self-Test: 5 Questions
Q1. How many lines of symmetry does a regular hexagon have?
Q2. Complete the pattern: 2, 5, 11, 23, 47, ___ , ___
Q3. Draw the line(s) of symmetry for the capital letter H.
Q4. A shape has a line of symmetry. One half has points at distances 2 cm, 3 cm, and 5 cm from the line. Where are the corresponding points on the other half?
Q5. Explain why a regular pentagon does NOT tessellate.
Answers
A1. A regular hexagon has 6 lines of symmetry — 3 through opposite vertices and 3 through midpoints of opposite sides.
A2. Rule: Multiply by 2 and add 1. 47 × 2 + 1 = 95. 95 × 2 + 1 = 191. Next terms: 95, 191.
A3. H has 2 lines of symmetry — one vertical (between the two vertical lines) and one horizontal (through the middle crossbar).
A4. The corresponding points are at the SAME distances on the other side of the line — 2 cm, 3 cm, and 5 cm from the line, but in the OPPOSITE direction.
A5. A regular pentagon has interior angles of 108°. For tessellation, the angles meeting at a point must sum to 360°. 108° × 3 = 324° (too little) and 108° × 4 = 432° (too much). Since no whole number of pentagons meets at exactly 360°, pentagons cannot tessellate.
