Answer key (for a parent)October week 2 · younger son1 / 6
Answers and reasoning
Tuesday is a new unit. If he says he has not learnt it, go back to question 1 and the example box
Tue 6 Oct — long division by two digits
1(printed) 71 remainder 2
2(1) 60 ÷ 20 = 3 (2) 40 ÷ 20 = 2 (the real quotients are 3 and 2 too)
3estimate 90 ÷ 30 = 3. 31 × 3 = 93, 97 − 93 = 4 → 3 remainder 4
4estimate 3 → 29 × 3 = 87 is over 86 → 2. 29 × 2 = 58, 86 − 58 = 28 → 2 remainder 28. Watch for leaving the estimate 3 as the answer.
5(1) 17 × 4 = 68, 85 − 68 = 17, same as the divisor → too small → 5 (17 × 5 = 85, remainder 0)
(2) 18 × 4 = 72 is over 55 → too big → 3 remainder 1 (18 × 3 = 54)
6estimate 3 → 17 × 3 = 51, 69 − 51 = 18, bigger than 17 → too small → 17 × 4 = 68, 4 remainder 1
7estimate 450 ÷ 50 = 9 → 52 × 9 = 468 is over 458 → 8. 52 × 8 = 416, 458 − 416 = 42 → 8 remainder 42
81 and 13 are smaller than 23. Estimate 140 ÷ 20 = 7 → 23 × 7 = 161 is too big → 23 × 6 = 138 → 6. Check he did not stop at "it does not go in". The quotient starts in the ones.
975 ÷ 21: estimate 4 gives 84 > 75, too big → 3, 21 × 3 = 63, 75 − 63 = 12. Bring down the 6 → 126. 126 ÷ 21 = 6 → 36
10the remainder 55 is bigger than the divisor 32 (a remainder must be smaller than 32), so the quotient is too small. 32 × 7 = 224, 247 − 224 = 23 → 7 remainder 23
11134 ÷ 24 = 5 remainder 14. The other 14 people ride too, so 6 buses (round up)
12100 ÷ 14 = 7 remainder 2 (14 × 7 = 98)
(1) 7 boxes, 2 left (2) the 2 left need a box too, so 8 boxes
13230 ÷ 35 = 6 remainder 20 (35 × 6 = 210)
(1) 6 pieces, 20 cm left (2) 35 − 20 = 15 cm
Wed 7 Oct — costume 1
1-184 ÷ 28 = 3 → one in 3 (estimate 90 ÷ 30 = 3)
1-2A: 88 ÷ 22 = 4 → one in 4. B: 90 ÷ 30 = 3 → one in 3. B shoots better (fewer shots for each one that goes in)
1-3estimate 60 ÷ 20 = 3 → about one in 3. 58 ÷ 19 = 3 remainder 1. The remainder 1 means 1 shot more than 19 × 3 = 57.
2-1300 + 400 + 200 = about 900 yen (really 909)
2-2estimate 300 ÷ 50 = 6. 47 × 6 = 282, 300 − 282 = 18 → 6 pieces, 18 yen change
2-3700 + 500 + 300 = 1,500 < 1,600 → enough (really 1,399). Even the over-estimate is under 1,600 yen, so it is certainly enough.
4estimate 270 ÷ 30 = 9 → 34 × 9 = 306 is too big → 34 × 8 = 272, 275 − 272 = 3 → 8 remainder 3
54,800 + 3,200 = about 8,000 (really 8,003). Did he say "about eight thousand" aloud?
687 ÷ 12 = 7 remainder 3. The other 3 ride too, so 8 boats
73 notes make only 3,000 yen, not enough. Round up to 4,000 → 4 notes
8estimate 60 × 7 = 420. 57 × 7 = 399, 412 − 399 = 13 → 7 remainder 13
9estimate 240 ÷ 40 = 6 → 36 × 6 = 216, 252 − 216 = 36, same as the divisor → too small → 36 × 7 = 252 → 7
Fri 9 Oct — rounding and estimating
1(printed) round down 4,200, round up 4,300, ordinary 4,300
2round down 7,000, round up 8,000, ordinary 7,000 (hundreds digit 2 → down)
3500 + 300 = about 800 yen (really 791)
45,200 − 2,900 = about 2,300 yen (really 2,307)
5(1) at least 295 and less than 305 (2) at least 295 and at most 304. 305 rounds to 310, hence "less than".
6at least 4,450 and less than 4,550. The largest whole number is 4,549 (4,550 rounds to 4,600)
7(1) no (under 110) (2) yes (3) no (140 is not less than 140) (4) yes
8235 ÷ 40 = 5 remainder 35. 5 buses would leave 35 behind → 6 buses (round up)
9700 ÷ 85 = 8 remainder 20 (85 × 8 = 680) → 8 sweets (round down)
10(1) hundreds digit 4 → 27,000 (2) the 2nd digit is 7 → up → 30,000
1180 × 30 = about 2,400 yen (really 2,418)
12400 + 200 + 300 + 200 = 1,100 is over 1,000 → the round-up cannot show it is enough. Adding exactly: 305 + 198 + 287 + 156 = 946 → it is enough. A round-up that is too big does not mean it really is not enough.
13B, 8,000. 4,000 + 4,000 = 8,000 (really 8,099). A and C have the wrong number of digits.
Sun 11 Oct — costume 2, corrections, puzzle
1-1the 3rd digit is 2 → down → 310,000
1-21,900,000 and 1,200,000. Bluesky is bigger by about 700,000 (really 627,447)
1-3190 ÷ 31 = 6 remainder 4 → about 6 times
2-1108 ÷ 12 = 9 (the estimate 10 is too big: 12 × 10 = 120)
2-2estimate 8 → 28 × 8 = 224, 252 − 224 = 28, same as the divisor → too small → 28 × 9 = 252 → 9
2-3250 × 3 = about 750 points. Exactly 252 × 3 = 756 points
4estimate 350 ÷ 50 = 7. 48 × 7 = 336, 345 − 336 = 9 → 7 remainder 9
5smallest 7,500, largest 8,499
6215 ÷ 32 = 6 remainder 23. The last 23 pages need another day → 7 days (round up)
7Check against the notebook.
870 − 12 = 58, 58 ÷ 2 = 29 is B. A = 29 + 12 = 41. A 41, B 29 (41 + 29 = 70, 41 − 29 = 12). No tick if there is no diagram.
928 is smaller than 36. Estimate 280 ÷ 40 = 7 → 36 × 7 = 252, 288 − 252 = 36, same as the divisor → 36 × 8 = 288 → 8
Stretch answers (for a parent)Stretch · younger son5 / 6
Stretch — answers and how to mark them
Not recorded. One sheet, or half of one, is fine — praise what got done
1 · Waste and recycling
What it testsSocietal context × ÷ two digits (per day) and estimating. PISA: retrieve and employ (1, 2) → interpret and evaluate (3). The 25 g a bottle is not needed.
1(printed) 392 ÷ 14 = 28 bottles
2Station 658 ÷ 14 = 47 (65 ÷ 14 → 4, 14 × 4 = 56, 65 − 56 = 9, 98 ÷ 14 = 7)
School 504 ÷ 14 = 36 (50 ÷ 14 → 3, 14 × 3 = 42, 50 − 42 = 8, 84 ÷ 14 = 6)
3Right. 47 + 36 + 28 = 111 bottles a day. 111 × 30 = 3,330, over 3,000 (111 is more than 100, so 30 days must give more than 3,000).
Full credit: 111 a day (more than 100), then 3,330 or "more than 100 × 30 = 3,000", and "right".
Partial: 111 correct, or only "more than 100" with no link to 30 days.
No credit: "not right", or no calculation.
2 · Pulse
What it testsScientific context × ×4 to a minute, ÷ two digits, what a remainder means. PISA: retrieve (1) → employ (2) → evaluate (3). The 24 beats after walking and the 24 °C room are not needed.
1(printed) 18 × 4 = 72 beats
21000 ÷ 72 = 13 remainder 64. In 13 minutes: 936 beats, 64 to go. It is not 1000 yet, so the count reaches 1000 in the 14th minute.
(Estimate: 1000 ÷ 70 is about 14.)
What the remainder means: round up. "13 minutes" would still be short of 1000.
3Right. Just after running: 38 × 4 = 152 beats. Twice the resting 72 is 144. 152 > 144.
Full credit: 152 and 144 worked out and compared, "right".
Partial: 152 correct but not compared. (Comparing in 15 seconds, 38 > 2 × 18 = 36, is also correct and earns full credit.)
No credit: "not right", or no calculation.
3 · Ways to connect batteries
What it testsScientific context × ÷ two digits (per second) and "how many times". PISA: retrieve and employ (1, 2) → evaluate (3). The car's 120 g and the 24 s for parallel are not needed (parallel is as fast as one battery, but lasts longer).
1(printed) 600 ÷ 24 = 25 cm
2600 ÷ 15 = 40 cm (15 × 4 = 60, so 60 ÷ 15 = 4; put the 0 back for 40)
3Not right. Twice 25 is 50. The series car's 40 cm falls short of 50. It is faster, but not twice as fast.
Full credit: compares 25 × 2 = 50 with 40: "not exactly twice".
Partial: only "it gets faster", or 40 > 25 without checking "twice".
No credit: "right, 40 is more than 25".
4 · The trip bus
What it testsSocietal context × what a remainder means (round up) and sharing a cost. PISA: retrieve (1) → employ (2) → evaluate (3). The 28 km is not needed.
2(1) 90 ÷ 40 = 2 remainder 10. The 10 left over must ride too, so 3 buses (2 buses hold only 80).
(2) 12,000 × 3 = 36,000 yen. 36,000 ÷ 90 = 400 yen (or see it as 3600 ÷ 9 = 400).
3Right. The pupils alone are 29 + 30 + 28 = 87. 87 ÷ 40 = 2 remainder 7. The 7 left over still need another bus, so it is still 3 buses.
Full credit: 87 (or 87 ÷ 40 = 2 remainder 7) and the 7 left over making it 3 buses.
Partial: only "2 buses are not enough", or 87 found with no conclusion about buses.
No credit: "not right", or "87 is less than 90, so 2 buses".
For a parent. In every unit, item 3 is about working out numbers and comparing them before giving a verdict. A reason with numbers in it is enough, whether the verdict is right or wrong. Nothing is recorded.