Proportional Reasoning - 2 | A

Question 1

Map Making Activity: Guide students to make a sketch of their classroom with an accurate scale (ratio of $1 : 50$ ). They should mark the location of various objects in the classroom like the teacher’s desk, blackboard, fans and lights, according to scale. Students can use appropriate symbols to represent different objects like fans, lights, tables, chairs, and so on.

Solution

Understand the Question

A scale of $1 : 50$ means that $1\text{ unit}$ on the map represents $50\text{ units}$ in the actual classroom.
To find any dimension on the map: $\text{Map dimension} = \dfrac{\text{Actual dimension}}{50}$
First convert all actual measurements to centimetres ( $\text{cm}$ ), scale them down by dividing by $50$ , and draw the layout using consistent symbols.

Step 1 · Measure Actual Classroom Dimensions

Assuming a typical classroom size:

\begin{aligned} L_{\text{actual}} &= 10\text{ m} = 10 \times 100\text{ cm} = 1000\text{ cm} \\[0.6em] W_{\text{actual}} &= 8\text{ m} = 8 \times 100\text{ cm} = 800\text{ cm} \end{aligned}

Step 2 · Calculate Scaled Classroom Dimensions

Using the scale ratio of $1 : 50$ :

\begin{aligned} L_{\text{map}} &= \dfrac{L_{\text{actual}}}{50} = \dfrac{1000\text{ cm}}{50} = 20\text{ cm} \\[0.6em] W_{\text{map}} &= \dfrac{W_{\text{actual}}}{50} = \dfrac{800\text{ cm}}{50} = 16\text{ cm} \end{aligned}

Step 3 · Scale Dimensions and Positions of Objects

Using one corner of the classroom as the origin $(0, 0)$ :

Blackboard:
- Actual width: $3\text{ m} = 300\text{ cm}$
- Scaled width: $\dfrac{300\text{ cm}}{50} = 6\text{ cm}$
- Scaled position: From $5\text{ cm}$ to $11\text{ cm}$ along the front wall
Teacher's Desk:
- Actual dimensions: $1.5\text{ m} \times 0.75\text{ m} = 150\text{ cm} \times 75\text{ cm}$
- Scaled dimensions: $\left(\dfrac{150\text{ cm}}{50}\right) \times \left(\dfrac{75\text{ cm}}{50}\right) = 3\text{ cm} \times 1.5\text{ cm}$
- Scaled position: $(4\text{ cm}, 2\text{ cm})$
Fans:
- Actual diameter: $1.2\text{ m} = 120\text{ cm}$
- Scaled diameter: $\dfrac{120\text{ cm}}{50} = 2.4\text{ cm}$
- Scaled center positions: Fan 1 at $(8\text{ cm}, 5\text{ cm})$ , Fan 2 at $(8\text{ cm}, 15\text{ cm})$
Lights:
- Actual dimensions: $0.6\text{ m} \times 0.15\text{ m} = 60\text{ cm} \times 15\text{ cm}$
- Scaled dimensions: $1.2\text{ cm} \times 0.3\text{ cm}$
- Scaled center positions: $\text{L1}(4\text{ cm}, 3\text{ cm})$ , $\text{L2}(12\text{ cm}, 3\text{ cm})$ , $\text{L3}(4\text{ cm}, 17\text{ cm})$ , $\text{L4}(12\text{ cm}, 17\text{ cm})$
Student Desks:
- Actual dimensions: $1.2\text{ m} \times 0.6\text{ m} = 120\text{ cm} \times 60\text{ cm}$
- Scaled dimensions: $2.4\text{ cm} \times 1.2\text{ cm}$
- Scaled positions: $\text{SD1}(2\text{ cm}, 6\text{ cm})$ , $\text{SD2}(6\text{ cm}, 6\text{ cm})$ , $\text{SD3}(2\text{ cm}, 10\text{ cm})$ , $\text{SD4}(6\text{ cm}, 10\text{ cm})$ , $\text{SD5}(2\text{ cm}, 14\text{ cm})$ , $\text{SD6}(6\text{ cm}, 14\text{ cm})$
Door:
- Actual width: $1\text{ m} = 100\text{ cm}$
- Scaled width: $\dfrac{100\text{ cm}}{50} = 2\text{ cm}$ (from $0\text{ cm}$ to $2\text{ cm}$ along the front wall)

Step 4 · Choose Symbols and Sketch the Map

Diagram 1

Classroom Walls: Thick rectangular boundary ( $20\text{ cm} \times 16\text{ cm}$ )
Blackboard: Thick rectangle on front wall
Teacher's Desk: Rectangle marked with 'T'
Fan: Circle with an 'X' inside
Light: Small rectangle marked with 'L'
Student Desk: Small rectangles
Door: Gap in the perimeter wall

Answer

A scaled classroom sketch measuring $20\text{ cm} \times 16\text{ cm}$ with all furniture and fixtures plotted at a scale of $1 : 50$ .

Common Mistakes

Unit Mismatch: Forgetting to convert measurements from metres ( $\text{m}$ ) to centimetres ( $\text{cm}$ ) before dividing by the scale factor of $50$ .
Inverse Scaling: Multiplying by $50$ instead of dividing by $50$ when converting actual dimensions into map dimensions.
Inconsistent Scale: Using different scales for room boundaries and internal objects. Every element must be scaled down by the exact same ratio of $1 : 50$ .