Multiply multi-digit numbers using partial products, the area model, and the standard algorithm, understanding them as the same computation.
NUM.MULTI_DIGIT_MULTIPLY“I can multiply multi-digit numbers and show where each partial product comes from.”
Compute 34 x 27 by area model and by the standard algorithm and match the four partial products.
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 by one-digit using partial products.
Two-digit by two-digit, area model matched to the algorithm.
Three-digit by two-digit, and multiplication of decimals by the same structure.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 4 | Multi-digit multiplication | intro | introduce | |
| Grade 4 | Division and remainders | advanced | review | ✓ |
| Grade 4 | Division and remainders | intermediate | review |
A rectangle partitioned by place value or by terms, with each sub-rectangle a partial product.
The same picture explains 23x47, 3/4 x 2/5, and (x+3)(x+5).
engine: partitioned rectangle with labelled regions
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
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: area_model
NUM.MULT.MISSING_ZEROOmits the placeholder zero in the second partial product row.
Repair: Label each row with the place value being multiplied.
NUM.MULT.CARRY_ADDAdds the carried digit before multiplying rather than after.
Repair: Write partial products in full before compressing to the algorithm.
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.
Multiplication is defined as equal groups in Grade 3 and connected immediately to arrays and area, with the distributive property used to derive unknown facts from known ones. Fluency with all products to 100 is a stated Grade 3 requirement.
Multiplication and division are introduced together as inverse views of the same equal-group picture, and the bar model carries "times as many" comparisons that would otherwise be language traps. Multi-digit work is done first with place-value discs, then compressed.
The tables are memorised early and thoroughly, but the emphasis quickly moves to using factor structure to simplify: rearranging a product, factoring out, and estimating. Division with remainder is treated as a topic in its own right with its own problem types.
Level C is multiplication tables and multi-digit multiplication as pure drill; Level D is long division. Both are taught by graded worksheet increments with a time standard, and division is reached earlier than in most school sequences because placement is by level rather than age.
The multiplication tables are learned through a formal chanted sequence (kuku) in Grade 2, but the meaning is established first through array lessons where the class compares ways of counting the same rectangle. The area model then carries multi-digit multiplication.
Tables are memorised by recitation and timed drill as a non-negotiable gate, and the long multiplication and long division algorithms are taught explicitly as procedures with worked examples, then spiralled through mixed practice for years.
Multiplication is laid out as bead chains and bead bars, so the multiples of seven are a physical distance and seven squared is a literal square. The checkerboard then handles multi-digit multiplication with place value colour-coded rather than remembered.
Multiplication is a route into number theory: factors, primes, and divisibility are pursued for their own sake, with puzzles that reward knowing 84 as 2 x 2 x 3 x 7 rather than as a table entry.
Times tables to 12 x 12 are a statutory Year 4 expectation with a national check, and the tables are built through arrays and scaling contexts before being drilled. The bar model handles "times as many" comparison problems.
Units move from equal groups to arrays to area, with students inventing partial-product strategies that are then connected to the standard algorithm in a synthesis rather than replaced by it.