70.5 Domain and Interpretation Errors

Domain and Interpretation Errors are the category of mistakes that occur after a candidate value has already been algebraically produced, covering ignored mathematical restrictions, accepted extraneous candidates, mismatched measurement units, ignored contextual requirements, and a misidentified final answer.

Denominator Restriction Ignored

Recognizing When a Zero-Denominator Restriction Was Overlooked

This diagnosis identifies work in which a candidate value making an original denominator equal to zero was accepted into the final answer, without that restriction ever being checked.

Why This Diagnosis Rechecks the Original Expression's Denominators

Because this restriction depends specifically on the original, unmodified expression's denominators, this diagnosis rechecks those original denominators directly, rather than relying on whatever form the expression had been reduced to by the end of the work.

Even-Root Domain Error Diagnosis

Recognizing When a Negative-under-an-Even-Root Restriction Was Overlooked

This diagnosis identifies work in which a candidate value making an expression beneath an even root negative was accepted, without that restriction ever being checked.

Why This Diagnosis Traces the Restriction Back to Its Origin

Because this restriction traces directly back to a specific root present in the original expression, this diagnosis identifies that specific root and confirms the candidate against exactly that restriction, rather than checking the candidate only against a later, transformed version of the expression.

Extraneous Candidate Accepted

Recognizing When a Candidate Was Never Tested against the Original Relation

This diagnosis identifies a candidate value that was accepted into the final answer without having been substituted back into the original relation to confirm it genuinely satisfies that relation.

Why This Diagnosis Applies This Test Directly

Because this is the exact test that candidate value verification is built around, this diagnosis applies that same original-relation substitution directly to the candidate in question, checking for the specific evidence that this test was actually performed.

Measurement Unit Mismatch

Recognizing When Units Were Combined Inconsistently

This diagnosis identifies a step in the work where two measurements expressed in different units were combined directly, without a conversion step bringing them into a consistent unit first.

Why This Diagnosis Traces Units alongside Numerical Values

Because a unit mismatch can occur even when every numerical calculation is performed correctly, this diagnosis tracks the unit attached to each quantity throughout the work, separately from tracking the numerical values themselves.

Contextual Sign Requirement Ignored

Recognizing When a Value with an Invalid Sign Was Accepted

This diagnosis identifies a final answer that violates a sign requirement implied by the real-world context of the problem, such as a negative value reported for a quantity that must be positive.

Why This Diagnosis Returns to the Original Problem's Context

Because this requirement comes from the practical situation the problem describes rather than from the mathematics alone, this diagnosis returns specifically to the original problem statement to confirm what sign, if any, the context actually requires.

Whole-Quantity Requirement Ignored

Recognizing When a Fractional Value Was Accepted for a Countable Item

This diagnosis identifies a final answer expressed as a fraction or decimal when the quantity it represents, such as a count of people or items, must be a whole number.

Why This Diagnosis Checks the Nature of the Quantity Represented

Because this requirement depends on whether the quantity represents something inherently indivisible, this diagnosis specifically examines what the variable represents in the original problem before judging whether a fractional answer was acceptable.

Answer Meaning Misidentified

Recognizing When the Final Value Answers the Wrong Question

This diagnosis identifies a final answer that is a mathematically correct value produced somewhere within the work, but does not actually correspond to the specific quantity the original problem asked to be found.

Why This Diagnosis Returns to the Original Question Directly

Because this error occurs even when every calculation is otherwise correct, this diagnosis sets the mathematics aside and returns directly to the original question, confirming precisely what quantity was asked for before checking whether the reported answer actually matches it.