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📖 Mathematics — Part 2

Std 8
Curriculum Book Guide
1 2 3 4 5 6 7
Chapters
1 Fractions in Disguise 2 The Baudhayana-Pythagoras Theorem 3 Proportional Reasoning - II 4 Exploring Some Geometric Themes 5 Tales by Dots and Lines 6 Algebra Play 7 Area
6
Chapter 6
Skill: 30%

Algebra Play

Algebra Play

Chapter at a Glance

This chapter introduces the playful side of algebra, using variables, expressions, and linear equations to deconstruct and explain mathematical puzzles, magic tricks, number patterns, and word problems. Students explore "Think of a Number" games, calendar grids, digit-rearrangement optimizations, and divisibility properties. By converting verbal instructions and grid patterns into algebraic expressions, they learn that algebra is not just a tool for finding unknowns, but also a method for explaining and verifying numerical patterns.

Key Definitions & Terminology

  • Variable (Letter-Number): A letter (like $x$, $y$, or $a$) used to represent an unknown number in expressions and equations.
  • Algebraic Expression: A mathematical phrase containing numbers, variables, and operation symbols (e.g., $2x + 4$).
  • Linear Equation: A mathematical statement asserting the equality of two algebraic expressions, where the variables are raised to the first power (e.g., $4a + 16 = 36$).
  • Number Pyramid: A grid arrangement of numbers where each block is the sum of the two blocks directly beneath it.
  • Virahāṅka-Fibonacci Sequence: A sequence of numbers starting with $1, 2, 3, 5, 8, \dots$, where each subsequent number is the sum of the two preceding numbers ($F_n = F_{n-1} + F_{n-2}$).
  • Multiplicand: The number that is to be multiplied by another (the multiplier).

Formulas, Rules & Properties

  • The "Think of a Number" Identity:
  • Verbal steps: (1) Think of $x$, (2) double it ($2x$), (3) add $4$ ($2x+4$), (4) divide by $2$ ($x+2$), (5) subtract the original number ($x$).
  • Algebraic identity:
    $$\frac{2x + 4}{2} - x = (x + 2) - x = 2$$
  • Calendar Grid Sum Rule:
  • For any $2 \times 2$ grid on a standard calendar with top-left date $a$:
    $$\text{Sum } (S) = a + (a + 1) + (a + 7) + (a + 8) = 4a + 16$$
  • Maximum Product Rule (3 Digits):
  • To form the largest possible product of the form $\text{Tens-Ones} \times \text{Multiplier}$ using three digits $p < q < r$ once:
    $$\text{Max Product} = (10q + p) \cdot r$$
    where $r$ (the largest digit) is the multiplier, and $qp$ (the other two digits in decreasing order) forms the multiplicand.
  • Divisibility Properties:
  • Reversed 2-Digit Difference: $(10a + b) - (10b + a) = 9(a - b)$, which is always divisible by $9$.
  • Reversed 2-Digit Sum: $(10a + b) + (10b + a) = 11(a + b)$, which is always divisible by $11$.
  • Cycled 3-Digit Sum: $(100a + 10b + c) + (100b + 10c + a) + (100c + 10a + b) = 111(a + b + c) = 37 \cdot 3(a+b+c)$, which is always divisible by $37$ and $3$.
  • Repeated 3-Digit Number: $abcabc = 1001 \cdot abc = 7 \cdot 11 \cdot 13 \cdot abc$, which is always divisible by $7, 11,$ and $13$.

Core Concepts & Topics

  • Algebra as a Proof Mechanism:
  • Verbal tricks are shown to be general rules through algebraic variables. Since the variable $x$ representing the starting number cancels out in the equation, the trick is guaranteed to work for any real number.
  • Binomial Coefficients in Number Pyramids:
  • Pyramids display binomial expansion patterns in their coefficients:
    • 3-row base ${a, b, c} \rightarrow$ top value is $a + 2b + c$.
    • 4-row base ${a, b, c, d} \rightarrow$ top value is $a + 3b + 3c + d$.
    • 5-row base ${a, b, c, d, e} \rightarrow$ top value is $a + 4b + 6c + 4d + e$.
  • Fibonacci Number Pyramids:
  • If the bottom row of an $n$-row pyramid contains the first $n$ numbers of the Virahāṅka-Fibonacci sequence, every block in the pyramid will be a Fibonacci number, and the top block will be $F_{2n-1}$.
  • Work-Rate and Financial Modeling:
  • Business margins are modeled using profit equations:
    $$\text{Profit} = \text{Revenue} - \text{Expenses} = (\text{Selling Price} \cdot \text{Quantity}) - (\text{Fixed Cost} + \text{Unit Cost} \cdot \text{Quantity})$$

Worked Examples

  • The Birthday Date Trick:
  • Problem: Shubham asks Mukta to think of a date, multiply the month $M$ by $5$, add $6$, multiply by $4$, add $9$, multiply by $5$, add the day $D$, and report the result. Mukta reports $1269$. Find her birthday.
  • Solution: The algebraic equation for the final answer is $100M + D + 165$.
    $$100M + D + 165 = 1269 \rightarrow 100M + D = 1104$$
    • Since $D \le 31$, $D = 04$ and $M = 11$.
    • Answer: November 4th.
  • Pyramid Unknown Solver (Page 138):
  • Problem: In a 3-row pyramid, the bottom row has $12, c, 8$. The middle row has $a, b$. The top row has $60$. Find $a, b,$ and $c$.
  • Solution: The system of equations is:
    $$a = 12 + c, \quad b = c + 8, \quad a + b = 60$$
    $$(12 + c) + (c + 8) = 60 \rightarrow 2c + 20 = 60 \rightarrow 2c = 40 \rightarrow c = 20$$
    • Thus, $a = 12 + 20 = 32$, and $b = 20 + 8 = 28$.
    • Answer: $a=32, b=28, c=20$.
  • Maximizing a Product (Figure it Out Q2, Page 144):
  • Problem: Arrange the digits $3, 5,$ and $9$ to form the largest product of the type $\text{Tens-Ones} \times \text{Multiplier}$.
  • Solution: The largest digit $9$ is the multiplier. The remaining digits $5$ and $3$ are written in descending order to form the multiplicand $53$.
    $$\text{Product} = 53 \times 9 = 477$$
  • Heads and Legs Puzzle (Figure it Out Q6, Page 145):
  • Problem: A farm has horses and hens. The total heads is $55$ and total legs is $150$. Find the count of each.
  • Solution 1 (Non-algebraic): If all $55$ animals were hens, there would be $55 \times 2 = 110$ legs.
    • The difference in legs is $150 - 110 = 40$.
    • Since a horse has $2$ more legs than a hen, the number of horses is $40 \div 2 = 20$.
    • The number of hens is $55 - 20 = 35$.
  • Solution 2 (Algebraic): Let $h$ be horses and $k$ be hens.
    $$h + k = 55 \rightarrow k = 55 - h$$
    $$4h + 2k = 150 \rightarrow 4h + 2(55 - h) = 150 \rightarrow 2h + 110 = 150 \rightarrow 2h = 40 \rightarrow h = 20$$
    • Hens $k = 55 - 20 = 35$.
  • Genie Coin Doubling Riddle (Karim's Dream, Page 147):
  • Problem: Karim double his coins each time he walks around a tree, but must pay the genie $8$ coins after each round. After three rounds, he is left with exactly $8$ coins (before paying the final $8$ coins). How many coins did he start with?
  • Solution: Let the initial coins be $C$.
    • After Round 1: $2C - 8$.
    • After Round 2: $2(2C - 8) - 8 = 4C - 24$.
    • After Round 3 (doubled): $2(4C - 24) = 8C - 48$.
    • Since this equals $8$:
      $$8C - 48 = 8 \rightarrow 8C = 56 \rightarrow C = 7\text{ coins}$$
    • Answer: Karim started with $7$ coins.

Practical Activities & Experiments

  • Calendar Magic Demonstration: Create a calendar page for the current month. Have a family member choose a $2 \times 2$ grid of dates and tell you their sum. Use the formula $a = \frac{\text{Sum} - 16}{4}$ to mentally find the top-left date, and then name all four dates in the grid to explain the algebra behind the trick.
  • Animals Legs Simulation: Cut out $55$ circles representing animal bodies. Use paper clips as legs (attaching $2$ for hens and $4$ for horses) to match a total of $150$ legs. Rearrange the clips to verify the algebraic solution of $20$ horses and $35$ hens.
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