Grandmother's Quilt

Chapter at a Glance

This chapter covers the concepts of perimeter (the boundary length of a 2D shape) and area (the region enclosed inside a 2D shape). Using quilt borders, table coverings, and leaf-tracing grid paper, students learn to count, calculate, and compare perimeters and areas. The chapter derives formulas for the areas and perimeters of squares and rectangles, demonstrates that shapes can have the same area but different perimeters (and vice versa), and includes division/multiplication word problems for fields and composite L-shaped figures.

Key Definitions & Terminology

Perimeter: The total distance around the outer boundary of a closed two-dimensional shape.
Area: The measure of the flat space enclosed inside a closed two-dimensional region.
Tiling (Tessellation): Covering a flat surface using repeating shapes so that there are no gaps and no overlapping pieces.
Unit Square: A reference square with sides of length $1\text{ unit}$, defining an area equal to $1\text{ unit square}$ (e.g. $1\text{ cm}^2$ or $1\text{ m}^2$).
Square Grid: A grid sheet composed of repeating unit squares, used to measure areas of irregular shapes.

Formulas, Rules & Properties

Square Formulas (with side length $S$):
$\text{Perimeter} = 4 \times S$
$\text{Area} = S \times S = S^2$
Rectangle Formulas (with length $L$ and breadth $B$):
$\text{Perimeter} = 2 \times (L + B) = 2L + 2B$
$\text{Area} = L \times B$
Tiling Rules:
Shapes like squares, rectangles, and triangles tile perfectly.
Circles do not tile because they leave gaps when placed next to each other.
Independence of Area and Perimeter:
Shapes with the same area can have different perimeters.
Shapes with the same perimeter can have different areas.

Core Concepts & Topics

Perimeters of Regular Shapes:
Calculating boundary lengths by multiplying the side length by the number of equal sides (triangle $= 3S$, hexagon $= 6S$).
Visualizing Area as a Row-Column Array:
Understanding that a patchwork grid of unit squares with length $L$ and breadth $B$ forms $L$ rows of $B$ squares, so the total area is $L \times B$ squares.
Measuring Irregular Areas:
Placing leaves or palm prints on a $1\text{ cm}$ square grid and using the boundary rule:
- Count squares covered half or more as $1\text{ cm}^2$.
- Ignore squares covered less than half.
Composite/Complex Shape Areas:
Splitting irregular or L-shaped figures into smaller rectangles, computing the area of each, and adding them.
Tile Expansion Game:
A turn-based strategy game where players roll a die, place that number of square tiles adjacent to each other, and track the perimeter until it reaches exactly $24\text{ units}$.

Worked Examples

Designing Rectangles with a Set Perimeter (e.g. $18\text{ cm}$):
The sum of length and breadth must be:
$$L + B = \text{Perimeter} \div 2 = 18 \div 2 = 9\text{ cm}$$
Possible integer rectangles and their resulting areas:
- $L = 8\text{ cm}, B = 1\text{ cm} \rightarrow \text{Area} = 8 \times 1 = \mathbf{8\text{ cm}^2}$
- $L = 7\text{ cm}, B = 2\text{ cm} \rightarrow \text{Area} = 7 \times 2 = \mathbf{14\text{ cm}^2}$
- $L = 6\text{ cm}, B = 3\text{ cm} \rightarrow \text{Area} = 6 \times 3 = \mathbf{18\text{ cm}^2}$
- $L = 5\text{ cm}, B = 4\text{ cm} \rightarrow \text{Area} = 5 \times 4 = \mathbf{20\text{ cm}^2}$ (largest area).
Rectangular Field Area:
Question: Find the area of a field with length $42\text{ m}$ and breadth $34\text{ m}$.
Solution: $\text{Area} = 42\text{ m} \times 34\text{ m} = \mathbf{1,428\text{ m}^2}$.
Finding Breadth from Area:
Question: A rectangular garden has an area of $64\text{ m}^2$ and a length of $16\text{ m}$. What is its breadth?
Solution: $\text{Breadth} = \text{Area} \div \text{Length} = 64 \div 16 = \mathbf{4\text{ m}}$.
Composite L-Shape Area:
Question: Find the area of an L-shape with heights and widths.
Solution: Split the L-shape into a vertical rectangle and a horizontal rectangle. For example, if it splits into a $12\text{ cm} \times 6\text{ cm}$ rectangle and a $20\text{ cm} \times 6\text{ cm}$ rectangle:
$$\text{Total Area} = (12 \times 6) + (20 \times 6) = 72 + 120 = \mathbf{192\text{ cm}^2}$$

Practical Activities & Experiments

Table Cover Tiling: Covering a classroom desk with notebooks, pencil boxes, or textbooks of equal size to see which items cover the surface with the fewest gaps.
Leaf and Palm Tracing: Tracing leaves and hands on graph paper and estimating their areas by counting grid squares.
Lace Bordering: Selecting ribbon lengths to cover the borders of rectangular blankets.
Grid Drawing Exercises: Drawing different shapes on grid paper that have the same area (e.g. $12\text{ cm}^2$) and comparing their perimeters.
Perimeter 24 Tile Game: Using physical square blocks to assemble shapes in turns, checking how adding tiles changes the perimeter.

📖 Mathematics

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