Expressions Using Letter-Numbers

Chapter at a Glance

This chapter introduces the transition from arithmetic to algebra, explaining how variables (referred to as letter-numbers) are used to represent unknown quantities and generalize mathematical patterns. By examining real-world situations like age differences (Aftab and Shabnam), business transactions (coconuts and jaggery costs), and geometric perimeters, the chapter demonstrates how formulas are constructed. It covers substituting numerical values into algebraic expressions, simplifying expressions by combining like terms, removing brackets, and using algebra to prove patterns (such as calendar puzzles and matchstick patterns).

Key Definitions & Terminology

• Letter-Number: A symbol or letter (such as $a, s, x$) used to denote a variable or unknown number in mathematical expressions.
• Algebraic Expression: A mathematical expression that combines numbers, operators, and at least one letter-number (e.g., $a + 3$, $2l + 2b$).
• Formula: An algebraic equation that expresses a general relationship between quantities (e.g., $p = 2l + 2b$).
• Like Terms: Terms within an algebraic expression that contain the exact same letter-numbers (e.g., $5c, 3c, 10c$ or $12n, -4n$). Like terms can be combined into a single simplified term.
• Unlike Terms: Terms that contain different letter-numbers (e.g., $18c$ and $11d$). Unlike terms cannot be combined.
• Simplified Form: An equivalent expression that has been reduced to contain the minimum possible number of terms.

Formulas, Rules & Properties

• Omission of the Multiplication Symbol:
• In algebra, the multiplication sign is omitted between numbers and letters, or between letters:
  • $4 \times n = 4n$
  • $a \times b = ab$
• The number (coefficient) is always written first, followed by the letters.
• Perimeters of Regular Polygons:
• Equilateral Triangle (sidelength $a$): $P = 3a$
• Square (sidelength $q$): $P = 4q$
• Regular Pentagon (sidelength $a$): $P = 5a$
• Regular Hexagon (sidelength $a$): $P = 6a$
• Rectangle (length $l$, breadth $b$): $P = 2l + 2b$
• Distributive Property Expansion:
• $a(x + y) = ax + ay$
• $a(x - y) = ax - ay$
• $(ax + by) - (cx + dy) = (a - c)x + (b - d)y$
• Endless Calendar Grid Positioning:
• In a grid with 4 columns, the number in row $r$ and column $c$ is:
  $$\text{Value} = 4(r - 1) + c$$
• Calendar diagonal sum rule:
• For a 2x2 calendar grid with top-left date $a$, the diagonal sums are always equal to $2a + 8$.
• Rope Cutting Fold Rule:
• Cutting a rope that is folded $r$ times once results in exactly $r + 2$ pieces.

Core Concepts & Topics

• Algebraic Representations of Relationships:
• Writing equations from word descriptions (e.g., Shabnam is 3 years older than Aftab $\implies s = a + 3$; conversely Aftab's age is $a = s - 3$).
• Substitution and Value Evaluation:
• Evaluating expressions by replacing variables with numerical values.
• Mending Mistakes: Identifying errors in substitutions, such as writing $3d = 36$ instead of $18$ when $d = 6$, or incorrectly adding unlike terms ($3a + 2b = 5$).
• Visualizing the Distributive Property:
• Using area models of rectangles to show that $4v + 3v = 7v$ and $12n - 4n = 8n$.
• Like vs. Unlike Terms:
• Grouping terms with identical variables to simplify expressions. For example, $5c + 3c + 10c$ simplifies to $18c$, while $18c + 11d$ cannot be simplified.
• Algebraic Modeling & Proofs:
• Calendar Puzzles: Using $a$ as the top-left corner of a 2x2 grid to show diagonal sum equivalence: $a + (a + 8) = 2a + 8$ and $(a + 1) + (a + 7) = 2a + 8$.
• Cross Shapes on Calendars: Proving that the sum of a calendar cross with center $a$ is $(a - 7) + (a - 1) + a + (a + 1) + (a + 7) = 5a$.
• Identifying repeating pattern positions:
• Traffic Signals: Sequences of Red ($4n - 3$), Green ($4n - 1$), and Yellow ($2n$).
• Row-Column coordinates: Dividing numbers by 4 to determine row and column coordinates in grid lists.

Worked Examples

• Ketaki's Laddu Business Cost:
• Problem: Coconuts cost ₹35 each, jaggery costs ₹60/kg. Find the formula for cost and calculate for 8 coconuts and 9 kg jaggery.
• Formula: $\text{Cost} = 35c + 60j$.
• Calculation: $35(8) + 60(9) = 280 + 540 = ₹820$.
• Krithika's Currency Notes:
• Problem: Find the total value of $x$ notes of ₹100, $y$ notes of ₹20, and $z$ notes of ₹5.
• Formula: $100x + 20y + 5z$.
• Example (8 of ₹100, 4 of ₹20, $z$ of ₹5): $8(100) + 4(20) + 5(z) = 880 + 5z$.
• Flour Mill grinding time:
• Problem: Mill takes 10s to start. Each kg of grain takes 8s to grind. Write the expression for $y$ kg grain.
• Expression: $10 + 8y\text{ seconds}$.
• Algebraic translations of statements:
• 5 more than a number: $d + 5$.
• 4 less than a number: $d - 4$.
• 2 less than 13 times a number: $13d - 2$.
• 13 less than 2 times a number: $2d - 13$.
• Tinkering Expressions Bracket mistakes (Mend the Mistake):
• $(4x + 3y) - (3x + 4y) \rightarrow 4x + 3y - 3x - 4y \rightarrow x - y$ (Incorrectly solved as $x + y$).
• $5 - (2 - 6z) \rightarrow 5 - 2 + 6z \rightarrow 3 + 6z$ (Incorrectly solved as $3 - 6z$).
• $4(2r + 3s + 5) \rightarrow 8r + 12s + 20$ (Incorrectly solved as $-20 - 8r - 12s$).
• Simplifying Grouped Expressions:
• Rental Furniture: Rents $x$ chairs (₹40 start, ₹6 back) and $y$ tables (₹75 start, ₹10 back).
• Expression: $(40x + 75y) - (6x + 10y) = 40x - 6x + 75y - 10y = 34x + 65y$.
• Radha's Cycling Practice:
• Problem: Radha cycles 5 km daily in week 1. She increases her daily run by $z$ km each subsequent week. Find total km cycled in 3 weeks.
• Expression: $7(5) + 7(5 + z) + 7(5 + 2z) = 35 + 35 + 7z + 35 + 14z = 105 + 21z\text{ km}$.
• Yahapur to Vahapur train:
• Problem: A train has 4 travel segments of $t$ minutes each and stops for 2 minutes at 3 intermediate stations. Find total travel time.
• Expression: $4t + 6\text{ minutes}$. (For $t=4\text{ mins}$, total travel time $= 4(4) + 6 = 22\text{ minutes}$).
• rope cutting folds:
• Problem: Find the pieces if a rope is folded 10 times and cut.
• Solution: pieces $= r + 2 = 10 + 2 = 12\text{ pieces}$.
• Matchstick Square pattern:
• Problem: Find the matchsticks required for $w$ squares in a linear row.
• Formula: $4 + 3(w - 1) = 3w + 1$. For $w=10$ squares, $30 + 1 = 31$ matchsticks.
• Traffic Signal Positions:
• Problem: Traffic signal cycles through Red ($4n - 3$), Green ($4n - 1$), and Yellow ($2n$). Identify colors at positions 90, 190, and 343.
• Solution:
  • Position 90: Even $\implies$ Yellow.
  • Position 190: Even $\implies$ Yellow.
  • Position 343: Odd and of form $4(86) - 1 \implies$ Green.
• Step Square pattern:
• Problem: In a grid step pattern of squares, Step 1 has 5 squares, Step 2 has 9, Step 3 has 13. Find Step 4, 10, 50, and general formulas for squares and vertices.
• Solution:
  • Squares formula: $5 + 4(n-1) = 4n + 1$. Step 4: 17, Step 10: 41, Step 50: 201 squares.
  • Vertices formula: $16n + 4$.
• Endless Grid row-column indexing:
• Problem: Find row and column for numbers 124, 147, 201 in a 4-column grid.
• Solution: Use $N = 4(r - 1) + c$.
  • 124: $124 \div 4 = 31$ remainder $0 \implies r = 31, c = 4$.
  • 147: $147 \div 4 = 36$ remainder $3 \implies r = 37, c = 3$.
  • 201: $201 \div 4 = 50$ remainder $1 \implies r = 51, c = 1$.

Practical Activities & Experiments

• Matchstick L-Pattern construction: Laying out matchsticks to build L shapes, finding linear equations to model materials.
• Calendar Grid sums check: Selecting various 2x2 grids on any calendar, calculating diagonal sums, and checking if they always equal $2a+8$.
• Matchstick Square row builds: Creating squares in rows, checking the transition of shared walls to understand why additional squares only need 3 matchsticks.
• Rope fold cuts: Folding cords or strings and cutting them to physically verify the $r+2$ parts formula.