Use the standard written algorithms with regrouping, understood as ten-for-one trades rather than digit rituals.
NUM.ADD_SUB_ALGORITHMS“I can add and subtract multi-digit numbers accurately and explain each regroup.”
Compute 4,003 − 1,847 and explain what the borrowing across zeros actually traded.
Fluency standard: 20 four-digit problems in 8 minutes, >=95% accurate
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.
Two-digit sums and differences without regrouping.
Three- and four-digit with regrouping in any column.
Subtraction across multiple zeros, and checking by inverse operation.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 2 | Mental strategies | intro | introduce | |
| Grade 2 | Multi-digit addition and subtraction | advanced | review | ✓ |
| Grade 2 | Multi-digit addition and subtraction | intermediate | review |
Nothing in the atlas depends on this yet.
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
Labelled discs (1, 10, 100, 0.1) moved between columns so the algorithm mirrors the manipulative.
Extends cleanly to decimals where blocks do not.
engine: column scene + disc actors + trade command
Colour-coded tiles marked 1, 10, 100, 1000 that abstract the golden beads one step.
Bridge from concrete quantity to written algorithm.
engine: tile grid with exchange rules
A dense page of graded exercises with a completion-time target and no scaffolding.
Automaticity, which frees working memory for reasoning.
engine: worksheet builder + print layout
Scene archetype: place_value_chart
NUM.ALG.SMALLER_FROM_LARGERSubtracts the smaller digit from the larger in each column regardless of position.
Repair: Trade with base-ten blocks so the shortfall is physical.
NUM.ALG.ZERO_BORROWCannot borrow across a zero.
Repair: Rename the whole number (400 = 39 tens and 10 ones) before subtracting.
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.
The problem-type taxonomy is explicit: add-to, take-from, put-together, compare, each with the unknown in three positions, and a grade is expected to master specific cells of that table. Fluency within 10 is required by Grade 1 and within 20 by Grade 2.
Number bonds do the conceptual work: a whole with two parts covers addition, both subtractions, and missing-addend problems as one picture. The make-ten strategy is taught as a named method with a drawn branching step rather than left to be discovered.
Addition and subtraction are met inside problems from the start, with the condition written out and schematised before computing. Comparison problems ("how many more") are given equal weight with joining problems rather than treated as harder.
Levels A and B are addition then subtraction as pure computation: +1 sheets, then +2 sheets, and so on, with the whole fact space covered by incremental worksheet sets and timed to a standard completion time. No word problems, no diagrams.
Addition with regrouping is taught through one problem such as 8 + 5, with several student decompositions compared on the board, and the make-ten method consolidated as the efficient one at the end. The comparison, not the demonstration, is the lesson.
Facts are drilled to automaticity on timed sheets, the column algorithm is taught explicitly with worked examples, and both then recur in mixed practice indefinitely. Regrouping is presented as a procedure to execute correctly rather than a trade to understand.
Addition begins as a physical composition of golden-bead quantities carried to a shared table, with exchange happening when ten of a category accumulate. The algorithm is met only after the exchange has been performed by hand many times.
Beast Academy leans on structure rather than facts: problems reward noticing that a long sum pairs into equal totals, or that compensation makes an awkward subtraction trivial. Speed follows from insight rather than from repetition.
Small-step sequences build bridging through ten with tens frames and part-whole models, and mental strategies are taught before the column method. Missing-number equations in every position are routine from Year 1.
Story-problem types are worked deliberately across their unknown positions, with student strategies elicited and connected to one another before any algorithm is named. Number talks give a regular slot for mental strategies to be compared.