An input-output table shows a consistent rule applied to each input to produce an output. Given inputs of 2, 5, 8 and outputs of 6, 15, 24, the rule is "multiply by 3." Students learn to identify the rule from given pairs, complete missing values, and express the rule in words. This is early function thinking -- the idea that a rule maps each input to exactly one output -- which is foundational for algebra. Input-output tables also reinforce arithmetic fluency as students test different operations to discover the pattern.
Start with single-operation rules (add 7, multiply by 4) and advance to two-step rules (multiply by 2 then add 1). Have students create their own input-output tables for classmates to solve. Use "function machines" as a metaphor: a number goes in, the machine applies the rule, a number comes out. Practice identifying rules from completed tables, then generating tables from stated rules.
An input-output table is built around a single hidden rule. Every input goes through the same rule to produce its output — no exceptions. Your job is to discover the rule from the given pairs, and then use it to fill in the missing values. Think of it as a function machine: a number drops in the top, the machine does something to it, and a result comes out the bottom. The machine never changes its behavior.
Start by looking at the relationship between one input and its output. Try the four operations. If the input is 5 and the output is 20, ask: did we add 15? Multiply by 4? These are both possible. Test your guess on a second pair to confirm. If input 3 gives output 12, "add 15" fails (3 + 15 = 18, not 12), but "multiply by 4" works (3 × 4 = 12). Always verify your rule with at least two pairs before using it to complete the table.
Multiplicative rules are easier to miss than additive ones because students naturally look at the difference between input and output first. Develop the habit of also testing: "Is the output a fixed multiple of the input?" For a table where inputs are 2, 4, 6 and outputs are 10, 20, 30, the difference between input and output varies (8, 16, 24), but the ratio is constant (5, 5, 5) — the rule is "multiply by 5." Always check both additive and multiplicative possibilities.
Two-step rules — like "multiply by 3 then add 2" — appear when neither simple operation works alone. Input 4, output 14: 4 × 3 = 12, then 12 + 2 = 14. Input 6, output 20: 6 × 3 = 18, then 18 + 2 = 20. When a single operation doesn't fit, look for a pattern in the remainders after removing the multiplicative component. Input-output tables are your first formal encounter with the idea that a rule maps each input to exactly one output — a concept that will be called a function when you reach algebra.