How to Teach Input-Output Tables (Function Tables)
A number goes in, a different number comes out. What happened? Input-output tables present this question, and answering it is one of the most important skills in the transition from arithmetic to algebra.
The concept: a rule machine
Think of a machine. You put a number in (input). The machine applies a rule. A new number comes out (output).
| Input | Output |
|---|---|
| 1 | 5 |
| 2 | 7 |
| 3 | 9 |
| 4 | 11 |
"What is the rule?" Look at the pattern: each output is the input × 2 + 3.
- 1 × 2 + 3 = 5 ✓
- 2 × 2 + 3 = 7 ✓
- 3 × 2 + 3 = 9 ✓
The rule is: output = 2 × input + 3 (or y = 2x + 3).
Key Insight: Finding the rule in an input-output table is exactly what algebra is about: expressing a relationship between two quantities. When your child can look at a table and write the rule, they are thinking algebraically — even if they have never written a formal equation.
How to find the rule
Step 1: Look at how each output relates to its input. Try simple operations:
- Is the output always the input plus something? (addition rule)
- Is the output always the input times something? (multiplication rule)
- Is it a combination? (two-step rule)
Step 2: Check the pattern difference. If the outputs increase by a constant amount, the rule involves multiplication by that constant rate.
Outputs: 5, 7, 9, 11 → increasing by 2 each time → the rule multiplies input by 2 (plus something).
Step 3: Test the rule on every row. A rule that works for one row might not work for others.
Types of rules
One-step rules:
| Input | Output | Rule |
|---|---|---|
| 2 | 6 | × 3 |
| 3 | 9 | × 3 |
| 5 | 15 | × 3 |
Two-step rules:
| Input | Output | Rule |
|---|---|---|
| 1 | 4 | × 3 + 1 |
| 2 | 7 | × 3 + 1 |
| 3 | 10 | × 3 + 1 |
Connection to graphing and functions
Every input-output table can be:
- Written as a rule: y = 3x + 1
- Graphed as points on the coordinate plane: (1,4), (2,7), (3,10)
- Described as a function: f(x) = 3x + 1
Input-output tables are the concrete entry point to all of these abstract ideas.
Common mistakes
Only checking one row: They find a rule that works for input 2 → output 6 (times 3!) but do not verify it works for other rows.
Missing two-step rules: They try "plus 3" and "times 2" separately but do not consider "times 2 plus 1." If simple rules do not work, try combinations.
Confusing input and output: They apply the rule backwards. Input goes in, rule transforms it, output comes out. If the rule is "times 3," input 4 gives output 12, not the other way around.
Input-output tables are the concrete version of algebraic functions. Find the rule by looking for patterns in how inputs become outputs, verify across all rows, and express the rule first in words, then as an equation. When your child can discover and express rules from tables, they are ready for formal algebra.
If you want a system that builds algebraic thinking through pattern recognition, input-output tables, and progressive equation work — that is what Lumastery does.