3/5. The wall always measures from the left edge, so the third piece selects the first three fifths: 3 of the 5 equal parts of one whole.

Halves are biggest (0.5 each), twelfths smallest (0.083̇ each). Every row is one whole, so the more pieces it is cut into, the smaller each piece is.

Every row ends at d/d (2/2, 3/3 … 12/12), and all of them equal 1 whole. A good moment to say that any number over itself is 1.

It meets an edge on the even rows (2, 4, 6, 8, 10, 12) and cuts through a piece on the odd rows (3, 5, 7, 9, 11). A row has a fraction equal to 1/2 exactly when its denominator is even.

3/4 is bigger by 1/12 (8/12 against 9/12). The orange band reaches further than the blue one, and the gap between the two dashed lines is the difference.

1/2, 2/4, 3/6, 4/8, 5/10 and 6/12 (six, counting 2/4 itself). Each denominator is double its numerator: multiplying the top and bottom by the same number keeps the value.

1/3 = 2/6 = 3/9 = 4/12, so four rows (3, 6, 9 and 12). 2/3 has the same four: 2/3, 4/6, 6/9, 8/12. Only rows that are multiples of 3 can show thirds.

Six: 3/4, 6/8, 9/12, 12/16, 15/20 and 18/24, one on each row that is a multiple of 4. With rows 1–12 there are three.

Each row cuts every piece of the row above in half, so one piece always equals two pieces of the row below: 1/2 = 2/4 = 4/8 = 8/16. The numerator and the denominator both double.

With rows 1–12 there are 34: every fraction in simplest form with a denominator of 7 or more (all the sevenths and elevenths, and fractions such as 3/8, 5/9, 7/10 and 5/12). A fraction has a partner only if the wall has another row that is a multiple of its simplest denominator.

Side by side, each third splits into exactly two sixths and each sixth into two twelfths, so the edges line up in a chain. Reordering rows into families (3, 6, 12) makes equivalence easy to see.

Thirds and quarters meet only at 0 and 1 (their highest common factor is 1). Quarters and sixths meet once, at 1/2. In general two rows share HCF − 1 edges between 0 and 1: thirds and twelfths share 2, sixths and twelfths share 5.

All three: 6/12 = 1/2, 4/12 = 1/3, 3/12 = 1/4. The simplest form is always on the highest row (the smallest denominator) that has an equal fraction.

8/12 = 6/9 = 4/6 = 2/3, and 2/3 is simplest. One step: divide the top and bottom by 4, the highest common factor. Two steps: ÷ 2 to get 4/6, then ÷ 2 again.

Going up, the first exact edge is on the eighths row: 6/8, which is not simplest. Keep going to 3/4 on the quarters row. The simplest form is the equal fraction with the smallest denominator, not the nearest one.

Twelfths: 1/12, 5/12, 7/12 and 11/12, four of them, because those numerators share no factor with 12. Tenths: 1/10, 3/10, 7/10 and 9/10, also four.

Elevenths have the most: all 10 fractions from 1/11 to 10/11, because 11 is prime (sevenths have 6). Halves have the fewest, just 1/2. Prime rows have d − 1; rows with many factors, such as 6 and 12, have few for their size (2 and 4).

2/3 is bigger: 9/15 < 10/15, a difference of 1/15. The fifths and thirds rows share no edges, so the gap is small but visible on the number line.

1/3 > 1/4, by 1/12. A bigger denominator means the whole is cut into more parts, so each part is smaller. This is the most common fraction comparison misconception.

The second is bigger each time, because its gap to 1 (1/5, 1/6, 1/7) is smaller. So 10/11 > 9/10: it is 1/11 away from 1 rather than 1/10. Comparing the gaps to a whole is a powerful strategy.

All three are less than 1/2, because double the numerator (6, 8, 10) is less than the denominator. 4/7, 5/9 and 6/11 are each just over 1/2.

The smallest is 1/132, between 1/12 and 1/11 (and between 10/11 and 11/12). Two fractions differ by at least 1 ÷ (product of the denominators), and 11 × 12 = 132 is the biggest product available.

Rows 9 to 12 are hardest to tell apart by eye: 1/11 and 1/12 differ by only 1/132 of the whole. Counting the pieces is more reliable than judging their width. Use 🔀 Mix up in ⚙ Rows for another go.

7/12 < 3/5 < 5/8 < 2/3 (0.583̇, 0.6, 0.625, 0.6̇). The closest pair is 7/12 and 3/5, which differ by 1/60.

They stop for 1/2, 1/4, 1/5, 1/8 and 1/10, whose denominators are made only of 2s and 5s. They recur for 1/3, 1/6, 1/7, 1/9, 1/11 and 1/12. The dots mark the repeating block: 1/6 = 0.16̇.

Each uses the same six digits, 1 4 2 8 5 7, in the same cycle, starting at a different place: 1/7 = 0.1̇42857̇, 2/7 = 0.2̇85714̇, 3/7 = 0.4̇28571̇.

Six: 1/3, 2/6, 3/9, 4/12 (all equal to 1/3, 33.3̇%), 3/8 (37.5%) and 4/11 (36.36̇%). 3/10 and 2/5 are exactly 30% and 40%, so they are not between.

Both are 0.3̇ = 33.3̇%. Equal fractions are the same number, so every way of writing them must match. 33% is only an approximation: 1/3 is a little more.

Eighths go up in 12.5% and twelfths in 8.3̇%. No row goes up in 15%: that would need 100 ÷ 15 = 6.6̇ pieces. 15% = 3/20, so it is three pieces of the twentieths row (rows 1–24).

5/8 = 0.625 = 62.5%. It sits exactly halfway between 1/2 (0.5) and 3/4 (0.75).

1/3 + 1/4 = 4/12 + 3/12 = 7/12. Both are renamed as twelfths, the first row that both thirds and quarters fit into exactly, so the pieces can be counted together.

1/2 + 1/3 = 5/6, on the sixths row. 1/2 + 1/4 = 3/4: quarters already fit into halves, so there is no need to go down to eighths. The common denominator is the lowest common multiple.

2/6 = 1/3, so it lights up on the thirds, sixths, ninths and twelfths rows. 3/6 = 1/2 lights up on every even row, and 4/6 = 2/3 on rows 3, 6, 9 and 12.

Four pairs of values: 1/6 + 5/6, 1/4 + 3/4, 1/3 + 2/3 and 1/2 + 1/2 (written in other ways too, such as 2/6 + 2/3). The second is always 1 minus the first: it fills the rest of the row.

2/3 + 1/2 = 4/6 + 3/6 = 7/6 = 1 1/6. The strip fills one whole and carries 1/6 on to a second line.

3/4 − 1/3 = 9/12 − 4/12 = 5/12. The 1/3 strip sits under the 3/4 strip and the gap is filled in: the difference is how much longer one is than the other.

1/2 − 3/4 = −1/4: the 3/4 strip overhangs the 1/2 strip by 1/4. Swapped, 3/4 − 1/2 = 1/4. Swapping a subtraction changes the sign of the answer; swapping an addition changes nothing.

3/4 = 1/2 + 1/4, 5/6 = 1/2 + 1/3, 7/12 = 1/3 + 1/4 (or 1/2 + 1/12), 2/3 = 1/2 + 1/6. All four work with two different unit fractions from the wall.

The claim is true: 1/a + 1/b = (a + b)/(ab). The wall uses the lowest common denominator, so its answer looks simpler whenever a and b share a factor: 1/4 + 1/6 = 5/12 rather than 10/24.

It stays put. 1/4 and 3/12 are the same number, so they sit at the same point on the line. Fractions are numbers, not just pieces.

5/8 is exactly halfway between 1/2 (4/8) and 3/4 (6/8), so it is equally near both.

2/5 is bigger by 1/15 (6/15 against 5/15). On the line both are measured from the same 0 on the same line, so a small gap is easier to see than when comparing pieces on two different rows.

Adding a positive amount always moves you to the right, so the sum is bigger than each part. A subtraction answer is to the left of the first number.

1 + 2 + … + 12 = 78. Up to 24 it is 300. The total for rows 1 to n is n(n + 1) ÷ 2, a triangular number.

1/2: 6. 1/3: 4. 1/4: 3. 1/5: 2. 1/6: 2. It is the number of multiples of the denominator up to 12, which is 12 ÷ d rounded down.

For example 1/2 + 1/3 + 1/6, 1/2 + 1/4 + 1/6 + 1/12, or 1/3 + 1/4 + 1/4 + 1/6. The panel adds exactly, so near misses such as 1/2 + 1/3 + 1/7 = 41/42 show up.

1, 2 and 3 pieces, and every colour totals 1/2. Different numbers of pieces, the same amount: equivalence again.

1 + 1/2 + … + 1/12 = 86021/27720 ≈ 3.10, just over 3. Without the whole it is about 2.10. Keep adding unit fractions and the total grows past any number, but very slowly.

45. Each place is a fraction in simplest form with a denominator from 2 to 12, and there are 1 + 2 + 2 + 4 + 2 + 6 + 4 + 6 + 4 + 10 + 4 = 45 of them. In order, they are the Farey sequence of order 12.

With labels, students can match numbers; without them they have to line up edges, which tests the idea that equal fractions are the same length.

Any piece that reaches the right value counts, so 6/8 is accepted for 3/4. Watch for students who always choose the fraction with the bigger denominator.

There is often more than one answer on the wall. Between any two different fractions there are infinitely many more (their mean, for a start), but not all of them are on this wall.

The simplest form is the equal fraction on the highest row, the one with the fewest, biggest pieces. Tapping an equal fraction that still simplifies gets the message “equal, but it simplifies further”.

2/5 is less than 1/2 on its own, so it can’t be 1/2 plus something more. The real answer is 5/6. Adding the tops and the bottoms always lands between the two fractions, never beyond the bigger one.

3/4 is bigger: 3/4 = 6/8, so it is double 3/8. With the same numerator, more pieces in the whole means each piece is smaller.

2/3 = 6/9, not 2/9: the pieces are different sizes. Counting shaded pieces only works when the pieces are the same size, so compare lengths.

Students have to count the pieces in the row for the denominator and the pieces selected for the numerator. Rows 9 to 12 are hard to tell apart by width, so counting is the honest method.

1/3 = 0.3̇ is bigger than 3/10 = 0.3, by 1/30. 0.3 is a common wrong answer for 1/3; the dot shows that the 3s never stop.

Two rows share a fraction strictly between 0 and 1 exactly when their denominators have a common factor bigger than 1. Thirds and sixths share 1/3 and 2/3; quarters and sixths share only 1/2; fifths and sevenths share nothing.

Every row must be a multiple of 6: with rows up to 24 that is 6, 12, 18 and 24, so the smallest is sixths. A row that shows halves and thirds must be a multiple of both 2 and 3.

1/3 + 1/2 = 5/6, and half of that is 5/12, on the twelfths row. The halfway point is the mean of the two fractions; it always exists, but it might need a denominator the wall doesn’t have.

Good questions start with a prediction and can be answered with the tool as it is. The share link means your partner starts from exactly the same wall.