Chapter 14 of 14 · Fractions
Sources
The public documents this volume rests on, with their address and the date on which each one was consulted.
What this chapter guarantees
Every claim in this volume that cannot be proved on a sheet of paper goes back to one of the entries below. That is the case of the order in which the notions are settled, of the etymology of the words denominator and numerator, of the central place given to the number line, and of the list of mistakes treated as named traps.
All these sources are public, free of charge, and can be consulted without an account on the day stated. None of them sits behind a paywall, and that is a condition of entry into this house: a book whose sources cannot be checked is not published here.
A word about licences: one is carried below only where it has been checked on the original page, and that is the case for three of the nine sources of this chapter. The other six could not be checked on that point on the day of consultation, and their line therefore carries no mention of a licence rather than an invented one. This changes nothing about consulting them, which is open and free of charge.
The sources of the content
These are the texts that fixed the progression, the vocabulary and the list of mistakes to be dealt with.
- Fractions et nombres décimaux au cycle 3, supporting resource for the mathematics curriculumÉduscol, ministry responsible for national education, France · accessed on August 9, 2026eduscol.education.fr/document/16510/download
- Programme de mathématiques pour le cycle 3Ministry of National Education, France · accessed on August 9, 2026www.education.gouv.fr/sites/default/files/programme-de-math-matiques-pour-le-cycle-3-439827.pdf
- Developing Effective Fractions Instruction for Kindergarten Through 8th Grade, evidence-based practice guide, report NCEE 2010-4039What Works Clearinghouse, Institute of Education Sciences, United States Department of Education · public domain · accessed on August 9, 2026ies.ed.gov/ncee/wwc/docs/practiceguide/fractions_pg_093010.pdf
| What the volume claims | Where it was checked |
|---|---|
| A fraction is built by a sharing into equal parts, then by the taking of a certain number of those parts. | The supporting resource on fractions and decimal numbers. |
| The denominator is the one that names, the numerator the one that counts. | The supporting resource, which gives the etymology of the two words. |
| The part becomes a counting unit, and a fraction reads as a repetition of that part. | The supporting resource, and the mathematics curriculum. |
| Fractions greater than the whole are settled early, and break down into a whole number plus a fraction. | The mathematics curriculum, section on fractions. |
| Locating a fraction on a number line is what gives it its standing as a number. | The mathematics curriculum, and the What Works Clearinghouse practice guide. |
| Placing a fraction between the two whole numbers that make it up is a frequent and documented mistake. | The What Works Clearinghouse practice guide, obstacle described in the recommendation on the number line. |
| Adding the numerators together and the denominators together is a frequent and documented mistake. | The What Works Clearinghouse practice guide, recommendation on calculation procedures. |
| Treating the whole part and the decimal part of a number with a decimal point separately is an obstacle settled by everyday language. | The supporting resource, section on mistakes and obstacles. |
The sources of the form
These are the manuals and the syntheses that fixed the chapter apparatus used here: announced objectives, worked examples, method boxes and a closing self-check.
- Prealgebra 2e, chapter on fractions, section Visualize FractionsOpenStax, Rice University · CC BY-NC-SA 4.0 · accessed on August 9, 2026openstax.org/books/prealgebra-2e/pages/4-1-visualize-fractions
- Manuels et Cahiers Sésamath, page Informations : licences et téléchargementsAssociation Sésamath · GNU FDL and CC BY-SA · accessed on August 9, 2026manuel.sesamath.net/?page=faq
- Organizing Instruction and Study to Improve Student Learning, practice guide of September 2007, and its recommendation on alternating worked examples with exercisesWhat Works Clearinghouse, Institute of Education Sciences, United States Department of Education · accessed on August 9, 2026ies.ed.gov/ncee/wwc/PracticeGuide/1
The sources of the table of correspondences
These are the official texts that establish, system by system, where the content of this volume is worked. The French curriculum cited above serves this table as well.
- Référentiel de mathématiques du tronc commun, year-by-year expectations from the first year of primary to the third year of secondaryFédération Wallonie-Bruxelles, Belgium · accessed on August 9, 2026ifpc-fwb.be/v5/documents/tc/refMATH.pdf
- Progression des apprentissages, Mathématique, primaire, document of 6 October 2009 completing the Programme de formation de l'école québécoiseMinistry responsible for education, Québec · accessed on August 9, 2026cdn-contenu.quebec.ca/cdn-contenu/education/pfeq/primaire/progressions-apprentissages/PFEQ-progression-apprentissages-mathematique-primaire.pdf
- Plan d'études romand, Mathématiques et Sciences de la nature, second cycle, objectives MSN 22 to 25, Nombres et OpérationsConférence intercantonale de l'instruction publique de la Suisse romande et du Tessin · accessed on August 9, 2026bdper.plandetudes.ch/uploads/per_pdf/Mathematiques_et_sciences_de_la_nature/Mathematiques/PER_print_MSN_22.pdf
What was taken up, and what was set aside
Taken up, and why
- The order of the notions from the public curriculum: meaning before technique, fractions greater than the whole early, the number line at the centre.
- The chapter apparatus of the open manuals: objectives with action verbs, a complete worked example, then an example leaving one step to the reader.
- The alternation between an example carried out in full and an exercise of the same kind, recommended by the 2007 synthesis on the organisation of study.
- The treatment of mistakes as traps named at the place where one falls into them, rather than as faults pointed out after the event.
Set aside, and why
- Any designation by a school year: it covers neither the same age nor the same curriculum from one country to another, and this house publishes in several languages.
- The non-commercial clause of the open manual consulted, which forbids the resale this house promises, and the share-alike clause of the French manuals consulted, which would impose its licence on any reuse of the text.
- Mind maps, often recommended for this subject: their effectiveness is not established by controlled trials, and a spatial object is not read aloud by a screen reader.
- Mnemonics that make a rule stick without its meaning: they hold for a few weeks and then produce wrong answers applied with confidence.
- The multiplication of two fractions by each other and division by a fraction: they call for a base that this volume settles but does not go beyond.
One source is missing, and its absence is stated rather than hidden: no complete mathematics manual under a free licence with no restrictive clause was found on the day of consultation in several of the languages this house publishes in. That is precisely the gap this volume sets out to fill, and it is also the reason why its text can be taken up again without asking permission.