Expand a(b + c) and reverse the process by factoring out a common factor, including with negative multipliers.
ALG.DISTRIBUTE“I can expand a bracket and factor a common factor back out.”
Expand −3(2x − 5) and factor 12x + 18y, checking each by re-expanding.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 6 | Expressions | intro | introduce | ✓ |
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
Unit, x, and x-squared tiles with signed colours, arranged into rectangles to factor and expand.
Completing the square is literally completing a square.
engine: snap-grid tile actors with rectangle validation
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
ALG.DIST.PARTIALMultiplies the first term in the bracket only.
Repair: Draw the area rectangle and check every region is covered.
ALG.DIST.NEG_SIGNDistributes a negative to only the first term.
Repair: Rewrite −3(2x − 5) as (−3)(2x) + (−3)(−5) explicitly.
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.
Grade 6 introduces variables as generalised numbers, with equivalence of expressions established by properties rather than by substitution. High school then adds the habit of choosing the form of an expression that reveals a property.
Algebra is delayed relative to arithmetic but arrives on a strong foundation, since bar models have already been doing algebraic work implicitly. The transition is often made explicit by drawing the bar and then labelling the unknown segment.
Letters appear early and are used for generalisation from primary school, so by the time formal algebra arrives the notation is familiar and the work is about structure — factoring, rewriting, and recognising form.
Level G moves straight from signed arithmetic into simplifying algebraic expressions, treating them as another computation form. Manipulation fluency is high and early; interpretation of what a variable means is not addressed.
Letters are introduced in 中1 as a way to write a general rule found by the class, so the first algebraic expression a student writes is one they discovered rather than one they were handed.
Rules for combining like terms and distributing are stated and drilled, with the pre-algebra book introducing notation systematically and Algebra 1 spiralling it through cumulative practice.
Algebraic identities are met physically long before symbolically: the binomial and trinomial cubes present (a+b)^3 as an object a child assembles years before writing it.
Expressions are approached as objects with structure to be exploited, with problems that reward seeing a difference of squares or a hidden common factor rather than expanding everything.
Algebraic notation begins in Year 6 and is developed through KS3 with function machines and area models, with heavy attention to the conventions of notation and to what a letter can stand for.
Expressions are built from contexts where two students wrote different-looking but equivalent expressions, making equivalence the question that motivates the properties.