Teacher copy: This page may include teaching prompts or answer keys. Answer keys in this public library can be viewed by anyone. Give learners a clean prompt, use checks as formative evidence, and change a case locally when prior access matters.
Keep this separate from learner prompts and fresh checks. Record first exact response | representation and reason | access | mathematical hint if any | next move. Accept equivalent correct methods. A blank response when access or time failed is not yet observed, not necessarily wrong. No check below has classroom validation or establishes a whole-year achievement judgement.
Daily exits
Day
Expected reasoning
Immediate next move if not yet
11
3/4−1/8=6/8−1/8=5/8, below3/4 on same whole.
Rebuild an eight-part strip; ask for a different subtraction.
12
4.75+2.60=7.35; 10.00−7.35=2.65; estimate7–8.
Line up tenths/hundredths and check with addition on fresh decimals.
13
1/2×3/4=3/8, an overlap smaller than3/4.
Shade a half of a four-part strip then name eighths.
14
2.4÷0.6=4, because four0.6 intervals total2.4.
Mark0,0.6,1.2,1.8,2.4; count gaps, not five markers.
16
60=2²×3×5, all leaves prime; 4×3×5 is numerically60 but 4 is not prime.
Split any composite leaf again and recheck84.
17
2³×3²=8×9=72; 3²=3×3.
Make repeated-prime cards, then evaluate a different product.
18
90/150=3/5 after dividing both by30; 150×3/5=90.
Pair equal prime leaves and try another common factor, then reduce.
19
√81=9 because 9²=81; 81=3⁴.
Check a proposed root by squaring it and distinguish base from exponent.
Thirty route keys and acceptance criteria
Route
Checkable result
D11-A
1/2=4/8; plus3/8 is7/8 of one strip. Result <1 and >1/2.
D11-B
5/6−1/3=5/6−2/6=3/6=1/2 hour.
D11-C
4/10 is false because denominators name different-size parts; 4/8+3/8=7/8, below1.
D12-A
4.75+2.60=7.35 m; 10.00−7.35=2.65 m; inverse 7.35+2.65=10.00.
D12-B
Used3.45+1.80=5.25 GB; remaining8.00−5.25=2.75 GB; sum back8.00.
D12-C
6.35+2.70=9.05 km; 8.105 is misaligned/incorrect, estimate6+3≈9 supports9.05.
D13-A
3-by-4 equal grid overlap6/12=1/2.
D13-B
2/5×3/4=3/10 of20 minutes =6 minutes; same whole period.
D13-C
2/5×3/4=6/20=3/10; 6/9 uses an unjustified denominator.
D14-A
2.4÷0.6=4 intervals; four0.6 lengths=2.4; five markers if endpoints included.
D14-B
1.5÷0.25=6 pieces;6×0.25=1.5.
D14-C
12.5%=1/8, and48÷8=6 panels.
D15-A, later
3/5+1/10=6/10+1/10=7/10, less than1.
D15-B, later
4.8÷0.8=6;6×0.8=4.8.
D15-C, later
20%=1/5;70÷5=14 tokens.
D16-A
Starting60 as6×10 or3×20 ends with two2s, one3, one5: 2²×3×5=60.
D16-B
84=4×21=(2×2)×(3×7)=2²×3×7.
D16-C
Split composite4 into2×2; 2²×3×5=60.
D17-A
Both72 trees yield 2³×3²;3 twos and2 threes.
D17-B
2³=2×2×2=8, 3²=9, product72; 2×3=6 is unrelated to exponent3.
Only matching prime leaves pair: shared2,3,5; one3 remains in90, one5 in150, hence3/5.
D18-C
Different divisors produce 9/5, which is not equal to90/150. Divide both by30 to get3/5;150×3/5=90.
D19-A
12²=144, √144=12; 12=2²×3, so 144=2⁴×3²;16×9=144.
D19-B
100=10²=(2×5)²=2²×5²; positive root10.
D19-C
81=3⁴=(3²)²=9²; positive square root9, not3.
D20-A, later
120=2³×3×5;8×3×5=120.
D20-B, later
13²=169; positive root13. 13 is prime if optional factor form requested.
D20-C, later
2³×5²=8×25=200, not150; true150=2×3×5².
Day 15 held-out key
Item
Exact answer and check
If not yet, next move
1
2/3+1/4=8/12+3/12=11/12, below1 and above2/3. Inverse 11/12−3/12=8/12=2/3.
Use one 12-part whole; recheck 1/2+1/3 on a fresh strip.
2
7.2÷0.3=72÷3=24 pieces; 24×0.3=7.2 units.
Draw groups on a simpler1.2÷0.3 model; then recheck a new decimal.
3
10% of80=8;5%=4;15%=12 cells. Or 0.15×80=12; whole80, 12/80=0.15.
Rebuild one tenth and half a tenth; recheck20% of60.
Day 20 held-out key
Item
Exact answer and check
If not yet, next move
1
108=4×27=(2×2)×(3×3×3)=2²×3³; 4×27=108. Every leaf prime.
Stop an easier tree only at primes; recheck54 or72.
2
196=14×14=14²; principal positive √196=14. Since14=2×7, 196=(2×7)²=2²×7²;4×49=196.
Use 12²=144 and13²=169 as benchmarks, then test14×14; recheck another square.
3
3²×5²=9×25=225, so proposed75 equation is false. 75=3×25=3×5².
Expand powers into repeated primes and multiply; recheck a fresh false equality.
Observation codes
Use I for independent mathematics with access provided, P if a content prompt supplied a step, R for a specific mismatch and N for not observed because access/time was missing. Preserve first work before feedback. Aural reading, screen reader, tactile blank aid or scribe is not itself a content hint; selecting the denominator, quotient or prime leaf is. Report one next move to families, not a permanent learner type.