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"ArithmeticResult" (Comparison Method)

"ArithmeticResult" (Comparison Method)

Compare mathematical expressions in a way suitable for exercises about arithmetic, allowing rearrangement but not arithmetic computation.

Details

  • The arithmetic comparison method considers two mathematical expressions to be equivalent if they match without doing any arithmetic operations, e.g. 1 + 1 2.
  • Answers are considered correct when the only differences are ordering of terms inside orderless operations like addition and multiplication.
  • The values of the key and answer can both be specified as held expressions Hold[expr] to prevent evaluation. Even when the values are held, reordering transformations are performed to determine equivalence during assessment. It is recommended to always hold the answer.
  • To require the order of terms in orderless operations to match the order of the answer key, use the "OrderlessMatch" setting: AssessmentFunction[key,<|"ComparisonMethod""ArithmeticResult","OrderlessMatch"False|>].
  • Values in "ArithmeticResult" answer keys are patterns and support Blank expressions.
  • The following table shows a comparison of "ArithmeticResult" with other comparison methods for a typical answer key of 2x+8 and several possible submitted answers:

Examples

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Basic Examples  (1)

Create an AssessmentFunction for an arithmetic question:

Wolfram Language code: af = AssessmentFunction[Hold[1 + Sqrt[2]], "ArithmeticResult"]

Non-computed addition is considered inequivalent and incorrect:

Wolfram Language code: af[Hold[1 + Sqrt[1 + 1]]]

Reordered terms are considered equivalent and correct:

Wolfram Language code: af[Hold[Sqrt[2] + 1]]

Scope  (3)

Create a simple addition question without holding the answer key:

Wolfram Language code: arithmetic = AssessmentFunction[1 + 2 + 3, "ArithmeticResult"]

The original question is not accepted as an answer:

Wolfram Language code: arithmetic[Hold[1 + 2 + 3]]

The computed result is correct:

Wolfram Language code: arithmetic[Hold[6]]

Create a radical simplification question:

Wolfram Language code: QuestionObject[StringForm["Simplify the expression `` so that no radicals are in the denominator", HoldForm[1 / Sqrt[2]]], AssessmentFunction[Hold[Sqrt[2] / 2], "ArithmeticResult"] ]

Require addition terms to have the same order as the answer key:

Wolfram Language code: ordered = AssessmentFunction[{Hold[a + b]}, <|"ComparisonMethod" -> "ArithmeticResult", "OrderlessMatch" -> False|>]

Try different answers:

Wolfram Language code: ordered[Hold[b + a]]
Wolfram Language code: ordered[Hold[a + b]]

Properties & Relations  (1)

Create an assessment function for an addition pattern with the order specified:

Wolfram Language code: asmf = AssessmentFunction[Hold[1 + 2 + _], <|"ComparisonMethod" -> "ArithmeticResult", "OrderlessMatch" -> False|>]

Answers must match the pattern, including the order of terms:

Wolfram Language code: asmf[Hold[1 + 2 + a]]
Wolfram Language code: asmf[Hold[3 + 1 + 2]]

With the "HeldExpression" comparison method, this is not possible. Using Hold in the answer key does not support patterns:

Wolfram Language code: AssessmentFunction[Hold[1 + 2 + _], <|"ComparisonMethod" -> "HeldExpression"|>][Hold[3 + 1 + 2]]
Wolfram Language code: AssessmentFunction[Hold[1 + 2 + _], <|"ComparisonMethod" -> "HeldExpression"|>][Hold[1 + 2 + 3]]

Using HoldPattern supports pattern matching but allows reordering:

Wolfram Language code: AssessmentFunction[HoldPattern[1 + 2 + _], <|"ComparisonMethod" -> "HeldExpression"|>][Hold[1 + 2 + 3]]
Wolfram Language code: AssessmentFunction[HoldPattern[1 + 2 + _], <|"ComparisonMethod" -> "HeldExpression"|>][Hold[3 + 1 + 2]]