Before you begin

Many people who calculate very well with whole numbers stop dead in front of a fraction. This is not a want of talent: it is that a fraction asks them to give up a habit that had always worked. With whole numbers, a longer number is a larger number, every number has a next one, and multiplying makes things bigger. Not one of those three habits survives fractions. A manual that fails to say so leaves the reader believing they have misunderstood, when in fact they are correctly applying a rule that no longer holds.

ODERSA publishes this volume because the subject is a threshold. Whoever does not hold fractions will hold neither decimal numbers, nor percentages, nor proportions, nor the scales on a map, nor the reading of a price per litre. What is missing is not the exercises: they exist everywhere. What is missing is a continuous text, free of charge, reusable without permission, which explains where the rules come from before setting them to work.

This book addresses an adult: someone who teaches, who supports a child, who runs a workshop, or who is taking up on their own what they never had the time to set down. It works just as well for a whole class as for one person alone with a pencil, a sheet of paper and a strip of paper.

The chapters are read in the order in which they are arranged. This is not a preference of layout: each chapter announces at its head what it assumes already held, and points back to the chapter of this book that settles it. A reader who starts with the calculation ends up applying rules whose origin they do not know, which is exactly the state this volume sets out to undo.

Each chapter opens on a list of what it will make you able to do, and closes on a summary. Between the two, the notion is first shown on a case, then named, then worked on an example carried out in full. The first examples of a chapter are taken all the way through; the last ones deliberately leave one step to be done, and that is the moment to pick up the pencil.

The exercises come next, from the closest to the lesson to the furthest from it. The last ones in a chapter call for something seen earlier in the book: this is done on purpose, because knowing how to recognise what you need is a skill in its own right. Each chapter ends with a short self-check, which takes up point by point the objectives announced at its head.

All the answers are gathered in the chapter of answers, at the end of the volume, and never under their question. They are written to be read after searching: they say what is being looked for, what is being used, why, and then the result. An answer read before the effort leaves no trace; the same answer read after a failed attempt is what teaches the most in the whole volume. The glossary, for its part, is consulted at any moment: it gives the exact sense of every technical word used here.

A strip of paper, a pair of scissors and a ruler are enough for everything this book asks you to handle. Nothing that is proposed here calls for an instrument, a piece of software or a purchase.

This volume owes its progression to the public texts that describe the teaching of fractions, and first of all to the supporting resources and curricula published in open access by the ministry responsible for education in France: that is where the order of the notions comes from, the idea of the part as a counting unit, and the etymology of the two words that name the terms of a fraction.

It owes to the research synthesis published in the public domain by the Institute of Education Sciences in the United States the central place given to the number line, as well as the list of the mistakes that come back most often, treated here as named traps rather than as faults.

It owes its form to the open manuals that came before it: the chapter apparatus of announced objectives, worked examples paired with an exercise, method boxes and a closing self-check is the one those houses have tested. All these references are named, with their address and the date on which they were consulted, in the chapter of sources.

This book is published under the Creative Commons Attribution 4.0 International licence: it may be copied, translated, adapted, printed and sold, provided that ODERSA and the address of the house are cited, that the licence is named, and that the changes made are stated. Only the editorial design remains reserved. The text itself belongs to whoever wants to make use of it.