Subject Enrichment Activity – Mathematics (Class 8) proportional reasoning

 

SUBJECT ENRICHMENT ACTIVITY – MATHEMATICS (CLASS 8)

(Ganita Prakash – Page 178)

Title of the Activity:

🧩 Binairo – A Logic Puzzle Using Proportional Reasoning

Binairo, also known as Takuzu, is a logic puzzle with simple rules. Binairo is generally played on a square grid with no particular size. Some cells start out filled with two symbols: here horizontal and vertical lines. The rest of the cells are empty. The task is to fill cells in such a way that: 1. Each row and each column must contain an equal number of horizontal and vertical lines. 2. More than two horizontal or vertical lines can’t be adjacent. 3. Each row is unique. Each column is unique. 
Puzzle: 

Solution 

 Solve the following Binairo puzzles: 

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Topic:

Proportional Reasoning & Logical Thinking (Patterns, Equality, Constraints)

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Aim:

To develop logical reasoning, proportional thinking, and pattern recognition skills by solving a Binaro puzzle using given mathematical rules.

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Learning Outcomes:

Students will be able to:

• Apply proportional reasoning (equal numbers in rows/columns)

• Identify and extend patterns logically

• Follow constraints systematically

• Improve problem-solving and critical thinking skills

• Work collaboratively and explain reasoning

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Materials Required:

• Ganita Prakash textbook

• Pencil and eraser

• Worksheet / notebook

• Ruler

• Coloured pencils (optional, for marking patterns)

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Introduction (Concept Overview):

Binaro (also called Takuzu) is a logic-based puzzle played on a square grid.
The grid is filled using two symbols only:

• Horizontal line (—)

• Vertical line (|)

The challenge is to fill the grid by following specific rules, ensuring balance, uniqueness, and logical consistency.

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Rules of the Binaro Puzzle:

1. Equal Proportion Rule:
   Each row and each column must contain an equal number of horizontal and vertical lines.

2. Adjacency Rule:
   No more than two identical lines (— or |) can be placed adjacent to each other horizontally or vertically.

3. Uniqueness Rule:

   • No two rows can be exactly the same.

   • No two columns can be exactly the same.

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Procedure:

1. Observe the partially filled grid carefully.

2. Count the number of cells in each row and column.

3. Since the grid is even-sized, divide the total number of cells by 2 to know how many of each symbol are needed.

4. Use the adjacency rule to avoid placing three identical symbols in a row or column.

5. Check row and column uniqueness after placing each symbol.

6. Continue until the grid is completely filled correctly.

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Observation Table:

Observation	Mathematical Reasoning
Equal symbols in each row	Proportional reasoning
No three identical symbols together	Pattern recognition
Rows are unique	Logical differentiation
Columns are unique	Combinatorial reasoning

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Solved Answers / Solutions (Explanation-Based):

Puzzle 1 (Left Grid): 

• Each row has 6 cells → 3 horizontal and 3 vertical lines.

• Missing symbols filled by checking adjacency.

• Final grid satisfies all three rules.

✔ Solution Valid:

• Equal count ✔

• No three adjacent ✔

• Rows & columns unique ✔

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Puzzle 2 (Middle Grid): 

• Several rows already had two consecutive symbols.

• Third symbol forced to be opposite to avoid violation.

• Final solution balanced proportionally.

✔ Solution Valid

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Puzzle 3 (Right Grid): 

• Columns used first to determine missing symbols.

• Uniqueness rule helped avoid repeating patterns.

• Final grid completed logically.

✔ Solution Valid

(Teacher Note: The solutions shown in the textbook image are correct and follow all three rules.)

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Result / Conclusion:

The Binaro puzzle can be solved successfully by applying proportional reasoning, logical constraints, and systematic checking. Each correct solution maintains balance, avoids repetition, and respects adjacency rules.

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Reflection (Student Responses – Sample):

• “I learned how proportions can be applied even without numbers.”

• “The uniqueness rule made the puzzle challenging but interesting.”

• “Logical elimination helped me decide which symbol to place.”

• “This activity improved my patience and thinking skills.”

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Higher Order Thinking Skills (HOTS):

Q1. Why must the grid size be even in a Binaro puzzle?

Answer:
Because each row and column must have equal numbers of two symbols. An odd number cannot be divided equally.

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Q2. What would happen if we allowed three symbols instead of two?

Answer:
The puzzle would become more complex. Proportional reasoning would involve ratios like 1:1:1, and uniqueness rules would be harder to maintain.

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Q3. Can Binaro be related to real-life situations?

Answer:
Yes. Examples include:

• Balanced team formation

• Timetable scheduling

• Computer binary systems (0 and 1)

• Resource allocation problems

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Q4. How is Binaro connected to proportional reasoning?

Answer:
Each row and column maintains a fixed ratio (1:1) between two symbols, which is the core idea of proportional reasoning.

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Extension Activity:

• Design your own 4×4 or 6×6 Binaro puzzle.

• Exchange puzzles with classmates and solve them.

• Try replacing symbols with colours or shapes.

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Assessment :

• ✔ Logical reasoning

• ✔ Accuracy of solution

• ✔ Explanation of steps

• ✔ Participation and collaboration

TEACHER ANSWER KEY

🔹 Explanation of Logic

• Each grid has an even number of cells, so symbols are split equally.

• If two identical symbols appear consecutively, the third must be opposite.

• If a row is close to completion, missing symbols are deduced proportionally.

• Uniqueness rule ensures only one valid solution.

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🔹 Final Solutions (as per textbook)

Puzzle 1 – Completed Grid

✔ Equal number of horizontal and vertical lines in every row and column
✔ All rows and columns unique
✔ No adjacency violations

(Matches the “Solution” grid shown in the textbook)

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Puzzle 2 – Completed Grid

✔ Proportional distribution maintained
✔ Logical placement using elimination method

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Puzzle 3 – Completed Grid

✔ All constraints satisfied
✔ No repeated rows or columns

Note for Teachers:
Students’ answers should exactly match the textbook solution grid. Any deviation violates at least one rule.

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🔟 Assessment Criteria (Teacher Use)

Criteria	Marks
Correct application of rules	4
Logical reasoning steps	3
Accuracy of final solution	2
Neatness & presentation	1
Total	10 Marks

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1️⃣1️⃣ Conclusion

This activity effectively integrates proportional reasoning with logical deduction, encouraging students to think mathematically beyond calculations. It strengthens analytical skills essential for higher mathematics.

MATHEMATICS SUBJECT ENRICHMENT ACTIVITY Class: VIII Chapter: NUMBER PLAY

 MATHEMATICS SUBJECT ENRICHMENT ACTIVITY

Class: VIII
Chapter / Theme: Number Play
Activity Title: Navakankari – Strategy, Counting & Logical Reasoning

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🔷 Topic

Number Play through the traditional Indian board game Navakankari (Sālu Mane Āṭa / Chār-Pār)

Navakankari Navakankari, also known as Sālu Mane Āṭa, Chār-Pār, or Navkakri, is a traditional Indian board game that is the same as ‛Nine Men’s Morris’ or ‛Mills in the West’. It is a strategy game for two players where the goal is to form lines of three pawns to eliminate the opponent’s pawns or block their movement. Gameplay 1. Each player starts with 9 pawns. The players take turns in placing their pawns on the marked intersections. An intersection can have at most one pawn. 2. Once all the pawns are placed, the players take turns to move one of their pawns to adjacent empty intersections to form lines of three. The line can be horizontal or vertical. 3. Once a player makes a line with their pawns they can remove any one of the opponent’s pawns as long as it is not a part of one of their lines. A player wins if the opponent has less than 3 pawns or is unable to make a move.

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🎯 Aim of the Activity

• To develop logical thinking, strategic planning, and numerical reasoning

• To understand patterns, counting, and combinations through gameplay

• To connect mathematics with Indian traditional games

• To enhance decision-making and problem-solving skills

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🧰 Materials Required

• Navakankari game board (printed/drawn)

• 9 pawns (coins) for each player (two different colors)

• Notebook & pencil

• Observation table worksheet

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📋 Prior Knowledge Required

• Counting

• Understanding of patterns

• Basic idea of turns and strategy

• Concept of horizontal and vertical alignment

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🧩 Description of the Game (Given)

Navakankari is a two-player strategy game similar to Nine Men’s Morris.
Each player places 9 pawns on the board intersections and tries to form a line of three pawns.

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🪜 Procedure / Steps

Phase 1: Placing Pawns

1. Two players take turns.

2. Each player places one pawn at an empty intersection.

3. No intersection can have more than one pawn.

4. This continues until all 18 pawns are placed.

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Phase 2: Moving Pawns

5. Players move one pawn at a time to an adjacent empty intersection.

6. Movement is allowed only along the lines drawn on the board.

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Phase 3: Making a Line

7. A line of three pawns can be formed horizontally or vertically.

8. When a player forms a line, they remove one opponent pawn (not part of a line).

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Winning Condition

9. A player wins if:

   • The opponent has less than 3 pawns, OR

   • The opponent cannot make a move

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📊 Observation Table (Sample)

Turn No.	Player	No. of Pawns	Lines Formed	Pawn Removed	Strategy Used
5	Player A	9	1	Yes	Blocking
8	Player B	8	0	No	Position control
12	Player A	7	2	Yes	Double line

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🔍 Observations

• Players who planned placements early had an advantage.

• Blocking opponent lines was as important as forming one’s own.

• Symmetry and balance helped in controlling the board.

• The game involves counting moves, predicting outcomes, and logical deduction.

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💭 Reflections (Student Thinking)

• I learned that numbers and patterns appear naturally in games.

• Every move affects future possibilities.

• Strategy is more important than speed.

• Mathematics is not only calculations but also thinking ahead.

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🔥 Higher Order Thinking Skills (HOTS)

1. Why is it important to prevent your opponent from forming a line?

2. Can you predict a win in advance by counting possible moves?

3. Is it better to attack or defend early in the game? Why?

4. What happens if one player makes careless placements?

5. How is this game related to logical reasoning and permutations?

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✅ Answers / Solutions (Teacher Support)

✔ Key Mathematical Learnings

• Counting: Tracking pawns and moves

• Patterns: Recognizing potential lines

• Strategy: Minimizing opponent options

• Decision-making: Choosing optimal moves

✔ Winning Strategy (Sample)

• Control the center intersections

• Create two possible lines at once (fork strategy)

• Block opponent before completing a line

• Reduce opponent pawns to below 3

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📌 Conclusion

Navakankari is an excellent example of how traditional games enhance mathematical thinking.
Through this activity, students understand number play, logical sequencing, and strategic reasoning in a fun and engaging way.

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📎 Suggested Extensions

• Draw the board and label intersections with numbers

• Count total possible lines of three

• Compare Navakankari with Tic-Tac-Toe or Chess

MATHEMATICS SUBJECT ENRICHMENT ACTIVITY Class: VIII Chapter: We Distribute, Yet Things Multiply

 

 MATHEMATICS SUBJECT ENRICHMENT ACTIVITY

Class: VIII
Chapter: We Distribute, Yet Things Multiply
Activity Title: Coin Conjoin – Flipping Triangular Coin Patterns

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🔷 Topic

Triangular Numbers, Patterns, and Optimization of Moves

Coin Conjoin Arrange 10 coins in a triangle as shown in the figure below on the left. The task is to turn the triangle upside down by moving one coin at a time. How many moves are needed? What is the minimum number of moves? A triangle of 3 coins can be inverted (turned upside down) with a single move, and a triangle of 6 coins can be inverted by moving 2 coins. The 10-coin triangle can be flipped with just 3 moves; did you figure out how? Find out the minimum possible moves needed to flip the next bigger triangle having 15 coins. Try the same for bigger triangular numbers. Is there a simple way to calculate the minimum number of coin moves needed for any such triangular arrangement?

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🎯 Aim of the Activity

• To explore triangular number arrangements using coins

• To develop logical reasoning and spatial visualization

• To find the minimum number of moves required to transform a pattern

• To identify patterns and general rules for larger numbers

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🧰 Materials Required

• 10, 15 or more identical coins

• Flat surface or chart paper

• Pencil and notebook

• Ruler (optional)

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🧠 Prior Knowledge

• Counting numbers

• Understanding triangular arrangements

• Basic reasoning and pattern recognition

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🧩 Activity Description (Given Situation)

Coins are arranged in the shape of a triangle.
The challenge is to turn the triangle upside down by moving one coin at a time, using the minimum number of moves.

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🪜 Procedure / Steps

Step 1: Forming the Triangle

1. Arrange 10 coins in a triangular shape (4 rows: 1, 2, 3, 4).

2. Observe the orientation (point facing upward).

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Step 2: Understanding Smaller Cases

3. Try flipping:

   • Triangle of 3 coins → 1 move

   • Triangle of 6 coins → 2 moves

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Step 3: Flipping the 10-Coin Triangle

4. Identify coins that remain in the same position in the inverted triangle.

5. Move only the coins that do not overlap with the final shape.

6. Count each coin shift as one move.

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Step 4: Extending the Pattern

7. Repeat the activity for:

   • 15 coins

   • 21 coins (optional)

8. Record observations.

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📊 Observation Table

Number of Coins	Rows	Moves Needed	Coins That Stay	Coins Moved
3	2	1	2	1
6	3	2	4	2
10	4	3	7	3
15	5	4	11	4

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✅ Answers / Solutions

✔ Solution for 10 Coins

• Minimum number of moves required = 3

• Reason: 7 coins overlap in both triangles

• Only 3 coins need repositioning

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✔ Solution for 15 Coins

• Minimum number of moves required = 4

• Coins form 5 rows

• Overlapping coins = 11

• Coins to move = 15 − 11 = 4

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🔍 Key Pattern Observed

For a triangle with n rows:

🔑 Minimum number of moves = n − 1

Rows (n)	Coins	Minimum Moves
2	3	1
3	6	2
4	10	3
5	15	4

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💭 Reflections (Student Response)

• I learned that not all coins need to be moved.

• Patterns help reduce effort and save time.

• Mathematics helps us find the best solution, not just any solution.

• Logical thinking is more important than trial and error.

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🔥 Higher Order Thinking Skills (HOTS)

1. Why do some coins remain fixed during inversion?

2. Can you predict the answer without physically moving coins?

3. How is this activity related to triangular numbers?

4. What happens as the number of coins becomes very large?

5. Can you design a similar puzzle with square numbers?

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📌 Conclusion

The Coin Conjoin activity shows how simple distribution leads to multiplication of ideas.
By observing patterns, students discover a general mathematical rule that reduces effort and increases efficiency.

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🧮 Mathematical Link

• Triangular Numbers:

  $$T_n = \frac{n(n+1)}{2}$$

• Optimization & reasoning

• Visual geometry

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✨ Extension Activities

• Predict moves for 21 coins

• Draw before–after diagrams

• Convert the rule into a formula

• Create your own coin puzzle

BINARIO GAME (PLAY WITH FUN)

Complete Solution

Study this solution, then try to solve it yourself!

🧩 Binairo Puzzle Game

Interactive Math Activity • Proportional Reasoning • Logic Challenge

|

Equal Count Error

Adjacency Error

Uniqueness Error

0

Moves

00:00

Time

100

Score

Game Rules

• Each row must have 3 red vertical lines and 3 blue horizontal lines
• Each column must have 3 red vertical lines and 3 blue horizontal lines
• No more than 2 identical symbols can be adjacent
• All rows must be unique from each other
• All columns must be unique from each other

How to Play

1. Drag red (|) or blue (-) symbols onto the yellow grid

2. Click a cell to cycle through symbols: Empty → Red → Blue → Empty

3. Follow all three rules to solve the puzzle

4. Click "Check" to verify your solution

5. Click "Show Solution" if you need help

6. Complete the puzzle to hear victory music!

Note: Gray cells are fixed clues - you cannot change them!