Input-Output Tables for 5th Grade: Find the Rule

Input-output tables show how one quantity (output) depends on another (input) according to a rule. In Grade 5, students complete input-output tables, find the rule that relates input to output, and use the rule to predict outputs for new inputs. This skill connects arithmetic to functions and helps students see relationships between variables, like “output = input × 3” or “output = input × 2 + 5.”

To find the rule, we look at how the output changes when the input changes. If output ÷ input is always the same, the rule is “multiply by that number.” If output − input is always the same, the rule is “add that number.” Some rules combine operations, like “multiply by 2, then add 5.” We can test a proposed rule by checking it against all given input-output pairs.

DETAILED EXPLANATION

Finding simple rules:

• If output = input × k (constant), the rule is “multiply by k.”

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• If output = input + k (constant), the rule is “add k.”

• If output = input − k, the rule is “subtract k.”

Finding two-step rules:

• Try “multiply then add”: output = input × a + b. Use two pairs to find a and b.

• Example: Input 2→6, Input 4→12. Output ÷ input: 6÷2=3, 12÷4=3. Rule: multiply by 3.

Example: Input 10→25, Input 12→29. Difference: 25−10=15, 29−12=17 (not constant). Try multiply+add: 10×2+5=25, 12×2+5=29. Rule: multiply by 2, add 5.

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WORKED EXAMPLES WITH STEP BY STEP SOLUTIONS

Example 1

Complete the table. Rule: Add 5.

Input: 3, 7, 10, ___

Output: 8, 12, 15, 20

Solutions:

Step 1: Verify the rule with the given pairs: \(3 + 5 = 8\) ✓, \(7 + 5 = 12\) ✓, \(10 + 5 = 15\) ✓.

Step 2: For output 20, we need the input such that input + 5 = 20. So input = \(20 – 5 = 15\).

Step 3: The missing input is 15.

Answer: Input = 15

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Example 2

Find the rule: Input 2 → 6; Input 4 → 12; Input 5 → 15

Solutions:

Step 1: Check output ÷ input: \(6 \div 2 = 3\), \(12 \div 4 = 3\), \(15 \div 5 = 3\).

Step 2: The quotient is always 3. The rule is multiply by 3 (or \(y = 3x\)).

Answer: Multiply by 3

Example 3

Input 10 → 25; Input 12 → 29; Input 15 → 35. Find the rule.

Solutions:

Step 1: Differences (output − input): 25−10=15, 29−12=17, 35−15=20. Not constant.

Step 2: Try “multiply then add”: For 10, what times 10 plus what gives 25? \(10 \times 2 + 5 = 25\) ✓. Check: \(12 \times 2 + 5 = 29\) ✓, \(15 \times 2 + 5 = 35\) ✓.

Step 3: The rule fits all pairs: multiply by 2, add 5 (or \(y = 2x + 5\)).

Answer: \(y = 2x + 5\) or multiply by 2 then add 5

Example 4

Find the rule: Input 1 → 4; Input 2 → 7; Input 3 → 10

Solutions:

Step 1: Output − input: 4−1=3, 7−2=5, 10−3=7. Not constant.

Step 2: Try multiply + add: 1×3+1=4 ✓, 2×3+1=7 ✓, 3×3+1=10 ✓.

Step 3: Rule: multiply by 3, add 1 (or \(y = 3x + 1\)).

Answer: Multiply by 3, add 1