Helping with maths homework when the method has changed
The most common frustration in helping a ten-year-old with maths is that you know the answer and cannot produce the working the school wants. That is not you being out of date, and it is not the school being awkward.
For most of these topics the answer is worth one mark and the method is worth three. Understanding what each method is for makes it much easier to help without accidentally teaching a shortcut that loses marks.
Long division is a place-value argument
The reason long division is still taught, when every phone can divide, is that it is one of the first sustained arguments a child constructs. Each step is small, each depends on the last, and the layout is the proof.
Dividing 9,452 by 7: seven goes into 9 once with 2 left. Bring down the 4 to make 24 — three sevens with 3 left. Bring down the 5 to make 35 — exactly five sevens. Bring down the 2 — no sevens, 2 left over. The answer is 1,350 remainder 2.
The check is the part worth teaching as a habit: 1,350 times 7 plus 2 is 9,452. It catches almost every arithmetic slip and it takes ten seconds, and children who acquire it early make far fewer careless errors for the rest of their schooling.
The remainder is always smaller than the divisor. If it is not, the last step was wrong — another useful self-check that costs nothing.
9,452 / 7
9 -> 1 seven, 2 left
24 -> 3 sevens, 3 left
35 -> 5 sevens, 0 left
2 -> 0 sevens, 2 left
Answer 1,350 r 2 check: 1,350 x 7 + 2 = 9,452
Work it out: Long Division Calculator →
Face value and place value are two different words
A question asking for the value of the 7 in 4,703 is testing one specific distinction. Its face value is 7 — the digit itself. Its place value is 700, because of where it sits. That is the whole question, and answering with the wrong one of the two is a very common lost mark.
The deeper point is that each column is ten times the one to its right, which is what makes this a base ten system. It is also why a zero has to be written even when there is nothing in that column: without the 0 in the tens, 4,703 becomes 473.
Expanded form — 4,000 + 700 + 3 — makes it visible, and it is worth writing out when a child is stuck. Most place-value confusion clears up the moment the number is broken apart.
Work it out: Place Value Calculator →
Rounding: look at one digit only
Rounding 4,763 to the nearest hundred means looking at the tens digit and nothing else. It is 6, which is five or more, so it rounds up to 4,800.
The mistake worth heading off is looking at more than one digit. 4,749 rounds to 4,700, because the deciding digit is 4 — even though 49 feels like it is nearly 50. Rounding is not cumulative and the digits after the deciding one are irrelevant.
It is also worth saying out loud that "five rounds up" is a convention rather than a fact. Exactly halfway is exactly halfway, so either answer is equally close; rounding up is simply what everyone agreed. Children find this reassuring rather than confusing, and it is honest.
Work it out: Rounding Calculator →
The unitary method is one idea
Find one first. That is the entire method, and it is what the name means. If five pens cost 60, then one costs 12, so eight cost 96. Two steps, always the same two steps.
It is worth insisting on the intermediate value being written down rather than jumped over, because the marks are attached to it — and because it makes the arithmetic easier, not harder.
The limit is worth mentioning to an older child: it only works when the relationship is proportional. Twice as many pens cost twice as much. Two people do not dig a hole in twice the time — they dig it in half. Spotting which situation is which is the actual skill the topic is building towards.
Work it out: Unitary Method Calculator →
Divisibility rules are worth knowing why
The rules themselves are quick: even numbers divide by 2, a digit sum divisible by 3 means the number is, the last two digits decide 4, and a 0 or 5 at the end decides 5.
The reason the digit-sum rule works is a genuinely satisfying piece of mathematics that a curious ten-year-old can follow. Ten leaves a remainder of one when divided by three, and so does a hundred, and a thousand, and every power of ten. So as far as three is concerned, a number is worth exactly the same as its digits added together. The same argument gives the rule for nine.
That kind of explanation is what turns a memorised list into something a child can rebuild if they forget it — which is much more valuable than the list itself, and considerably more interesting to teach.
Work it out: Divisibility Rules Calculator →
Questions
Why does the school want a particular method?
Because the marks are mostly in the working. The answer is typically worth one mark and the method three, so a correct answer arrived at by a shortcut can still score badly.
How do I check a long division answer?
Multiply the answer by the divisor and add the remainder — it should give back the original number. 1,350 times 7 plus 2 is 9,452.
What is the difference between face value and place value?
Face value is the digit itself; place value is what it is worth where it sits. In 4,703 the 7 has a face value of 7 and a place value of 700.
Does 4,749 round to 4,800?
No — 4,700. Only the digit immediately below the rounding place matters, and that is a 4. Rounding never works cumulatively.
Why does 5 round up?
By convention. It is exactly halfway, so either answer is equally close, and rounding up is what was agreed. It is worth saying so — children find the honesty easier than a false rule.
What is the unitary method?
Find the value of one, then multiply. Five pens for 60 means one costs 12, so eight cost 96. It only works where the relationship is genuinely proportional.
Why does adding the digits test for divisibility by 3?
Because every power of ten leaves a remainder of one when divided by three. So as far as three is concerned, a number is worth the same as its digits added up. The same argument gives the rule for nine.