In and Out Boxes: Complete Function-Machine Rule Tables

Algebra and Function Practice

An in-and-out box, also called a function machine, takes an input number, applies a rule, and produces an output number. The rule is often written in words such as add 4, subtract 2, multiply by 3, or divide by 5. To complete a table, apply the same rule to every input. Work left to right or top to bottom, and record each output in the matching row. This lesson focuses only on completing rule tables, not on matching rules or graphing functions.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Read the rule and the table structure

A function-machine table usually has two columns: input and output. The rule box states what happens to each input. For example, a rule of add 4 means output equals input plus 4. A rule of multiply by 3 means output equals input times 3.

Before calculating, check that each input has one matching output row. If the table is partially filled, use a completed row to confirm the rule. Then apply the rule to every blank output. Keep the same order and units throughout the table.

1. Identify the rule written in the rule box.
2. Write the rule as a short equation, such as output = input + 4.
3. Apply the rule to the first input and record the output.
4. Repeat for each remaining input, checking one row at a time.

Apply one-step rules accurately

Most in-and-out tables use one operation. For addition and subtraction, keep place value aligned. For multiplication and division, recall the fact or use a simple check. If the rule is divide by 5, the input should be divisible by 5 for a whole-number output unless the table allows remainders.

Use simple checkable arithmetic. For example, if the rule is multiply by 3 and the input is 7, the output is 21 because 7 times 3 equals 21. Check by reversing: 21 divided by 3 equals 7.

1. For addition, add the rule number to the input.
2. For subtraction, subtract the rule number from the input.
3. For multiplication, multiply the input by the rule number.
4. For division, divide the input by the rule number and check by multiplying back.

Use completed rows to verify the rule

When a table already has some outputs, test the rule on those rows. If the rule is subtract 2 and a completed row shows input 9 with output 7, the rule fits because 9 minus 2 equals 7. If a row does not fit, re-read the rule before filling blanks.

Verification prevents copying an error down the column. After completing all outputs, spot-check two rows by reversing the operation. This is especially useful for division and subtraction.

1. Choose a completed row and apply the rule to its input.
2. Compare your result with the given output.
3. If they match, continue filling blank outputs.
4. After finishing, reverse-check at least two rows.

Record answers clearly

Write each output in the same row as its input. Keep digits legible and align columns so the table can be checked quickly. If a rule produces a two-digit output, write both digits without crowding the next column.

For handwriting, use a consistent convention: form digits with clear strokes, keep number height between the baseline and midline, and leave a small space between columns. Letter styles may vary, but digits and operation signs should be unambiguous.

1. Place each output directly beside its input.
2. Keep digit height between baseline and midline.
3. Leave a small gap between the input and output columns.
4. Review the table from top to bottom for missing entries.

Worked example

A function machine has the rule add 4. Complete the outputs for inputs 3, 6, 10, and 12.

Four practice exercises

Common mistakes

• Applying the rule in the wrong direction, such as subtracting when the rule says add. Write the rule as an equation before calculating, then apply it exactly to each input.
• Skipping a row or placing an output beside the wrong input. Work one row at a time and point to the input before writing its output.
• Using a completed row that does not match the rule. Test the rule on a completed row first; if it fails, re-read the rule box.
• Forgetting to check division by multiplying back. After dividing, multiply the output by the rule number to confirm the input.

Self-check rubric

• Applies the stated rule to every input without changing the operation.
• Places each output in the same row as its matching input.
• Uses correct arithmetic for all four outputs.
• Shows a reverse check on at least one row.
• Writes digits legibly using a consistent baseline-to-midline height; valid alternative digit styles are accepted if unambiguous.

Questions and answers