Function Machines — Input & Output Rules (Grades 5–6)
A function machine takes an input number, applies a rule, and gives an output — like a rule "multiply by 3, then add 1."
Understanding function machines — input/output rules
Put in 2, get out 7. The same rule always gives the same output for a given input. Reading a table of inputs and outputs, you can work out the hidden rule by asking "what was done to every input to get its output?"
Seeing it in action
Worked example
A machine uses a steady "multiply then add" rule. It turns 2→7, 3→10, and 4→13. What is the rule, and what does 5 give?
Each output is input ×3 + 1 (2·3+1 = 7, 3·3+1 = 10, 4·3+1 = 13 — all three confirm it).
So 5 gives 5·3 + 1 = 16.
Apply ×3 + 1 to move from input to output.
A common mistake, and the fix
Reading the rule from one pair only.
2 → 7 fits “add 5” as well as “multiply by 3, then add 1”, so a rule found from one pair can be wrong. Test the rule on every pair: “add 5” turns 3 into 8, not 10, so it fails, while “multiply by 3, then add 1” gives 3 → 10 and 4 → 13.
Try a few
Rule "×2": input 6
Rule "+4": input 9
A "multiply then add" machine gives 1→5, 2→8, 3→11. Rule?
and 4 → 14.
Function Foundry
A calm pre-algebra game for function machines, inverse operations, and solving for an unknown.
Try it in Numeris.
A short lesson, then practice with feedback on every answer.
Put the reasoning on paper.
Try six fraction, angle, perimeter and area questions with explained answers, for grades 4–6.
Free printable · 3 pagesText version (screen-reader friendly): Draw it. Explain it. Math sample
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