Chapter 2 of 14 · Fractions
Chapter 1. Sharing a whole into equal parts
A fraction is born of a sharing into equal parts: this chapter settles the reference whole, the part, and the two numbers that write a fraction.
When whole numbers are no longer enough
Take a piece of string and a ruler graduated in centimetres. As long as the string measures seven centimetres, eight centimetres or twelve centimetres, whole numbers are enough. But if its end falls between two graduations, no whole number gives its length. What is needed then is a new kind of number, able to point at something larger than seven and smaller than eight. Fractions were born of that need, and they serve nothing else: they measure what whole numbers leave between them.
The gesture that makes a fraction takes two steps. You take a whole, and you share it into parts that are all equal. Then you take a certain number of those parts. The result of these two steps is a fraction, and the fraction reads as the report of what has just been done.
This chapter asks for nothing more than knowing how to count and knowing how to fold a sheet of paper in two. All the rest of the book rests on it, and it is better to hold it truly than to hold it quickly.
What this chapter makes you able to do
Four gestures, and each one can be checked on a sheet of paper.
- Recognise a sharing into equal parts and tell it apart from any old cutting up.
- Name the reference whole of a situation before writing anything at all.
- Write and read a fraction from a sharing described in words or drawn.
- Point out the denominator and the numerator of a fraction, and say what each one brings.
This chapter is the first of the volume: it assumes no other chapter of this book has been read. It assumes only that you can count and write whole numbers.
Sharing into equal parts, and the condition that matters
Cut a strip of paper into four pieces at random. You obtain four pieces, and nothing more: none of those pieces is a quarter of the strip, because they are not the same size. Start again by folding the strip in two, then in two again, and cut along the folds. This time the four pieces lie exactly on top of one another: each one is a quarter of the strip.
The condition of equality is not a detail of vocabulary, it is what makes the word quarter usable. If the parts are unequal, saying that one of them is taken tells you nothing about any quantity. The first gesture in front of a fraction is therefore always to check two things: which whole are we talking about, and are the parts really equal.
Once the sharing is done, each part receives a name that comes from the number of parts. A whole shared into two gives halves, into three thirds, into four quarters, into five fifths, into ten tenths, into a hundred hundredths. Beyond two, the name is always built the same way: it is the ordinal number of the sharing, as in fifth, sixth or tenth. Only the half stands outside that construction, and quarter is a second name for the fourth, the one everyday speech prefers.
The reference whole matters as much as the sharing itself. Half a page of an exercise book and half a sports field are both written 1/2, and they do not cover the same surface. A fraction never states a quantity on its own: it states a quantity in relation to a whole, and that whole has to be known.
The words of this chapter
Every one of these words will be used exactly as it stands here until the end of the volume.
- The whole
- What is shared, and what the fraction relates to. A strip of paper, a quantity of marbles, a length, a duration, a sum of money. Nothing obliges the whole to be a single object: fifteen marbles can be the whole.
- The part
- One of the pieces obtained by the sharing. All the parts of one and the same sharing have the same size, and it is that equality which allows them to be named.
- The denominator
- The number written under the bar, or after the slash. It says into how many equal parts the whole has been shared, and it therefore gives the part its name. Its name comes from Latin and means the one that names.
- The numerator
- The number written above the bar, or before the slash. It says how many of those parts are taken. Its name means the one that counts.
- The unit fraction
- A fraction whose numerator is 1: a single part of the sharing. 1/2, 1/3, 1/4 and 1/10 are unit fractions. They are the brick every other fraction is made of.
Worked example. Writing the fraction of a sharing described in words
The problem: a pie is shared into eight equal parts, and five parts are served. What fraction of the pie has been served?
- Name the whole
The whole is the entire pie, before any sharing. That is what the fraction will relate to, and not what is left in the dish. - Note the number of parts of the sharing
The pie is shared into eight equal parts. That number, eight, is the one that names the part: each part is an eighth of a pie. It goes under the bar. - Note the number of parts taken
Five parts have been served. That number, five, counts the parts taken. It goes above the bar. - Write and read the fraction
The fraction served is 5/8, and it reads five eighths. Put plainly: the pie was shared into eight equal parts, and five of them were taken. - Check that the answer makes sense
Five parts taken out of eight available: that is more than half, and less than the whole pie. The answer is therefore plausible, and this check takes two seconds.
Reading a fraction the other way round
The same reasoning works in reverse, and that is what makes it possible to draw what a fraction describes. In front of 3/5, you read the bottom number first: five, so the whole is shared into five equal parts. You then read the top number: three, so three of them are taken. A rectangle shared into five equal bands with three of them coloured in represents exactly 3/5 of the rectangle.
This going back and forth between the writing and the figure is the only useful drill at the start. A fraction you can draw is a fraction you have understood; a fraction you can only copy out is of no use at all.
Here are four sharings, each described in three different ways. The three columns of one row say the same thing, and you have to be able to go from any one of them to the other two.
Now make the journey yourself on a fresh case: that is the exercise along the way placed under the table, and its answer is waiting at the end of the volume, like all the others.
| Written form | Read aloud | What was done |
|---|---|---|
| 1/2 | one half | The whole is shared into two equal parts, one of them is taken. |
| 2/3 | two thirds | The whole is shared into three equal parts, two of them are taken. |
| 3/4 | three quarters | The whole is shared into four equal parts, three of them are taken. |
| 7/10 | seven tenths | The whole is shared into ten equal parts, seven of them are taken. |
Exercise along the way, chapter 1. The missing row of the table
Set in the course of the chapter: do it before going on. The bottom number is the one of the sharing, the top number is the one of the parts taken.
- A ribbon is shared into six equal parts, and four of them are taken. Write the fraction, say it aloud, then write the row of the table that corresponds to this sharing.
A trap to know about from now on: a fraction is not two numbers written one above the other, it is a single number. The reflex of treating the numerator and the denominator as two independent whole numbers is one of the most frequent mistakes in the whole subject, and it will later produce wrong answers that look as though they follow a rule. Always read a fraction as one piece: three quarters, and not three then four.
Exercise 1.1. Recognising a sharing into equal parts
Answer in writing, in one sentence per question. Fold a strip of paper if that helps you.
- A loaf of bread is cut into three pieces: one big and two small. Can the big piece be called a third of the loaf? Justify your answer.
- A sheet of paper is folded in two, then each half is folded in two. How many parts are obtained, and what is each one called?
- A garden is shared into six plots of the same area. What is one plot called in relation to the whole garden?
Exercise 1.2. Writing a fraction from a situation
Write the fraction asked for with a slash, then write it out in words.
- A ribbon is shared into ten equal parts, and three of them are taken. What fraction of the ribbon is taken?
- A bar of chocolate has twelve identical squares. Five of them are eaten. What fraction of the bar has been eaten?
- An hour is shared into four equal parts of fifteen minutes. What fraction of an hour do three of those parts make?
Exercise 1.3. Naming the whole, and seeing why it matters
These three questions are about the reference whole. Answer by naming explicitly what is being shared.
- A class has twenty pupils; five of them are absent. What is the reference whole, and what fraction of the class is absent?
- Two people each claim to have eaten half a pizza, and yet one of them has eaten far more than the other. How is that possible?
- A litre of water is poured into five identical glasses, filled to the brim. What fraction of the litre does one glass hold, and what fraction do three glasses hold?
Self-check, chapter 1
Answer without turning back. If an answer does not come, the matching objective is to be taken up again before the next chapter.
- What has to be checked before calling one of the four pieces of a cut object a quarter?
- In the fraction 7/9, which of the two numbers names the part, and which one counts it?
- Why can two people both write 1/2 while pointing to very different quantities?
What to remember from this chapter
- A fraction is the report of two gestures: sharing a whole into equal parts, then taking a certain number of those parts.
- The denominator, under the bar, says into how many parts the whole is shared: it names the part.
- The numerator, above the bar, says how many parts are taken: it counts.
- Without equal parts there is no fraction, only pieces.
- A fraction means nothing as long as the reference whole is not known.