Fractions on a number line

Activities to try

1. 1

   Three marks make fourths: partition and place on an interactive line

   Why it works: A fraction 1/b comes from cutting one unit into b equal segments (needing b−1 interior marks), and a/b is the point at the end of the a-th segment. The unit stays 'one whole' even when the line gets longer, so 3/4 lives between 0 and 1, not between 3 and 4. Equivalent fractions are the same point under different labels.

   1. On a 0-to-1 line, set a 'cut this unit into ___ parts' control to 4 and watch exactly THREE marks drop in, splitting the line into four equal segments.
   2. Say aloud how many marks made four parts (three, not four), and colour each of the four segments, counting SEGMENTS rather than marks.
   3. Drop a bead on the third mark; if the line reports a fraction, check it announces '3/4' and explain why that point is 3/4.
   4. Extend the line from 0-to-1 to 0-to-4, keeping each whole-number unit the same length (or redraw it that way), and notice the 3/4 bead stays three-quarters of the way from 0 to the first '1', nowhere near the '3'.
   5. On a second track below, lined up with the same 0 and 1 and labelled in tenths, slide a 1/2 bead until it lands exactly on 5/10, and state that the two labels name the same point.
2. 2

   Place and reason on paper lines: pre-marked, landmark, then unmarked

   Why it works: Placement is driven by magnitude and distance from landmarks, not by counting ticks. Children who can partition an unmarked line themselves have internalised that the unit is divided into equal segments, and they keep fractions within the correct whole-number interval.

   1. On a 0-to-1 line already divided into four equal segments (fourths), place 1/2, 1/4 and 3/4 by counting equal segments from 0.
   2. On a minimally-labelled line showing only 0, 1/2, 1, 1½ and 2, place a concrete fraction such as 1¾ using landmarks — reasoning that it sits just below 2 (between 1½ and 2) — rather than counting every mark.
   3. On an UNMARKED 0-to-1 line, decide how many segments you need, make the b−1 marks yourself, then place the fraction.
   4. Answer landmark-reasoning questions such as 'Is 1/12 closer to 1/4 or to 0?' and 'Is 3/4 closer to 0 or to 1?', explaining using distance.
   5. On a 0-to-4 line, place both 3 and 4, then separately place 3/4, and say in your own words why 3/4 does not go between 3 and 4.
3. 3

   Bridge from the shape model and compare unit fractions

   Why it works: A fraction names one number and has a single location on the line; the area (shape) model and the length (line) model agree. More equal parts of the same unit means each part is smaller, so 1/8 is closer to 0 than 1/3 — the denominator counts how many parts split ONE whole.

   1. Shade 3/4 of a fraction strip exactly as long as the 0-to-1 unit, starting the shading at one end, then lay that end at 0 along the line so the shaded part reaches a point; name that point as the number 3/4.
   2. Partition one unit into thirds, then partition an identical unit into eighths, and compare the lengths of a single segment in each.
   3. Place 1/3 and 1/8 on the same 0-to-1 line and decide which is further from 0, explaining using segment length.
   4. State in your own words whether a fraction such as 3/4 is 'a number' or 'just a piece of a shape', and point to where it lives on the line.
   5. Reference check: label 0, 1/4, 1/2, 3/4, 1 on a line and explain for each point which segment's end it marks.