Weeks 02–05 — younger son, Grade 4 / Number
Four moves, repeated — and how to undo them
There is almost nothing to memorise in long division. You repeat the same four moves, once for each digit.
Where it goes wrong is usually not the sum but 'where does the next digit go'. That is why only that step is coloured in the figure below.
12 ÷ 4 can be read two ways.
① Share between 4 — how many each? → 3 (this finds the amount per one)
② Put them in groups of 4 — how many groups? → 3 (this finds the number of groups)
Same sum, different question. In a word problem, decide which one is being asked before you calculate anything.
145 ÷ 6 = 24 remainder 1. That much is arithmetic. The *answer* depends on what was asked.
'Six each — how many children get some?' → 24 (the leftover cannot be given)
'Six each for everyone — how many sweets are needed?' → 150 (buy more, because one is not enough)
'Cars holding 6 — how many for 145 people?' → 25 (one person still needs a car)
The same '24 remainder 1' becomes 24 or 25. Read the question again after you calculate.
In □ x 6 = 342, the □ is 342 ÷ 6 = 57.
Multiplication turned around is division — and the other way too:
□ ÷ 8 = 27 means □ = 27 x 8 = 216.
Always put your answer back into the original. 57 x 6 = 342 ✓ 216 ÷ 8 = 27 ✓
96 ÷ 4
Tens 9 ÷ 4 = 2 remainder 1 → write 2
4 x 2 = 8 9 − 8 = 1 bring down the 6 → 16
Ones 16 ÷ 4 = 4 → write 4
Answer 24 (check: 4 x 24 = 96 ✓)
175 ÷ 6
Tens 17 ÷ 6 = 2 remainder 5 → write 2
6 x 2 = 12 17 − 12 = 5 bring down the 5 → 55
Ones 55 ÷ 6 = 9 remainder 1 → write 9
Answer 29 remainder 1
Check 6 x 29 + 1 = 175 ✓ Do not forget to add the remainder back.
618 ÷ 6
Hundreds 6 ÷ 6 = 1 → write 1, nothing left, bring down the 1
Tens 1 ÷ 6 = 0 remainder 1 → write the 0, bring down the 8 → 18
Ones 18 ÷ 6 = 3 → write 3
Answer 103
People write 13. That is the answer with the tens zero left out. Checking catches it at once: 6 x 13 = 78.
□ x 7 = 259
□ = 259 ÷ 7 = 37
Check 37 x 7 = 259 ✓
84 ÷ 3 = 28 (3 x 28 = 84 ✓)
175 ÷ 5 = 35 (5 x 35 = 175 ✓)
238 ÷ 7 = 34 (7 x 34 = 238 ✓)
There is a remainder.
437 ÷ 8 = 54 remainder 5
Check 8 x 54 + 5 = 432 + 5 = 437 ✓
Watch the middle of the answer.
624 ÷ 6 = 104
In the tens, 2 ÷ 6 = 0 remainder 2, so write the zero. Leaving it out gives 14.
□ = 342 ÷ 6 = 57 (57 x 6 = 342 ✓)
This time it is the reverse of the reverse.
□ = 27 x 8 = 216 (216 ÷ 8 = 27 ✓)
Division turned around is multiplication.
One division, two different answers.
(a) 145 ÷ 6 = 24 children, 1 sweet left over
(b) 25 x 6 = 150 needed. 150 − 145 = buy 5 more
(a) throws the remainder away; (b) says the remainder is not enough, so buy more. One calculation, two answers — which is why you re-read the question.
28 players, 5 to a team.
28 ÷ 5 = 5 remainder 3
Five teams, and 3 players left. What happens to those three?
・they sit out → 5 teams
・they join existing teams → 5 teams of unequal size
・they form another team → 6 teams, one of only 3
The maths stops here; the rest is a decision. When a problem has no single answer, saying how you decided *is* the answer.
1 MB is roughly 1000 KB.
How many MB is 7500 KB? 7500 ÷ 1000 = 7.5 MB
If a photo is 3 MB, how many fit in 120 MB of space?
120 ÷ 3 = 40 photos
With big numbers you can estimate from the zeros alone: 12 ÷ 3 = 4, then put one zero back to get 40.
92 biscuits shared equally between 8 people.
92 ÷ 8 = 11 remainder 4
11 each, with 4 left over.
Share those 4 between 8 people and each gets 1⁄2 a biscuit. This is where fractions come from. The Grade 5 topic 'fractions and division' is this leftover, continued.
1,250,000 experience shared between 5.
1,250,000 ÷ 5 = 250,000
Do 125 ÷ 5 = 25 first, then put the four zeros back.
Between 10 it is 125,000. Double the sharers, halve the share. That is inverse proportion, which arrives properly in Grade 5.
Not the real questions — see the box, and the link to the official paper.
What this question is really testing：Whether the remainder can be judged from the situation. This shape appears every year. The arithmetic is easy; the marks are lost on deciding whether to round up or down.
⑴ 137 ÷ 40 = 3 remainder 17
⑵ 4 buses
⑶ A good explanation
"Three buses carry only 40 x 3 = 120 people, so 17 are left behind. Those 17 still need a bus, so when there is a remainder you add one more bus, making 4."
Writing 'round the remainder up' is not enough. *Why* you round up — because you cannot leave 17 children behind — is the point of the question.
The same numbers can go the other way. 'How many 40 cm pieces can you cut from a 137 cm ribbon?' is 3. Seventeen centimetres is not a piece, so that one rounds down.