Symmetry and Patterns
1. What Is Symmetry?
A shape has LINE SYMMETRY if it can be FOLDED along a line so that the two halves match EXACTLY.
'The fold line is called the LINE OF SYMMETRY or MIRROR LINE.'
Examples:
- A butterfly has ONE line of symmetry (down the middle of its body).
- A square has FOUR lines of symmetry.
- A circle has INFINITE lines of symmetry.
'Symmetry is like looking in a mirror. If the two halves look the SAME, the shape is SYMMETRICAL.'
| Shape | Number of Lines of Symmetry |
|---|---|
| Circle | Infinite |
| Square | 4 |
| Rectangle | 2 |
| Equilateral Triangle | 3 |
| Isosceles Triangle | 1 |
| Scalene Triangle | 0 |
| Regular Pentagon | 5 |
| Regular Hexagon | 6 |
2. Symmetry in Letters
Some letters of the alphabet have line symmetry. Some have NONE.
Letters with Vertical Symmetry (fold left to right):
A, H, I, M, O, T, U, V, W, X, Y
'Fold these letters DOWN THE MIDDLE. The left half matches the right half.'
Letters with Horizontal Symmetry (fold top to bottom):
B, C, D, E, H, I, K, O, X
'Fold these letters ACROSS THE MIDDLE. The top half matches the bottom half.'
Letters with NO Symmetry:
F, G, J, L, N, P, Q, R, S, Z
'These letters do NOT have a fold line that gives two matching halves.'
| Letter | Vertical Symmetry | Horizontal Symmetry | Both |
|---|---|---|---|
| A | Yes | No | No |
| H | Yes | Yes | Yes |
| O | Yes | Yes | Yes |
| X | Yes | Yes | Yes |
| S | No | No | No |
| Z | No | No | No |
3. Symmetry in Numbers
| Number | Symmetry | Lines |
|---|---|---|
| 0 | Yes | Vertical and Horizontal |
| 1 | No | — |
| 2 | No | — |
| 3 | Yes | Horizontal |
| 4 | No | — |
| 5 | No | — |
| 6 | No | — |
| 7 | No | — |
| 8 | Yes | Vertical and Horizontal |
| 9 | No | — |
'Symmetrical numbers: 0, 3, and 8. All others have NO line symmetry.'
4. Reflective Symmetry
REFLECTIVE SYMMETRY means one half of a shape is the MIRROR IMAGE of the other half.
'If you place a mirror on the line of symmetry, the reflection looks EXACTLY like the other half.'
Drawing Symmetrical Shapes:
- Draw ONE half of the shape.
- Count the distance of each point from the mirror line.
- Mark the SAME distance on the OTHER side.
- Connect the points to complete the shape.
5. Patterns in Shapes
SHAPE PATTERNS follow a rule that REPEATS.
Repeating Patterns:
▲ ■ ▲ ■ ▲ ■ — What comes next? 'This pattern alternates between TRIANGLE and SQUARE. Next shape is ▲.'
Growing Patterns:
■, ■■, ■■■, ■■■■ — What comes next? 'Each term adds ONE more square. Next term has 5 squares: ■■■■■'
Pattern Rules:
- Find what REPEATS (the CORE of the pattern).
- Identify the RULE (alternating, growing, shrinking, rotating).
- PREDICT the next term.
| Pattern | Rule | Next Term |
|---|---|---|
| ○ ○ ● ○ ○ ● | Repeats every 3: ○ ○ ● | ○ |
| 1, 2, 4, 7, 11 | Add 1, then 2, then 3, then 4 | 16 (add 5) |
6. Patterns in Numbers
Even and Odd Patterns:
Even: 2, 4, 6, 8, 10... (add 2 each time) Odd: 1, 3, 5, 7, 9... (add 2 each time)
'Even numbers END in 0, 2, 4, 6, or 8. Odd numbers END in 1, 3, 5, 7, or 9.'
Skip Counting Patterns:
- Count by 5s: 5, 10, 15, 20, 25...
- Count by 10s: 10, 20, 30, 40, 50...
- Count by 100s: 100, 200, 300, 400...
Arithmetic Patterns:
A sequence where the SAME number is added (or subtracted) each time.
| Sequence | Rule | Next Three |
|---|---|---|
| 3, 6, 9, 12, 15 | Add 3 | 18, 21, 24 |
| 50, 45, 40, 35 | Subtract 5 | 30, 25, 20 |
| 10, 20, 30, 40 | Add 10 | 50, 60, 70 |
Geometric Patterns:
A sequence where each term is MULTIPLIED by the same number.
| Sequence | Rule | Next Two |
|---|---|---|
| 2, 4, 8, 16 | Multiply by 2 | 32, 64 |
| 3, 9, 27, 81 | Multiply by 3 | 243, 729 |
7. Tessellation
TESSELLATION is a pattern of shapes that FIT TOGETHER WITHOUT gaps or overlaps.
'Tiles on a bathroom floor form a TESSELLATION. The shapes fit perfectly with NO empty spaces.'
Shapes That Tessellate:
| Shape | Can it Tessellate? |
|---|---|
| Square | Yes |
| Rectangle | Yes |
| Equilateral Triangle | Yes |
| Regular Hexagon | Yes |
| Regular Pentagon | No |
| Circle | No |
'Circles CANNOT tessellate because they leave CURVED gaps between them.'
Real-Life Tessellations:
- Floor and wall tiles
- Honeycomb (hexagons)
- Brick walls
- Snakeskin patterns
- Islamic art and architecture
8. Common Mistakes
- Drawing a line through a shape that is NOT a line of symmetry: 'A line of symmetry must divide the shape into TWO EXACT halves. If the halves don't match, it is NOT a line of symmetry.'
- Thinking ALL triangles have symmetry: 'Only ISOSCELES triangles have 1 line of symmetry. EQUILATERAL have 3. SCALENE have NO lines of symmetry.'
- Confusing symmetry with patterns: 'Symmetry is about FOLDING. Patterns are about REPETITION. They are different concepts!'
- Missing the pattern rule: 'When finding the next term in a number pattern, always check the DIFFERENCE between consecutive terms first.'
9. Key Facts to Remember
- 'A line of symmetry divides a shape into two EXACTLY matching halves.'
- 'A circle has INFINITE lines of symmetry.'
- 'Regular polygons have as many lines of symmetry as they have sides.'
- 'A tessellation has NO gaps and NO overlaps between shapes.'
- 'Only regular polygons with angles that fit together at 360° can tessellate.'
10. Self-Test
Q1: How many lines of symmetry does a rectangle have?
Q2: Which of these letters has vertical symmetry? F, M, S, Z
Q3: Draw the next three shapes in this pattern: ● ○ ● ○ ● ○
Q4: Find the next two numbers: 5, 10, 15, 20, ___, ___
Q5: Find the next two numbers: 64, 32, 16, 8, ___, ___
Q6: Can a regular pentagon tessellate? Why or why not?
Q7: Name an object in nature that shows symmetry.
Q8: How many lines of symmetry does an equilateral triangle have?
Answers:
A1: 2 lines of symmetry (vertical and horizontal). A2: M (fold vertically — both sides match). A3: ● ○ ● (alternating black and white circles). A4: 25, 30 (add 5 each time). A5: 4, 2 (divide by 2 each time). A6: No. The interior angle of a regular pentagon (108°) does not divide 360° evenly. A7: A butterfly, a leaf, a starfish, a snowflake (any valid example). A8: 3 lines of symmetry.
