Fun with Symmetry — NCERT Solutions

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:

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:

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.