In fifth grade, students analyze two related number patterns simultaneously and identify the relationship between them. Given the rule "add 3" starting from 0 and the rule "add 6" starting from 0, the sequences are (0, 3, 6, 9, 12) and (0, 6, 12, 18, 24). Students observe that each term in the second sequence is twice the corresponding term in the first. They form ordered pairs from corresponding terms -- (0,0), (3,6), (6,12), (9,18), (12,24) -- and plot them on the coordinate plane, discovering they form a straight line. This is the earliest encounter with the idea that a relationship between two quantities can be expressed as a rule and visualized as a graph.
Generate paired sequences from two rules, record in a two-column table, then plot as ordered pairs. Ask students to describe the relationship between the paired values. Use real-world contexts: "If you earn $3 per hour and your friend earns $6 per hour, how do your total earnings compare after 1, 2, 3, 4, 5 hours?" Observe that the plotted points form a line.
You've worked with sequences before — rules like "add 3" that generate a chain of numbers (0, 3, 6, 9, ...). You've also used input-output tables to track how one quantity changes with another, and you've plotted ordered pairs on a coordinate grid. This topic brings all three together: when you run two sequences side by side and pair their corresponding terms, you're creating a relationship between two quantities that you can describe, tabulate, and graph all at once.
Start with two rules: "add 3" and "add 6," both beginning from 0. The first sequence is 0, 3, 6, 9, 12... and the second is 0, 6, 12, 18, 24... Line them up in a table — the first pair of corresponding terms is (3, 6), the second is (6, 12), the third is (9, 18). Look across each row: the second number is always exactly double the first. That doubling relationship isn't a coincidence — it comes from the fact that the second rule adds twice as much per step as the first rule does. The relationship between the outputs mirrors the relationship between the rules that generated them.
When you plot those ordered pairs on a coordinate plane — (0, 0), (3, 6), (6, 12), (9, 18), (12, 24) — they fall in a straight line. This is the geometric signature of a constant ratio between two quantities: every time the first quantity increases by a fixed amount, the second increases by a fixed amount too. A straight line through the origin (0, 0) means both sequences started at 0 with a fixed multiplicative relationship between their rules. If the second rule were "add 6" but started at a different value, the points would still be collinear — but the line wouldn't pass through the origin.
This is the earliest form of algebraic thinking: the idea that a rule connecting two quantities can be described in words ("the second is always double the first"), captured as specific pairs in a table, and seen as a geometric pattern on a graph. These three representations — rule, table, graph — all encode the same underlying relationship. When you study functions in later courses, you'll express the same idea with an equation like y = 2x, and you'll work with all three representations again in much greater depth. The foundation being built here is learning to move fluidly between them and recognizing that the pattern is the same thing no matter which form it appears in.