A fraction wall is a stack of bars of the same length, each split into equal parts, so that edges lining up vertically mark equivalent fractions. With it you can see at a glance that one half is the same as three sixths (1/2 = 3/6), which fraction is greater and what a sum comes to. If you would rather have a drawing made with your own fraction, the visual fractions tool has a wall view.
| You want | Look at | Section |
|---|---|---|
| An equivalent fraction | Another span that ends at the same edge | How to read equivalences |
| Which is greater | Which one reaches further right | How to compare using the edges |
| To add | A row that holds both partitions | Worked example |
| To simplify | The row with the fewest pieces that matches | How to read equivalences |
Worked example: adding 1/2 + 1/3 on the sixths row
You want to know how much one half and one third of the same bar make together.
- Find a row that holds both partitions. One half equals 3 sixths and one third equals 2 sixths, so the sixths row works.
- Shade the first addend. 1/2 takes up 3 cells of the sixths row.
- Shade the second one right after it. 1/3 takes up 2 more cells.
- Count the covered cells. That is 3 + 2 = 5 cells out of 6.
Result: 5/6 of the bar.
Check: one cell is missing to reach the right edge, and 1/6 + 5/6 = 1.
Why the rows have to be the same length
A fraction describes a part of a whole, and the wall can be trusted only if that whole is the same in every row. The left edge is 0, the right edge is 1, and each cut falls at the same value in every row that shares it. If you draw one strip shorter than another, half of one no longer lines up with half of the other and the columns stop being comparable. The guide on fractions of the same whole goes into that requirement.
A simple wall can have rows of 2, 3, 4, 6 and 12 parts, because 12 is a multiple of all those denominators and the cuts line up. Ninths do not fit on a strip split into 12, and eighths need a length that is a multiple of 8, like the 24 cm strip below.
How to read equivalences
Two fractions are equivalent when their spans end at the same edge, and the value of the span does not change even though the pieces do. Find the fraction you know, go straight down, and note which spans in other rows end there.
| Fraction | Row | Span of the whole |
|---|---|---|
| 1/2 | halves | 50% |
| 2/4 | quarters | 50% |
| 3/6 | sixths | 50% |
| 6/12 | twelfths | 50% |
All four spans end at the same edge, the middle of the bar. Multiplying the numerator and denominator by the same number is multiplying by 1, which is why the drawing and the arithmetic agree. Thirds form another family, such as 2/3 = 4/6 = 8/12, and quarters another, such as 3/4 = 6/8 = 9/12. To simplify, go the other way: 10/12 ends where a span of 5 pieces on the sixths row ends, so it simplifies to 5/6. If no row with fewer pieces matches at that edge, the fraction is already in lowest terms. To generate or check a particular pair, use the equivalent fractions calculator.
How to compare using the edges
To compare, find a row that holds both fractions (twelfths for 2/3 and 3/4) and see which one reaches further right: 8/12 against 9/12. The second span covers one more twelfth, so 2/3 is less than 3/4.
Before drawing, estimate with the benchmarks 0, one half and 1 and rule out impossible orders. The guide on benchmark fractions explains that estimate. When you have a long list, order fractions applies the same rule to all of them at once, and compare fractions settles a pair with cross multiplication.
How to add on the wall
Adding requires a row that contains the edges of both fractions, and that row is the one for the least common multiple (LCM) of the denominators, or a multiple of it. With thirds and quarters it is twelfths; with halves and thirds it is sixths, as in the example above. The numerators are counted as cells, and the denominator is the name of the row.
The guide on why denominators are not added covers this mistake in more depth, and add fractions does the calculation with denominators the wall does not reach.
What the wall cannot show
The wall holds only the denominators you drew. With sevenths, elevenths or thirteenths there is no row to look at, and the common denominator has to be calculated. A wall of a single unit shows only values from 0 to 1, so an improper fraction or a mixed number falls outside it. To draw several units, the guide on drawing fractions explains which model to choose in each case. A wall drawn by eye, with unequal parts, produces false equivalences. With large denominators, use it to check a result rather than to find one.
How to draw one by hand
Choose a strip whose length can be divided by 12 in whole centimeters. At 24 cm, each twelfth is 2 cm, and the eighths also come out exact, at 3 cm.
- Draw the row for the whole unit, with no cuts.
- On the halves row, mark a cut at 12 cm.
- On the thirds row, mark cuts at 8 and 16 cm.
- On the quarters row, mark cuts at 6, 12 and 18 cm.
- On the sixths row, mark cuts at 4, 8, 12, 16 and 20 cm.
- On the twelfths row, mark a cut every 2 cm.
Always measure from the same end and never space the cuts by eye, because any drift breaks the alignment. If in class or at home you would rather work with material you can touch, Cuisenaire rods use the same idea with colored bars, and the whole changes depending on the rod you choose.
Practice with the wall
- Write a fraction in twelfths that is equivalent to 1/4.
- Which is greater, 5/12 or 3/6?
- Add 1/6 + 3/4 on the twelfths row.
See the answers
- 3/12, because 12 ÷ 4 = 3 and 1 × 3 = 3.
- 3/6, which is 6/12 and is greater than 5/12.
- 11/12, because 2/12 + 9/12 = 11/12.
Sources
The English Wikipedia article Fraction says that multiplying the numerator and denominator of a fraction by the same non-zero number gives an equivalent fraction, and that one way to compare or add fractions with different denominators is to find a common denominator (accessed October 17, 2026). That is the idea behind the wall: two spans that end at the same edge are the same amount with a different name. The wall is just a visual way to see what is written there with numbers.