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Mathematics: Class 5 — Chapter 10: Symmetrical Designs

10.1 Alphabet Cutout (Reflection Symmetry)

Using the concept of Reflection Symmetry and Lines of Symmetry, paper cutouts of letters can be made easily by folding and cutting paper.

10.1.1 One Line of Symmetry (Vertical)

The letter A has a vertical line of symmetry. It can be cut out by drawing only half of the letter:

  • Fold a paper in half.
  • Draw half of the letter A along the fold.
  • Cut along the outline.
  • Open the paper to see the full letter A.

10.1.2 Two Lines of Symmetry (Vertical and Horizontal)

The letter H has two lines of symmetry. It can be cut out by drawing only one-fourth of the letter:

  • Fold the paper into one-fourth (once vertically, once horizontally).
  • Draw one-fourth of the letter H along the fold.
  • Cut along the outline.
  • Open the paper to see the full letter H.

10.1.3 Activity: Analyzing Symmetry in Letters

Analyze the letters: E, N, X, T, K, V, O

  • Horizontal line of symmetry: E, X, K, O
  • Vertical line of symmetry: X, T, V, O
  • Both vertical and horizontal lines of symmetry: X, O
10.2 Let Us Make a Windmill Firki (Rotational Symmetry)

A windmill firki can be constructed from a square piece of paper by folding diagonally, cutting halfway to the center, and folding alternate corners to the center.

10.2.1 Rotational Symmetry of a Firki

When we rotate a firki, we observe its orientation at different turns:

  • A firki has Rotational Symmetry because it looks exactly the same after a $\frac{1}{4}$ turn, $\frac{1}{2}$ turn, $\frac{3}{4}$ turn, and a full turn.
10.3 Rotational Symmetry in Letters and Digits

An object has Rotational Symmetry if it looks the same after being rotated by less than a full turn around its center.

10.3.1 Rotational Symmetry in Letters

  • The letter H has rotational symmetry because it looks the same when rotated by a half turn ($\frac{1}{2}$ turn).
  • Other shapes, like a firki, look the same at multiple fractional turns ($\frac{1}{4}$, $\frac{1}{2}$, and $\frac{3}{4}$ turn).

10.3.2 Symmetry in Digits and Numbers

Consider the digits: $1, 2, 3, 4, 5, 6, 7, 8, 9, 0$

  • Reflection Symmetry: Digits like $1, 3, 8, 0$ have reflection symmetry.
  • Rotational Symmetry: Digits like $0, 8$ (and sometimes $1$ depending on font style) look the same after a $\frac{1}{2}$ turn.
  • Both Symmetries: Digits like $0$ and $8$ exhibit both reflection and rotational symmetries.
  • Symmetrical Numbers: Numbers like $11$ and $1001$ exhibit both reflection and rotational symmetry.
10.4 Making Symmetrical Designs

Symmetrical designs can be created and modified by combining geometric shapes like squares and equilateral triangles.

10.4.1 Modifying Designs for Rotational Symmetry

  • A design can be modified by adding shapes so that it looks the same after a $\frac{1}{2}$ turn.
  • Further modifications can make the design look the same after a $\frac{1}{4}$ turn.
  • Designs can also be checked for reflection symmetry by drawing lines of symmetry across them.
10.5 Let Us Explore: Block Printing

Block printing is a traditional craft of Rajasthan where artisans use carved wooden blocks to print beautiful, repetitive patterns on fabric.

10.5.1 Symmetry in Patterns

  • Wooden blocks are printed repeatedly (e.g., 4 times) to create a composite design.
  • Some composite designs look the same after every $\frac{1}{4}$ turn, exhibiting rotational symmetry.
  • Other designs may have only reflection symmetry, only rotational symmetry, or both.