Decide whether two expressions are equivalent for all values, and choose the form that reveals a desired property.
ALG.EQUIVALENT_EXPRESSIONS“I can prove two expressions are equivalent and choose the most useful form.”
Show 2(3x + 5) − 4 and 6x + 6 are equivalent, and say which form makes the y-intercept obvious.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 6 | Expressions | intro | introduce | ✓ |
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
Paired columns of inputs and outputs with a difference column exposing the rule.
Constant differences vs constant ratios distinguishes model families.
engine: table widget with computed delta column
A worked solution containing a deliberate mistake that the learner must find, name, and repair.
Diagnoses misconceptions directly and builds critique skill.
engine: step list with a poisoned node in the DAG
Scene archetype: algebra_tiles
ALG.EQUIV.ONE_VALUEChecks one substitution and declares equivalence.
Repair: Require a structural argument, or at minimum several values.
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.