## SAT Math -- Curriculum Overlay

This addendum applies only to Digital SAT Math questions. Use it for SAT-specific
rules that are narrower than the base and type guidance.

---

## SAT Math Item Format

Determine the item format before scoring.

**MCQ**
- Must have exactly 4 answer choices labeled A, B, C, D.
- More or fewer than 4 choices fails `specification_compliance`.
- "All of the above" and "None of the above" are invalid SAT Math choices and fail
  `specification_compliance`.

**SPR / grid-in**
- Has no populated `answer_options`.
- Answer must be numeric: integer, decimal, fraction, or an acceptable numeric range.
- Variable expressions, words, or lists of unrelated values fail `specification_compliance`.
- With no distractors present, set `distractor_quality = 1.0` unless another rule creates a
  distractor issue.

---

## Stimulus Requirements

Most Algebra, Advanced Math, and Geometry items are self-contained; a stimulus is optional.

For PSDA items that reference a table, graph, scatterplot, histogram, dot plot, box plot,
bar graph, line graph, or similar data display, the display is required. If the item says
"use the table," "based on the scatterplot," "the figure above/below," etc. and no matching
stimulus exists, fail `stimulus_quality`.

When a data display or geometry figure is present, it must be internally consistent and
match the question text. Contradictory labels, axes, values, angle/side labels, or a figure
that reveals the asked-for unknown fail `stimulus_quality`.

---

## SAT Math Metric Rules

### factual_accuracy

Verify the keyed answer by solving the problem. A mathematically wrong key fails
`factual_accuracy`.

No two answer choices may be algebraically equivalent. Equivalent choices create an
ambiguous key and fail `factual_accuracy`.

Notation must be consistent enough to preserve meaning. Pure style mismatch belongs under
`clarity_precision`; a variable/quantity contradiction between stem and options fails
`factual_accuracy`.

### distractor_quality

For MCQ items, each distractor must map to a plausible mathematical error, not just a
nearby number. A distractor with no traceable error pathway fails `distractor_quality`.
Common SAT Math error paths include sign errors, order-of-operations errors, misapplied
formulas, solving for the wrong quantity, unit confusion, inverse/reciprocal errors, and
confusion between related algebraic forms.

Choices should have comparable notation style and simplification. A choice that telegraphs
or hides the key by being uniquely simplified, formatted, or precise fails
`distractor_quality`.

### curriculum_alignment

The item must exercise the stated SAT Math substandard, not merely describe a concept or ask
for formula recall. It must stay inside the College Board assessment boundary for that
substandard. Examples:
- A.1 should not introduce a second variable as the target skill.
- AM.1 should rewrite expressions, not solve equations.
- PSDA.3 should not fit a scatterplot model; that belongs to PSDA.4.
- G.2 should not test trigonometric ratios; that belongs to G.3.

Course boundary rules:
- PSDA.7 is SAT-only; fail `curriculum_alignment` for PSAT 8/9 or PSAT 10.
- PSDA.6 applies to SAT, PSAT/NMSQT, and PSAT 10; fail for PSAT 8/9.
- G.3 and G.4 are SAT-only; fail for PSAT 8/9.

---

## Difficulty Calibration

Use Curriculum API difficulty definitions when provided. Otherwise apply the SAT-specific
guidance below.

### Algebra (SAT.M.A.1-A.5)

**A.1 Linear equations in one variable**
- Easy: straightforward one-step equations.
- Medium: multi-step equations, equation creation from context, or coefficient/constant interpretation.
- Hard: no/one/infinite-solution conditions, parameters, or complex contextual analysis.

**A.2 Linear equations in two variables**
- Easy: direct substitution or y-intercept/coefficient meaning.
- Medium: contextual coefficient interpretation or connecting equation forms.
- Hard: multi-condition line creation or simultaneous algebraic/graphical reasoning.

**A.3 Linear functions**
- Easy: direct function evaluation.
- Medium: y-intercepts, slopes/constants in context, or rules from input/output pairs.
- Hard: multi-step rate interpretation or chained/multi-representation reasoning.

**A.4 Systems of two linear equations**
- Easy: simple integer systems.
- Medium: fractional/decimal coefficients, contexts, or several basic steps.
- Hard: complex elimination, special cases, or algebraic/graphical connection.

**A.5 Linear inequalities**
- Easy: direct substitution in an inequality.
- Medium: creating/interpreting inequalities from constraints.
- Hard: tabular/algebraic/graphical connections or multi-step contextual inequalities.

### Advanced Math (SAT.M.AM.1-AM.3)

**AM.1 Equivalent expressions**
- Easy: polynomial operations, common-factor factoring, difference of squares.
- Medium: trinomial factoring or recognizing useful structure.
- Hard: rational expressions, rational exponents/radicals, or factoring beyond common cases.

**AM.2 Nonlinear equations and systems**
- Easy: standard/factored quadratics, discriminant, quadratic formula, completing the square.
- Medium: nonstandard quadratics, simple absolute-value/rational/radical equations, nonlinear systems.
- Hard: solution-count reasoning, parameters, or multi-step rational/radical structure.

**AM.3 Nonlinear functions**
- Easy: evaluating or identifying a simple graph point/intercept/initial value.
- Medium: creating/interpreting quadratic or exponential models and parameters.
- Hard: roots in context, solving for inputs from outputs, or multi-representation nonlinear reasoning.

### PSDA (SAT.M.PSDA.1-PSDA.7)

College Board does not define per-standard difficulty tiers for PSDA. Use:
- Easy: direct procedure/formula from a clear display.
- Medium: multi-step reasoning, combining displays/concepts, or basic statistical interpretation.
- Hard: conditional probability, design judgment, multi-variable inference, growth-model comparison,
  margin of error, or conceptual transfer/reversal.

Do not fail a single-step PSDA item as too easy when the challenge is a genuine conceptual
reversal, such as interpreting a decimal multiplier as the opposite percent change or
undoing a percent increase with an asymmetric decrease.

### Geometry and Trigonometry (SAT.M.G.1-G.4)

College Board does not define per-standard difficulty tiers for Geometry. Use:
- Easy: direct single-step formula or theorem application with clear labels.
- Medium: multi-step formula selection, scale factors, combined properties, special triangles,
  or basic trig ratios.
- Hard: k/k^2/k^3 scale reasoning, completing the square for circles, deriving trig values
  by similarity, radian arc/sector reasoning, or multi-theorem synthesis.
