Break a number into parts and rebuild it, including the ten-and-some-ones structure of the teens.
NUM.COMPOSE_DECOMPOSE“I can break a number into parts and put it back together in more than one way.”
Give three different part–part–whole splits of 9, then show 14 as ten and four.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Kindergarten | Parts of numbers | intro | introduce | ✓ |
A whole with two or three parts joined by arcs, used for part-whole reasoning in either direction.
Fact families become one picture instead of four sentences.
engine: node-and-arc scene with value slots
A 2x5 grid that makes "ten" a single perceptual chunk and shows how far a number is from ten.
Makes make-a-ten strategies visible instead of memorised.
engine: grid scene with slot occupancy + fill animation
Units, rods, flats, and cubes traded ten-for-one to make regrouping physical.
Regrouping is a trade, not a superscript "1".
engine: stackable prop actors with trade animation
Scene archetype: number_bond
NUM.DECOMP.ONE_WAYBelieves a number has a single correct decomposition.
Repair: Collect every split the class finds and display them together.
One row per instructional system. Rows marked cluster carry authored treatment for this family of topics; rows marked template are derived from the system’s general pattern and are not topic-specific research. Confidence is recorded on every row.
Place value is developed as an explicit progression: teens as ten-and-some-ones in Kindergarten, bundling in Grade 1-2, and in Grade 4 the deliberate statement that each place is ten times the one to its right, which is then extended rightward into decimals in Grade 5.
Place-value discs are preferred to base-ten blocks because the same manipulative extends to decimals: a disc labelled 0.1 behaves exactly like a disc labelled 10. Trading ten-for-one is rehearsed physically before any written algorithm.
Positional notation is taught as a system with a stated rule rather than as a set of column names, and comparison of multi-digit numbers is justified place by place. Numbers into the millions appear earlier than in the US sequence.
Place value is not taught as a concept at all; it appears implicitly through columns on the worksheet when multi-digit addition begins around Level A-B. The learner infers the column structure from the layout rather than from an explanation.
The base-ten structure is made explicit through the Japanese number words themselves, which say "ten-one" and "two-ten" and so encode the structure the diagram is trying to show. Lessons exploit this rather than working against irregular English names.
Place value is defined, named, and drilled through expanded-form exercises that recur in mixed practice for years. Expanded notation is written out explicitly long after the child could do without it.
The golden beads present the hierarchy physically: a unit bead, a ten bar, a hundred square, a thousand cube, so a child holds the thousandfold difference in two hands. The stamp game then abstracts the same structure into colour-coded tiles, and the dot game abstracts it again.
Place value is treated as a base system among possible base systems, with problems that expose why ten is a choice. This pays off later in number-theory work on divisibility and digit puzzles.
Place value opens every year group as the first unit, using part-whole models and Gattegno charts, with variation exercises where only one digit changes so the effect of position is isolated.
Units are built around comparing representations of the same number, so children argue about whether a drawing, a numeral, and a bundle show the same amount. The standard notation is named in the synthesis after those arguments.