Algebra - Expressions with Variables

Introduction to Variables: A variable is a letter or symbol that represents an unknown quantity whose value can change. In Grade 6, we often use $x, y, z, l,$ or $n$. Visually, imagine a variable as a 'mystery box' or an empty container where you can place different numbers at different times.

Constants: A constant is a value that never changes, such as $5, -12, \frac{3}{4},$ or $100$. Unlike variables, which are like shifting clouds, constants are like solid rocks that remain the same size and shape regardless of the situation.

Algebraic Expressions: An expression is a mathematical phrase created by combining variables and constants using operations like addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$). For example, $3x + 5$ is an expression. Visualizing an expression is like looking at a recipe where some ingredients are fixed (constants) and some can be adjusted (variables).

Terms and Coefficients: In an expression like $4y + 7$, the parts separated by $+$ or $-$ signs are called 'terms'. Here, $4y$ and $7$ are terms. In the term $4y$, the number $4$ is called the numerical coefficient of the variable $y$. You can visualize this as a tree diagram: the whole expression is the trunk, terms are the main branches, and coefficients/variables are the smaller twigs.

Forming Expressions from Phrases: Word problems are translated into algebraic expressions by identifying keywords. For instance, '8 more than $p$' becomes $p + 8$, and 'the product of $x$ and 10' becomes $10x$. Think of this as translating English sentences into a shorthand mathematical code.

Variables in Geometry Rules: Variables allow us to write general rules for shapes. If $s$ represents the side of a square, the perimeter is $s + s + s + s = 4s$. Visually, if you look at a square with four sides labeled $s$, the variable $s$ acts as a placeholder for any side length you choose.

The Distributive Property: This property shows how multiplication relates to addition, written as $a(b + c) = ab + ac$. Visualizing this involves an area model: a large rectangle with width $a$ and length divided into two parts $b$ and $c$. The total area ($a(b+c)$) is the sum of the two smaller rectangles ($ab$ and $ac$).