{
  "id": "math.fractions.compare-by-gap",
  "uri": "https://open-misconceptions.github.io/miscon-data/m/math.fractions.compare-by-gap",
  "uuid": "c2797bf8-5aff-4019-98c3-3462a3c42f71",
  "version": "0.1.1",
  "status": "draft",
  "trust": "low",
  "title": "Comparing fractions by the gap between numerator and denominator",
  "statement": "The size of a fraction is decided by how many pieces are missing from the whole; fractions with the same gap between numerator and denominator are equal.",
  "kind": "misapplied-analogy",
  "domain": "math.fractions",
  "about": [
    {
      "scheme": "CCSS",
      "code": "4.NF.A.2",
      "uri": "http://corestandards.org/Math/Content/4/NF/A/2/"
    }
  ],
  "level_band": [
    "primary",
    "middle"
  ],
  "locale": "en",
  "evidence_patterns": [
    {
      "item_shape": "compare two fractions with the same numerator-denominator gap",
      "signature": "declares the fractions equal because each is the same number of pieces short of one whole",
      "example": {
        "item": "Which is larger, 5/6 or 7/8?",
        "expected": "7/8",
        "response": "They are the same"
      },
      "notes": "The gap is an additive comparison; the learner ignores that eighths are smaller than sixths."
    },
    {
      "item_shape": "compare two fractions with different gaps where the smaller gap belongs to the smaller fraction",
      "signature": "picks the fraction with the smaller numerator-denominator gap as larger regardless of denominators",
      "example": {
        "item": "Which is larger, 2/3 or 7/10?",
        "expected": "7/10",
        "response": "2/3"
      },
      "notes": "Gaps are 1 and 3; the learner reasons 'two thirds is only one piece away'."
    }
  ],
  "discriminators": {
    "vs_slip": "Ask the learner to explain. Gap thinkers say 'both are one away from a whole' or 'this one is closer'. A slip comes with no such rule and is not repeated on 3/4 vs 9/10.",
    "vs": {
      "math.fractions.compare-whole-number-components": "On 5/6 vs 7/8 component comparison picks 7/8; gap thinking says equal.",
      "math.fractions.bigger-denominator-bigger-fraction": "On 1/3 vs 1/8 gap thinking compares the size of one piece and often answers correctly; the denominator rule picks 1/8."
    }
  },
  "relations": {
    "conflicts_with": [
      {
        "external": "http://corestandards.org/Math/Content/4/NF/A/2/",
        "label": "CCSS 4.NF.A.2: compare two fractions with different numerators and different denominators"
      }
    ],
    "resolved_by": [
      {
        "external": "http://corestandards.org/Math/Content/3/NF/A/3/",
        "label": "CCSS 3.NF.A.3: explain equivalence of fractions and compare fractions by reasoning about their size"
      },
      {
        "external": "http://corestandards.org/Math/Content/4/NF/A/2/",
        "label": "CCSS 4.NF.A.2: compare two fractions with different numerators and different denominators"
      }
    ],
    "confusable_with": [
      "math.fractions.compare-whole-number-components",
      "math.fractions.bigger-denominator-bigger-fraction"
    ]
  },
  "alignments": [
    {
      "scheme": "CCSS",
      "code": "4.NF.A.2",
      "uri": "http://corestandards.org/Math/Content/4/NF/A/2/",
      "relation": "about"
    }
  ],
  "provenance": {
    "sources": [
      {
        "type": "paper",
        "citation": "Pearn, C., & Stephens, M. (2004). Why you have to probe to discover what Year 8 students really think about fractions. In I. Putt, R. Faragher & M. McLean (Eds.), Mathematics Education for the Third Millennium: Towards 2010 (Proceedings of MERGA 27, pp. 430-437). MERGA.",
        "note": "Names and documents 'gap thinking' in Year 8 students."
      },
      {
        "type": "paper",
        "citation": "Clarke, D. M., & Roche, A. (2009). Students' fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72(1), 127-138.",
        "doi": "10.1007/s10649-009-9198-9",
        "note": "Classifies gap thinking among fraction comparison strategies and shows it is not a robust strategy."
      }
    ],
    "origin": "llm-drafted",
    "notes": "Drafted from the cited literature for the Open Misconceptions seed pack; statement, examples and discriminators are original text. Awaiting maintainer review."
  },
  "history": {
    "changelog": [
      {
        "version": "0.1.1",
        "date": "2026-09-05",
        "change": "Schema 0.2 migration: about/alignments carry a free-string scheme (corestandards.org URLs relabelled CCSS where present); trust computed from reviews[] (none yet, so low); no change to the statement or evidence."
      }
    ]
  },
  "license": "CC-BY-4.0"
}