The short chains
Ten chains, one for each number from 1 to 10. Each short chain is made of that-number bead-bars (the 5-chain is five 5-bars strung together, totalling twenty-five beads; the 6-chain is six 6-bars totalling thirty-six beads; the 10-chain is ten 10-bars totalling one hundred beads).
The child lays out the short chain in a straight line on a small work rug. They walk alongside it with a pointer, counting each bead. Then they skip-count by placing small number labels at the end of each bead-bar: for the 5-chain, "5, 10, 15, 20, 25". The skip-counts are the multiplication table for that number.
The square of each number is formed by folding the chain. Five 5-bars laid out straight is a chain; stacked as five rows of five is a square. The square is literally 5×5 in beads. The child sees that the chain, folded, makes the square.
A short chain set (all ten chains plus labels) costs £30-60 new. The labels are small laminated cards; a DIY set is practical.
The long chains
Two chains. The 100-chain is a hundred unit-beads strung in ten 10-bars. The 1000-chain is a thousand unit-beads strung in one hundred 10-bars. Both are considerably longer than any other Montessori material.
The 100-chain, laid out, is about two and a half metres long. A UK kitchen or a long hallway can accommodate it. The child walks alongside and counts the beads in tens, placing a small label at each 10-bar: 10, 20, 30, ..., 100. The chain of 100, when folded, forms the 100-square (a hundred-square, same as the Golden Bead material). The child has now seen that a hundred is a chain, and a chain of a hundred is a square of ten-by-ten.
The 1000-chain is roughly ten to twelve metres long depending on bead size. A UK family typically lays it out down a hallway and through two rooms. The child walks alongside and counts the beads in hundreds: 100, 200, 300, ..., 1000. The chain of 1000, folded, forms the 1000-cube.
This is a memorable physical experience. A four- or five-year-old's encounter with the 1000-chain rolled out through the house is the sort of thing that gets photographed. More importantly, it gives the child an embodied sense of what "a thousand" is. Thousand is no longer an abstract digit; it is the length of a walk.
Skip counting and multiplication tables
The short chains are where the multiplication tables emerge naturally.
A child walking the 2-chain counts "2, 4, 6, 8, 10, 12, 14, 16, 18, 20". That is the two times table. A child on the 3-chain counts "3, 6, 9, 12, 15, 18, 21, 24, 27, 30". The three times table. The same for 4, 5, 6, 7, 8, 9 and 10.
The child does not memorise the table from a sheet. They count the beads, with the labels placed along the chain, and the pattern emerges from the counting. After a few repetitions of the 6-chain, the child knows 6, 12, 18, 24, 30 because they have said these numbers walking along a physical object that justifies them.
The UK Year 4 times-tables check (a statutory assessment in state schools) expects children to know the 2s through 12s by age nine. A Montessori child who has worked through the short chains at five or six meets this expectation organically; there is no separate "learn the times tables" task, because the tables have already been walked.
Cubes and cube chains
The short chains can be extended into cube work. A 3-chain folded becomes a 3-square (3×3). Three 3-squares stacked become a 3-cube (3×3×3 = 27).
A dedicated cube chain material exists for each number: the 3-cube chain is three 3-squares strung together, totalling twenty-seven beads. The child lays out the cube chain; counts the beads; folds them into squares; stacks the squares into a cube. They have built the cube of 3.
This extends the sensorial experience into indirect preparation for later power-of-a-number work (x², x³) in secondary maths.
UK floor space
A practical note: the 1000-chain requires twelve metres of clear floor. Most UK homes can just about manage this if the main walls of a ground floor are cleared or if a hallway and a room can be used together. Laying the chain out is a weekend project, often done once or twice.
The 100-chain is easier; two and a half metres fits in most living rooms. The short chains need less than a metre each.
For families with very limited floor space, the 1000-chain can be worked with in two halves: a 500-bead portion laid out, walked, counted, then the next portion. This loses some of the "one long walk" experience but preserves the pedagogical content.
Common home mistakes
Not rolling out the 1000-chain. The chain of a thousand is the point; having the material and never laying it out misses what it is for. Even once a term is enough; the memory of the walk lasts.
Introducing the chains in the wrong order. Short chains come first, at around four to five. Long chains come later, at around five to six, after the child has some decimal-system sense from the Golden Beads.
Using the chains as decoration. The chains are striking-looking and often sit strung on shelves without being walked. Treat the shelf display as storage, not as work.
Skipping the cube work. The folding and stacking of the squares into cubes is the part that links to secondary algebra later. Doing only the linear counting loses half the material.
Forgetting the labels. The small number labels placed at each bead-bar along the chain are what turn a walk into a skip-count. Without the labels, the child is just pointing at beads.
A real family's chain year
A dad we will call Marcus introduced the short chains to his daughter when she was five. He had bought a second-hand set of all ten short chains and the 100-chain for £50.
Months one to four: short chains in ascending order, one or two per month. She walked the 2-chain, then the 3-chain, up to the 10-chain. By month four she had counted all ten short chains several times and had the two, three, five and ten times tables fluently without having been taught them.
Month five: the 100-chain. Laid out on a Saturday morning across the sitting-room rug. She walked alongside, counted in tens, placed the labels. Folded it into the 100-square alongside their Golden Bead hundred-squares; noticed that they matched. Counted the whole thing twice more that week.
Month seven: Marcus bought a 1000-chain on eBay for £45 and laid it out one Sunday along the upstairs hallway and through two bedrooms. His daughter walked the whole length twice, placing the hundred labels. She wanted to do it again the following weekend. He laminated the photograph of the walk and it hangs in her room.
By the end of year one with the chains, she had the full times tables to ten, understood the square and cube operations physically and had embodied the magnitudes 100 and 1000.