Input-Output Tables

For Elementary Students

What Is an Input-Output Table?

An input-output table is like a machine! You put a number in (input), the machine follows a secret rule, and a number comes out (output).

Think about it like this: It's like a vending machine — you put in money (input), it does something inside, and you get a snack (output)!

Your job: Figure out what the machine is doing!

Finding the Rule: Look for Patterns!

Step 1: Look at how the output changes

Input:  1 → 2 → 3 → 4
       +1  +1  +1  (input goes up by 1 each time)

Output: 5 → 7 → 9 → 11
       +2  +2  +2  (output goes up by 2 each time!)

Pattern: When input goes up by 1, output goes up by 2!

Step 2: Test the pattern

• Input 1 → multiply by 2 = 2... but output is 5! We need to add 3!
• Rule: Multiply by 2, then add 3

Check:

• Input 1: 1 × 2 + 3 = 5 ✓
• Input 2: 2 × 2 + 3 = 7 ✓
• Input 3: 3 × 2 + 3 = 9 ✓

The rule is: multiply by 2, then add 3!

Simple One-Step Rules

Sometimes the rule is just ONE operation!

Example 1: Multiply by 3

Pattern: Each output is input × 3!

Example 2: Add 5

Pattern: Each output is input + 5!

Using the Rule to Find Missing Numbers

Example: Rule is "multiply by 4"

Use the rule:

• Input 2: 2 × 4 = 8
• Input 5: 5 × 4 = 20
• Input 7: 7 × 4 = 28

For Junior High Students

What Is an Input-Output Table?

An input-output table (also called a function table) shows pairs of numbers. A rule (or function) transforms each input into an output. Your job is to figure out the rule.

Vocabulary:

• Input — the starting value (often called x)
• Output — the result value (often called y)
• Rule — the mathematical operation that connects input to output
• Function — a relationship where each input has exactly one output

Goal: Find the rule that transforms input into output.

In this case: output = 2 × input + 3

Verification: 2(1)+3=5, 2(2)+3=7, 2(3)+3=9 ✓

Finding the Rule

Step 1: Look at the Change (Finding the Coefficient)

Check how the output changes as the input increases by 1.

Example:

Pattern: When input goes up by 1, output goes up by 2

This 2 is the coefficient (the multiplier) in our rule.

Partial rule: output = 2 × input + ?

Step 2: Find the Constant (Starting Adjustment)

Use any row to find what number to add (or subtract).

Using input = 1:

• 2 × 1 = 2, but output is 5
• Difference: 5 − 2 = 3
• We need to add 3

Complete rule: output = 2 × input + 3

Step 3: Verify with All Rows

Test with every input to make sure the rule works:

• Input 1: 2(1) + 3 = 5 ✓
• Input 2: 2(2) + 3 = 7 ✓
• Input 3: 2(3) + 3 = 9 ✓
• Input 4: 2(4) + 3 = 11 ✓

Rule confirmed!

Simple Rules (One Operation)

Sometimes the rule is a single operation:

Type 1: Multiplication

Pattern: output = 3 × input

Type 2: Addition

Pattern: output = input + 6

Type 3: Subtraction

Pattern: output = input − 4

Two-Step Rules (Multiply/Divide, Then Add/Subtract)

Most rules involve two operations.

Example:

Step 1: Find the rate of change

• From input 2 to 5 (increase of 3): output goes from 9 to 18 (increase of 9)
• Rate: 9 ÷ 3 = 3

Step 2: Build the rule

Guess: output = 3 × input + ?

Step 3: Find the constant using input = 2

• 3 × 2 = 6, but output is 9
• Difference: 9 − 6 = 3

Rule: output = 3 × input + 3

Step 4: Verify with input = 10

• 3(10) + 3 = 30 + 3 = 33 ✓ Correct!

Non-Uniform Input Tables

Sometimes inputs don't increase by 1 each time. Use ratios to find the multiplier.

Example:

Step 1: Calculate change

• Input changes: 2 → 4 (+2), 4 → 6 (+2)
• Output changes: 11 → 19 (+8), 19 → 27 (+8)

Step 2: Find the multiplier

• For every +2 in input, we get +8 in output
• Rate: 8 ÷ 2 = 4

Rule so far: output = 4 × input + ?

Step 3: Find constant using input = 2

• 4 × 2 = 8, output is 11
• Constant: 11 − 8 = 3

Rule: output = 4 × input + 3

Verify: 4(6) + 3 = 24 + 3 = 27 ✓

Using Function Notation

In algebra, we write rules using function notation:

Instead of: "output = 2 × input + 3"

We write: f(x) = 2x + 3

Where:

• f(x) means "function of x" (the output)
• x is the input
• 2x + 3 is the rule

Reading it: "f of x equals 2x plus 3"

Using it: To find f(5), substitute 5 for x:

• f(5) = 2(5) + 3 = 10 + 3 = 13

Real-Life Applications

Earnings: "Paid $15 per hour plus $20 bonus"

• Rule: pay = 15 × hours + 20

Temperature conversion: Celsius to Fahrenheit

• Rule: F = 1.8 × C + 32

Taxi fare: "$3 base fare plus $2 per mile"

• Rule: fare = 2 × miles + 3

Input-output tables are an early introduction to functions — a concept you'll use all through algebra and beyond.

Identifying Rule Types

Linear rules: y = mx + b (constant rate of change)

• Example: y = 3x + 2

Nonlinear rules: Rate of change isn't constant

• Example: y = x² (squaring)

Notice output changes: +3, +5, +7 (not constant!)

Tips for Finding Rules

Tip 1: Always check if the rate of change is constant first

Tip 2: Start with the simplest possible rule (one operation) and add complexity if needed

Tip 3: Verify your rule with ALL the input-output pairs, not just one

Tip 4: If inputs don't increase by 1, calculate the ratio of changes

Tip 5: For nonlinear patterns, look for squares, cubes, or other operations

Practice