Symmetrical Designs

Given: We need to identify capital letters that have a horizontal line of symmetry (i.e., the top half is a mirror image of the bottom half).

Concept: A horizontal line of symmetry divides a letter into two equal halves — top and bottom — that are mirror images of each other.

Working: Checking each capital letter:
- B: top half mirrors bottom half ✓
- C: top half mirrors bottom half ✓
- D: top half mirrors bottom half ✓
- E: top half mirrors bottom half ✓
- H: top half mirrors bottom half ✓
- I: top half mirrors bottom half ✓
- K: top half mirrors bottom half ✓
- O: top half mirrors bottom half ✓
- X: top half mirrors bottom half ✓

Answer: The letters that have a horizontal line of symmetry are: B, C, D, E, H, I, K, O, X

(Note: The exact set depends on the specific font/style shown in the textbook. Students should draw a horizontal line through the middle of each letter and check if both halves match.)

Given: We need to identify capital letters that have a vertical line of symmetry (i.e., the left half is a mirror image of the right half).

Concept: A vertical line of symmetry divides a letter into two equal halves — left and right — that are mirror images of each other.

Working: Checking each capital letter:
- A: left half mirrors right half ✓
- H: left half mirrors right half ✓
- I: left half mirrors right half ✓
- M: left half mirrors right half ✓
- O: left half mirrors right half ✓
- T: left half mirrors right half ✓
- U: left half mirrors right half ✓
- V: left half mirrors right half ✓
- W: left half mirrors right half ✓
- X: left half mirrors right half ✓
- Y: left half mirrors right half ✓

Answer: The letters that have a vertical line of symmetry are: A, H, I, M, O, T, U, V, W, X, Y

(Students should draw a vertical line through the centre of each letter and check if both halves match.)

Given: We need to find letters that have both a vertical AND a horizontal line of symmetry.

Concept: A letter has both symmetries if it looks the same when folded along a vertical line AND also when folded along a horizontal line.

Working: From the previous two answers:
- Letters with horizontal symmetry: B, C, D, E, H, I, K, O, X
- Letters with vertical symmetry: A, H, I, M, O, T, U, V, W, X, Y
- Letters in BOTH lists: H, I, O, X

Answer: The letters that have both vertical and horizontal lines of symmetry are: H, I, O, X

These letters can be cut out by folding the paper into one-fourth (quarter) because they have two lines of symmetry.

Given: A firki (windmill) made from a square paper with four blades. A dot is marked on it to track orientation.

Concept: Rotational symmetry — a shape has rotational symmetry if it looks the same after being rotated by a certain angle about its centre.

Working:
- A firki has 4 identical blades arranged equally around the centre.
- After a $\frac{1}{4}$ turn (90°): one blade moves to where the next blade was → the firki looks the same (but the dot has moved).
- After a $\frac{1}{2}$ turn (180°): the firki looks the same.
- After a $\frac{3}{4}$ turn (270°): the firki looks the same.
- After a full turn (360°): the firki looks exactly the same, including the dot position.

Note: The dot helps us track that the firki looks the same in shape/design at each of these turns, even though the dot itself moves.

Answer: Yes, the firki looks the same after $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, and a full turn. It has rotational symmetry at every $\frac{1}{4}$ turn.

Given: A table showing letters H, i, and others, with columns for $\frac{1}{4}$ turn, $\frac{1}{2}$ turn, $\frac{3}{4}$ turn, full turn, and rotational symmetry.

Concept: A letter has rotational symmetry if it looks the same (ignoring the dot marker) after rotating it by less than a full turn.

Working through the letters:

Letter H:
- $\frac{1}{4}$ turn: ✗ (does not look the same)
- $\frac{1}{2}$ turn: H ✓ (looks the same)
- $\frac{3}{4}$ turn: ✗ (does not look the same)
- Full turn: H ✓
- Rotational symmetry: **Yes, at $\frac{1}{2}$ turn (already given in table)

Letter I (capital I):**
- $\frac{1}{4}$ turn: ✗
- $\frac{1}{2}$ turn: I ✓ (looks the same)
- $\frac{3}{4}$ turn: ✗
- Full turn: I ✓
- Rotational symmetry: **Yes, at $\frac{1}{2}$ turn

Letter O:**
- $\frac{1}{4}$ turn: O ✓
- $\frac{1}{2}$ turn: O ✓
- $\frac{3}{4}$ turn: O ✓
- Full turn: O ✓
- Rotational symmetry: **Yes, at every $\frac{1}{4}$ turn

Letter X:**
- $\frac{1}{4}$ turn: X ✓
- $\frac{1}{2}$ turn: X ✓
- $\frac{3}{4}$ turn: X ✓
- Full turn: X ✓
- Rotational symmetry: **Yes, at every $\frac{1}{4}$ turn

Answer:** Students should fill the table by rotating (or tracing) each letter and checking at each turn. Letters like H, I, N, O, S, X, Z have rotational symmetry. Letters that do NOT look the same at any turn less than a full turn do not have rotational symmetry.

Given: The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (in standard printed form).

Concept: A digit has reflection symmetry if there exists a line (vertical or horizontal) such that one half is the mirror image of the other half.

Working (using standard printed digit forms):
- 0: Has both vertical and horizontal lines of symmetry ✓
- 1: Has a vertical line of symmetry ✓
- 2: Generally no line of symmetry ✗
- 3: Has a horizontal line of symmetry ✓
- 4: No line of symmetry ✗
- 5: No line of symmetry ✗
- 6: No line of symmetry ✗
- 7: No line of symmetry ✗
- 8: Has both vertical and horizontal lines of symmetry ✓
- 9: No line of symmetry ✗

Answer: Digits with reflection symmetry: 0, 1, 3, 8

(Note: This depends on the font/style. In some fonts, 3 has a horizontal line of symmetry. Students should check the digits as printed in their textbook.)

Given: The digits 0–9 in standard printed form.

Concept: A digit has rotational symmetry if it looks the same after a rotation of less than 360° about its centre.

Working:
- 0: Looks the same after $\frac{1}{2}$ turn ✓
- 1: Looks the same after $\frac{1}{2}$ turn (in some fonts) ✓
- 2: Does not look the same after any partial turn ✗
- 6: Looks like 9 after $\frac{1}{2}$ turn — not the same digit ✗
- 8: Looks the same after $\frac{1}{2}$ turn ✓
- 9: Looks like 6 after $\frac{1}{2}$ turn — not the same ✗

Answer: Digits with rotational symmetry: 0, 8 (look the same after $\frac{1}{2}$ turn)

(In some interpretations, 1 and 6/9 as a pair are included, but strictly speaking, only 0 and 8 look the same digit after rotation.)

Given: The digits 0–9.

Concept: We need digits that appear in both the reflection symmetry list and the rotational symmetry list.

Working:
- Digits with reflection symmetry: 0, 1, 3, 8
- Digits with rotational symmetry: 0, 8
- Digits in both lists: 0 and 8

Answer: The digits that have both rotational and reflection symmetry are: 0 and 8

- 0: Has vertical and horizontal lines of symmetry (reflection) AND looks the same after $\frac{1}{2}$ turn (rotation).
- 8: Has vertical and horizontal lines of symmetry (reflection) AND looks the same after $\frac{1}{2}$ turn (rotation).

Given: The numbers $||$ (which represents 11) and $|00|$ (which represents 1001).

Concept: We check each number as a whole for reflection symmetry (mirror image looks the same) and rotational symmetry (looks the same after $\frac{1}{2}$ turn).

**For the number 11 (written as $||$):**
- Reflection symmetry:
- Vertical line of symmetry: Yes — left half mirrors right half ✓
- Horizontal line of symmetry: Yes — top mirrors bottom ✓
- Rotational symmetry: After $\frac{1}{2}$ turn, $||$ still looks like $||$ ✓
- Answer for 11: Both (c) rotational and reflection symmetry

**For the number 1001 (written as $|00|$):**
- Reflection symmetry:
- Vertical line of symmetry: Yes — 1001 reversed is 1001 ✓
- Horizontal line of symmetry: Each digit (1, 0, 0, 1) has horizontal symmetry ✓
- Rotational symmetry: After $\frac{1}{2}$ turn, 1001 still reads as 1001 ✓
- Answer for 1001: Both (c) rotational and reflection symmetry

Final Answer: Both numbers $||$ (11) and $|00|$ (1001) have (c) both rotational and reflection symmetries.