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The LibraryODERSA publishing house
An ODERSA resource · Knowledge programmeA book only goes online once it is whole and read by someone else.

Chapter 3 of 14 · Fractions

Chapter 2. Counting with parts

The part becomes a counting unit: it is repeated, it goes beyond the whole, and every fraction is bracketed between two consecutive whole numbers.

A new unit for counting

Nothing obliges you to stop at the whole. Once a whole has been shared into three equal parts, the part obtained exists in its own right: it is a third, and nothing prevents you from taking one, two, three, four or ten of them. Counting in thirds amounts to changing the counting unit, exactly as you move from units to tens. Two thirds is one third plus one third. Four thirds is one third repeated four times.

This way of seeing changes everything. As long as a fraction is seen as a slice of cake, taking four thirds of a cake is absurd: there are not four thirds in a cake. Seen as a count, the same writing raises no difficulty at all: two cakes cut into thirds are needed in order to take four thirds, and what you then obtain is one whole cake plus one third.

Both readings are correct and they serve different ends. The reading by sharing serves to understand what a fraction measures. The reading by counting serves to calculate, to go beyond the whole and to bracket. This chapter settles the second one.

What this chapter makes you able to do

Four gestures, all of them checkable on a graduated strip of paper.

  • Break a fraction down into a sum of unit fractions.
  • Write a fraction greater than the whole, and say what it represents.
  • Break a fraction greater than the whole down into a whole number plus a fraction smaller than the whole.
  • Bracket a fraction between two consecutive whole numbers.

What has to have been worked before this chapter

A fraction is a sum of identical parts

The first change of outlook consists in writing a fraction as an addition. Three fifths is one fifth plus one fifth plus one fifth. This writing seems useless as long as you stay on simple cases, and it becomes the main tool as soon as there is calculating to do: it explains why fractions with the same denominator can be added, and why those without the same denominator cannot be added directly.

The unit fraction is therefore the brick. The denominator chooses the brick: fifth, tenth, hundredth. The numerator says how many bricks are stacked. Two fractions made of the same brick are compared and added as easily as whole numbers, because counting fifths asks for nothing more than counting.

One immediate consequence deserves to be set down straight away: when the numerator reaches the denominator, all the parts of the sharing are taken, and the fraction is worth the whole. Three thirds are worth one, eight eighths are worth one, a hundred hundredths are worth one. If the numerator goes beyond the denominator, the fraction goes beyond the whole.

The words of this chapter

These three expressions will come back in every chapter of calculation.

Counting unit
The part chosen for counting. When the denominator is 5, the counting unit is the fifth, and every quantity in the problem is said in fifths.
Fraction smaller than the whole
A fraction whose numerator is smaller than its denominator: not all the parts have been taken. 3/4 and 7/10 are two of them.
Fraction greater than the whole
A fraction whose numerator is larger than its denominator: more than one whole was needed in order to take those parts. 7/4 and 12/10 are two of them.

Worked example. Breaking 7/4 down into a whole number and a fraction

The problem: write 7/4 in the form of a whole number plus a fraction smaller than the whole, then bracket 7/4 between two consecutive whole numbers.

  1. State the counting unit
    The denominator is 4: we count in quarters. The fraction 7/4 is therefore seven quarters, that is to say one quarter repeated seven times.
  2. Look for how many quarters make one whole
    Four quarters make one whole, since all the parts of the sharing are taken. That is the only fact needed here.
  3. Take away one complete whole
    Out of seven quarters, we take four quarters, which make one whole. Seven minus four are left, that is to say three quarters. Can a second whole be taken away? No: three quarters are not enough to make four quarters.
  4. Write the breakdown
    Seven quarters are therefore worth one whole plus three quarters, which is written 7/4 = 1 + 3/4.
  5. Deduce the bracket from it
    Since 7/4 is worth 1 plus something smaller than 1, this number is larger than 1 and smaller than 2. The bracket is written 1 < 7/4 < 2. The breakdown therefore gives the bracket with no further calculation, and that is why it is done first.

The same gesture, faster

Once the reasoning is understood, it shortens. Bracketing a fraction amounts to looking for how many whole ones fit inside it, which is exactly a division of the numerator by the denominator. For 17/5, we look for how many times five fits into seventeen: three times, since three times five is fifteen, and two are left. So 17/5 is worth 3 + 2/5, and the bracket is 3 < 17/5 < 4.

Take this shortcut up again on 23/6: that is the exercise along the way below, and its answer is at the end of the volume.

One particular case remains to be noted: when the division comes out exactly, there is no remainder, and the fraction is worth a whole number. That is the case of 12/4, which is worth exactly 3. It is not bracketed between two whole numbers: it is one.

Exercise along the way, chapter 2. The shortcut on 23/6

Set in the course of the chapter: do it before going on, with the method of the example 17/5.

  1. Break 23/6 down into a whole number plus a fraction, then give the bracket: how many times does six fit into twenty-three, and what is left?

A classic trap: believing that a fraction is always smaller than 1, because the first examples met are slices of cake. A fraction whose numerator goes beyond its denominator is perfectly legitimate, and it will turn up in every addition calculation of chapter 6. Do not correct it by swapping the two numbers: 7/4 and 4/7 are two different numbers, one larger than 1 and the other smaller.

Exercise 2.1. Breaking down into unit fractions

Write each fraction as a sum of unit fractions, then say in one sentence what that means.

  1. Break 4/5 down into a sum of unit fractions.
  2. Break 3/8 down into a sum of unit fractions.
  3. Which fraction is equal to one tenth repeated nine times?

Exercise 2.2. Recognising what goes beyond the whole

For each fraction, say whether it is smaller than 1, equal to 1, or larger than 1, and justify in one sentence.

  1. Classify 5/6, 6/6 and 9/6 in relation to 1.
  2. A strip of paper is shared into thirds. How many thirds are needed to cover two whole strips?
  3. Can 5/4 be taken from a single pie? Explain what this writing calls for.

Exercise 2.3. Breaking down and bracketing

Write each fraction in the form of a whole number plus a fraction smaller than the whole, then give its bracket between two consecutive whole numbers.

  1. Break down and bracket 11/3.
  2. Break down and bracket 25/8.
  3. What happens when the same method is applied to 20/5? What becomes of the bracket?

Self-check, chapter 2

Answer without turning back. An answer that does not come points to the objective to take up again.

  1. What is a fraction worth when its numerator is equal to its denominator, and why?
  2. Which operation between the numerator and the denominator gives the bracket of a fraction between two whole numbers directly?
  3. Why is it said that the denominator chooses the counting unit?

What to remember from this chapter

  • A fraction is a sum of unit fractions: 3/5 is worth one fifth plus one fifth plus one fifth.
  • The denominator chooses the counting unit, the numerator counts the units.
  • Numerator equal to the denominator: the fraction is worth 1. Numerator larger: it goes beyond 1.
  • Every fraction breaks down into a whole number plus a fraction smaller than the whole, by dividing the numerator by the denominator.
  • That breakdown gives at once the bracket of the fraction between two consecutive whole numbers.