Mathematics Subject Enrichment Activity Chapter: Area Theme: Transforming Shapes without Changing Area (Śulba-Sūtras)

 

📘 Mathematics Subject Enrichment Activity

Chapter: Area

Theme: Transforming Shapes without Changing Area (Śulba-Sūtras)

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🔷 Topic

Area Preservation through Geometric Dissection and Rearrangement

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🔷 Aim

To understand that area remains the same when a figure is cut and rearranged into another shape, and to explore ancient Indian (Śulba-Sūtras) and Euclidean methods for transforming geometric figures.

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🔷 Materials Required

• Geometry box (ruler, compass, protractor)

• Colour papers / chart paper

• Scissors and glue

• Pencil & eraser

• Colour pencils

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🔷 Learning Outcomes

Students will be able to:

• Apply area formulas correctly

• Perform geometric dissections

• Appreciate the contribution of Śulba-Sūtras

• Develop visual–spatial reasoning

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🔷 Activity 1

Convert a Trapezium into a Rectangle of Equal Area

Procedure

1. Draw trapezium ABCD.

2. Extend non-parallel sides to form congruent triangles.

3. Cut triangles ∆AHI and ∆DGI, ∆BEJ and ∆CFJ.

4. Rearrange pieces to form rectangle EFGH.

Observation

• Cut triangles are congruent.

• Rearranged figure is a rectangle.

• Area remains unchanged.

Conclusion

Dissection preserves area even though shape changes.

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🔷 Activity 1(b)

Construct a Trapezium of Area 144 cm²

Solution (Example):
Take parallel sides = 18 cm, 6 cm
Height = 12 cm

$$\text{Area} = \frac{1}{2}(18+6)\times12 = 144 \text{ cm}^2$$

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🔷 Activity 2

Convert an Isosceles Trapezium into a Rectangle

Method:
Cut off two congruent triangular ends and shift them inward to form a rectangle.

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🔷 Activity 3

Rectangle → Rhombus (Equal Area)

Idea:
Cut along diagonals and rearrange to form a rhombus with same base × height.

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🔷 Activity 4

Rhombus → Rectangle (Śulba-Sūtras Method)

Observation:
Height of rhombus becomes breadth of rectangle → area preserved.

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🔷 Activity 5

Rectangle → Isosceles Triangle (Śulba-Sūtras)

Method:
Cut rectangle along diagonal and join halves to form triangle.

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🔷 Activity 6

Area Comparison

(a) Equilateral Triangle vs Square (same side length a)

• Triangle area = $\frac{\sqrt{3}}{4}a^2$

• Square area = $a^2$

✅ Square has greater area

(b) Two equilateral triangles vs square

$$2\times \frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{2}a^2 < a^2$$

✅ Square still has greater area

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🔷 Activity 7

Triangle → Rectangle (Śulba-Sūtras)

Method:
Duplicate triangle, join base-to-base → rectangle.

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🔷 Activity 8

Isosceles Triangle → Rectangle (Simpler Method)

Hint Used:
∆ADB and ∆ADC become two halves of a rectangle.

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🔷 Activity 9

Rectangle of Twice the Area of a Triangle

Method 1:
Duplicate triangle → rectangle

Method 2:
Extend height or base proportionally

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🔷 Activity 10

Quadrilateral of Half the Area of Another Quadrilateral

Method:
Divide quadrilateral by diagonal → take one triangle → rearrange.

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🔷 Observations (Common)

• Shape may change, area does not

• Dissection relies on congruency

• Visual reasoning is essential

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🔷 Reflections (Student Writes)

• I learnt that area depends on base and height, not shape.

• Ancient mathematicians used logical geometric methods.

• Dissection made geometry easier to understand.

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🔷 HOTS (Higher-Order Thinking)

1. Can every polygon be converted into a rectangle of equal area?

2. Why are triangles most useful for dissection?

3. How is this concept used in land measurement?

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🌟 Mathematical Heritage Link

Śulba-Sūtras (800–500 BCE) show that Indian mathematicians understood area conservation centuries before modern geometry.

1️⃣ Trapezium → Rectangle (Equal Area)

Labels to show on diagram (for students):

• Trapezium ABCD

• Height h

• Parallel sides AB, CD

• Cut triangles: ∆AHI ≅ ∆DGI, ∆BEJ ≅ ∆CFJ

• Rearranged rectangle EFGH

• Same height h

📝 Key Note for Students:

Pieces are rearranged, not resized → area remains same

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2️⃣ Isosceles Trapezium → Rectangle

Highlight:

• Equal non-parallel sides

• Two congruent triangles cut and shifted

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3️⃣ Rectangle → Rhombus (Śulba-Sūtras)

Label:

• Same base

• Same height

• Area = base × height (unchanged)

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4️⃣ Triangle → Rectangle (Śulba-Sūtras)

Student Hint on Diagram:

Two identical triangles → rectangle

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🧠 PART B: Competency-Based Worksheet (with Answer Key)

🔹 Section A: Understanding

1. What happens to the area when a shape is dissected and rearranged?
   Answer: Area remains unchanged.

2. Name the ancient Indian texts that discuss area transformation.
   Answer: Śulba-Sūtras.

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🔹 Section B: Application

3. A trapezium has parallel sides 12 cm and 8 cm, height 10 cm.
   Find its area.
   Answer:

$$\frac{1}{2}(12+8)\times 10=100{\text{ cm}}^2$$

4. If this trapezium is converted into a rectangle, what will be the area of the rectangle?
   Answer: 100 cm²

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🔹 Section C: Reasoning

5. Why are triangles commonly used in dissection methods?
   Answer:
   Because triangles are the simplest polygons and can easily form other shapes.

6. Explain why a rectangle formed from a triangle has the same area.
   Answer:
   No part is removed or added; only rearranged.

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🔹 Section D: HOTS

7. Can a circle be converted into a rectangle using exact dissection? Why?
   Answer:
   No, because curved boundaries cannot be exactly rearranged into straight edges.

8. If the height of a triangle is doubled, what happens to its area?
   Answer:
   Area doubles.

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🔹 Section E: Assertion–Reason

Assertion (A): Area depends on shape.
Reason (R): Rearrangement changes area.

Correct Answer: ❌ Both A and R are false.

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📘 PART C: Teacher Rubric Page (Assessment)

Criteria	Excellent (4)	Good (3)	Satisfactory (2)	Needs Improvement (1)
Concept of Area	Clear & accurate	Mostly clear	Partial	Incorrect
Diagrams	Neat, labelled, colourful	Labelled	Rough	Missing
Mathematical Reasoning	Logical & clear	Mostly logical	Limited	Incorrect
Application of Śulba-Sūtras	Correct & explained	Correct	Partial	Not shown
Reflection	Deep insight	Relevant	Minimal	Missing

Total Marks: ____ / 20

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🌟 Teacher Tip (Optional Page Note)

This activity integrates history of mathematics, geometry, and visual reasoning, aligning with NEP 2020 competency-based learning.

MATHEMATICS SUBJECT ENRICHMENT ACTIVITY Chapter: Proportional Reasoning – II Topic: Representation of Data Using Pie Chart

 

📘 MATHEMATICS SUBJECT ENRICHMENT ACTIVITY

Chapter: Proportional Reasoning – II

Topic: Representation of Data Using Pie Chart (Proportional Reasoning)

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🎯 Aim

To collect data on favourite sports of Class 8 students and represent the data proportionally using a pie chart, and to identify:

• The sport liked by maximum students

• The sport liked by minimum students

• Sports liked by equal number of students

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🧰 Materials Required

• Notebook

• Pen / Pencil

• Scale

• Compass

• Protractor

• Colour pencils / sketch pens

• Graph paper

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📊 Data Collection

A survey was conducted among 40 classmates of Class 8 to find their favourite sport.

Collected Data Table

Sport	Number of Students
Cricket	14
Football	10
Badminton	6
Volleyball	6
Basketball	4
Total	40

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🧮 Calculations (Proportional Reasoning)

Formula used:

$$\text{Angle of sector} = \frac{\text{Number of students}}{40} \times 360^\circ$$

Sport	Students	Fraction	Angle
Cricket	14	14/40	126°
Football	10	10/40	90°
Badminton	6	6/40	54°
Volleyball	6	6/40	54°
Basketball	4	4/40	36°

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🎨 Colourful Pie Chart

(Students should draw this pie chart neatly using the above angles and colour each sector differently.)

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🧑‍🏫 Procedure

1. Conduct a survey among 40 classmates.

2. Record the data in a table.

3. Convert the data into fractions of the total.

4. Calculate angles for each sport using proportional reasoning.

5. Draw a circle using a compass.

6. Mark angles with a protractor.

7. Colour each sector and label it clearly.

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👀 Observations

• Cricket occupies the largest sector.

• Basketball occupies the smallest sector.

• Badminton and Volleyball have equal-sized sectors.

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📝 Conclusion

• Maximum favourite sport: Cricket

• Minimum favourite sport: Basketball

• Same preference: Badminton and Volleyball

• Pie charts help us understand proportional relationships visually.

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💭 Reflection (Student Learning)

• I learned how fractions and ratios are used in real-life data.

• I understood the connection between angles and proportions.

• Drawing a pie chart improved my data handling and reasoning skills.

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🧠 Teacher’s Note (Assessment Link)

✔ Uses proportional reasoning
✔ Integrates data handling
✔ Encourages real-life application
✔ Supports visual learning

Competency-Based Questions (CBQs) perfectly aligned with
Class 8 – Ganita Prakash (Part 2)
Chapter: Proportional Reasoning – II
based on the Favourite Sports Pie Chart Activity.

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🧠 Competency-Based Questions

(Data Handling & Proportional Reasoning)

Case Study

A survey was conducted among 40 Class 8 students to find their favourite sport.
The data collected is shown below:

Sport	Students
Cricket	14
Football	10
Badminton	6
Volleyball	6
Basketball	4

The data is represented using a pie chart.

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🔹 Level 1: Understanding & Interpretation

1. What fraction of the students like Football?
a) 1/4
b) 1/5
c) 1/8
d) 3/10

2. Which sport is liked by the maximum number of students?

3. Name the sports which have equal representation in the pie chart.

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🔹 Level 2: Application of Proportional Reasoning

4. If the total number of students is 40, what angle represents Cricket in the pie chart?

5. The angle representing Basketball is:
a) 36°
b) 54°
c) 72°
d) 90°

6. If 1 student represents 9°, verify whether the angle for Badminton is correct.

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🔹 Level 3: Analysis & Reasoning

7. Compare the sectors of Cricket and Football.
Which is larger and why?

8. If 5 more students start liking Basketball, how will the angle of the Basketball sector change?

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🔹 Level 4: Higher-Order Thinking (HOTS)

9. If the number of students increases to 60 but the ratio of preferences remains the same, find the new angle for Cricket.

10. Why is a pie chart more suitable than a bar graph for showing proportional data in this activity?

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🔹 Assertion–Reason Type

11. Assertion (A): The total angle of all sectors in a pie chart is 360°.
Reason (R): A pie chart represents the whole data set as a circle.

a) Both A and R are true and R is the correct explanation of A
b) Both A and R are true but R is not the correct explanation of A
c) A is true but R is false
d) A is false but R is true

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🔹 Real-Life Application

12. How can this method of data representation help a school decide which sports facilities to improve?

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📝 Teacher Answer Key (Brief)

1. a)

2. Cricket

3. Badminton and Volleyball

4. 126°

5. a)

6. 6 × 9° = 54° ✔

7. Cricket, because it has more students

8. Angle increases proportionally

9. 126° (ratio unchanged)

10. Shows part-to-whole relationship clearly

11. a)

12. Helps in decision-making using data

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🎯 Competencies Assessed

✔ Data Interpretation
✔ Proportional Reasoning
✔ Mathematical Communication
✔ Critical Thinking
✔ Real-life Application