Algebra - Introduction to Algebraic Expressions

Variables and Constants: A variable is a symbol, usually a letter like $x$, $y$, or $n$, that represents an unknown number or a value that can change. A constant is a fixed number that does not change, such as $5$, $-10$, or $3.5$. Visually, you can imagine a variable as an empty box where different numbers can be placed, while a constant is a solid, unchangeable block.

Terms, Coefficients, and Operators: An algebraic expression is made of 'terms' separated by plus or minus signs. In the term $7x$, $7$ is the 'coefficient' (the numerical factor) and $x$ is the variable. If a term is just a number like $+12$, it is a constant term. Visually, think of an expression like $4x - 3y + 5$ as a train where each carriage is a separate term ($4x$, $-3y$, and $+5$).

Like and Unlike Terms: Like terms are terms that have the exact same variables raised to the same powers, such as $3a$ and $10a$. Unlike terms have different variables or different exponents, such as $2x$ and $2x^2$. Visually, you can group like terms by drawing circles around all '$x$' terms and squares around all '$y$' terms to see which ones can be combined.

Simplifying Expressions: To simplify an expression, you combine like terms by adding or subtracting their coefficients while keeping the variable part the same. For example, $5x + 2x$ becomes $7x$. Visually, this is like putting all the 'apple' blocks in one pile and all the 'orange' blocks in another.

Evaluating Expressions: This is the process of finding the numerical value of an expression by substituting specific numbers for the variables. For example, to evaluate $x + 5$ when $x = 3$, you replace $x$ with $3$ to get $3 + 5 = 8$. Visually, think of the variable as a placeholder or a 'window' where you slot in a specific number to see the final result.

The Distributive Property: This property allows you to multiply a single term by every term inside a set of parentheses, written as $a(b + c) = ab + ac$. Visually, imagine an arrow drawn from the number outside the bracket to each individual term inside, showing that the multiplier is shared across all components within the group.

Algebraic Translation: This involves converting word problems into math symbols. For instance, 'five more than a number' translates to $n + 5$, and 'twice a number' translates to $2n$. Visually, you can create a 'translation table' where words like 'sum' or 'increased by' map to the $+$ sign, and 'product' maps to the $\times$ sign.