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Jaka EduTech

📖 Mathematics

Std 5
Books Book Guide
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Chapters
1 We the Travellers — I 2 Fractions 3 STATUE! Angles as Turns 4 Making Sums Equal 5 Far and Near (Measuring Distance) 6 The Dairy Farm 7 Shapes and Patterns 8 Check! Check! (Weight and Capacity) 9 Division Facts 10 Alphabet Cutout (Rotational Symmetry) 11 Grandmother's Quilt 12 Racing Seconds 13 Animal Jumps (Factors and Multiples) 14 Maps and Locations 15 Data Through Pictures
6
Chapter 6
Skill: 30%

The Dairy Farm

The Dairy Farm

Chapter at a Glance

This chapter covers multiplication through concrete, visual, and algebraic methods. Using dairy farm tasks (counting milk yield from Gir cows, shipping ghee, and billing fodder) and community event planning (estimating waste composting and tournament financing), students explore standard multiplication strategies. The chapter covers the commutative and zero properties, mental math techniques (doubling/halving, nearest multiples), three formats for two-digit multiplication (grid method, partial products, and standard algorithm), exponential growth (the King's Reward puzzle), and logical coordinate row-column calculations.

Key Definitions & Terminology

  • Commutative Property of Multiplication: The mathematical rule stating that changing the order of factors does not change the product ($A \times B = B \times A$).
  • Doubling and Halving: A mental math strategy where one factor is multiplied by $2$ and the other is divided by $2$ to simplify the multiplication (e.g. $16 \times 25 = 8 \times 50 = 400$).
  • Nearest Multiple Strategy: A strategy where you multiply by a nearby round number (like $20$ or $100$) and then adjust by adding or subtracting the remaining group (e.g., $14 \times 21 = 14 \times 20 + 14$).
  • Exponential Growth (The King's Reward): A pattern of growth where a quantity is multiplied by a constant factor in each time interval, leading to rapid increases.
  • Mandi / Cooperative: A collective business system where local farmers sell their yields (milk, ghee) to wholesalers.

Formulas, Rules & Properties

  • Commutative Property:
    $$A \times B = B \times A$$
  • Zero Property:
    $$A \times 0 = 0 \times A = 0$$
  • Multiplication by Powers of 10:
  • Multiplying by $10$ moves digits $1$ place value left, appending $0$ on the right.
  • Multiplying by $100$ moves digits $2$ place values left, appending $00$.
  • Multiplying by $1,000$ moves digits $3$ place values left, appending $000$.
  • Doubling & Halving Property:
    $$A \times B = \left(\frac{A}{2}\right) \times \left(2B\right)$$
  • Adjacent Product Rules:
  • $A \times (B + 1) = (A \times B) + A$
  • $A \times (B - 1) = (A \times B) - A$

Core Concepts & Topics

  • Solving Multi-Step Clue Puzzles:
  • Finding a two-digit mystery number by filtering out contradictory and redundant clues from a list of properties.
  • Mental Math Strategies:
  • Using doubling and halving for factors ending in $5$, $25$, or $50$.
  • Applying nearest-multiple adjustments for factors ending in $1$ or $9$.
  • Three Multi-Digit Multiplication Formats:
  • Grid Method (Nida's Solution): Setting up a box to multiply split place values (hundreds, tens, ones) individually and summing the fields.
  • Expanded Partial Products (Kanti's & John's Solutions): Multiplying place values step-by-step and adding them in columns.
  • Standard Algorithm (Mili's Father's Method): Carrying digits over to adjacent place value columns during multiplication.
  • Real-World Word Problems:
  • Dairy Cooperative: Calculating the total milk yield from $453$ Gir cows giving $13\text{ l}$ daily ($453 \times 13 = 5,889\text{ l}$).
  • Compost Savings: Computing earnings from selling $150\text{ kg}$ compost at $\text{Rs. } 24\text{/kg}$ ($150 \times 24 = \text{Rs. } 3,600$).
  • Virat Kohli vs. Sachin Tendulkar: Adding runs to find totals.
  • Exponential Growth (The King's Reward):
  • Comparing three rewards over $7$ days:
    • Choice 1 (Double 5): $5 \times 2^7 = 640$ coins.
    • Choice 2 (Triple 3): $3 \times 3^7 = 6,561$ coins.
    • Choice 3 (Quintuple 1): $1 \times 5^7 = 78,125$ coins (highest reward).
  • Row-Column Seating Calculations:
  • Solving overall crowd sizes using relative front/behind and left/right coordinates.

Worked Examples

  • Two-Digit Clue Mystery:
  • Problem: Identify the number using: greater than 8, multiple of 9, less than 50, ones digit even, tens digit odd, not a multiple of 4 or 11.
  • Solution: Multiples of 9 below 50 are ${9, 18, 27, 36, 45}$. Filter by even ones digit and odd tens digit: leaves $18$ and $36$. Since it is not a multiple of 4, the number is $18$.
  • Doubling and Halving:
  • $25 \times 12 = 50 \times 6 = 300$
  • $16 \times 45 = 8 \times 90 = 720$
  • Nearest Multiple Adjustments:
  • $4 \times 19 = 4 \times 20 - 4 = 80 - 4 = 76$
  • $14 \times 21 = 14 \times 20 + 14 = 280 + 14 = 294$
  • Ghee Sales Revenue:
  • Question: A cooperative sells $125\text{ kg}$ ghee for $\text{Rs. } 574\text{/kg}$. Find the monthly earnings.
    Solution: $125 \times 574 = 574 \times (100 + 20 + 5) = 57,400 + 11,480 + 2,870 = \text{Rs. } 71,750$.
  • Mala's Book Purchase:
  • Question: Mala buys $18$ books sold at $3$ books for $\text{Rs. } 150$. She has $\text{Rs. } 20$ left. How much did she start with?
    Solution: Unit cost $= 150 \div 3 = \text{Rs. } 50$. Total spent $= 18 \times 50 = \text{Rs. } 900$. Started with $= 900 + 20 = \text{Rs. } 920$.
  • Women's Football Tournament Finance:
  • Question: A club sells $57$ tickets at $\text{Rs. } 115$ each and registers $3$ teams at $\text{Rs. } 1,599$ each. Ground rent is $\text{Rs. } 1,750$ and food/water costs $\text{Rs. } 1,129$. Find the net collections and expenses.
    Solution:
    • Net revenue $= (57 \times 115) + (3 \times 1599) = 6,555 + 4,797 = \text{Rs. } 11,352$.
    • Expenses $= 1,750 + 1,129 = \text{Rs. } 2,879$.
  • Ananya's Row-Column Position:
  • Question: Ananya sits on a ground. There are $12$ rows of students in front of her and $17$ behind her. There are $18$ students to her right and $22$ to her left. How many total students are on the ground?
    Solution:
    • Total rows $= 12 \text{ (front)} + 17 \text{ (behind)} + 1 \text{ (Ananya)} = 30$ rows.
    • Students per row $= 18 \text{ (right)} + 22 \text{ (left)} + 1 \text{ (Ananya)} = 41$ students.
    • Total students $= 30 \times 41 = 1,230$ students.

Practical Activities & Experiments

  • Number Grid Game: Drawing a board with cells ${100, 25, 5, 10, 2, 36, 12, 4, 3}$ and competing to represent integers from 0 to 20 using only two numbers and one operation.
  • Compost Audit: Charting organic kitchen waste output at home over a month and calculating the potential weight and financial value of compost generated in a year.
  • King's Gold Simulation: Drawing a chart with three columns to calculate daily gold totals for the three ministers to visualize exponential growth.
  • Seating Layout Grid: Using graph paper to model and count row-column coordinate problems.
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