Canonical Gymnasium Mathematics: Sek I Numbers / Terms / Algebra Audit

Snapshot: 2026-04-05

Purpose:

• review the current canonical Sek I Zahlen / Terme / Algebra inventory before more bundeslandwise widening
• use already reviewed lower-secondary state evidence to test whether the current canonical cuts are pedagogically stable
• define the next canonical work packages for Sek I Zahlen / Terme / Algebra

Scope

In scope:

• the canonical Sek-I numbers / terms / algebra topic surface in
• curricula/DE/Gymnasium/canonical/DE_DEU_S_GYM_CANONICAL_MATHEMATIK.de.json
• the already reviewed lower-secondary source and mapping evidence from
• HB
• HH
• HE
• MV
• NI
• NW
• RP
• SL
• BB
• BE
• ST
• SN
• BW
• BY

Out of scope:

• learner-facing composition views
• quadratic-function packaging beyond the algebra/function boundary

Reviewed source evidence

Hessen (HE)

Observed signal:

• HE provides the mature baseline for arithmetic, term work, equations, proportional reasoning, and later counting/probability bridges
• the Hessen source lane suggests that arithmetic fluency, algebraic term work, and equation solving should not collapse into one diffuse broad package

Bremen (HB)

Observed signal:

• HB already exposes broad lower-secondary arithmetic/algebra anchors in J5/6, J7/8, and J9
• the Bremen lane is currently useful for package validation, but still too broad to force many new algebra atoms on its own

Hamburg (HH)

Observed signal:

• HH confirms a stable lower-secondary progression with broad J6, J8, and J10 anchors
• the reviewed Hamburg function corridors also indicate that proportional reasoning, linear equations, and algebraic modelling sit directly on the border between algebra and functions

Schleswig-Holstein (SH)

Observed signal:

• SH provides the first additional lower-secondary pressure test after the initial HB / HH / HE pass
• the lane separates
• variables / terms
• proportionality / rule-of-three
• linear equations
• linear systems while keeping
• quadratic equations
• exponential equations attached to the function-side progression
• this is exactly the kind of reviewed evidence that tests whether a visible A5 is really needed

Sachsen (SN)

Observed signal:

• SN now contributes seven explicit reviewed lower-secondary strips:
• K8 Arbeiten mit Termen und Gleichungen
• K7 Arbeiten mit rationalen Zahlen
• K6 Arbeiten mit gebrochenen Zahlen
• K6 Vernetzung: Anteile
• K5 Arbeiten mit natuerlichen Zahlen
• K5 Gemeine Brueche und Dezimalzahlen
• K10 Vernetzung: Zinsrechnung
• together they confirm
• the visible A1 arithmetic / number-foundation package
• the visible A2/A4 boundary between term work and equation work
• the still-broad attachment of first percentage notation, shares, and the later percentage / simple-interest residue to the existing A3 bridge package
• SN still does not force a visible A5; its reviewed lower-secondary algebra material, now including the lineare-Gleichungssysteme residue inside K8 Funktionen und lineare Gleichungssysteme and the later powers / roots strip inside K9 Funktionen und Potenzen, fits the current A1-A4 surface without demanding a new shared late-algebra package

Sachsen-Anhalt (ST)

Observed signal:

• ST now contributes a first reviewed lower-secondary corridor on the broad JG 5/6 snapshot lane
• the corridor groups
• natural numbers
• fractions
• rational numbers / gebrochene Zahlen
• first equations and inequalities
• this confirms the early A1 arithmetic surface and the first A4 entry point without forcing a separate visible late-algebra package

Rheinland-Pfalz (RP)

Observed signal:

• RP now contributes a first reviewed lower-secondary algebra corridor on the explicit Klassenstufen 7 und 8 lane
• the Rheinland-Pfalz source separates
• Rationale Zahlen
• Prozent- und Zinsrechnung
• this makes RP a useful pressure test for whether the visible A1/A3 surface can absorb a later number-extension strip next to a broad percent/simple-interest bridge without forcing a new late-algebra package

Mecklenburg-Vorpommern (MV)

Observed signal:

• MV now contributes reviewed lower-secondary algebra corridors on the explicit Klassen 5/6 and Klasse 7 lanes
• the Mecklenburg-Vorpommern Klassen 5/6 source now separates
• Natuerliche Zahlen und Teilbarkeit
• Brueche und Dezimalbrueche
• the Mecklenburg-Vorpommern Klasse 7 source still separates
• Prozent- und Zinsrechnung
• a broad strip across ganze Zahlen, rationale Zahlen, and Quadratwurzel
• this makes MV a useful pressure test for whether the visible A1 arithmetic surface can absorb an early foundations lane before the later A3 bridge and number-extension surface appear, without forcing a new early-algebra package

Saarland (SL)

Observed signal:

• SL now contributes a first reviewed lower-secondary algebra corridor on the explicit Klassenstufe 7 lane
• the Saarland source separates
• Rationale Zahlen
• Terme, Gleichungen und Ungleichungen
• Prozentrechnung
• this makes SL a useful pressure test for whether the later number-extension surface, the broad percentage bridge, and a still-mixed terms/equations strip can coexist without forcing either a sharper visible A2/A4 split or a new A5

Thueringen (TH)

Observed signal:

• TH now contributes a first reviewed lower-secondary algebra corridor on the explicit Klassenstufen 5/6 lane
• the Thueringen source separates
• Natuerliche Zahlen
• Gebrochene Zahlen
• this makes TH a useful pressure test for whether the visible early A1 arithmetic surface can absorb a clean 5/6 foundations lane without forcing a new early-algebra package
• TH now also contributes a first reviewed broad Klassenstufen 7/8 algebra corridor
• the archived 7/8 source still exposes only one shared Arithmetik/Algebra strip
• this confirms that the shared broad Sek-I algebra surface can absorb a reviewed wide 7/8 lane without forcing a sharper canonical split from source granularity alone
• TH now also contributes a first reviewed broad Klassenstufen 9/10 algebra corridor
• the archived 9/10 source still exposes only one shared Arithmetik/Algebra strip
• this confirms that the same shared broad later Sek-I algebra surface can absorb a reviewed wide 9/10 lane without forcing a sharper canonical split from source granularity alone

Current canonical numbers / terms / algebra inventory

The canonical graph is already materially seeded here.

Important current package questions:

1. arithmetic fluency and number representations
2. variables, terms, and term transformations
3. proportional reasoning, percentage, and rule-of-three style bridges
4. equations, solvability, and early algebraic modelling

Audit judgment

The canonical Sek-I algebra topic is not missing a foundation. The main risk is that the current packaging may still mix arithmetic fluency, term manipulation, proportional reasoning, and equations too coarsely for topic-first nationwide work.

This is therefore first a packaging-and-boundary audit, not first a missing-content audit.

Findings

1. Arithmetic fluency should stay visible as its own lower-secondary package surface

The reviewed state evidence suggests that:

• early number representations
• operation fluency
• later number-set extensions

should not be silently merged with later symbolic algebra work.

2. Term work and equation work should not be treated as the same package

The current topic row already points to this risk explicitly:

• transforming and structuring terms
• solving equations and reasoning about solvability

belong together, but they are not the same pedagogical package.

3. Proportional reasoning needs an explicit boundary against both arithmetic and functions

Proportionality, percentage, and rule-of-three style reasoning:

• touches arithmetic
• touches algebra
• touches functions

This boundary should be decided explicitly in the canonical cut instead of drifting between topics.

4. Linear equations and early modelling likely need a clearer visible package

The reviewed lower-secondary lanes suggest that:

• equations
• ratio equations
• simple statements about solvability
• first algebraic modelling

should probably sit in a visible package rather than only as scattered atoms.

5. The algebra/function boundary must stay explicit

Quadratic functions, function representations, and late Sek-I modelling should not silently swallow the algebra audit. This topic should clarify the algebraic side first and only then reconnect to the function topic.

Proposed canonical work packages

For Sek I Zahlen / Terme / Algebra, use these work packages:

1. A1 Arithmetic fluency and number representations
2. basic operations
3. secure number representations

4. later number-set extensions where they still serve algebra entry

5. A2 Variables, terms, and symbolic transformations

6. variable meaning
7. term construction and interpretation

8. equivalent transformations

9. A3 Proportional reasoning, percentages, and ratio bridges

10. proportional / inverse proportional reasoning
11. percentage contexts

12. rule-of-three style bridges

13. A4 Equations, solvability, and algebraic modelling

14. linear equations
15. ratio equations
16. statements about solvability

17. first modelling translations

18. A5 Linear systems and late Sek-I equation extensions

19. only where the reviewed evidence really forces a visible separate package

Canonical design step executed (2026-04-01)

The first canonical packaging pass is now in place.

Inserted visible package surface in the canonical graph:

1. A1 arithmetic fluency and number representations
2. A2 variables, terms, and equivalent transformations
3. A3 proportional reasoning, percentages, and rule-of-three bridges
4. A4 equations, solvability, and algebraic modelling

Boundary decisions executed:

• the mixed Strukturen und funktionale Zusammenhaenge (Sek I) corridor was narrowed to the function side
• variables, core term work, and equation packaging were pulled back into Zahlen, Terme und Algebra (Sek I)
• proportional reasoning stays visible as a bridge package instead of disappearing either into arithmetic or into the function topic
• A5 remains explicitly open: linear systems still live inside A4 until mapping pressure proves they need their own visible package

Residue pressure test executed (2026-04-01, SH)

The Schleswig-Holstein and Sachsen lower-secondary lanes do not force a visible separate A5.

Judgment:

• lineare Gleichungssysteme fit the existing late equation atom surface cleanly
• quadratische Gleichungen do not come in SH as a broad late-algebra continuation, but as part of the quadratic-function boundary package
• Exponentialgleichungen sit even more clearly on the J10 function continuation side

Interpretation:

• the reviewed SH lane argues against inventing one broad visible A5 that would bundle together
• linear systems
• quadratic equations
• exponential equations
• the current canonical split remains the cleaner design:
• algebra keeps linear equations, solvability, modelling, and linear systems in A4
• quadratic equations remain attached to the quadratic boundary package
• exponential equations remain attached to the J10 function continuation

Current conclusion:

• do not open a visible A5 now
• only reopen this question if another reviewed lower-secondary lane shows a genuinely shared late-algebra corridor that sits clearly outside both A4 and the function-side packages

Recommendation

Do not widen more bundesland mappings on Sek-I algebra until the revised A1-A4 packaging has been pressure-tested against the reviewed lower-secondary lanes.

That pressure test has now reached a stable interim judgment:

• no visible A5 package is currently justified
• the Sachsen K10 interest, K6 shares, K8 terms-and-equations, K7/K6/K5 early-number and fraction corridors, and the first Sachsen-Anhalt JG 5/6 foundations corridor confirm that the visible A1, A3, and A2/A4 split is stable enough for coverage work
• further lower-secondary work should treat A5 as a reopen-only-if-forced residue, not as an active design task

Additional reviewed close-out evidence (BW, BY, BE, BB, NI, NW)

1. BW now pressure-tests the algebra surface across the full reviewed lower-secondary pilot subset:
2. Klassen 5/6 carry the foundation strip across Zahl-Variable-Operation
3. Klassen 7/8 now include a broad algebra widening with percent / interest work, formula rearrangement, and roots
4. Klassen 9/10 now keep explicit variable-term and power-rule follow-ons on the shared algebra and late-number surfaces
5. BY now adds a reviewed Sek-I algebra lane on the shared J5-J10 spine:
6. the Bavaria M8 tranche now exact-resolves connected fraction-term and linear-system leaves on the existing A1-A4 surface
7. the reviewed M9 tranche adds roots and quadratics without forcing a visible separate A5
8. NI now carries a fully mapped lower-secondary pilot snapshot with the arithmetic base and imported algebra strip closed on source-goal level, plus explicit right-triangle / similarity and quadratics follow-ons outside the core algebra cut.
9. NW now carries an explicit lower-secondary prerequisite strip with quantity relations, rule-of-three basics, rational-number / term / linear-equation prerequisites, while the remaining broad lower-secondary parents are closed on the shared prerequisite and function/algebra surfaces.
10. BE and BB do not yet expose narrower dedicated algebra corridors beyond the shared J7-J10 framework and the first reviewed functions lane, but both now keep their remaining broad lower-secondary algebra parents stably mapped on the shared canonical Sek-I algebra surface.

Interpretation:

• BW, BY, NI, and NW now give corridor-level evidence that the frozen A1-A4 cut survives broader lower-secondary comparison lanes
• BE and BB resolve as broad anchor lanes for this topic rather than as evidence for another shared algebra package

Sachsen K9/K8/K7/K6/K5 lower-secondary algebra corridor set extended with powers, fractions, decimals, and the K8 function/LGS boundary (2026-04-03)

Outcome:

1. SN now confirms the visible A1 arithmetic package, the broad A3 shares/percentage bridge, the later number-extension package for powers and roots, the A2/A4 package boundary, and the choice to keep first linear-system residue inside A4 on reviewed lower-secondary lanes
2. the corridor set still does not justify a separate visible A5 package
3. the algebra topic should therefore stay in coverage mode rather than reopening package design

Current next concrete step

1. keep the canonical A1-A4 package cut frozen
2. use the next unresolved lower-secondary state lane to pressure-test early-number and fraction coverage before reopening algebra packaging

Close-out judgment (2026-04-05)

The current Sek-I numbers / terms / algebra sweep can now be treated as closed.

Why this is now strong enough:

1. the accepted visible package surface stays at A1-A4
2. the reviewed BW/BY/MV/NI/NW/RP/SH/SL/SN/ST lanes do not force a visible A5
3. the broader BE/BB lower-secondary algebra parents are now resolved as stable anchor evidence on the shared Sek-I algebra surface rather than as open package debt
4. the remaining tension stays at source-granularity residue or at the algebra/function boundary, not at a missing shared lower-secondary algebra corridor

Exit criteria for this audit

This topic audit is complete when:

1. the canonical subpackage boundaries above are either accepted or revised
2. the canonical graph has a stable Sek-I algebra packaging
3. reviewed BW, BY, MV, NI, NW, RP, SH, SL, SN, and ST evidence can be described as aligned to the revised packaging or intentionally broader because of source granularity, while BE and BB are explicitly accepted as resolved broad anchor lanes
4. it is clear whether late lower-secondary linear-system / extension material deserves a visible A5 package

Sachsen-Anhalt JG 9 powers/logarithms corridor connected (2026-04-03)

• the ST JG9 strip Potenzen und Logarithmen now points to the existing later number-extension surface
• the reviewed ST lane therefore keeps logarithm residue on the broad late-number bridge instead of forcing an earlier visible logarithm package before the J10 function branch

Sachsen-Anhalt JG 7/8 rational-numbers/roots corridor connected (2026-04-03)

• the ST JG7/8 strip Rationale Zahlen und Wurzeln now points to the existing later number-extension surface
• the reviewed ST lane therefore confirms that the rational-number / root residue can stay on the broad late-number bridge without forcing a separate visible roots package

Sachsen-Anhalt JG 7/8 percentage corridor connected (2026-04-03)

• the ST JG7/8 strip Prozentrechnung now points to the broad visible A3 bridge
• the reviewed ST lane therefore confirms that later percentage residue can stay on the existing proportionality / percentages / rule-of-three surface without forcing a narrower separate package

Sachsen-Anhalt JG 7/8 equations/inequalities corridor connected (2026-04-03)

• the ST JG7/8 strip Gleichungen und Ungleichungen now points to the visible A4 surface
• the reviewed ST lane therefore confirms that this mixed equations/inequalities residue can stay on the existing solvability / algebraic-modelling corridor without forcing a visible A5

Sachsen-Anhalt JG 7/8 variables corridor connected (2026-04-03)

• the ST JG7/8 strip Arbeiten mit Variablen now points to the visible A2 surface
• the reviewed ST lane therefore confirms that this variable/term residue can stay on the existing variables / terms / equivalent-transformations corridor without forcing a new separate package

Rheinland-Pfalz Klassenstufen 7 und 8 algebra corridor connected (2026-04-03)

• the broad RP Klassenstufen 7 und 8: Zahl und Zahlbereiche anchor now carries a reviewed split into Rationale Zahlen and Prozent- und Zinsrechnung
• Rationale Zahlen points to the visible later number-extension surface
• Prozent- und Zinsrechnung points to the broad percent/simple-interest bridge
• the RP lane therefore confirms the visible A1/A3 surface without forcing a new visible A5 or another late-algebra package

Saarland Klassenstufe 8 algebra corridor connected (2026-04-04)

• the broad SL Klassenstufe 8: Zahl und Operation anchor now carries a first reviewed corridor on the shared Sek-I algebra surface
• the current Saarland grade-8 source still exposes only one shared algebra strip
• this confirms the shared broad Sek-I algebra surface can absorb an explicit reviewed class-8 lane without forcing a sharper state-local split

Saarland Klassenstufe 9 algebra corridor connected (2026-04-04)

• the broad SL Klassenstufe 9: Zahl und Operation anchor now carries a first reviewed corridor on the shared Sek-I algebra surface
• the current Saarland grade-9 source still exposes only one shared algebra strip
• this confirms the shared broad Sek-I algebra surface can absorb an explicit reviewed class-9 lane without forcing a sharper state-local split

Saarland Klassenstufe 10 algebra corridor connected (2026-04-04)

• the broad SL Klassenstufe 10: Zahl und Operation anchor now carries a first reviewed corridor on the shared Sek-I algebra surface
• the current Saarland grade-10 source still exposes only one shared algebra strip
• this confirms the shared broad Sek-I algebra surface can absorb an explicit reviewed class-10 lane without forcing a sharper state-local split