# Optional extensions · two more routes for each day

The [25-minute core](LESSONS.md) is complete without these. Use **one** A/B extension when the learner wants more practice or the timetable offers another 10–12 minutes. At home, no printing, screen, purchase or parent marking is required; a family can opt out. Paper, fingers, spoken reasoning, AAC, a tactile strip or a school-approved tool are enough. Do not use real family spending, distances, records or child names. The [teacher key](teacher/TEACHER-KEY.md) gives results and next moves.

## Day 11 · Bound the answer

- **A · Near-ten comparison:** Without exact multiplication first, place `39×21` between 39×20 and 40×21. Then calculate the product and explain your bounds.
- **B · Scale detective:** A fictional answer for `62×15` is **93**. Use 60×15 to prove the scale error, then repair it with an exact route.

## Day 12 · Zero does a job

- **A · Placeholder build:** Compare `504×3` with `54×3` using place cards. Calculate both and explain why the inserted zero is not harmless.
- **B · Spoken partitions:** An adult reads `706×4`. Name the hundreds, tens and ones partials aloud or in AAC before giving the total.

## Day 13 · Tens row

- **A · Missing row:** A child writes `84×27 = 84×20 + 84×7` but records the first part as 168. Repair and finish the product.
- **B · Reverse check:** Find `46×32` two ways: 46×(30+2) and (40+6)×32. Show why both agree.

## Day 14 · Strategy comparison

- **A · Near-twenty:** Calculate `214×19` as 214×20−214. State a reasonable rounded check.
- **B · Convenient quarter:** Find `25×48` with 25×(4×12) or 25×(50−2). Explain which route you prefer and why.

## Day 15 · After the check

- **A · Error story:** Repair `132×14=1,320+428=1,748`. Name the wrong partial product, then say whether the final result survives. Use a new problem, not the check.
- **B · Make a near estimate:** Invent a two-digit multiplier for 119 so the answer is near 2,400; give one exact example and justify the estimate.

## Day 16 · Money without private data

- **A · Fictional paper costs:** 14 made-up sets at $29 each, with a $420 limit. Estimate, calculate and find the exact difference.
- **B · Missing price:** A model total for 12 items is $456. Find a possible equal unit price; say why a real price claim would need a source.

## Day 17 · Two checks

- **A · Fact-family card:** Calculate `306×8`; write both related divisions and a nearby benchmark. Check that the product still fits the scale.
- **B · Wrong inverse:** Someone claims `2,730÷7=390` checks `390×7=2,370`. Find the mismatch and correct the product or quotient as needed.

## Day 18 · Unknowns with meaning

- **A · Missing factor:** Solve `?×18=900`, then substitute and explain why `?=5` misses a place.
- **B · Two equivalent routes:** Find `?` in `24×?=1,008` by splitting 1,008 or trying nearby multiples; verify exactly.

## Day 19 · Missing criterion

- **A · Same amount:** Compare `12×50` and `20×30`. The totals match. Name **two different possible constraints** that could favour different layouts, without asserting either constraint was given.
- **B · Cost cannot be guessed:** A note says 28 packs of 15 and 21 packs of 20 have the same count. Compute the count; explain why neither plan can be called cheaper without prices.

## Day 20 · After the check

- **A · Teach the method:** Make an original 2-digit multiplication card for a fictional art display, solve it, then deliberately write one wrong-place partial product for a partner to repair. Keep the fresh check unseen.
- **B · Explain the decision:** Make two different equal-total arrangements and one explicit bundle-count limit. Show both products and say which meets your limit; ask what remains unknown.

**Original resource rights:** © NeuroForgeIO Pty Ltd 2026, SubjectNest, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/). Credit, link and indicate changes.