Chapter 13 of 14 · Fractions
Reference tables
Four tables: the names of the parts, the common fractions with their decimal notation, the first multiples, and the place of this volume in four French-language school systems.
What these tables are for
The first three tables hold no new notion: they gather, in a form that can be consulted, results already established in the chapters. Keeping them to hand during an exercise is not cheating, it is economy of attention: what has to be searched for is the reasoning, not the list of the multiples of seven. The fourth one is of another kind: it does not serve to calculate, but to place this volume within the school systems of the French-speaking world.
They also serve as a check. A result that contradicts one of these tables is wrong, and that can be seen in an instant. The values they carry have all been established by the methods of the book, and each of them can be found again without them.
| The whole is shared into | Each part is called | The unit fraction is written |
|---|---|---|
| 2 parts | one half | 1/2 |
| 3 parts | one third | 1/3 |
| 4 parts | one quarter | 1/4 |
| 5 parts | one fifth | 1/5 |
| 6 parts | one sixth | 1/6 |
| 10 parts | one tenth | 1/10 |
| 100 parts | one hundredth | 1/100 |
| 1000 parts | one thousandth | 1/1000 |
Beyond two, the name of the part is always built in the same way: it is the ordinal number of the sharing, as in sixth, tenth or hundredth. The spelling adjusts here and there along the way: five gives fifth, nine gives ninth, twelve gives twelfth, and twenty gives twentieth. Only the half escapes the construction entirely, and quarter is a second name for the fourth, the one everyday speech prefers: those are the two names to be remembered as they stand.
| Fraction | Decimal fraction with denominator one hundred | Decimal notation |
|---|---|---|
| 1/2 | 50/100 | 0.5 |
| 1/4 | 25/100 | 0.25 |
| 3/4 | 75/100 | 0.75 |
| 1/5 | 20/100 | 0.2 |
| 2/5 | 40/100 | 0.4 |
| 1/10 | 10/100 | 0.1 |
| 3/10 | 30/100 | 0.3 |
| 1/3 | no exact writing in hundredths | no finite decimal notation |
The last row of this table is not an oversight. A fraction can be written with a decimal point and a finite number of digits when, in its simplest form, its denominator divides a power of ten. That is the case of 2, 4, 5, 8, 10, 20, 25, 50 and 100; it is not the case of 3, 6, 7 or 9. The condition really does bear on the simplest form: 3/6 simplifies into 1/2 and is therefore worth 0.5, whereas its starting denominator divided no power of ten. Chapter 8 explains why, and forgetting it leads to writing false equalities.
| Number | Its first multiples |
|---|---|
| 2 | 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 |
| 3 | 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 |
| 4 | 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 |
| 5 | 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 |
| 6 | 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72 |
| 7 | 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84 |
| 8 | 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 |
| 9 | 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108 |
| 10 | 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120 |
| 12 | 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144 |
Using the table of multiples
- Step 1
Find the two denominators of your calculation, and go to their two rows.
- Step 2
Run through both lists side by side, and stop at the first number that appears in both: that is the smallest common denominator.
- Step 3
If no shared number appears in the lists, simply multiply the two denominators together: the product always works, at the cost of simplifying the result at the end.
Where this volume sits in French-language school systems
This volume carries a stage, and that stage says what the book does within a progression: it settles the basic gestures of fractions, without assuming anything known about the subject. It does not say the age of whoever reads it, and that is deliberate. A school year covers neither the same age nor the same curriculum from one country to another, and a book set on a single system would be wrong everywhere else.
The table that follows exists for the adult who is preparing a lesson and who needs to know where this content is placed at home. It brings the content of this volume alongside the official texts of four French-language systems, open and checked at their address. Each of those texts is cited in the chapter of sources, with the date of consultation.
It reads as a landmark, never as a verdict. An adult taking up fractions again has no school year, and a child ahead of or behind the table is neither one nor the other: it is the gestures of chapter 1 that say where to begin, not this page.
| System | Years concerned | What the official text places there |
|---|---|---|
| France | The three years of cycle 3: cours moyen première année, cours moyen deuxième année, classe de sixième. | Fractions with denominators up to 20 in the first year, up to 60 in the second; bracketing between two whole numbers, comparison, addition and subtraction; the third year adds the link with division, the equality of fractions and multiplication by a whole number. |
| Belgium, Fédération Wallonie-Bruxelles | From the third to the sixth year of primary school in the tronc commun. | Thirds, sixths and eighths, equivalent fractions and simplifying in the third year; tenths, twentieths and hundredths, arranging in order and entry into decimal numbers in the fourth; fractions greater than the unit in the fifth; the transformation of a fractioned magnitude into an equivalent one in the sixth. |
| Québec | The six years of primary school, the progression being set out year by year. | The table of fractions covers the role of the numerator and of the denominator, reading and writing them, comparison with 0, with one half and with 1, the equivalence of two fractions, arranging in order with equal denominators and then with equal numerators, and locating them on a number line. |
| French-speaking Switzerland | The second cycle of the Plan d'études romand, years 5 to 8. | Positive rational numbers enter the working number domain in the seventh and eighth years; the progression names the comparison and the ordering of unit fractions and of fractions with the same denominator. |
This table covers only the four systems whose official texts could be opened and read at their address on the day stated in the chapter of sources. Other French-speaking countries and territories teach the same notions under other year names: their correspondence is not written here, because the house does not publish an equivalence it has not checked. This is the only place in this volume where national class names appear, and that is intended: everywhere else, the book speaks of what it settles, never of the age of whoever reads it.
What these tables do not replace
- The reasoning: a table gives a result, it never says why that result is right.
- The check by estimating: a value read in a table may have been read on the wrong row.
- The general method of putting over the same denominator, which works even when the numbers fall outside these lists.
- An official curriculum: the table of correspondences places this volume, it does not say what a system expects of a pupil at a given date.