Patterns and Function - Functional Relationships

A Pattern is a series of numbers, shapes, or objects that follow a specific, predictable rule. In a visual pattern, you might see a sequence like one square, then three squares, then five squares, showing a 'growing' pattern where two squares are added each time.

Functional Relationships describe how one quantity (the input) changes to become another quantity (the output) based on a consistent rule. Imagine a 'Function Machine' where you drop a number into the top, a rule like $+5$ happens inside, and a new number pops out the bottom.

The Rule is the mathematical instruction that connects the input to the output. For example, if the input is $3$ and the output is $12$, the rule could be $\times 4$ or $+9$. You must check other pairs in the pattern to confirm which rule is consistent.

An Input-Output Table (or T-Chart) is a visual tool used to organize functional relationships. It consists of two columns: the left column represents the 'Input' (often labeled $x$) and the right column represents the 'Output' (often labeled $y$). Each row shows a pair that follows the same rule.

Growing Patterns are sequences that increase in a predictable way. Visually, this might look like a staircase where the first step has $1$ block, the second has $2$ blocks, and the $n^{th}$ step has $n$ blocks. The relationship is between the 'position' and the 'number of items'.

Inverse Operations are used to find the input if you only know the output. If the function rule is $+10$, you can work backward by using the inverse operation $-10$. If the rule is $\times 2$, the inverse is $\div 2$.

Variables are symbols, like a empty box $\square$ or a letter $n$, used to represent an unknown number in a functional relationship. For example, in the rule $n + 5 = \text{Output}$, $n$ represents any input number you choose.