# Math

> The topic pack for arithmetic, percentages, algebra, probability, and money math. Your agent reads it for lessons with numbers and formulas.

Extra guidance for lessons where the learner works with numbers and formulas. It adds to `reference/teaching-guide.md`; where they disagree, the teaching guide wins.

## 1. One method, not a chapter

* Teach **one method or relationship**: "Find a percentage change", "Add fractions with different denominators", "Solve a one-step equation". Not "Fractions".
* Write the `objective` as a calculation the learner can do afterwards: "Work out the new price after a 15% discount".
* **Hook:** ask for a quick estimate or gut answer that most people get wrong. "A price rises 50%, then falls 50%. Is it back where it started?" The first concept brick explains why.

## 2. Concepts: worked example first

* **Concrete numbers before the general rule.** Work one example with real numbers, then show the formula it follows.
* **One operation per line.** In a worked example, each step does one thing, so the learner can see where each number comes from. Put the key formula in a `math` visual; use `\( … \)` for inline math in text.
* **Name the quantity before the symbol:** "the interest rate, `\( r \)`", not just `\( r \)`.
* **Keep numbers friendly.** Pick numbers that make the idea visible and the arithmetic easy (10%, 200, `\( \tfrac{1}{4} \)`). Learners have no calculator in the player, and a messy sum hides the idea.
* **Show why, briefly.** One sentence on why the method works ("dividing by the original amount turns the change into a share of where you started") beats a rule to memorise.

## 3. Practice that fades the support

After each concept brick, take support away one step at a time:

1. **Finish a worked example:** a `blank` where the learner picks the missing step or term from chips.
2. **Solve a full problem** with the same structure: a `choice` with mistake-based options.
3. **Vary it:** new numbers, a new context, or the problem run backwards ("the discounted price is $68; what was the original?").

How the question bricks fit math:

| Type       | Natural for…                                                                                       |
| ---------- | -------------------------------------------------------------------------------------------------- |
| `choice`   | The final answer; which formula applies; which statement is true                                   |
| `blank`    | A missing step, term, or sign in a formula or worked solution (`"numeric": true` for a number gap) |
| `hotspot`  | Finding the wrong line in a worked solution: one `data-hotspot` element per line in an HTML visual |
| `sort`     | Classifying: rational or irrational, linear or not, which expressions equal `\( 2(x + 3) \)`       |
| `order`    | Steps of a procedure where order matters: solving an equation, order of operations                 |
| `match`    | Equivalent forms: fraction ↔ decimal ↔ percent, expression ↔ simplified form                       |
| `compare`  | Two solution methods, or two charts of the same data: which is right, or clearer                   |
| `scenario` | Choosing which calculation a real decision needs (loan offers, unit prices)                        |

## 4. Wrong options from real errors

Every numeric `choice` should have 3–4 options, each produced by a specific, common slip, with feedback that names the slip and shows the right step. Mistakes worth building on:

* **Wrong base for a percentage:** a 25% discount undone by adding 25%; percent change measured from the new value.
* **Percent vs percentage points:** 4% → 5% is 1 point, but a 25% increase.
* **Fractions:** adding numerators and denominators (`\( \tfrac{1}{2} + \tfrac{1}{3} = \tfrac{2}{5} \)`); forgetting to scale the numerator too.
* **Order of operations:** working left to right through `\( 2 + 3 \times 4 \)`.
* **Signs:** subtracting a negative; moving a term across `\( = \)` without changing its sign.
* **Distributing powers:** `\( (a + b)^2 = a^2 + b^2 \)`.
* **Probability:** adding probabilities that should be multiplied, or the reverse.
* **Averages:** mean where the median is meant, or averaging averages of unequal groups.
* **Units and scale:** mixing cm and m; squaring a length but not its unit; one step short.

The right answer shouldn't stand out: it shouldn't be the only round number, always the middle value, or the longest option.

## 5. Accuracy

* **Work every answer yourself, step by step, before writing the options.** Then check each wrong option: is it really wrong, and does it really come from the slip its feedback names?
* **State rounding and units in the prompt** ("to the nearest dollar", "in cm²"), and make sure exactly one option is right after rounding.
* Don't simplify away conditions that matter (`\( x \neq 0 \)`, "assuming the rate stays the same").

## 6. Notation

* Inline math goes in `\( … \)`; a formula on its own line goes in a `math` visual. A `$` is always a plain dollar sign.
* Use `\times` or `\cdot` for multiplication, never `*`, and `\frac{a}{b}` (or `\tfrac` inline) for fractions.
* **Follow the lesson's language conventions:** a decimal comma where it's normal (`de`, `fr`, `es`, `pt-BR`…). Inside TeX, write it as `3{,}5` so it doesn't get extra space.
* **Graphs and diagrams** are inline SVG: label the axes, color with theme classes, and add `direction='ltr'` to the `<svg>` (see the teaching guide's Visuals section).