Interpret a x b as a groups of b, as an array, and as an area, and recognise when a situation is multiplicative.
NUM.MULTIPLICATION_MEANING“I can draw a picture of a multiplication and explain what each number counts.”
Draw 4 x 6 as equal groups, as an array, and as a rectangle, and say what changes and what does not.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 2 | Equal groups and arrays | intro | introduce | ✓ |
Objects in rows and columns showing multiplication as repeated equal groups.
Commutativity becomes a quarter turn.
engine: grid of prop instances with row/column highlight
Discrete movable objects arranged, grouped, and recounted to establish cardinality and conservation.
Separates "how many" from "the last word I said".
engine: prop_catalog instances + drag layer
Rectangular bars segmented to show parts inside a whole, with the unknown drawn as a labelled gap.
Word problems become a picture you can read the equation off.
engine: segmented bar actor with labelled brace
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
Scene archetype: array_grid
NUM.MULT.ADDITIVEAdds the two factors instead of multiplying.
Repair: Build the array physically and count the total.
NUM.MULT.ALWAYS_BIGGERBelieves multiplying always makes a number larger.
Repair: Delay-proof by returning to this claim when multiplying by a fraction.
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.