miscon:math.fractions.divide-makes-smaller · draft · trust low · v0.1.1 · kind overgeneralization

Division always makes smaller

Dividing always gives a result smaller than the number you started with, so dividing by a fraction cannot make a number bigger.

Stable URI: https://open-misconceptions.github.io/miscon-data/m/math.fractions.divide-makes-smaller
UUID: 72884d31-1a9a-418f-92ee-f073eef02008
Domain: math.fractions
Level band: primary, middle, secondary, adult
Locale: en
About: CCSS 5.NF.B.7 http://corestandards.org/Math/Content/5/NF/B/7/
CCSS 6.NS.A.1 http://corestandards.org/Math/Content/6/NS/A/1/

Evidence patterns

Pattern 0: predict whether a quotient will be larger or smaller than the dividend, without calculating

predicts the quotient of a number divided by a proper fraction is smaller than the number

ItemWithout calculating, is 6 / 1/2 more or less than 6?

ExpectedMore (it is 12)

ResponseLess

Pattern 1: evaluate a whole number divided by a unit fraction

halves the number instead of doubling it

Item6 / 1/2 = ?

Expected12

Response3

Discriminators

vs a slip. A slip is one wrong prediction. The belief is stated as a rule ('dividing makes it smaller') and holds for decimals too; it often coexists with math.fractions.multiply-makes-bigger.

vs math.fractions.multiply-makes-bigger. Mirror belief; test each separately.

vs math.fractions.invert-dividend. Ask the learner to predict before calculating. Procedure bugs give a wrong number but often a correct prediction of direction; this belief gets the direction wrong.

vs math.fractions.division-is-sharing-only. Sharing-only believers say 6 / 1/2 'makes no sense' or reinterpret it; this belief answers confidently with a smaller number.

Relations

conflicts_with: CCSS 5.NF.B.7: divide unit fractions by whole numbers and whole numbers by unit fractions http://corestandards.org/Math/Content/5/NF/B/7/
resolved_by: CCSS 5.NF.B.7: divide unit fractions by whole numbers and whole numbers by unit fractions http://corestandards.org/Math/Content/5/NF/B/7/
CCSS 6.NS.A.1: interpret and compute quotients of fractions http://corestandards.org/Math/Content/6/NS/A/1/
confusable_with: math.fractions.invert-dividend
math.fractions.division-is-sharing-only

Alignments

CCSS 5.NF.B.7 http://corestandards.org/Math/Content/5/NF/B/7/ (about)
CCSS 6.NS.A.1 http://corestandards.org/Math/Content/6/NS/A/1/ (about)

Provenance

Origin: llm-drafted

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
Implicit partitive model of division: the quotient must be smaller than the dividend.
Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25. doi:10.2307/749817
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
Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909-918. doi:10.1037/edu0000025

Drafted from the cited literature for the Open Misconceptions seed pack; statement, examples and discriminators are original text. Awaiting maintainer review.

Review status

Status draft, trust low (computed from the reviews below against the reviewer registry).

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Formats

Raw JSON
CASE 1.1 package (CFItem 72884d31-1a9a-418f-92ee-f073eef02008)
Source on GitHub

Cite

miscon:math.fractions.divide-makes-smaller
https://open-misconceptions.github.io/miscon-data/m/math.fractions.divide-makes-smaller