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.

Given: We need to find multi-digit numbers with various types of symmetry.

Concept:
- For a number to have reflection symmetry (vertical): it must read the same forwards and backwards (palindrome), using only digits that themselves have vertical symmetry (0, 1, 8).
- For a number to have rotational symmetry ($\frac{1}{2}$ turn): it must look the same upside down, using digits that look the same or swap correctly when rotated (0→0, 1→1, 8→8, 6→9, 9→6).

2-digit numbers:
- Reflection symmetry only: 11 (vertical symmetry, reads same both ways)
- Rotational symmetry only: 69 (6 becomes 9 and 9 becomes 6 when rotated $\frac{1}{2}$ turn)
- Both symmetries: 11, 88, 00 (trivial)

3-digit numbers:
- Reflection symmetry: 101, 111, 181, 808, 818, 888
- Rotational symmetry: 609 (rotated $\frac{1}{2}$ turn → 609), 619, 689, 906, 916, 986, 808, 818, 888, 111
- Both symmetries: 111, 808, 818, 888

4-digit numbers:
- Reflection symmetry: 1001, 1111, 1881, 8008, 8118, 8888
- Rotational symmetry: 1001, 1691, 1961, 6009, 6119, 6699, 9006, 9116, 9669, 8008, 8118, 8888
- Both symmetries: 1001, 8008, 8118, 8888, 1111

Summary Table:

| Digits | Symmetry Type | Examples |
|--------|--------------|----------|
| 2-digit | Reflection only | 11 |
| 2-digit | Rotation only | 69, 96 |
| 2-digit | Both | 11, 88 |
| 3-digit | Both | 111, 808, 888 |
| 4-digit | Both | 1001, 8008, 8888 |

Given: A design is shown (image not visible, but we solve based on the concept).

(a) Does the design have rotational symmetry?

Concept: A design has rotational symmetry if it looks the same after rotating it by some angle less than 360° about its centre.

Working: Look at the design and rotate it mentally (or using tracing paper) by $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$ turns.

Answer: Students should observe the given design carefully. If the design does NOT look the same at any partial turn → No. If it does → Yes (state at which turn).

(b) Modify the design to have $\frac{1}{2}$ turn symmetry:

Concept: For $\frac{1}{2}$ turn (180°) symmetry, every part of the design must have a matching part exactly opposite to it through the centre.

Method:
- Take the original design.
- Add a copy of each shape, rotated 180° about the centre of the design.
- The new design will look the same after a $\frac{1}{2}$ turn.

Answer: Draw the original design, then add its 180°-rotated copy on the opposite side. (Students draw this in their notebook.)

(c) Modify the design to have $\frac{1}{4}$ turn symmetry:

Concept: For $\frac{1}{4}$ turn (90°) symmetry, the design must look the same after every 90° rotation. This means the design must have 4-fold rotational symmetry.

Method:
- Take the design from part (b).
- Add copies rotated by 90° and 270° about the centre.
- Now the design has 4 identical parts, one in each quadrant.

Answer: Draw the design with 4-fold symmetry. (Students draw this in their notebook.)

(d) Do the new designs have reflection symmetry?

Concept: A design with $\frac{1}{2}$ turn symmetry may or may not have reflection symmetry. A design with $\frac{1}{4}$ turn symmetry often has 4 lines of symmetry (vertical, horizontal, and two diagonal).

Answer:
- The design with $\frac{1}{2}$ turn symmetry: Check if a mirror line exists. If shapes are symmetric about a line → Yes, draw that line.
- The design with $\frac{1}{4}$ turn symmetry: Typically has 4 lines of symmetry — one vertical, one horizontal, and two diagonal lines passing through the centre.

Students should draw the lines of symmetry on their designs in their notebooks.

Given: Designs are shown in the textbook (images not fully visible; solving conceptually).

Concept:
- $\frac{1}{2}$ turn = 180° rotation about the centre.
- $\frac{1}{4}$ turn = 90° rotation about the centre.

For the first design (appears to be a simple shape or pattern):

Working: Rotate the design 180° about its centre. If every part of the design maps onto another part of the design → it has $\frac{1}{2}$ turn symmetry.

Answer: Students should use tracing paper or mentally rotate the design.
- If the design has 2-fold symmetry: Yes, it looks the same after $\frac{1}{2}$ turn.
- If the design has 4-fold symmetry: Yes, it looks the same after both $\frac{1}{4}$ and $\frac{1}{2}$ turns.

General guidance:
- A design with $\frac{1}{4}$ turn symmetry automatically also has $\frac{1}{2}$ turn symmetry.
- A design with $\frac{1}{2}$ turn symmetry does NOT necessarily have $\frac{1}{4}$ turn symmetry.

Given: A square divided into smaller parts (grid). We need to colour it with two colours so it has $\frac{1}{4}$ turn (90°) rotational symmetry.

Concept: For a design to look the same after every $\frac{1}{4}$ turn, it must have 4-fold rotational symmetry. This means if we rotate the design by 90°, it maps onto itself.

How to colour:
- Divide the square into 4 quadrants.
- Whatever pattern you colour in one quadrant, rotate it 90° for the next quadrant, and so on.
- Use Colour 1 and Colour 2 in a pattern that repeats every 90°.

Example (for a 4×4 grid):
$$\begin{array}{|c|c|c|c|}\hline A & B & B & A \\ \hline B & A & A & B \\ \hline B & A & A & B \\ \hline A & B & B & A \\ \hline\end{array}$$
where A = Colour 1, B = Colour 2. This pattern looks the same after every $\frac{1}{4}$ turn.

How many times does it look the same during a full turn?

Working: If the design has $\frac{1}{4}$ turn symmetry, it looks the same at:
- $\frac{1}{4}$ turn (90°) → 1st time
- $\frac{1}{2}$ turn (180°) → 2nd time
- $\frac{3}{4}$ turn (270°) → 3rd time
- Full turn (360°) → 4th time

Answer: The shape looks the same 4 times during a full turn.

Reflection symmetry:

A design with 4-fold rotational symmetry often also has reflection symmetry.

Answer: Yes, such designs typically have 4 lines of symmetry:
1. Vertical line through the centre
2. Horizontal line through the centre
3. Diagonal line from top-left to bottom-right
4. Diagonal line from top-right to bottom-left

Students should draw all 4 lines of symmetry on their coloured design.

Given: A shape made by combining squares and equilateral triangles (image shown in textbook).

Concept:
- Reflection symmetry: A line exists such that one half is the mirror image of the other.
- Rotational symmetry: The shape looks the same after rotating by some angle less than 360°.

Reflection Symmetry:

Working: Look at the shape and check if a mirror line can be drawn such that both halves match.

Answer: (Based on the typical shape shown — a square with triangles on sides)
- If triangles are placed symmetrically → Yes, the shape has reflection symmetry.
- Lines of symmetry: Typically vertical and/or horizontal lines through the centre, depending on the arrangement.
- Students should draw the line(s) on their shape.

Rotational Symmetry:

Working: Rotate the shape and check at which turn it looks the same.

Answer:
- If the shape has triangles on all 4 sides of a square → it has 4-fold rotational symmetry (looks the same at $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, and full turn).
- If triangles are only on 2 opposite sides → it has 2-fold rotational symmetry (looks the same at $\frac{1}{2}$ turn).

Both symmetries:
- If the shape has both a line of symmetry AND rotational symmetry → Yes, it has both.

Sorting designs into 3 categories:
1. Only rotational symmetry: Designs that look the same when rotated but have no mirror line (e.g., a pinwheel/firki shape).
2. Only reflection symmetry: Designs that have a mirror line but do not look the same when rotated (e.g., letter A).
3. Both symmetries: Designs like a square with triangles on all sides, letter H, digit 8, etc.

Given: Images of wooden blocks and their corresponding prints are shown. One matching is already done as an example.

Concept: When a wooden block is pressed onto fabric, it creates a print. The print is the mirror image (reflection) of the block design, because the block is pressed face-down.

Working:
- Observe the design on each wooden block.
- The print will be the mirror/reflected image of the block.
- Match each block to the print that is its mirror image.

Method for students:
1. Look at the shape/pattern on the wooden block.
2. Mentally flip it (as if pressing it onto paper).
3. Find the print that matches this flipped image.
4. Draw a line connecting the block to its matching print.

Answer: Students should draw lines matching each block to its correct print based on the reflection relationship. (Since the actual images are not visible in the OCR, students must refer to their textbook and match accordingly, keeping in mind that the print is the mirror image of the block.)

Given: Design A has $\frac{1}{4}$ turn rotational symmetry. We need to determine the rotational symmetry of Design B.

Concept:
- A design made by using a block 4 times (rotating 90° each time) → $\frac{1}{4}$ turn symmetry.
- A design made by using a block 2 times (rotating 180°) → $\frac{1}{2}$ turn symmetry.
- The number of times the block is used determines the order of rotational symmetry.

Working: Looking at Design B (made from a wooden block used in a specific arrangement):
- If the block is used 2 times (placed at 0° and 180°) → the design looks the same after every $\frac{1}{2}$ turn.
- If the block is used 4 times → $\frac{1}{4}$ turn symmetry.

Answer:

The design B looks the same after every $\boxed{\dfrac{1}{2}}$ turn. This design has rotational symmetry.

(Note: The exact answer depends on the image of Design B. If Design B uses the block twice in opposite orientations, it has $\frac{1}{2}$ turn symmetry. Students should verify with the textbook image.)

Given: Various shapes are shown on the border of the page (images not fully visible in OCR).

Concept:
- Reflection symmetry: A shape has reflection symmetry if a line can be drawn through it such that both halves are mirror images.
- Rotational symmetry: A shape has rotational symmetry if it looks the same after rotating by some angle less than 360°.

General guidelines for students:

For each shape on the border:

Step 1 — Check reflection symmetry:
- Try to draw a vertical, horizontal, or diagonal line through the shape.
- If one half is the mirror image of the other → put a ✓ mark.

Step 2 — Check rotational symmetry:
- Try rotating the shape by $\frac{1}{4}$, $\frac{1}{2}$, or $\frac{3}{4}$ turn.
- If it looks the same at any of these turns → put a * mark.

Common shapes and their symmetry:

| Shape | Reflection Symmetry | Rotational Symmetry |
|-------|--------------------|-----------------------|
| Circle | ✓ (infinite lines) | * (any angle) |
| Square | ✓ (4 lines) | * ($\frac{1}{4}$ turn) |
| Equilateral triangle | ✓ (3 lines) | * ($\frac{1}{3}$ turn) |
| Rectangle | ✓ (2 lines) | * ($\frac{1}{2}$ turn) |
| Regular hexagon | ✓ (6 lines) | * ($\frac{1}{6}$ turn) |
| Scalene triangle | ✗ | ✗ |
| Isosceles triangle | ✓ (1 line) | ✗ |

Answer: Students should look at each specific shape on the border of their textbook page and mark ✓ for reflection symmetry and * for rotational symmetry based on the above guidelines.