Number Machines

Forward Processing (Finding the Output)

• Step 1: Identify the starting input value.
• Step 2: Apply the first operation in the sequence to the input.
• Step 3: Use the result of the previous step as the input for the next operation.
• Step 4: Repeat until all boxes are cleared to find the final output.

Inverse Processing (Finding the Input)

• Step 1: Start with the known output value.
• Step 2: Move backward through the machine (from right to left).
• Step 3: Apply the inverse operation for each box. For example, if the box says '+ 10', you must '- 10'. If it says '$\times 3$', you must '$\div 3$'.
• Step 4: The final result after reversing all steps is the original input.

• Order of Operations: The sequence in a number machine is strictly left-to-right. This differs from standard algebraic expressions where BIDMAS/BODMAS rules apply unless brackets are used to enforce a specific order.

• Single vs. Multi-stage: A single-stage machine represents a simple operation (e.g., $x + 5$), while a multi-stage machine represents a composite function (e.g., $3(x + 5)$).

• The Reverse Order Rule: When working backward to find an input, you must not only use inverse operations but also reverse the order of the boxes. The last operation performed in the forward direction is the first one reversed in the backward direction.

• Verification: Always check your 'inverse' answer by running it through the machine in the forward direction to see if it produces the original output.

• Algebraic Conversion: If an exam asks for an expression, write the input as $x$ and build the expression step-by-step. For a machine that adds 3 then multiplies by 4, the expression is $4(x + 3)$, not $4x + 3$.

• Ignoring the Chain: A common mistake is applying all operations to the original input separately rather than using the intermediate results. Each box 'consumes' the result of the previous one.

• Inverse Sign Errors: Students often forget that the inverse of 'subtracting a negative' is 'adding a negative' (which is effectively subtraction), or they confuse squaring with doubling.

• Missing Brackets: When converting a machine to an algebraic expression, failing to use brackets for addition/subtraction steps that occur before multiplication/division is a frequent error.