miscon:math.fractions.compare-by-gap · draft · trust low · v0.1.1 · kind misapplied-analogy

Comparing fractions by the gap between numerator and denominator

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.

Stable URI
https://open-misconceptions.github.io/miscon-data/m/math.fractions.compare-by-gap
UUID
c2797bf8-5aff-4019-98c3-3462a3c42f71
Domain
math.fractions
Level band
primary, middle
Locale
en
About
CCSS 4.NF.A.2 http://corestandards.org/Math/Content/4/NF/A/2/

Evidence patterns

Pattern 0: compare two fractions with the same numerator-denominator gap

declares the fractions equal because each is the same number of pieces short of one whole

ItemWhich is larger, 5/6 or 7/8?
Expected7/8
ResponseThey are the same

The gap is an additive comparison; the learner ignores that eighths are smaller than sixths.

Pattern 1: compare two fractions with different gaps where the smaller gap belongs to the smaller fraction

picks the fraction with the smaller numerator-denominator gap as larger regardless of denominators

ItemWhich is larger, 2/3 or 7/10?
Expected7/10
Response2/3

Gaps are 1 and 3; the learner reasons 'two thirds is only one piece away'.

Discriminators

vs a 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.

vs 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
CCSS 4.NF.A.2: compare two fractions with different numerators and different denominators http://corestandards.org/Math/Content/4/NF/A/2/
resolved_by
CCSS 3.NF.A.3: explain equivalence of fractions and compare fractions by reasoning about their size http://corestandards.org/Math/Content/3/NF/A/3/
CCSS 4.NF.A.2: compare two fractions with different numerators and different denominators http://corestandards.org/Math/Content/4/NF/A/2/
confusable_with
math.fractions.compare-whole-number-components
math.fractions.bigger-denominator-bigger-fraction

Alignments

Provenance

Origin: llm-drafted

  1. 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.
    Names and documents 'gap thinking' in Year 8 students.
  2. 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
    Classifies gap thinking among fraction comparison strategies and shows it is not a robust strategy.

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

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Cite

miscon:math.fractions.compare-by-gap
https://open-misconceptions.github.io/miscon-data/m/math.fractions.compare-by-gap