Distinguish definition from postulate from theorem, build a valid deductive chain, and disprove a claim with a counterexample.
GEO.PROOF_STRUCTURE“I can write a valid proof and disprove a false claim with a counterexample.”
Write a two-column proof of the vertical angle theorem, and disprove a false converse with a counterexample.
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.
Complete a partially filled proof by supplying reasons.
Write a full two-column proof from a marked diagram.
Choose a proof strategy, including indirect proof, and construct it unaided.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Geometry (Grade 9-10) | Foundations and constructions | advanced | review | ✓ |
| Geometry (Grade 9-10) | Foundations and constructions | intermediate | review | |
| Geometry (Grade 9-10) | Foundations and constructions | intro | introduce |
Statements paired with reasons, with the reason bank supplied and the order to be chosen.
Proof taught as structure, gradeable step by step.
engine: DAG-backed statement ordering widget
A directed graph of claims and justifications showing multiple valid routes to the conclusion.
Matches our DAG solver natively; rewards alternative routes.
engine: existing Socratic DAG visualiser
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: proof_graph
GEO.PROOF.ASSUME_CONCLUSIONUses the statement being proved as a reason.
Repair: Trace each reason back to a given, definition, postulate, or earlier theorem.
GEO.PROOF.PICTURE_EVIDENCEJustifies a step by how the diagram looks.
Repair: Require a named reason for every claim; appearance is never a reason.
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.
Congruence is defined through rigid motions, so the criteria are established rather than assumed, and constructions are required as a way of grounding the theorems. Proof is expected but the format is left open.
Geometric reasoning is developed through angle-property chains with written justification at each step, building toward proof gradually rather than through a dedicated proof course.
Formal geometry is a named multi-year subject with theorems proved in sequence and students expected to reproduce proofs and construct new ones. Compass-and-straightedge construction is core work rather than illustration, and proof arrives earlier than in the US.
Geometry and proof are essentially outside the Kumon math sequence, which stays on the computation and algebra track. A Kumon student may reach calculus levels without ever writing a geometric proof.
Proof is introduced in lower secondary through lessons where several students propose different justifications for the same claim and the class evaluates which are complete. The board record makes the logical structure visible.
The two-column proof is the organising format, taught explicitly with a bank of definitions, postulates, and theorems to be memorised, and proofs practised in graded sets from fill-in-the-reason to fully open.
Geometric reasoning is built from constructive materials and the geometry cabinet, with equivalence and area relationships demonstrated by rearranging physical pieces — an informal proof by dissection rather than a written argument.
Proof is treated as ordinary mathematical writing rather than a special format: arguments are written in prose, multiple approaches are compared for elegance, and hard configuration problems drive the need for rigour.
Reasoning chains with stated angle facts run through KS3, with formal proof concentrated at GCSE higher tier including circle theorems and congruence proofs.
The Geometry course builds proof from transformation arguments, with students first convincing a partner informally and then tightening the argument, and proof formats introduced as tools rather than requirements.