Mystery Matrix The Transport Museum

Chapter at a Glance

This chapter covers multiplication and division with two-digit multipliers, divisors, and large products using the theme of a visit to the Transport Museum. Students learn to construct mathematical times-tables (from $11$ to $20$) by splitting factors into $10$ and single units, or using equal grouping and doubling. They explore arithmetic properties of multiples of $10$ and $100$, solve division problems with remainders, and calculate travel capacities across various transit models (toy trains, double-decker buses, rescue flights, and traditional snake boats).

Key Definitions & Terminology

Mystery Matrix: A multiplication grid puzzle where outer header numbers must be solved based on intersecting row and column products.
Remainder: The integer value left over after dividing when the dividend is not a multiple of the divisor.
Vande Bharat Mission: A major aviation repatriation mission during the COVID-19 pandemic to bring back stranded citizens.
Vallam Kali: The traditional seasonal snake-boat race held in Kerala.
Punjab Mail: The oldest long-distance express train of Indian Railways, commencing its first journey on October 12, 1912.
Hop-on Hop-off Bus: A double-decker sightseeing bus used by tourists in cities like Goa.

Formulas, Rules & Properties

The Split-and-Sum Multiplication Rule:
To multiply a number by a two-digit factor, decompose the factor into tens and ones:
$$A \times 15 = A \times (10 + 5) = (A \times 10) + (A \times 5)$$
The Equal Grouping and Doubling Rule:
To multiply by an even factor, halve the factor, multiply, and double the result:
$$A \times 14 = A \times (7 + 7) = 2 \times (A \times 7)$$
Multiplying by Multiples of 10 and 100:
$N \times 10 = N \text{ Tens} = N0$
$N \times 20 = N \times 2 \text{ Tens} = (2 \times N)0$
$N \times 100 = N \text{ Hundreds} = N00$
$N \times 200 = N \times 2 \text{ Hundreds} = (2 \times N)00$
Division Equality Principle:
$$\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$$
Where $0 \le \text{Remainder} < \text{Divisor}$.

Core Concepts & Topics

Constructing Tables ($11$ to $20$):
Splitting columns into $10$ and $5$ to build the times-15 table.
Comparing times-15 to times-5 tables, noting that the differences are multiples of 10.
Multiplication and Division by Multiples of 10 and 100:
Exchanging tokens to group 10 hundreds into 1 thousand ($1000$).
Solving division word problems (e.g. $800 \text{ bands} \div 4 \text{ per packet} = 200 \text{ packets}$).
Museum Word Problems (Multi-digit Multiplication and Division):
Toy Train Coach Allocation: $15$ coaches seating $14$ children each can carry $15 \times 14 = 210$ children.
Coach Shortage with Remainder: Distributing $324$ children into $14$-seater coaches:
$$324 \div 14 = 23 \text{ coaches with } 2 \text{ children remaining}$$
Thus, $24$ coaches are needed to seat everyone.
Vande Bharat Rescue Flights: $64$ flights carrying $152$ people each:
$$64 \times 152 = 9728\text{ total passengers}$$
Vallam Kali Snake Boats: $960$ participants, with $64$ rowers per boat:
$$960 \div 64 = 15\text{ boats}$$
Sack Packing & Division:
Packing $60\text{ kg}$ or $600\text{ kg}$ rice in $10\text{ kg}$ or $100\text{ kg}$ sacks.
Travel Budgeting and Capacity Matching:
Calculating ticket prices for bus-boat transfers to Chilika Lake.
Solving ticket change combinations for Bujji, Munna, Balu, Chinnu, and Sansu.

Worked Examples

Grid Mystery Matrix:
Problem: Solve the header digits for row products ${56, 54, 63}$ and column products ${32, 42, 45, 21}$.
Solution: Factors are derived by finding common divisors (e.g. column $21$ and row $63$ share factor $7$).
Vande Bharat Grid Multiplication ($64 \times 152$):
Split $64 \rightarrow (60 + 4)$ and $152 \rightarrow (100 + 50 + 2)$:
- $60 \times 100 = 6000$; $60 \times 50 = 3000$; $60 \times 2 = 120$
- $4 \times 100 = 400$; $4 \times 50 = 200$; $4 \times 2 = 8$
- Sum: $6000 + 3000 + 120 + 400 + 200 + 8 = 9728$.
Punjab Mail Historical Data:
Question: On its first run in 1912, the train had $6$ coaches: $3$ passenger coaches carrying $96$ people each, and $3$ goods coaches. How many people travelled?
Solution: $3 \times 96 = 288$ passengers.
Question: Today, the train runs with $24$ coaches carrying $72$ passengers each. Find the total capacity.
Solution: $24 \times 72 = 1728$ passengers.
Goa Double-Decker Sightseeing Bus:
Question: Lower deck has $15$ seats, upper deck has $10$ seats, each seat fits $2$ passengers. How many total seats are available? Are there enough seats for Amala, her $35$ classmates, and $6$ teachers?
Solution: Total seats $= (15 + 10) \times 2 = 50$ seats. Total passengers $= 1 \text{ (Amala)} + 35 + 6 = 42$ people. Since $42 \le 50$, there are enough seats.
Chilika Lake Dolphin Trip Cost:
Question: $8$ people travel. Bus ticket costs $\text{Rs. } 60$ per person. The boat hire for $8$ people is $\text{Rs. } 1200$. How much is spent per person?
Solution: Boat cost per person $= 1200 \div 8 = \text{Rs. } 150$. Total per person $= 60 + 150 = \text{Rs. } 210$.
Chinnu's Coins ticket change:
Ticket cost $= \text{Rs. } 750$.
- Bujji ($\text{Rs. } 200$ notes) $\rightarrow$ pays $4$ notes ($\text{Rs. } 800$, gets $\text{Rs. } 50$ change).
- Munna ($\text{Rs. } 50$ notes) $\rightarrow$ pays $15$ notes ($\text{Rs. } 750$, no change).
- Balu ($\text{Rs. } 20$ notes) $\rightarrow$ pays $38$ notes ($\text{Rs. } 760$, gets $\text{Rs. } 10$ change).
- Chinnu ($\text{Rs. } 5$ coins) $\rightarrow$ pays $150$ coins ($\text{Rs. } 750$, no change).
- Sansu ($\text{Rs. } 2$ coins) $\rightarrow$ pays $375$ coins ($\text{Rs. } 750$, no change).
Vehicle Match-up Capacity:
$75\text{ Cycles} \rightarrow 75\text{ people}$ (capacity: $1$ per cycle)
$20\text{ Minibuses} \rightarrow 400\text{ people}$ (capacity: $20$ per minibus)
$103\text{ Cars} \rightarrow 412\text{ people}$ (capacity: $4$ per car)
$52\text{ Autos} \rightarrow 156\text{ people}$ (capacity: $3$ per auto)
$30\text{ Aeroplanes} \rightarrow 4560\text{ people}$ (capacity: $152$ per plane)
$12\text{ Train coaches} \rightarrow 864\text{ people}$ (capacity: $72$ per coach)

Practical Activities & Experiments

Table Construction Craft: Constructing a 15-times grid table on cardboard, splitting columns at $10$ and $5$ to demonstrate mental summation.
Double-Decker Bus Roleplay: Drawing a layout grid of lower and upper decks, and using coins to represent passengers filling seats.
Token Exchange Game: Exchanging ten $100$-tokens for one $1000$-token to practice grouping during multiplication.
Snake Boat Crew math: Setting up groups of counters to model dividing $960$ participants into $64$-member boat crews.

📖 Mathematics

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