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.)*