What the insets are, physically
A wooden stand holding ten metal circles. Each circle is about 12cm across, cut from steel or iron, painted red.
The first circle is one whole, unbroken. The second is divided into two equal halves. The third into three equal thirds. The fourth into four quarters. And so on, up to the tenth, divided into ten equal tenths. Each piece has a small knob on top; the child lifts pieces out by their knobs, places them in various arrangements, returns them.
The frames are green. The metal inserts are red. The colour scheme mirrors the Metal Insets (the language-area pencil-control material, which is also red in green frames) by a kind of visual pun: both are "metal insets"; both are worked with in similar ways. Many Montessori families use the same pencil-control rotations with the fraction insets as with the language insets, tracing around pieces and hatching them.
A commercial set costs £80-150 in the UK. Second-hand sets often £50-90. DIY is possible with plywood and careful cutting, though the metal precision of a commercial set is hard to match.
The first work: wholes and parts
The first meeting with the insets is simply taking them apart and putting them back together.
Invite the child. Carry the stand to the work rug. Take out each circle's pieces and lay them on the rug, jumbled. Then reassemble each circle, piece by piece.
The child learns, physically, that two halves make a whole. That three thirds make a whole. That four quarters make a whole. And so on. The vocabulary (half, third, quarter, fifth, sixth, seventh, eighth, ninth, tenth) is introduced through the three-period lesson alongside.
A child who has done this work for a fortnight knows, physically, that a fraction is a division of a whole into equal parts. This is the fundamental definition of a fraction; it is laid in the child's hands before they see a fraction written on paper.
Equivalence
The second major work. Starting from around six.
The child takes two quarters from the fourths circle. They lay them next to one half from the halves circle. They see that the two pieces are the same size; two quarters equals one half.
Extended: three sixths equals one half. Two tenths equals one fifth. Five tenths equals one half. The child discovers equivalences by fitting pieces together and noticing which groups match which.
A variation: the child is given one piece (say, a quarter) and asked to find other combinations that equal it. They might find that two eighths equals a quarter; or that five twentieths (if a twentieths inset is available; not standard) equals a quarter.
Equivalence work takes months and leads naturally into the written representation of equivalent fractions (1/2 = 2/4 = 3/6 = 5/10). The written representation is just a notation for what the child has already handled.
Operations
Addition, subtraction, multiplication and division of fractions all follow.
Addition. "One half plus one quarter." The child takes the half piece and the quarter piece, places them together. They see that the combined pieces fill three quarters of the whole. Written: 1/2 + 1/4 = 3/4.
The child discovers the rule for adding fractions (common denominator, add numerators) by doing many additions physically and noticing the pattern. They are not taught the rule; they derive it.
Subtraction. "Three quarters minus one half." The child starts with the three quarters piece, removes the half, sees what remains (one quarter). Works similarly for subtraction with borrowing (where the piece to subtract is larger than the immediately-available piece).
Multiplication. "Two times one third." Two copies of the third piece laid together. Results in two thirds.
Division. "One half divided by two." One half piece; the child finds a piece that, taken twice, equals the half (one quarter). One half divided by two equals one quarter.
The work extends progressively. By seven or eight, a child working steadily with the fraction insets can do all four operations on fractions with confidence. They meet the written forms (1/2 × 3/4 = 3/8) as descriptions of what they have already done with pieces.
Decimal fractions and the bridge
A related material: the decimal fraction inserts. Less commonly owned in UK home Montessori but worth naming. A wooden stand with cards or wooden pieces for 1, 0.1, 0.01, 0.001, representing decimal fractions as place values.
The child works out that 1/10 = 0.1; 1/100 = 0.01; 1/1000 = 0.001. This links the earlier Golden Bead work (which was about whole numbers and the decimal system) to the fraction work (which is about parts of wholes). The two strands meet.
The decimal fraction material is £30-60 from UK suppliers; optional. Many home families do the work with the standard fraction insets plus paper-and-pencil decimal introduction.
Common home mistakes
Using printable paper circles instead of metal insets. Printable paper circles wear out and cannot be fitted together precisely. The metal is the point; the physical manipulation is what teaches.
Introducing operations before wholes-and-parts is consolidated. The child must see that "two halves is one whole" physically many times before they can meaningfully add "one half plus one quarter". Several weeks of wholes-and-parts before moving to equivalence.
Using formal fraction language too soon. "Denominator" and "numerator" come after the child has done the physical work. Early language is "the bottom number tells you how many parts the whole is broken into; the top number tells you how many pieces you have". Formal terminology later.
Mixing the stand with other red-green-metal materials. The Metal Fraction Insets and the language Metal Insets look similar; storing them side-by-side confuses. Clear labelling and separate shelves.
Treating the insets as a toy rather than a work material. The pieces are strong and not prone to breaking; children sometimes use them as throwing objects or to build non-fraction structures. A clear "this is the fraction work; it stays on the rug" boundary helps.
A real family's fraction year
A family we will call the Achiengs bought the Metal Fraction Insets for their daughter's sixth birthday. The daughter had worked with the Golden Beads and Stamp Game for over a year; whole-number arithmetic was consolidated.
September: introduction and wholes-and-parts. The daughter worked with the stand twice a week for about twenty minutes. By November she could name and identify all ten fractions.
December to February: equivalence. She discovered that two quarters equals a half on her own (without prompting) during a session in January. She then sat and found equivalences for most of the other fractions over a couple of weeks.
March to June: addition and subtraction. Combining pieces, subtracting pieces, recording the operations on paper alongside.
July and August: multiplication and division. Slower; the Achiengs did shorter sessions daily during the summer holidays.
By September the following year, the daughter could do fraction addition, subtraction, multiplication and division on paper with full underlying understanding. The insets were still on the shelf and she returned to them when she hit a harder problem.
Cost: £85 for the insets. Time: about two sessions a week of twenty minutes each for nine months. Outcome: a seven-year-old with rock-solid fraction skills.