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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 4 of 14 · Fractions

Chapter 3. A fraction is a number

Placing a fraction on a number line gives it its standing as a number, and reveals three habits of whole numbers that stop being true.

Leaving the slice of cake behind

As long as a fraction remains a part of something, it is not yet a number: it is the description of a cut-up object. The move to the standing of a number happens on the day the fraction is placed on a number line, between two whole numbers, just as 7 or 12 are placed there. This gesture is the most important one in the volume, and it is also the one that public research on the teaching of fractions puts at the centre, as the chapter of sources says.

A number line is a line on which a point 0 has been marked, then a point 1, then 2, 3, and so on at equal intervals. The distance from 0 to 1 is the unit. Placing a fraction on that line amounts to sharing the interval that runs from 0 to 1 into as many parts as the denominator indicates, then moving forward by as many parts as the numerator indicates.

Once that is done, a fraction stops being an operation waiting to happen: it is a point, therefore a number, which has one place and one only.

What this chapter makes you able to do

Four gestures, all of them possible with a ruler and a pencil.

  • Mark up a number line for a given denominator.
  • Place a fraction smaller or greater than the whole on a number line.
  • Read off the fraction pointed to by a point already marked on a number line.
  • Name three habits of whole numbers that stop being true with fractions.

What has to have been worked before this chapter

Mark up, then move forward

The gesture always breaks down in the same way. You find the unit, that is to say the interval from 0 to 1. You share that interval into as many equal parts as the denominator indicates, which calls for drawing a number of marks equal to the denominator less one: three marks to obtain four quarters, nine marks to obtain ten tenths. Then you move forward, starting from 0, by the number of parts the numerator indicates.

For a fraction greater than the whole, nothing changes in the method: you simply carry on moving forward beyond 1. The sharing into quarters continues between 1 and 2, between 2 and 3, and so on, with parts that are always the same size. This is where the breakdown of the previous chapter earns its keep: knowing that 7/4 is worth 1 plus 3/4, you know before drawing anything that the point looked for lies between 1 and 2, three quarters of the way along.

Sharing a segment into equal parts without measuring is done with a set of parallel lines evenly spaced: you lay the segment across them, and the lines cut it into equal parts. A sheet of ruled paper is enough, and this guide saves you from having to divide lengths with a ruler.

Worked example. Placing 5/3 on a number line

The problem: draw a number line from 0 to 3 and place on it the point that corresponds to 5/3.

  1. Set down the whole numbers
    We draw the line and mark 0, then 1, 2 and 3, keeping the intervals the same length. Those intervals are the units.
  2. Predict where the point will fall
    The denominator is 3, so we count in thirds. Five thirds are worth three thirds plus two thirds, that is to say 1 + 2/3. The point looked for is therefore between 1 and 2, and closer to 2 than to 1.
  3. Share each unit
    We share each interval into three equal parts, by drawing two marks per interval. All the parts obtained have the same length, whichever unit they happen to fall in.
  4. Move forward by five parts
    Starting from 0, we move forward by five parts. The first three lead exactly to 1; the next two lead to the point looked for.
  5. Check against the prediction
    The point obtained is indeed between 1 and 2, two thirds of the way along the interval. The prediction made at the second step is confirmed: the drawing taught nothing new, it checked. That is the order to keep, because a wrong prediction is then spotted straight away.

Three habits that stop being true

The number line makes three breaks with whole numbers visible, and it is those breaks that explain most lasting mistakes. They are observed, they are not proved here, and it is enough to have seen them once.

First break: a number no longer has a next one. After the whole number 7 comes 8, with nothing between the two. After 1/2, it is impossible to name the number that comes just after, because between 1/2 and any larger number you can always find another one. Between 1/2 and 3/4, for instance, lies 5/8.

Second break: the number written with the largest digits is not necessarily the largest. On the line, 1/100 is almost stuck to 0 while 1/2 is far away, halfway to 1. A large denominator makes small parts, and therefore a small number.

Third break: multiplying no longer always makes things bigger. Multiplying 8 by 1/2 gives 4, that is to say less than 8, whereas multiplying by a whole number has always made things bigger up to now. This point is worked in chapter 7, but the line already lets it be sensed: as soon as you multiply by a number smaller than 1, you move back instead of forward.

Now place two fractions on one and the same line yourself: that is the exercise along the way below, and its answer is at the end of the volume.

Exercise along the way, chapter 3. Two points on the same line

Set in the course of the chapter: the drawing is done like the one in the worked example.

  1. Place 5/4 and 1/4 on one and the same number line from 0 to 2, then say which of these two fractions is the closer to 1.

A mistake observed regularly, and documented by studies on the teaching of fractions: placing 3/4 between 3 and 4 on the number line. It comes from reading the two numbers as two separate whole numbers. The remedy is one question to be asked before any drawing: how many whole ones fit inside this fraction? Three quarters is less than one whole, so the point lies between 0 and 1, and nowhere else.

Exercise 3.1. Marking up and placing

Draw each number line asked for on a sheet of paper, then describe in writing where the point falls.

  1. Draw a number line from 0 to 1 shared into fifths, and place 3/5. How many marks did you draw between 0 and 1?
  2. Draw a number line from 0 to 2 shared into quarters, and place 7/4. Between which whole numbers does this point lie?
  3. On a number line marked in tenths, place 4/10 and 9/10. Which of the two points is the closer to 1?

Exercise 3.2. Reading a point already placed

Each situation describes a number line and a point. Give the matching fraction and justify your answer.

  1. A line runs from 0 to 1, shared into eight equal parts. A point is marked at the fifth mark after 0. Which fraction does it point to?
  2. A line runs from 0 to 3, each unit being shared into three equal parts. A point is placed two parts after the number 2. Which fraction does it point to?
  3. A line is marked in sixths, and a point falls exactly on the number 1. Which fraction with denominator 6 points to this point?

Exercise 3.3. The breaks with whole numbers

These questions cross this chapter with the two before it. Answer in one or two sentences.

  1. Give a fraction lying between 1/2 and 1, then explain how you found it.
  2. Between 1/100 and 1/2, which is the closer to 0? Justify without calculating, by thinking about the size of the parts.
  3. A pupil claims: since 9 is larger than 2, then 1/9 is larger than 1/2. Where is the mistake, and what has to be looked at instead?

Self-check, chapter 3

Answer without turning back, then compare with the answer.

  1. How many marks have to be drawn between 0 and 1 in order to obtain sevenths?
  2. Which question has to be asked before placing a fraction, so as to avoid putting it between the wrong whole numbers?
  3. Name one habit of whole numbers that stops being true with fractions, and give an example.

What to remember from this chapter

  • A fraction is a number: it occupies one point and one only on a number line.
  • To place it, the unit is shared into as many parts as the denominator, then you move forward by the number of parts the numerator indicates.
  • Obtaining parts between two whole numbers calls for drawing one mark fewer than there are parts: four marks for five parts, six for seven.
  • The breakdown into a whole number plus a fraction says in advance between which whole numbers the point falls.
  • Three habits of whole numbers fall away: no next number, no reading of the largest by the largest digit, and multiplying no longer always makes things bigger.