# How Many Squares?

## Overview

This chapter introduces the concepts of **area** and **perimeter** using intuitive activities such as comparing stamps, footprints, hand shapes, and constructing puzzles with squares. It focuses on developing spatial understanding, estimating areas, exploring shapes, and solving problems through reasoning and creativity rather than formulas.

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## Key Topics Covered

### 1. Exploring Rectangles with Equal Area

* Students use **12 unit squares** to create different rectangles.
* All rectangles have the same area (12 square units) but different **perimeters**.
* **Perimeter** is defined as the length of the boundary.

### 2. Comparing Areas: Stamp Activity

* Learners measure different **postage stamps** by counting 1 cm × 1 cm squares.
* Identify:

* Stamps with the **largest** and **smallest** areas.
* Stamps with **equal** areas.
* Area differences between largest and smallest stamps.

### 3. Estimating and Comparing Areas

* Children estimate and compare areas of:

* Their **footprint** vs. the book page.
* Two ₹5 notes vs. a ₹100 note.
* A ₹10 note vs. 100 square cm.
* Encourages visual estimation and discussion.

### 4. My Hand and My Footprint

* Students trace their **hand** and **footprint** on squared paper.
* Calculate area by counting squares within the traced outline.
* Compare areas with classmates to explore differences.

### 5. Animal Footprint Comparison

* Actual-size footprints of animals like **dog**, **hen**, **rhino**, and **tiger** are compared.
* Students estimate areas and relate them to their own footprints.

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### 6. Try Triangles

* Focus on **triangles** that are **half of a rectangle**.
* Students observe that a triangle occupying half the area of a rectangle has:

* Area = ½ × Area of the rectangle.
* Recognize symmetry and develop their own strategies for finding area.

### 7. Creating Triangles of a Given Area

* Task: Draw different **triangles** with an area of **10 square cm** inside a rectangle.
* Reinforces idea that **shapes can differ in form but share the same area**.



### 8. Complete the Shape

* Students complete partial shapes to achieve a **target area** (e.g., 4 sq. cm or less than 2 sq. cm).
* Encourages visualization and symmetry.
* Can use both **straight and curved lines** to make the shape.

### 9. Puzzles with Five Squares (Pentominoes)

* Students create different shapes using **five unit squares**.
* Explore:

* Number of unique shapes possible.
* Shapes with **maximum** and **minimum** perimeters.
* All shapes have the **same area** but different perimeters.

### 10. Dividing Rectangles into Triangles and Other Shapes

* Tasks involve drawing:

* One line to create **two equal triangles**.
* One line to make **two equal rectangles**.
* Two lines to make **one rectangle and two equal triangles**.
* Develops understanding of area and division of shapes.

### 11. Tiling and Patterns

* Students build a **10×6 rectangle** using all 12 pentomino shapes.
* Multiple arrangements possible (over **2000**).
* Encourages **trial-and-error**, creativity, and **spatial reasoning**.

### 12. Game Time: Pentomino Challenge

* Two players take turns placing pentomino shapes on a **chessboard**.
* Shapes must **not overlap**.
* Last person to place a piece wins.

### 13. Make Your Own Tile

* Start with a **square of side 3 cm**.
* Modify by adding/removing **triangles** to/from its sides.
* Calculate the **area** of modified tile.
* Create **floor patterns** using repeated tiles (tiling).
* Explore which shapes **tile** well without gaps and which do not.

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## New Terms and Definitions

| Term | Simple Definition |
| --------- | --------------------------------------------------------------- |
| Area | The space a shape covers. Count squares to find it. |
| Perimeter | The total length around a shape. |
| Square cm | A square that is 1 cm long and 1 cm wide. Used to measure area. |
| Rectangle | A shape with 4 sides and 4 right angles. |
| Triangle | A shape with 3 sides. |
| Tiling | Repeating a shape to cover a surface without any gaps. |
| Pentomino | A shape made by joining 5 squares edge-to-edge. |
| Footprint | The shape or outline left by a foot. |

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## Practice Questions

### Easy (3)

1. **What is the area of a rectangle made using 12 unit squares?**
→ **12 square cm**
*Explanation: Each square is 1 cm², and there are 12.*

2. **If a square is 1 cm on each side, what is its area?**
→ **1 square cm**
*Explanation: 1 cm × 1 cm = 1 cm²*

3. **Which has a larger area – 5 stamps of 4 cm² or 3 stamps of 6 cm²?**
→ **5 stamps of 4 cm² = 20 cm²; 3 stamps of 6 cm² = 18 cm² ⇒ 5 stamps**
*Explanation: 20 > 18*

### Medium (2)

4. **You trace your hand on squared paper and count 38 squares. What is the area of your hand?**
→ **38 square cm**
*Explanation: Count of unit squares = area.*

5. **A triangle covers exactly half of a rectangle of area 16 square cm. What is the area of the triangle?**
→ **8 square cm**
*Explanation: ½ × 16 = 8*

### Difficult (3)

6. **Two shapes have the same area of 20 square cm. One is a square and one is a rectangle. Will their perimeters be the same?**
→ **No**
*Explanation: Shapes can have same area but different boundary lengths.*

7. **You make a shape with 5 squares and it has a perimeter of 12 cm. Can you make another shape with 5 squares but perimeter 14 cm?**
→ **Yes**
*Explanation: Rearranging squares changes perimeter, not area.*

8. **How many 2 cm × 2 cm squares will fit into a 10 cm × 6 cm rectangle?**
→ **15 squares**
*Explanation: Area of large rectangle = 60 cm²; area of each square = 4 cm²; 60 ÷ 4 = 15*

### Very Difficult (2)

9. **You create a tile with area 9 cm². Can this tile be repeated without gaps to cover a 36 cm² floor?**
→ **Yes, 4 tiles needed**
*Explanation: 36 ÷ 9 = 4*

10. **A triangle is made from half of a rectangle of unknown area. If the triangle’s area is 12 cm², what is the rectangle’s area?**
→ **24 square cm**
*Explanation: Triangle = ½ × rectangle ⇒ 12 = ½ × x ⇒ x = 24*

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