Pattern and Function - Introduction to Variables and Simple Equations

Number Patterns and Sequences: A pattern is a sequence of numbers or shapes that follows a specific rule. For example, in the sequence $3, 6, 9, 12, ...$, the rule is to add $3$ to the previous term. Visually, you can imagine this as a series of towers made of blocks, where each new tower is exactly $3$ blocks taller than the one before it.

Variables as Unknowns: A variable is a letter or symbol (like $x$, $n$, or $y$) used to represent a number we do not know yet. Think of a variable as a 'mystery box' or a 'placeholder' in a mathematical sentence. For instance, in $x + 5 = 10$, the $x$ is the mystery value that makes the statement true.

Algebraic Expressions: An expression is a mathematical phrase that combines numbers, variables, and operation symbols (like $+$, $-$, $\\times$, $\\div$). Unlike an equation, an expression does not have an equals sign. For example, $n + 4$ is an expression. Visually, this is like saying 'take any number of marbles and add $4$ more'.

Simple Equations: An equation is a mathematical sentence which states that two expressions are equal using the $=$ sign. Visualize a balance scale: the left side of the equation must weigh exactly the same as the right side. If the scale is balanced, the relationship is true.

The Balance Method: To solve an equation, we must keep the balance scale level. Whatever operation we perform on one side of the equation, we must perform the exact same operation on the other side. For example, if we subtract $5$ from the left side, we must also subtract $5$ from the right side to maintain equality.

Function Machines and Rules: A function is a rule that describes the relationship between an input and an output. Imagine a machine where you drop a number ($Input$) into the top, the machine follows a 'Rule' (like $\\times 2$), and a new number ($Output$) comes out the bottom. If the rule is $Output = Input + 10$, an input of $5$ results in an output of $15$.

Inverse Operations: Inverse operations are 'opposite' operations that undo each other. Addition is the inverse of subtraction ($+$ and $-$), and multiplication is the inverse of division ($\\times$ and $\\div$). We use these to isolate a variable (get it by itself) to solve equations.