Fractions on a number line (age 8+)

Activities to try

1. 1

   Build fourths: three cuts, not four marks

   Why it works: Four equal parts require three interior cuts (b equal parts need b−1 interior marks); a/b is measured as a LENGTH from 0 (a jumps of 1/b), not as a count of tick marks. Drawing four equally spaced interior marks cuts the unit into five equal parts (fifths), so calling them fourths mislabels the pieces — making the tick-counting error visible so the child can correct it.

   1. Start with a bare line labelled only 0 and 1 — the 0–1 segment is the unit.
   2. To make 'fourths', first try the common mistake: drop FOUR equally spaced marks inside the 0–1 segment. Notice this cuts the unit into FIVE equal parts, each one a fifth (1/5) — the far interior mark sits at 4/5. So you have actually made fifths, not fourths: a sign something has gone wrong.
   3. Correct to THREE interior marks so the unit is cut into exactly four equal parts (quarters); the marks land at 1/4, 2/4, 3/4, and a fourth quarter-length jump lands exactly on 1 (4/4).
   4. State the rule in your own words: b equal parts need b−1 interior marks.
   5. Place 3/4 by jumping the quarter-length three times from 0, landing one quarter-length short of 1.
2. 2

   Read the point, then place the fraction

   Why it works: Reading a given point (interpret) and placing a given fraction on a blank line (produce) are both needed; a fraction is the same size / the same point however it is represented.

   1. On a 0–1 line already divided into 6 equal parts, with a dot on the 4th mark from 0, say what fraction the dot shows and explain how you know (four 1/6-lengths from 0 = 4/6).
   2. On the same line, find and mark 1/2 (the 3rd mark, i.e. 3/6) and explain why it sits halfway.
   3. Then take a blank 0–1 line and place 3/4 yourself: partition into 4 equal parts, count 3 parts from 0, land three-quarters of the way to 1.
   4. Talk through each step aloud as you place it.
   5. Also try ordering several fractions along one line to compare their positions.
3. 3

   Which segment is the unit?

   Why it works: The unit is the 0–1 segment, not the whole drawn picture; with 0 and 1 kept in place, a/b keeps its position no matter how far the line is drawn. Lines extending past 1 let children show, and teachers address, errors such as placing 3/4 at or beyond 1, or 'between 3 and 4'.

   1. On a line that extends past 1 (to 2 or beyond), first find and mark where 1 is.
   2. Place 3/4 and check it sits below 1, NOT at three-quarters of the whole drawn line.
   3. Answer: if this line went all the way to 2 instead of 1, would 3/4 move? Explain why not.
   4. State that everything is measured against the fixed 0–1 length, whatever the drawing shows.