Fun with Symmetry

Given: A sheet of paper is folded in half, colour is dropped at the centre fold, and the paper is pressed so the colour spreads on both halves.

Concept: When a shape or design can be divided into two mirror-image halves, it is called a symmetrical design. The dividing line is called the line of symmetry.

Answer:
- Yes, the ink-blot pattern is a symmetrical pattern because both halves are mirror images of each other.
- The line of symmetry is drawn along the fold line — the crease made when the paper was folded in half.
- This line is called the line of symmetry (also known as the mirror line or line of reflection).

Given: Figures 3, 4, and 5 show different stages of folding a paper airplane.

Concept: A line of symmetry divides a figure into two identical halves that are mirror images of each other.

Answer: In each of Fig. 3, Fig. 4, and Fig. 5, the paper is folded symmetrically. The line of symmetry runs vertically down the centre (along the central fold/crease) of each figure. Mark a vertical dotted line along the middle fold in each figure.

Given: Fig. 8 shows the completed paper airplane.

Concept: Count the number of ways the shape can be folded so that both halves match exactly.

Answer: The completed paper airplane has 1 line of symmetry — the vertical line running along the central fold from the nose to the tail of the plane.

Given: Fig. 8 is the completed symmetrical paper airplane.

Concept: When a mirror is placed along the line of symmetry, the reflection of one half reproduces the other half exactly.

Answer:
- Place the mirror vertically along the central fold line (the line of symmetry) of Fig. 8.
- Yes, the reflection of the right half will look exactly the same as the left half, because the airplane is symmetrical about that central line.

Activity: This is a hands-on activity. Fold the paper airplane as shown in the steps and fly it. Observe how it moves through the air.

Given: A symmetrical paper airplane flies smoothly.

Concept: Symmetry in an airplane ensures that both wings are equal in size and shape, providing balanced lift and drag on both sides.

Answer: If there is no line of symmetry, the two wings will be unequal. This will cause unbalanced forces on the two sides, making the plane tilt or spin to one side. The plane will not fly straight and will likely crash quickly. Symmetry is important for stable flight.

Activity: Make a paper airplane where the two halves are not mirror images — for example, fold one wing more than the other, or make one wing larger. This creates an asymmetrical plane.

Expected Observation:
- The symmetrical plane flies in a straight path and stays in the air for a longer time.
- The asymmetrical plane veers to one side, spins, or dips quickly and falls sooner.

Conclusion: The symmetrical plane flies better and for a longer duration because balanced wings provide equal lift on both sides.

Activity: Discuss with classmates:
- The symmetrical plane flew straighter and longer.
- The asymmetrical plane was unstable.
- Symmetry is important not just in paper planes but in real aircraft, birds' wings, and many objects that need to move in a balanced way.

Given: Paper is folded twice (once horizontally, once vertically). A straight cut is made at the folded corner, and two small squares are cut on two sides.

Concept: When paper is folded twice and cut, each cut is reflected across both fold lines, so one cut produces multiple holes/cuts when unfolded.

Answer:
- The straight cut at the corner (the fully folded corner = centre of the original paper): when unfolded, this produces a diamond/square shaped hole at the centre of the paper.
- The two square cuts on the sides: because the paper is folded, each side cut appears on both sides (reflected). When unfolded, the square cuts on the edges become rectangular notches or square holes symmetrically placed on all four sides of the paper.
- The overall unfolded design will be symmetrical about both the horizontal and vertical fold lines.

Given: Paper is folded once. Two cuts are made in the middle of the folded paper (creating a slit/tab shape).

Concept: When a folded paper is cut and unfolded, the cut edges are reflected across the fold line.

Answer:
- When the paper is folded once and two parallel cuts are made in the middle, a rectangular flap/tab is created.
- When unfolded, the two cuts on the folded paper become 4 cuts in total (each cut is mirrored).
- The resulting shape (the main paper with the cuts opened) will have the original 4 sides of the rectangle plus the additional edges created by the cuts.
- The shape will have 8 sides (the original rectangle's 4 sides plus 4 new edges from the 2 cuts reflected on both halves).

Given: A square sheet of paper is folded twice (once horizontally, once vertically), bringing all four corners together.

Concept: Folding twice means any cut is reflected across both fold lines, appearing 4 times when unfolded.

Answer:
- After folding the paper twice, the centre of the original paper is now at the folded corner (the corner where all layers meet).
- To make a square hole at the centre, cut a small square shape at the folded corner (the corner that represents the centre of the original paper).
- Number of cuts required: 2 (two straight cuts — one horizontal and one vertical — at the folded corner to remove a small square piece).
- When unfolded, these 2 cuts will produce a square hole exactly at the centre of the paper due to the symmetry of the double fold.

Given: Half of a design is drawn on one side of a line of symmetry (mirror line).

Concept: To complete a symmetrical design, the other half must be the mirror image of the given half. Each point on the given half must be reflected to the same distance on the other side of the line of symmetry.

Steps to complete the design:
1. Identify the line of symmetry (the dotted/bold line shown).
2. For each point or part of the design on one side, find its mirror image on the other side — it should be the same distance from the line of symmetry.
3. Connect the reflected points to complete the design.
4. The completed design should look identical on both sides of the line of symmetry.

Answer: Draw the mirror image of the given half on the other side of the line of symmetry to complete each design. (Actual drawing to be done by the student in the book.)

Given: Various shapes are shown along the border of the page.

Concept: A shape is symmetrical if it has at least one line of symmetry — a line along which the shape can be folded so that both halves match exactly.

Steps:
1. Draw each shape carefully on the dot grid by joining the dots.
2. For each shape, check if it can be folded to give two matching halves.
3. If yes, it is symmetrical — draw the line(s) of symmetry.

General answers for common shapes:
- Square: Symmetrical — has 4 lines of symmetry (2 through midpoints of opposite sides, 2 through opposite corners).
- Rectangle: Symmetrical — has 2 lines of symmetry (through midpoints of opposite sides).
- Equilateral triangle: Symmetrical — has 3 lines of symmetry.
- Isosceles triangle: Symmetrical — has 1 line of symmetry (through the apex to the midpoint of the base).
- Scalene triangle: Not symmetrical — no line of symmetry.
- Circle: Symmetrical — has infinite lines of symmetry.
- Irregular shapes: Generally not symmetrical.

Answer: Draw each shape on the dot grid, identify whether it is symmetrical, and draw the fold lines (lines of symmetry) accordingly. (Actual drawing to be done by the student.)

Given: Shape A is a basic shape (such as a right-angled triangle or half-shape). Different complete shapes are shown that can be obtained by placing a mirror at different positions.

Concept: When a mirror is placed along a line of symmetry of a shape, the shape and its reflection together form a new, larger symmetrical shape.

Answer:
- To get a square or rectangle: Place the mirror along the vertical or horizontal edge of shape A.
- To get a larger triangle: Place the mirror along the hypotenuse (slanted side) of shape A.
- To get a parallelogram or rhombus: Place the mirror at a diagonal to shape A.
- For each resulting shape shown in the book, place the mirror along the edge of shape A that, when reflected, produces that shape.

*(Since the actual figures are not visible, the student should physically place a small mirror along each edge/side of shape A and observe which resulting shape matches the ones shown. Mark the mirror position with a dotted line.)*

Given: Digits 0 to 9 are shown. We need to find which digits look the same when reflected in a mirror.

Concept: A digit has the same mirror image (when the mirror is placed vertically on one side) if it is symmetrical about a vertical axis.

Digits with the same mirror image (vertical mirror):
$$0, 1, 8$$
(These digits look the same when reflected left-to-right in a vertical mirror.)

Note: The digit $3$ has the same mirror image when the mirror is placed horizontally. Digits like $0$, $1$, $8$ are symmetric about a vertical axis.

4-digit numbers with the same mirror image:
Using only the digits $0$, $1$, $8$:
- The mirror is placed vertically to the right of the number.
- Examples of 4-digit numbers whose mirror image reads the same:
- $1001$, $1111$, $1881$, $8008$, $8118$, $8888$, $1001$, $0110$ (not valid as 4-digit), $1001$, $8008$, $1111$, $8118$, $1881$, $8888$, $1001$, $8008$
- Valid 4-digit numbers (not starting with 0): $1001,\ 1111,\ 1881,\ 8008,\ 8118,\ 8888$
- The mirror is kept vertically on the right side of the number.
- You can make many such numbers — as many combinations of $0$, $1$, $8$ as possible in 4 digits (first digit $\neq 0$).

Given: A 3-digit number near 120 whose mirror image is the same.

Concept: For a number's mirror image to be the same, it must use only digits that are symmetric about a vertical axis: $0, 1, 8$.

Working:
- 3-digit numbers near 120 using digits from $\{0, 1, 8\}$:
- $100, 101, 108, 110, 111, 118, 180, 181, 188$
- Numbers near 120: $118$ (closest to 120 using symmetric digits)
- Also $111$ is near 120.

Answer: The number is $\mathbf{111}$ or $\mathbf{118}$.
- $111$ is near 120 and uses only the digit $1$ which is symmetric.
- $118$ is also near 120 and uses digits $1$ and $8$, both symmetric.
- The mirror is kept vertically on the right side of the number.
- When reflected, $111 \rightarrow 111$ and $118 \rightarrow 811$ — wait, $118$ reflected gives $811$ which is different.
- So the correct answer is $\mathbf{111}$ (mirror image is $111$, same number). The mirror is placed vertically to the right of the number.

Activity:
- Choose digits from $\{0, 1, 8\}$ only.
- Form numbers (e.g., $1001$, $8008$, $1881$, $180081$) whose mirror image reads the same.
- Write a clue such as: *'I am a 4-digit even number. My mirror image is the same as me. What am I?'* (Answer: $8008$ or $1001$, etc.)
- Challenge friends to find the number and identify where the mirror is placed (vertically on the right side).

Observation about Ambulance:
- The word AMBULANCE on the front of an ambulance is written in mirror writing (laterally inverted / reversed).
- Reason: When a driver looks in the rear-view mirror, the mirror reverses the writing. So the reversed writing in the mirror appears as the correct word AMBULANCE. This helps drivers immediately recognise an ambulance behind them and give way.

Identifying the mirror words:

| Written Word | Mirror Reading | Place Mirror |
|---|---|---|
| 9AT | TAB / TAP (looks like 'TAB' or 'TAP' — the 9 is mirror of a letter) | Vertically on the right side |
| CYM | MYC / looks like 'MYC' — or it could read GYM | Vertically on the left side |
| WOW | WOW (W and O are symmetric, so WOW reads the same in mirror) | Vertically on either side |
| H3H | H3H (H and 3 are symmetric about horizontal axis, so it reads the same) | Horizontally below |

Detailed answers:
- 9AT: Place the mirror vertically on the left — it reads TAB or TAP (9 is mirror image of a 'q' or 'b' shape).
- CYM: Place the mirror vertically on the left — it reads MYC or GYM (if C is used as mirror of a reversed C).
- WOW: Place the mirror vertically on either side — it reads WOW (same word, as W and O have vertical symmetry).
- H3H: Place the mirror horizontally (below or above) — it reads H3H (same, as H and 3 have horizontal symmetry).

Activity: Try writing your own name or words in mirror writing and challenge friends to read them using a mirror.

Given: Half of each shape is drawn with a mirror line (line of symmetry) shown.

Concept: To complete a symmetrical shape, reflect every point of the given half to the other side of the mirror line at the same perpendicular distance.

Steps:
1. Identify the mirror line (line of symmetry) for each shape.
2. For each point/corner of the given half, measure its distance from the mirror line.
3. Mark the corresponding point at the same distance on the other side of the mirror line.
4. Join the reflected points in the same order to complete the shape.
5. The completed shape should be identical on both sides of the mirror line.

Answer: Complete each shape by drawing its mirror image on the other side of the given line of symmetry. (Actual drawing to be done by the student in the book.)

Given: Various regular and irregular polygons are shown.

Concept:
- A regular polygon with $n$ sides has exactly $n$ lines of symmetry.
- An irregular polygon may have fewer or no lines of symmetry.
- Lines of symmetry can be checked by folding the traced shape.

Answers for common shapes:

| Shape | Number of Sides | Lines of Symmetry |
|---|---|---|
| Equilateral Triangle | 3 | 3 |
| Square | 4 | 4 |
| Rectangle | 4 | 2 |
| Regular Pentagon | 5 | 5 |
| Regular Hexagon | 6 | 6 |
| Regular Octagon | 8 | 8 |
| Circle | Infinite (curved) | Infinite |
| Irregular shape | Varies | 0 or 1 |

Key Rule: For a regular polygon with $n$ sides:
$$\text{Number of lines of symmetry} = n$$

Method to verify: Trace each shape on paper, cut it out, and fold it in different ways. Each fold that makes both halves match exactly is a line of symmetry.

Answer: Trace each shape, fold it, and count the number of fold lines that give matching halves. Record the number of sides and lines of symmetry for each shape as shown in the table above.

Given: Tiling patterns made of repeated shapes.

Concept: In a tiling pattern, a repeating unit (tile) is the smallest shape or group of shapes that, when repeated by sliding (translating), flipping, or rotating, creates the entire pattern without gaps or overlaps.

Steps:
1. Look at the pattern carefully.
2. Find the smallest unit that repeats to make the whole pattern.
3. This unit is the tile (repeating unit).
4. Continue the pattern by repeating this unit in the same way.

Answer:
- Identify the repeating unit in each pattern (e.g., a single rectangle, a pair of rectangles, an L-shape, etc.).
- Continue the pattern by placing the same tile next to the existing ones using the same arrangement (slide/flip/rotate as needed).
- Ensure there are no gaps and no overlaps.

*(Actual continuation of the pattern to be drawn by the student in the book.)*

Answer: This depends on the tiles designed by the student. Common shapes used to make tiles include:
- Squares
- Rectangles
- Triangles (equilateral, right-angled, isosceles)
- Hexagons
- Parallelograms
- Trapeziums
- Combinations of two or more of the above shapes.

Write the names of the shapes you actually used in your tile design.

Concept: A tile is symmetrical if it has at least one line of symmetry.

Answer (general):
- A tile made of a square is symmetrical — 4 lines of symmetry.
- A tile made of a rectangle is symmetrical — 2 lines of symmetry.
- A tile made of an equilateral triangle is symmetrical — 3 lines of symmetry.
- A tile made of an irregular shape may not be symmetrical — 0 lines of symmetry.

For your own designed tile:
1. Trace the tile on paper.
2. Try folding it in different ways.
3. Each fold that gives two matching halves is a line of symmetry.
4. Draw these fold lines on the tile as dotted lines.

*(Actual answer depends on the student's tile design.)*

Activity Steps:
1. Choose two or more shapes (e.g., a square and a triangle, or two rectangles).
2. Join them together to form a new tile shape.
3. Trace this tile shape repeatedly in your notebook.
4. Arrange the traced tiles so that they fit together perfectly — no gaps between tiles and no overlaps.
5. This creates a tessellation (tiling pattern).

Example: Join a square and an equilateral triangle to make an 'arrow' or 'house' shape. Repeat this shape to tile a surface.

*(Actual drawing to be done by the student in the notebook.)*

Given: Shapes that can tile a surface (tessellate) are shown.

Observations:
1. The shapes fit together perfectly — there are no gaps and no overlaps between them.
2. This is called tessellation or tiling.
3. Some shapes (like squares, equilateral triangles, regular hexagons) tessellate on their own.
4. Other shapes need to be flipped, rotated, or slid to fit together.
5. The shapes may look different in different positions (standing, sleeping, flipped) but they are all the same shape and size.
6. By changing the colour or position of the same tile, many different and creative patterns can be made.

Conclusion: Tessellation is the art of covering a flat surface using one or more geometric shapes with no gaps or overlaps. It is seen in floor tiles, brick walls, honeycombs, and many natural and man-made patterns.

Given: A tile shape (designed by the student or given, e.g., Mini's cat-shaped tile).

Concept: A floor pattern (tessellation) is created by repeating a tile shape to cover a surface completely with no gaps and no overlaps.

Steps:
1. Take your tile shape.
2. Trace it on a large sheet of paper.
3. Place the tile next to the first tracing — slide it, flip it, or rotate it so it fits perfectly.
4. Continue tracing the tile in all directions.
5. Make sure there are no gaps (empty spaces) and no overlaps (two tiles on top of each other).
6. Colour the pattern creatively.

Answer: The floor pattern is ready when the entire surface is covered with the repeated tile shape. *(Actual drawing to be done by the student.)*

Discussion questions:
- How many shapes have been used?
- Would the design look the same if only one colour is used?
- Try creating different designs using the same shapes with different colours.

Given: Steps to make a cat-shaped tile from a square piece of paper by cutting and sliding.

Concept: This is an example of how a square (which tessellates) can be transformed into a creative shape (cat) that still tessellates, because the piece cut from one side is added to another side — the total area and fitting property are preserved.

Steps:
1. Take a square piece of paper.
2. Mark the cat's head and ears on the top edge.
3. Cut along the outline of the head and ears.
4. Slide the cut piece down to align with the bottom edge of the square.
5. Tape the pieces together — now you have a cat-shaped tile.
6. Trace the cat tile repeatedly on a large sheet of paper.
7. Arrange the tracings so they fit together with no gaps and no overlaps.
8. Draw the cat's face and colour the tiles creatively.

Key Observation: Even though the tile is now cat-shaped (not a simple square), it still tiles perfectly because the cut piece was only moved (slid) from one side to another — the overall shape still fits together like the original square.

Answer: The catty wall is a tessellation made of cat-shaped tiles. *(Actual activity to be done by the student.)*

Project Guidelines:

Step 1 — Nature Walk:
- Go to a nearby park or around your school with a teacher or parent.
- Collect fallen leaves, petals, and flowers (do not pluck from plants).
- Observe patterns on tree bark, flower petals, spider webs, honeycombs, etc.

Step 2 — Categorise Leaves:
- Symmetrical leaves: Most leaves have a central vein (midrib) that acts as a line of symmetry. Fold the leaf along the midrib — if both halves match, it is symmetrical. Examples: mango leaf, banana leaf, rose leaf.
- Non-symmetrical leaves: Some leaves have unequal halves. Example: begonia leaf.
- Make two columns in your project file: *Symmetrical* and *Non-symmetrical*, and paste or draw the leaves in the correct column.

Step 3 — Designs and Patterns:
- Arrange leaves and flowers in repeating patterns (tessellations).
- Create rangoli-style symmetrical designs.

Step 4 — Greeting Card:
- Apply paint or ink to a leaf and press it on paper to make a leaf imprint.
- Use dry flowers to decorate the card.
- Write a message on the card.

Step 5 — Animals/Designs:
- Use leaves of different shapes and sizes to create animals (e.g., a fish using an oval leaf, a butterfly using two leaves).
- Paste them in your project file and label them.

Conclusion: Nature is full of symmetry and patterns. Most leaves, flowers, and animals show symmetry, which makes them beautiful and balanced.