Answer key (for a parent)October week 3 · younger son1 / 6
Answers and reasoning
No new unit. If his estimate misses, tell him that fixing it is the method
Tue 13 Oct — long division by two digits
252 → 50, estimate 7. 52 x 7 = 364, 365 − 364 = 1 → 7 remainder 1
378 ÷ 31 = 2 (62) r 16, bring down the 2: 162 ÷ 31 = 5 (155) r 7 → 25 remainder 7
464 ÷ 27 = 2 (54) r 10, bring down 5: 105 ÷ 27 = 3 (81) r 24 → 23 remainder 24. Check 27 x 23 + 24 = 621 + 24 = 645 ✓. 24 is smaller than 27.
528 x 5 = 140, 170 − 140 = 30. 30 is 28 or more → 5 is too small. With 6: 28 x 6 = 168, left 2 → 6 remainder 2
622 x 7 = 154 is bigger than 150 → 7 is too big. With 6: 22 x 6 = 132, left 18 → 6 remainder 18
7135 ÷ 42 = 3 remainder 9. The 9 left over still need a seat, so 4 buses. If he says 3, ask what happens to the 9.
8300 ÷ 18 = 16 remainder 12 → 16 full boxes, 12 mandarins left
9The mistake: the remainder 39 is bigger than the divisor 26, so the quotient is too small. 143 ÷ 26 = 5 remainder 13 (26 x 5 = 130)
10A 7 cars (the 6 left over still need a car) B 6 boxes (6 sweets left cannot fill a box) C 6 pieces and 6 cm left
11450 ÷ 35 = 12 remainder 30 (35 x 12 = 420) → 12 pieces, 30 cm left
1223 x 15 + 8 = 345 + 8 = 353 (353 ÷ 23 = 15 remainder 8 ✓)
13The remainder can be 0 up to 26 (27 − 1). With the largest remainder 26: 27 x 12 + 26 = 324 + 26 = 350
Wed 14 Oct — costume 1: Pokémon or everyday life
1-1215 ÷ 35 = 6 remainder 5 (35 x 6 = 210). 5 HP is left → 7 hits
1-2Estimate 300 ÷ 50 = 6 → about 6 hits. 312 ÷ 48 = 6 remainder 24 (48 x 6 = 288) → 7 hits
1-3Luckiest: 49 → 360 ÷ 49 = 7 remainder 17 → 8 hits. Unluckiest: 40 → 360 ÷ 40 = 9 hits (exact). So 8 to 9 hits
2-1540 ÷ 45 = 12 km (45 x 12 = 540). More than the estimate of 450 km ✓
2-238 x 10 = 380 and 456 is more → more than 10 km per litre. 456 ÷ 38 = 12 km
2-3200 ÷ 13 = 15 remainder 5. 15 L goes 195 km, 5 km short → 16 L
467 ÷ 32 = 2 (64) r 3, bring down 8: 38 ÷ 32 = 1 r 6 → 21 remainder 6
526 x 6 = 156, 208 − 156 = 52. 52 is twice 26 → 6 + 2 = 8. 26 x 8 = 208 → 8 (remainder 0)
6A 1500 ÷ 40 = 37 r 20 → about 37 (38 is fine too). B 1500 ÷ 38 = 39 remainder 18 (38 x 39 = 1482) → 39 sweets, 18 yen left
7tens digit 5 → up → 6,500
8at least 4,250 and less than 4,350 (4,350 itself rounds to 4,400, hence "less than")
9400 + 300 + 600 = 1,300 → less than 1,500 → enough (really 1,152 yen)
Fri 16 Oct — rounding and estimating, word problems
1(printed) 700 < 800 → enough
2A 500 + 800 + 300 = 1,600. That is over 1,500, so we cannot say "enough". B 470 + 740 + 250 = 1,460 ≤ 1,500 → enough (really 1,452 yen)
3A 90 ÷ 10 = about 9 boxes. B 87 ÷ 12 = 7 remainder 3 → the 3 left need a box → 8 boxes
430 x 50 = about 1,500. Exactly 32 x 48 = 1,536
5at least 3,550 and less than 3,650
6smallest 24,500, largest 25,499 (25,500 becomes 26,000)
7Town A 12,480 → 12,000; town B 12,520 → 13,000. 12,500 is the border and the two towns lie on either side of it.
8Estimate 1,000 ÷ 50 = 20. 1,000 ÷ 48 = 20 remainder 40 → 20 sweets, 40 yen left
9A 316 ÷ 50 = about 6 buses. B 316 ÷ 45 = 7 remainder 1 → 8 buses. C Rounding 45 up to 50 makes each bus look bigger, so the count comes out low; and one pupil is left over, who needs one more bus.
1080 x 30 = about 2,400 yen. Really 78 x 29 = 2,262 yen. The estimate is bigger (both numbers were rounded up)
1160 x 7 = 420, 150 x 3 = 450 → 420 + 450 = about 870 yen (really 847; under 1,000)
12The number is at least 115 and less than 125 (115 to 124). 115 ÷ 4 = 28 remainder 3 → 29 groups. 124 ÷ 4 = 31 → 31 groups. 29 to 31 groups
13Estimate 500 x 20 = 10,000. 100,032 is ten times too big → wrong. 528 x 19 = 10,032
Sun 18 Oct — costume 2, corrections, puzzle
1-1636 ÷ 12 = 53 yen (12 x 53 = 636)
1-2A 658 ÷ 14 = 47 yen, B 1,150 ÷ 25 = 46 yen → B is 1 yen cheaper. The estimate (50 yen each) cannot decide it. When the gap is small, calculate properly.
1-31,000 ÷ 38 = 26 remainder 12 (38 x 26 = 988) → 26 bottles, 12 yen change
2-1Estimate 372 ÷ 20 = 18 remainder 12 → about 18 to 19 hits. 372 ÷ 24 = 15 remainder 12 (24 x 15 = 360) → 16 hits
2-2372 ÷ 48 = 7 remainder 36 (48 x 7 = 336) → 8 hits, which is half of the normal move's 16. (372 ÷ 24 = 15.5 and 372 ÷ 48 = 7.75 are exactly half, and both are rounded up.)
2-3Biggest 29 → 250 ÷ 29 = 8 remainder 18 → 9 hits. Smallest 21 → 250 ÷ 21 = 11 remainder 19 → 12 hits. So 9 to 12 hits
490 ÷ 37 = 2 (74) r 16, bring down 5: 165 ÷ 37 = 4 (148) r 17 → 24 remainder 17
5hundreds digit 5 → up → 39,000
6100 ÷ 12 = 8 remainder 4 → 8 full boxes, 4 pencils left. To hold all of them the 4 left need a box too → 9 boxes
7Check against the notebook. If he can say where it went wrong, that is enough.
858 − 12 = 46, 46 ÷ 2 = 23 → B has 23, A has 23 + 12 = 35 (35 + 23 = 58 ✓). Look at his bars: are the two equal after the 12 is removed?
9The remainder 40 is bigger than the divisor 32. 200 ÷ 32 = 6 remainder 8 (32 x 6 = 192)
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 · Buses for the trip
What it testsSocial × ÷ two digits, remainders, estimating. PISA: employ (1, 2) → evaluate (3). A remainder needs one more bus. The 52,000-yen fare is not needed.
236 → 40, 35 → 40, 37 → 40, 33 → 30. 40 + 40 + 40 + 30 = about 150.
Exactly 36 + 35 + 37 + 33 = 141 (9 away from the estimate).
3Not right. 141 ÷ 45 = 3 remainder 6 (45 × 3 = 135). Three buses carry only 135, so the 6 left over cannot ride. So 4 buses.
Full credit: 141 ÷ 45 = 3 r 6, or 45 × 3 = 135, and "6 cannot ride → 4 buses".
Partial: just "4 buses", or stops at "only 135 fit".
No credit: "enough, because 141 is less than 150".
2 · Moon shapes and time in the sky
What it testsScience × time arithmetic, ÷ two digits (700 ÷ 14), "per day". PISA: employ (4, 5) → evaluate (6). The half-moon column and "a cycle in about 30 days" are not needed.
55:50 pm → midnight is 6 h 10 min, midnight → 6:20 am is 6 h 20 min. 6 h 10 min + 6 h 20 min = 12 h 30 min.
6Not right. Day 0, 6:10 am → day 14, 5:50 pm is 11 h 40 min = 700 minutes. 700 ÷ 14 = 50 minutes. About 50 minutes later each day, not 30.
Full credit: 700 minutes (or 11 h 40 min) divided by 14 days gives 50, and "not 30".
Partial: gets 11 h 40 min, or says "later than 30 minutes" without reaching 50.
No credit: "right", or ignores the sources.
3 · The gym's electricity bill
What it testsSocietal context × ÷ two digits, estimating the quotient, "per day". PISA: retrieve/employ (7) → employ (8) → evaluate (9). The May bar and the 48 lights are not needed.
8Guess: 50,000 ÷ 30 = 1,666…, so about 1,700 yen.
Long division: 52,700 ÷ 31 = 1,700 yen (check: 31 × 1,700 = 52,700). Close to the guess.
(Any guess between 1,600 and 1,800 is fine.)
9Not right. August: 18,600 ÷ 31 = 600 yen. 600 × 3 = 1,800 yen, which is more than July's 1,700 yen. (August is a little more than a third of July.)
Full credit: gets 600, compares 1,800 with 1,700 and says "it passes → not right".
Partial: 600 correct, or tripled but not compared.
No credit: "right, summer holiday so it is low" with no calculation.
4 · How often the heart beats
What it testsScientific context × ÷ two digits, multiplying big numbers, rounding. PISA: employ (10) → employ/formulate (11) → evaluate (12). The "after walking" column is not needed.
1172 × 60 = 4,320 (one hour). 4,320 × 24 = 103,680. The thousands digit is 3, so round down: about 100,000.
(Another way: one day = 1,440 minutes, 72 × 1,440 = 103,680.)
12Right. After running: 324 ÷ 2 = 162 beats a minute. Twice resting = 72 × 2 = 144. 162 is more than 144.
Full credit: gets 162 and 144, compares them, says "right".
Partial: gets 162 (or 144) but does not compare.
No credit: decides with no calculation.