Replace a number with a nearby benchmark to a chosen place, and use estimates to predict and check answers.
NUM.ROUND_ESTIMATE“I can round to any place and use an estimate to check whether an answer is sensible.”
Round 4,738 to the nearest hundred, then estimate 4,738 x 21 and say whether 99,498 is plausible.
Depth is a column, not a second topic. The same topic is taught at these depths in different years; the path table below places each rung.
Round two- and three-digit numbers to the nearest ten or hundred using a number line.
Round to any place, including decimals, and estimate sums and products.
Choose a precision deliberately and state error bounds on the estimate.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 3 | Rounding and estimation | intro | introduce | |
| Grade 4 | Place value to a million | advanced | review | ✓ |
| Grade 4 | Place value to a million | intermediate | develop |
A measured line for locating, comparing, ordering, and jumping between numbers.
One representation carries whole numbers, fractions, integers, and irrationals.
engine: axis scene + tick generator + hop actor
An unscaled line where the learner chooses landmarks and records their own jumps.
Exposes the strategy instead of hiding it in an algorithm.
engine: freeform axis with learner-placed marks
Scene archetype: number_line
NUM.ROUND.RULE_ONLYApplies "5 or more round up" to the wrong digit.
Repair: Locate the number between the two benchmarks on a number line first.
NUM.ROUND.CHAINRounds repeatedly (4,748 -> 4,750 -> 4,800).
Repair: Round once, directly from the original number.
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.