# Teacher copy · Year 6 maths answers and next moves

Keep this page away from the [clean learner prompts](../LEARNER.md), [practice choices](../DAILY-CHOICES.md) and [fresh checks](../STUDENT-CHECKS.md). Record `first response | mathematical reason | access route | content hint if any | next move`. Read-aloud, scribe, AAC, pointing or tactile supports may preserve the mathematical construct; placing a factor or fraction tick for the learner does not. One item is not an achievement-standard decision.

## Daily exits · expected evidence

| Day | Expected response | If not yet, teach then recheck on a different example |
| ---: | --- | --- |
| 11 | `16=4×4` square and composite; factors 1,2,4,8,16. `1=1×1` square but neither prime nor composite; 2 prime. | Rebuild 9 and 10 arrays, asking for factor counts before labels. |
| 12 | 23 has only `1×23`; small possible divisors 2,3,4 fail, and `5×5>23` means no new pair starts at 5. | List pairs for 21, then check fresh 19 systematically. |
| 13 | Any positive `n²>1` has distinct positive factors `1,n,n²`, so is composite; e.g. 25=`5×5`. 1 is the exception. | Pair an `n×n` array with its factor list and recheck 9. |
| 14 | 36 has `6×6` but also `4×9`; `12×15=180` through valid regrouping. | Compare a square array with another rectangle for 16. |
| 16 | On one 0–1 whole: `1/4=3/12 < 1/3=4/12 < 1/2=6/12`. | Fix endpoints and equal gaps on a new 12-part whole; use bars. |
| 17 | `3/4=9/12 > 2/3=8/12` by `1/12`. `2/4=1/2` is one mark. | Overlay 12 equal cells and recheck 1/3 versus 1/2. |
| 18 | `5/4=15/12 < 3/2=18/12` on one 0–2 scale, gap `3/12=1/4`; 1 is gap12, 2 gap24. | Label 0,1,2 and all equal spaces before naming fractions. |
| 19 | `2/4=1/2=6/12` occupies one mark. Denominator-only rule is repaired with same whole and equal-part model. | Give 1/3 versus 1/4 on one unit strip and require a marked reason. |

## Thirty routes · examples or precise acceptance criteria

| Route | Response to check |
| --- | --- |
| D11-A | 12 distinct pairs `1×12,2×6,3×4`; 16 square `4×4` and composite because also `2×8`, plus 1 and 16 factors. |
| D11-B | 1 square/neither prime nor composite; 2 prime/not square; 12 composite/not square; 16 square/composite. Factors justify each. |
| D11-C | `2×8` is a different 16-cell rectangle; square describes a possible `4×4` equal-sided array, while composite describes having >2 factors. |
| D12-A | 18 pairs `1×18,2×9,3×6`, no rotated duplicates as new pairs; after 3, trial 4 fails and 5²>18. |
| D12-B | 20: `1×20,2×10,4×5`; 21: `1×21,3×7`. Both allow >1 by >1 rows; e.g. 4×5 and3×7. |
| D12-C | 23 indivisible by 2 (odd), 3 (digit sum 5), 4 (not a multiple); next possible small side 5 has `5²=25>23`; only 1,23 factors. |
| D13-A | 9,25,49; for each factors include `1,n,n²` respectively `1,3,9`; `1,5,25`; `1,7,49`, so square/composite. |
| D13-B | 1 square/neither; 9 square/composite; 17 prime; 36 square/composite; 49 square/composite. Correct factor or no-small-factor witness. |
| D13-C | Example 16=`4×4` square but also `2×8`, so composite, not prime. 1=`1×1` is square but has one positive factor and belongs to neither category. |
| D14-A | 28 allows `4×7` (not square); 29 has only `1×29` (prime, not >1 by >1); 36 allows `6×6` square and `4×9` rectangle. |
| D14-B | 24 with six across gives four rows (`4×6`); other distinct layouts include `2×12`, `3×8`, `1×24`. Six-across condition selects 4×6. |
| D14-C | `12×15=(3×4)×(3×5)=(3×3)×(4×5)=9×20=180`; 36 includes `6×6` and `4×9`. |
| D15-A, later | 14=`2×7`, composite not square; 19 prime after checking 2,3,4, with 5²>19. |
| D15-B, later | 81=`9×9` square, composite (also `3×27`); 1=`1×1` square, neither prime nor composite. |
| D15-C, later | 26=`2×13`, composite; no whole-number square root since `5²=25<26<36=6²`, so not square. |
| D16-A | `1/4=3/12`, `1/3=4/12`, `1/2=6/12`; positions/gap order use one fixed whole. |
| D16-B | In 12 equal-duration steps, quarter=3 steps, third=4, half=6; least quarter, most half. |
| D16-C | In one 12-beat cycle, quarter cue after beat3, third after beat4, half after beat6; timing order 3<4<6. |
| D17-A | `2/4=1/2=6/12` one mark, `2/3=8/12`, `3/4=9/12`; latter exceeds former by 1/12. |
| D17-B | Same-length shelves: 3/4=9/12 >2/3=8/12 by 1/12 of one shelf. No real capacity claim. |
| D17-C | Equivalent twelfths 8/12 and9/12 refute the guess; 3/4 is larger by1/12 when whole same. |
| D18-A | Gap15=`5/4=15/12`, gap18=`3/2=6/4=18/12`; gap3=`3/12=1/4`. |
| D18-B | Two 12-step cycles: 5/4 cycles is step15; 3/2 is step18; gap3 equal steps=`1/4` cycle. |
| D18-C | Same unit: 5/4=15/12,3/2=18/12, so latter longer by3/12=1/4 unit. |
| D19-A | `1/4=3/12 < 1/2=6/12 < 2/3=8/12`, source is one shared 0–1 whole. |
| D19-B | `1/3=4/12 <1/2=6/12` on same whole. For **unit fractions of the same whole**, more equal parts means each part is smaller; do not generalise to arbitrary numerators. |
| D19-C | `2/4=1/2=6/12` coincide; `3/4=9/12` lies right by3/12=1/4 of one whole. |
| D20-A, later | `4/3=16/12 <3/2=18/12`, difference `2/12=1/6`. Both between 1 and2. |
| D20-B, later | On same 12-panel mural 1/3=4 panels;3/4=9 panels; latter is5 panels more, or difference5/12 whole. |
| D20-C, later | Both invented fractions are from halves/thirds/quarters and between0 and2; equal scale, exact equivalent positions and justified order/equality. Accept varied pairs; recheck arithmetic. |

## Day 15 held-out check · exact key

| Item | Accepted mathematics | What it diagnoses |
| ---: | --- | --- |
| 1 | 27 distinct positive pairs `1×27`, `3×9`; factors 1,3,9,27. Composite; not square (`5²=25<27<36=6²`). | Complete factor list and square test. |
| 2 | 37 is prime: odd, digit sum10 not divisible by3, not ending0/5; 4 and6 divisibility would imply2/3. Since `6²=36<37<49=7²`, all possible smaller factors through6 have been excluded. Factors are only1,37. | Systematic proof rather than guess. A correct explicit trial list 2–6 also works. |
| 3 | 64=`8×8` square. It is also composite, e.g. `2×32` (or4×16); factors1,2,4,8,16,32,64. | Overlapping properties. |
| 4 | 33 works as `3×11`;31 does not because it is prime (not divisible by2,3 or5 and next possible factor6 would exceed its square-root bound). `1×31` fails both dimensions >1. | Property used for a contextual constraint. |

## Day 20 held-out check · exact key

| Item | Accepted mathematics | What it diagnoses |
| ---: | --- | --- |
| 1 | Same 24-tile whole: `1/3=8` tiles, `1/2=12`, `3/4=18`. On 0–1: `1/3=4/12` at tick4, `1/2=6/12` at tick6, `3/4=9/12` at tick9; so `1/3<1/2<3/4`. | Shared whole, counts, equal-gap position and order. |
| 2 | `5/3=20/12` is tick20; `7/4=21/12` is tick21. Thus `7/4>5/3` by `1/12` of one unit. Both lie between1 and2 on the **same** 0–2 line. | Equivalent names beyond1, consistent scale and exact gap. |
| 3 | A uses `24÷3=8` tiles; B uses `12÷2=6` tiles, so B does **not** use more tiles. `1/2>1/3` as fractions **of the same whole** is still true; the panel wholes differ here. | Distinguish proportion from count across different whole sizes. |

## Evidence codes and immediate next moves

For each check row, write the learner's exact reason plus `I` (independent mathematics with available access), `P` (content prompt supplied), `R` (specific mismatch), or `N` (not observed because access/time was missing). Do not collapse items into one percentage or permanently label a learner.

- **Prime/composite confusion:** reconstruct 1,2 and a new square 9 by positive factor count; recheck 19 versus21.
- **Square treated as opposite of composite:** place 25 or49 in both property columns and demand a side pair and a second factor.
- **Prime decision by appearance:** show a systematic small-divisor table and where to stop; recheck a number not in the model.
- **Fraction ticks counted as pieces:** mark endpoints first, then count equal *gaps*; compare 1/3 and1/4 on a new 12-gap line.
- **Different wholes compared as if identical:** overlay equal-size unit strips first, then return to a context with two unequal panels and calculate actual counts.
- **Equivalent names at different points:** overlay bars or say both twelfth names, then re-mark a clean line.
- **Access route unavailable:** provide the full text/tactile/large-print route later and mark not yet observed; do not score a blank as a conceptual error.

Family feedback should name one next mathematical move with a concrete example. This is an internal formative guide, not a validated test or formal achievement-standard scale. Original SubjectNest staff key © NeuroForgeIO Pty Ltd 2026, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/).