Mathematics: Class 5 — Chapter 7: Shapes and Patterns

To make a basic paper mat, you need a base colored paper ($30\text{ cm} \times 20\text{ cm}$) and eight paper strips of two different colors ($3\text{ cm}$ wide and longer than $20\text{ cm}$).

• Fold the base paper in half along the longer side.
• Cut vertical slits from the closed end, leaving a gap of $3\text{ cm}$ at the top.
• Weave the strips through the slits in alternating patterns.

Weave patterns can be described mathematically by the sequence of going over and under the slits:

• Pattern 1:
  • Row 1: 1 under, 1 over, 1 under, 1 over, ... (repeat)
  • Row 2: 1 over, 1 under, 1 over, 1 under, ... (repeat)
• Pattern 2:
  • Row 1: 2 over, 1 under, 2 over, 1 under, ... (repeat)
  • Row 2: 2 under, 1 over, 2 under, 1 over, ... (repeat)
• Pattern 3:
  • Row 1: 2 over, 1 under, 2 over, 1 under, ... (repeat)
  • Row 2: 1 under (do not repeat), 3 over, 3 under, 3 over, 3 under, ... (repeat)
  • Row 3: 2 under, 1 over, 2 under, ... (repeat)
  • Row 4: 1 over (do not repeat), 3 under, 3 over, ... (repeat)

A regular shape is a geometric figure where all sides are equal in length and all angles are equal.

• Regular Pentagon (5-sided): Does not tessellate. When placed around a point, they leave gaps.
• Equilateral Triangle (3-sided): Tessellates. Six equilateral triangles fit perfectly around a point without gaps or overlaps ($6 \times 60^\circ = 360^\circ$).
• Square (4-sided): Tessellates. Four squares fit perfectly around a point ($4 \times 90^\circ = 360^\circ$).
• Regular Hexagon (6-sided): Tessellates. Three regular hexagons fit perfectly around a point ($3 \times 120^\circ = 360^\circ$).
• Regular Octagon (8-sided): Does not tessellate on its own because its interior angle is $135^\circ$, which does not divide $360^\circ$ evenly.

Tessellations can also be formed by combining different shapes:

• Octagons and Squares: Regular octagons can tile a plane if the gaps are filled with small squares.
• Squares and Triangles: A combination of squares and equilateral triangles can form a stable tiling pattern.

Triangles can be classified based on the equality of their sides and angles:

• Isosceles Triangle: Has at least 2 equal sides and 2 equal angles.
• Equilateral Triangle: Has all 3 sides equal and all 3 angles equal (each angle is $60^\circ$).
• Scalene Triangle: Has no equal sides and no equal angles.

A quadrilateral is a 4-sided polygon. Key types include:

• Rhombus: A quadrilateral with all four sides of equal length. It can be divided into four congruent right-angled triangles.
• Kite: A quadrilateral with two pairs of equal-length adjacent sides (Side 1 = Side 2, and Side 3 = Side 4).
• Parallelogram: A quadrilateral whose opposite sides are equal and parallel. Opposite angles are also equal.
• Rectangle: A special type of parallelogram where all four angles are equal and are right angles ($90^\circ$).

A tangram is a traditional puzzle consisting of flat shapes (called tans) that are put together to form various designs. Analyzing tangram pieces helps in understanding spatial relationships, angles, and side lengths.

If you draw two perpendicular diameters of a circle and join their four endpoints in order, you form a square (or a rectangle if the diameters are not perpendicular).

By dividing a circle's circumference into equal parts (e.g., 24 points) and connecting them using specific rules (such as connecting point $n$ to point $n+10$), intricate geometric patterns and star polygons can be created.

A net is a 2D pattern that can be folded to form a 3D solid. A cube has 11 distinct nets.

When a large solid cube made of $27$ small unit cubes ($3 \times 3 \times 3$) is painted red on the outside, the distribution of painted faces on the individual unit cubes is as follows:

• Three faces painted red: $8$ cubes (located at the corners).
• Two faces painted red: $12$ cubes (located along the edges, excluding corners).
• One face painted red: $6$ cubes (located at the center of each of the 6 faces).
• No faces painted red: $1$ cube (located at the very center/core of the large cube).