Patterns and Algebra - Introduction to Variables and Algebraic Expressions

Understanding Variables: A variable is a letter, such as $x$, $y$, or $n$, that represents an unknown quantity or a value that can change. Visually, you can think of a variable as a placeholder or a 'mystery box' where different numbers can be placed inside.

Algebraic Expressions: An algebraic expression is a mathematical phrase that combines numbers, variables, and operators (like $+$, $-$, $\times$, and $\div$). Unlike an equation, an expression does not have an equals sign ($=$). For example, $3x + 5$ is an expression, which can be visualized as three groups of an unknown amount plus five individual units.

Terms, Coefficients, and Constants: In an expression like $4a + 7$, $4a$ and $7$ are called terms. The number $4$ is the coefficient (the number multiplying the variable), $a$ is the variable, and $7$ is the constant (a value that remains the same). You can visualize this as a train where each car represents a separate term.

Substitution and Evaluation: Evaluating an expression means replacing the variable with a given number and performing the calculation. If $x = 4$, then the expression $2x + 3$ becomes $2(4) + 3$, which equals $11$. This is like swapping a label on a container for the actual items inside.

Simplifying Expressions by Combining Like Terms: Like terms are terms that have the same variable part. To simplify, we add or subtract their coefficients. For example, $3x + 2x = 5x$. Visually, this is similar to grouping objects: if you have 3 apples and 2 apples, you have 5 apples ($3a + 2a = 5a$), but you cannot combine 3 apples and 2 oranges ($3a + 2b$).

Translating Word Problems: Words can be translated into algebraic language. 'The sum of' implies addition ($+$), 'the difference' implies subtraction ($-$), 'the product' implies multiplication ($\times$), and 'the quotient' implies division ($\div$). For example, 'six more than a number $n$' is written as $n + 6$.

The Distributive Property: This property allows you to multiply a single term by two or more terms inside a set of parentheses. It is written as $a(b + c) = ab + ac$. Visually, this can be represented by an area model where the total area of a large rectangle is the sum of the areas of two smaller rectangles within it.