With Cuisenaire rods, a fraction is the length of one bar divided by the length of the whole you choose. If the whole is 6, the 3 and 2 rods make a row of 5 units, and that is 5/6. Each whole allows only the denominators that divide it, and the table below collects them.
This guide works with bars lined up by length. To see the same principle with columns in a wall, look at the fraction wall; to draw bars and circles for any fraction, the guide to drawing fractions and the tool do that job.
Which fractions each whole allows
Rods measure 1 to 10 units and are named by length: the 4 rod is four units long. For a rod to split into n equal parts made of whole rods, n must divide the whole and each part must measure 1 to 10 units. The whole can be a single bar or a strip of several, such as the 10 and 2 rods, which make a whole of 12.
| Whole | Exact parts | Not possible with this whole |
|---|---|---|
| 2 | halves (1) | thirds, fourths, fifths, sixths |
| 3 | thirds (1) | halves, fourths, fifths, sixths |
| 4 | halves (2), fourths (1) | thirds, fifths, sixths |
| 5 | fifths (1) | halves, thirds, fourths, sixths |
| 6 | halves (3), thirds (2), sixths (1) | fourths, fifths |
| 7 | none of the five | halves, thirds, fourths, fifths, sixths |
| 8 | halves (4), fourths (2) | thirds, fifths, sixths |
| 9 | thirds (3) | halves, fourths, fifths, sixths |
| 10 | halves (5), fifths (2) | thirds, fourths, sixths |
| 12 (strip 10 + 2) | halves (6), thirds (4), fourths (3), sixths (2) | fifths |
The 6 and the 12 are the most useful wholes: they bring halves and thirds together, and the 12 adds fourths and sixths. If the number of parts does not divide the whole, change the whole before forcing the drawing: with a whole of 6, fourths would measure 1.5 units, and that rod does not exist.
Adding one half and one third with the whole 6
The whole 6 works because 6 is the least common multiple of 2 and 3. It is the reference example of this guide.
Worked example: 1/2 + 1/3 over a whole of 6
With the 6 rod as the whole, what length do one half and one third make together?
- Choose the whole. The 6 rod.
- Find the half. Two copies of the 3 rod fit in the whole, so the half is the 3 rod.
- Find the third. Three copies of the 2 rod fit, so the third is the 2 rod.
- Line them up in a row. 3 + 2 = 5 units.
- Read the fraction against the whole. 5 units out of 6 is 5/6.
Result. 5/6 of the whole.
Check. The row of 5 units leaves a single gap of 1 in the whole.
The same rod with a different whole
A rod has no fraction of its own: it gets one from the whole. The 6 rod fills a whole of 6 completely and is worth 1, but it takes up half of a whole of 12. No bar moved; the unit changed. The guide to the same whole explains why two fractions can be compared or added only when they share that whole.
Comparing two thirds and three fourths with the whole 12
With the whole 12 there is a rod for thirds and for fourths. Two thirds are two 4 rods, which is 8 units. Three fourths are three 3 rods, which is 9 units. Since 8 is less than 9, the bar for three fourths sticks out by one unit.
8/12 < 9/12
The comparison needs the same whole and nothing else. The fraction number line puts the two results at neighboring points, and the equivalent fractions tool confirms each conversion to twelfths.
A 15-minute lesson with rods
The sequence moves from naming to operating without changing material. Each pair needs one set of rods.
- Minutes 1 to 3. Each pair makes the whole 6 with one rod and finds which rod is the half and which is the third.
- Minutes 4 to 7. They change the whole to 12 with the 10 and 2 rods, and name the 6, 4, 3 and 2 rods.
- Minutes 8 to 11. They add the half and the third over the whole 6 and write down the length and the fraction.
- Minutes 12 to 15. They compare the two fractions over the whole 12 and explain which bar sticks out.
One final question closes the lesson: what changes if the whole goes from 6 to 12? The expected answer is that the bars keep their length and the fractions change. This sequence is a suggested plan, not the result of a classroom trial.
Practice
Three exercises with different wholes
- With the whole 8, which rod is 1/4?
- With the whole 10, what is 1/2 + 1/5?
- With the whole 12, what is 3/4 − 1/6?
Show the answers
- The 2 rod, because four copies fill 8.
- 5 + 2 = 7 units out of 10, which is 7/10.
- 9 − 2 = 7 units out of 12, which is 7/12.
To make more exercises with answers, the fraction worksheet generator lets you choose the operation, and fraction practice checks your own answers.
Sources
The Wikipedia article Cuisenaire rods describes a set of rods from 1 cm to 10 cm long, calls Georges Cuisenaire a Belgian primary school teacher, says Caleb Gattegno popularized the set in the early 1950s, and lists fractions among the topics taught with it (accessed October 17, 2026).