Introduction:

In Lesson 1.1, you learned to recognize patterns and figure out their rules. Now we go further: representing the same pattern in multiple ways. The four main representations are words (description), tables (organized data), graphs (visual plot), and equations (symbolic formula). Being able to move smoothly between these forms is called representation fluency. It helps you check your work, see connections, and understand relationships on a deeper level.

When you convert from one representation to another, you're testing whether your understanding is solid. For example, if you build a table from an equation and then graph those points, the graph should match what the equation predicts. If it doesn't, you know something went wrong. This back‑and‑forth checking is how mathematicians verify their work.

Let's break down how to move between representations. Words → Table: Describe the rule in plain language, then compute values step by step. Table → Graph: Plot each ordered pair (n, T) on a coordinate plane. If the pattern is linear, the points form a straight line. Graph → Equation: For linear patterns, find the slope m (rise over run) and the y‑intercept b (value when n = 0), then write T(n) = m·n + b. Equation → Words: Translate the symbols back into plain English — "Start at b and add m each step."

Real‑world connection: Engineers represent the same bridge design as blueprints (visual), load calculations (equations), and material lists (tables). Being fluent in all forms ensures accuracy and communication across teams.

Concrete Representations (with tables)

Linear representation — "start at 3, add 4 each time"

Words: Start at 3 and add 4 each time.

Step n 1 2 3 4 5 6 7
T(n) 3 7 11 15 19 23 27

How to get slope from the table: Pick two rows, say (2,7) and (5,19):

m = (19 − 7)⁄(5 − 2) = 12⁄3 = 4

Exponential representation — "start at 250, increase by 12% each step"

Words: Start at 250 and increase by 12% each step.

Step n 1 2 3 4 5 6
T(n) 250 280 313.6 351.23 393.38 440.59