{
  "id": "math.fractions.multiply-makes-bigger",
  "uri": "https://open-misconceptions.github.io/miscon-data/m/math.fractions.multiply-makes-bigger",
  "uuid": "04246969-9c77-41b0-a67b-b48be9b2f5a3",
  "version": "0.1.1",
  "status": "draft",
  "trust": "low",
  "title": "Multiplication always makes bigger",
  "statement": "Multiplying always gives a result larger than the number you started with, so multiplying by a fraction cannot make a number smaller.",
  "kind": "overgeneralization",
  "domain": "math.fractions",
  "about": [
    {
      "scheme": "CCSS",
      "code": "5.NF.B.5",
      "uri": "http://corestandards.org/Math/Content/5/NF/B/5/"
    }
  ],
  "level_band": [
    "primary",
    "middle",
    "secondary",
    "adult"
  ],
  "locale": "en",
  "evidence_patterns": [
    {
      "item_shape": "predict whether a product will be larger or smaller than a factor, without calculating",
      "signature": "predicts the product of a number and a proper fraction is larger than the number",
      "example": {
        "item": "Without calculating, is 8 x 3/4 more or less than 8?",
        "expected": "Less",
        "response": "More"
      }
    },
    {
      "item_shape": "choose the operation for a word problem with a fractional multiplier",
      "signature": "rejects multiplication because the answer must be smaller, and divides instead",
      "example": {
        "item": "A recipe uses 2/3 of a 6-cup bag of flour. How much flour is used? Which operation?",
        "expected": "6 x 2/3 = 4",
        "response": "6 / 2/3 (or 6 / 2 = 3)"
      }
    }
  ],
  "discriminators": {
    "vs_slip": "A slip is one wrong prediction. The belief is stated as a rule and generalises to decimals (0.5 x 8 'must be more than 8'); it often coexists with math.fractions.divide-makes-smaller.",
    "vs": {
      "math.fractions.divide-makes-smaller": "Mirror belief; test each separately with prediction items for multiplication and division by a proper fraction."
    }
  },
  "relations": {
    "conflicts_with": [
      {
        "external": "http://corestandards.org/Math/Content/5/NF/B/5/",
        "label": "CCSS 5.NF.B.5: interpret multiplication as scaling"
      }
    ],
    "resolved_by": [
      {
        "external": "http://corestandards.org/Math/Content/5/NF/B/5/",
        "label": "CCSS 5.NF.B.5: interpret multiplication as scaling"
      },
      {
        "external": "http://corestandards.org/Math/Content/5/NF/B/4/",
        "label": "CCSS 5.NF.B.4: multiply a fraction or whole number by a fraction"
      }
    ]
  },
  "alignments": [
    {
      "scheme": "CCSS",
      "code": "5.NF.B.5",
      "uri": "http://corestandards.org/Math/Content/5/NF/B/5/",
      "relation": "about"
    }
  ],
  "provenance": {
    "sources": [
      {
        "type": "paper",
        "citation": "Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3-17.",
        "doi": "10.5951/jresematheduc.16.1.0003",
        "note": "Implicit repeated-addition model of multiplication: the product must exceed the multiplicand."
      },
      {
        "type": "paper",
        "citation": "Bell, A., Swan, M., & Taylor, G. (1981). Choice of operation in verbal problems with decimal numbers. Educational Studies in Mathematics, 12(4), 399-420.",
        "doi": "10.1007/BF00308139"
      },
      {
        "type": "paper",
        "citation": "Graeber, A. O., Tirosh, D., & Glover, R. (1989). Preservice teachers' misconceptions in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 20(1), 95-102.",
        "doi": "10.5951/jresematheduc.20.1.0095"
      },
      {
        "type": "paper",
        "citation": "Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909-918.",
        "doi": "10.1037/edu0000025",
        "note": "Direction-of-effect items show adults and children expecting multiplication to increase."
      }
    ],
    "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"
}