miscon:math.fractions.add-across · reviewed · trust high · v1.1.0 · kind overgeneralization

Adding fractions across numerators and denominators

Fractions are added by adding the numerators together and adding the denominators together.

Likely origins: whole-number bias (numerator and denominator treated as independent whole numbers) and overgeneralisation from the multiplication rule.

Stable URI: https://open-misconceptions.github.io/miscon-data/m/math.fractions.add-across
UUID: 20f1dd60-ea8a-4e14-ab41-12addc94b751
Domain: math.fractions
Level band: primary, middle
Locale: en
About: CCSS 5.NF.A.1 http://corestandards.org/Math/Content/5/NF/A/1/

Evidence patterns

Pattern 0: add two fractions with unlike denominators

response numerator = a + c and response denominator = b + d for a/b + c/d, with no attempt at a common denominator

Item1/2 + 1/3

Expected5/6

Response2/5

Strongest single indicator: the response is smaller than at least one addend, which the learner rarely notices.

Pattern 1: add two fractions with like denominators

response numerator = a + c and response denominator = b + b for a/b + c/b; denominators are added even though they match

Item1/4 + 2/4

Expected3/4

Response3/8

Discriminates from math.fractions.add-numerators-keep-denominator, which answers like-denominator items correctly.

Discriminators

vs a slip. A slip appears on one item and not on the next of the same shape, and the learner self-corrects when asked to check with a diagram or with 1/2 + 1/2. A holder of this belief applies the rule consistently across unlike- and like-denominator items and defends 1/2 + 1/2 = 2/4 as correct.

vs math.fractions.add-numerators-keep-denominator. Give a like-denominator item. Add-across produces a doubled denominator (1/4 + 2/4 = 3/8); add-numerators-keep-denominator produces the correct 3/4 and only fails on unlike denominators.

Relations

conflicts_with: CCSS 5.NF.A.1: add and subtract fractions with unlike denominators http://corestandards.org/Math/Content/5/NF/A/1/
resolved_by: CCSS 5.NF.A.1: add and subtract fractions with unlike denominators http://corestandards.org/Math/Content/5/NF/A/1/
CCSS 3.NF.A.1: understand a fraction 1/b as one part of a whole partitioned into b equal parts http://corestandards.org/Math/Content/3/NF/A/1/
confusable_with: math.fractions.add-numerators-keep-denominator

Alignments

CCSS 5.NF.A.1 http://corestandards.org/Math/Content/5/NF/A/1/ (about)
Add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions. CASE Network GUID for this item to be added when the CASE Network export of CCSS Math is confirmed.

Provenance

Origin: literature

Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4(4), 379-426. doi:10.1207/s15516709cog0404_3
Frames systematic errors as stable procedural bugs, distinct from slips; the add-across rule is the canonical fraction example in the repair-theory tradition.
Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52. doi:10.1207/s15326985ep4001_3
Whole number bias: treating numerator and denominator as separate whole numbers.
Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273-296. doi:10.1016/j.cogpsych.2011.03.001
Reports independent whole-number operations on numerators and denominators as the most common fraction-arithmetic error.
Ashlock, R. B. (2010). Error Patterns in Computation: Using Error Patterns to Help Each Student Learn (10th ed.). Allyn & Bacon.
Practitioner catalogue of computational error patterns including this one.

Statement and examples written for Open Misconceptions. No text reused from any proprietary item bank.

Review status

Status reviewed, trust high (computed from the reviews below against the reviewer registry).

kind	by	date	scope	verdict	notes
human	Vikram Maram (github:vikram-learnco)	2026-09-05	statement, evidence, discriminators, sources	accept	Reviewed in session against the design worked example; statement trimmed to the belief; cause moved to notes.
model	claude-fable-5-1	2026-09-05	statement, evidence, discriminators, sources	accept
attested	source 0: Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs i	2026-09-05	statement	accept	Brown & VanLehn (1980): add-across as the canonical fraction bug in repair theory.
attested	source 2: Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of	2026-09-05	statement	accept	Siegler, Thompson & Schneider (2011): independent whole-number operations on numerators and denominators as the most common fraction-arithmetic error.

Formats

Raw JSON
CASE 1.1 package (CFItem 20f1dd60-ea8a-4e14-ab41-12addc94b751)
Source on GitHub

Cite

miscon:math.fractions.add-across
https://open-misconceptions.github.io/miscon-data/m/math.fractions.add-across