How to explain input and output to a child
Imagine a magic machine that you put a number into, and it always changes the number the same way. The number you put in is the input. The number that comes out is the output. The secret rule is what the machine does to every input to make the output.
Worked examples of input and output
Example 1 (upper elementary): single-operation input-output rule
Suppose a function machine uses this rule: "add 5." That means every output is the input plus 5.
- Input 2: apply the rule 2 + 5 = 7, so the output is 7.
- Input 4: 4 + 5 = 9, so the output is 9.
- Input 10: 10 + 5 = 15, so the output is 15.
An input-output table for this rule might look like:
Inputs: 2, 4, 10
Outputs: 7, 9, 15
The pattern is that each output is 5 more than its input.
Example 2 (middle/high school): using an input-output table and function notation
Suppose a function is defined by the rule "multiply by 3, then subtract 2." In function notation, this can be written as f(x) = 3x − 2, where x is the input and f(x) is the output.
- For input x = 1: f(1) = 3(1) − 2 = 3 − 2 = 1, so the output is 1.
- For input x = 3: f(3) = 3(3) − 2 = 9 − 2 = 7, so the output is 7.
- For input x = 5: f(5) = 3(5) − 2 = 15 − 2 = 13, so the output is 13.
The input-output table is:
Inputs x: 1, 3, 5
Outputs f(x): 1, 7, 13
If these ordered pairs (1, 1), (3, 7), and (5, 13) are graphed on a coordinate plane, they line up in a straight line, so this is a linear function.
Going deeper with input and output
In later grades, input and output are used to describe functions, which are relationships where each input has exactly one output. Students move from simple rules like "add 4" to more complex rules that involve multiplication, exponents, or combinations of operations.
Input-output tables help students see patterns and decide what type of function might fit, such as a linear function or an exponential function. From a table, students can write a function rule, use function notation like f(x), and then graph ordered pairs to visualize how the inputs and outputs are related.
As work becomes more advanced, input-output pairs can describe sequences, such as an arithmetic sequence, or more complex functions built from a polynomial. The core idea stays the same: a consistent rule connects each input to its output.
Common mistakes with input and output
- Mixing up the rule: A student might see inputs 2, 4, 6 and outputs 5, 7, 9 and say the rule is "add 3 to the input each time" instead of "add 3 to get from input to output." The correct rule is that each output equals the input plus 3.
- Using a different rule for each pair: A student might explain 2 → 6 as "add 4" and 3 → 9 as "add 6," missing that both follow "multiply by 3." The rule must be the same for every input-output pair in a function.
- Reversing input and output: In a table with 3 → 11 and 5 → 19, a student might treat 11 and 19 as inputs when writing a rule. Inputs are the starting values, and outputs are the results after applying the rule.
Where input and output show up in everyday life
Input and output ideas appear any time a number is changed in a predictable way. For example, a game might give 10 points for each level completed, so the number of levels is the input and the total points is the output. A store might charge the same price per snack, so the number of snacks is the input and the total cost is the output.