Algebra - Use of Variables in Common Rules

A variable is a literal number or a letter like $x, y, z, l, m$ or $n$ that represents a value that is not fixed. It is used in rules and formulas to represent any number in a set of possibilities.

Perimeter of a Square: In geometry, a square has four equal sides. If we represent the length of one side with the variable $s$, the rule for the perimeter $P$ is found by adding the four sides ($s + s + s + s$), which simplifies to $P = 4s$. Visually, this represents the total length of the outer boundary of the square.

Perimeter of a Rectangle: A rectangle has two equal lengths ($l$) and two equal breadths ($b$). To find the total boundary length, we add all sides: $l + b + l + b$. This is expressed as $2l + 2b$ or $2(l + b)$. You can visualize this as walking around a rectangular field and covering each dimension twice.

Commutative Property of Addition: This arithmetic rule states that the order in which two numbers are added does not change the result. Using variables $a$ and $b$ to represent any two numbers, we write the rule as $a + b = b + a$. For example, $5 + 10$ gives the same result as $10 + 5$.

Commutative Property of Multiplication: Similar to addition, the product of two numbers remains the same regardless of their order. Using variables $a$ and $b$, the rule is expressed as $a \times b = b \times a$. This can be visualized as an array of dots; whether you look at it as $a$ rows of $b$ or $b$ rows of $a$, the total count remains the same.

Distributive Property: This rule explains how multiplication is distributed over addition. If we have three numbers $a, b,$ and $c$, the rule is $a \times (b + c) = (a \times b) + (a \times c)$. Imagine a large rectangle with width $a$ and length $(b + c)$; its total area is equal to the sum of the areas of two smaller rectangles ($a \times b$ and $a \times c$).

Pattern Rules using Variables: Variables are used to generalize patterns. For instance, if forming one 'L' shape requires 2 matchsticks, two 'L' shapes require 4, and three 'L' shapes require 6, we can generalize that for $n$ shapes, the number of sticks required is $2n$.