Introduction:

A pattern is a situation where values change in a predictable way. The goal is to find the rule that generates the pattern and use it to answer questions without guessing. You will see patterns in many forms: lists of numbers (sequences), tables, graphs, and equations. Being able to move among these forms is a core math skill. It lets you check your work and understand the relationship itself, not just the numbers.

A linear pattern changes by the same amount each step. We call that constant change the rate of change or common difference. For example, "start at 7 and add 5 each step" gives 7, 12, 17, 22, 27, … The "+5" is the rate of change. Linear patterns model situations like hourly pay (each hour adds the same money), taxi fare with a per‑kilometre rate, or saving a fixed amount each week. Graphically, linear patterns make straight lines. In formulas we often write y = m·x + b, where m is the rate (slope) and b is the start value (y‑intercept).

An exponential pattern does not add the same amount; instead it multiplies by the same factor each step. For example, "start at 200 and multiply by 1.1 each step" gives 200, 220, 242, 266.2, … This models compound interest, bacterial growth, radioactive decay, or viral shares online. Exponential graphs curve upward (growth) or downward (decay). In formulas we often write T(n) = a · r⁽ⁿ⁻¹⁾, where a is the starting value and r is the constant factor (growth or decay rate).

Not every pattern is linear or exponential. If the differences themselves change in a regular way (for example +3, +5, +7, +9 …), we may have quadratic growth — associated with parabolas — where the second differences are constant. For now, recognize "constant difference → linear", "constant ratio → exponential", and "constant second difference → quadratic."

Concrete Representations (with tables)

Linear example — "start at 2, add 3 each time" (rule: y = 3x − 1)

• Difference between values: +3 each time (constant)
• Equation check: y = 3(4) − 1 = 11 (check)
• Graph: straight line rising by 3 per step

Exponential example — "start at 500, multiply by 1.2 each time" (rule: T(n) = 500 · 1.2⁽ⁿ⁻¹⁾)

• Ratio between values: ×1.2 each time (constant factor)
• Equation check: T(4) = 500 · 1.2³ = 864 (check)
• Graph: curve that gets steeper as n increases

Quadratic "feel" — "differences increase by 2"

Values: 7, 12, 19, 28, 39, 52, …

Differences: +5, +7, +9, +11, +13 (second differences +2) → not linear, not exponential; this suggests quadratic behavior.