Keep this page away from the clean learner prompts, practice choices and fresh checks. 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.