# Workable teacher models and low-cost materials

All counts and contexts are invented for mathematics. They are **not** actual transport, sport, shop, charity, school or family records. A shared whole/unit must stay fixed when comparing options. Required materials are paper, pencil and a way to make place columns; counters and the [A4 mats](print/TEXT-ALTERNATIVES.md) are optional. Give a child numbers in text, enlarged print or accessible spoken/AAC form. If a learner chooses a calculator under school policy, ask for a prior estimate, the operation entered and a reasonableness explanation.

## Four explicit models to revisit

1. **Estimate before exact.** A fictional stand has 48 seats in each of 19 drawn rows. Round to 50 and 20 for a loose benchmark of about 1,000 seats. Exact `48×19 = 48×(20−1) = 960−48 = 912`. This model does **not** tell how many people attended or whether a real stand is safe. `912` is below 1,000; the difference is sensible because both rounded factors went up.
2. **One-digit partition.** `324×6 = (300×6)+(20×6)+(4×6) = 1,800+120+24 = 1,944`. Estimate `300×6 = 1,800`; exact is a little higher. When using a written algorithm, keep each place value; do not write a carried digit as a free-standing unit.
3. **Two-digit partial products.** `128×23 = (128×20)+(128×3) = 2,560+384 = 2,944`. A rough anchor `130×20=2,600` is low because 23 is greater than 20; it is an order-of-magnitude check, not the exact answer. Use the [partial-products mat](print/partial-products-mat.pdf) to retain the zero in twenty groups.
4. **Compensation and inverse check.** `247×18 = 247×(20−2) = 4,940−494 = 4,446`. Check: `4,446÷18 = 247` (or `18×247 = 4,446`). Estimate `250×18=4,500`, so 4,446 is plausible. This is efficient *here* because 18 is near 20; a child may choose a different correct, inspectable strategy.

## Teaching phrases that prevent common errors

- `Estimate` means a justified nearby value or bound, **not** permission to replace a tight decision with a loose guess. A budget of $470 for 28 invented kits at $16 each requires the exact $448 to find $22 remaining; `about $450` only checks scale.
- `×20` means two tens of groups: `128×20=2,560`, not `256`. A zero placeholder marks the place, but the explanation is the value of 20.
- If an option makes the same total under a different grouping, compare the **stated** constraint as well. `24×36 = 32×27 = 864`, but if a fictional workspace allows at most 30 bundles, the 27-bundle plan fits while the 36-bundle plan does not. No price is implied.
- An inverse fact checks a product but can repeat a memorised error if copied. Ask the child to show a second route or a bound; for `468×7=3,276`, `3,276÷7=468` and `470×7=3,290` are different checks.
- `24×?=720` can be found by `720÷24=30` and checked by `24×30=720`. The unknown is a group count in a particular model, not a mysterious symbol.

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