Pattern and Function - Expressing Patterns as Rules

A pattern is a sequence of numbers or shapes that follows a specific logical rule. For example, in a growing pattern of triangles where the first step has $3$ sides and the second has $6$, you can visualize the pattern as adding a new triangle at every step.

An Input-Output table, often called a T-chart, is used to organize the relationship between two sets of numbers. The 'Input' (usually $x$) represents the position or step number, while the 'Output' (usually $y$) represents the value at that position. Visually, this is shown as two vertical columns labeled $x$ and $y$.

The 'Common Difference' is the constant amount added or subtracted to get from one term to the next. In a sequence like $5, 10, 15, 20$, the common difference is $+5$. On a number line, this looks like equal-sized jumps between points.

A 'Position-to-Term' rule is a mathematical formula that allows you to calculate the output directly from its position number without knowing the previous term. This is much more efficient than a 'Term-to-Term' rule when trying to find the $100^{th}$ term.

Geometric patterns use shapes to represent functions. For example, a pattern of squares where each new step adds one square to each side can be visualized as an expanding cross or 'L' shape. The rule describes how the total number of squares relates to the step number.

The 'Zero Term' or 'Adjustment' is the value the output would have if the input were zero. In the rule $y = 2x + 1$, the $+1$ is the adjustment. Visually, on a graph, this corresponds to where the line would cross the vertical $y$-axis.

Function machines are a visual way to represent rules. Imagine a box where an input number goes in, a specific operation happens (like $\times 3$ then $+ 2$), and a resulting output number comes out. The 'Rule' is the logic inside the machine.