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Crib Sheet Our daily guide to educational jargon, teaching methods and the strange things children may bring back from school as homework. If there is a particular aspect of your child's education you wish explaining, use the POL Ask an Expert service. Arithmetic Basic arithmetic is so easy for most of us that it's only when we try to explain it to someone that we realise that it's actually quite complicated. In this Crib Sheet, I'll try to explain not only how to do adding up, taking away, multiplying and dividing, but also why you do what you do. If you need a refresher about place value (Hundreds, Tens and Units), please check out the Crib Sheet before you read further. Mistakes After each explanation, there's a discussion of the kinds of mistakes your child might make. Obviously, some mistakes are just that - errors of computation (the sort of thing that happens when you add two and two and make five). Practice, and knowing the number bonds up to twenty off by heart will help with this. However, sometimes children don't understand what they are doing, and make systematic errors in their work. It's important to catch these early. No amount of practice will help if a child doesn't grasp the underlying concepts. Sometimes, it's possible to spot what's going on just from looking at a page of work - but this can be misleading. This is where you, as a parent, can help a lot, since you can go through the work with your child in a calm, relaxed way, and really find out what the problems are. Addition
TU 7 9 + ---- 16 ---- The way this simple sum is written masks an important part of the process - 'carrying'. Seven plus nine equals sixteen... what it really means is: seven Units plus nine Units equals one lot of Ten plus six extra Units. Since you can't have a 'Ten' in the Units column, you 'carry it over' into the Tens column. Usually, it's most helpful to write the 'carry' down, so you don't forget to add it in.
TU 7 19 + ---- 16 ---- Here's a more complicated example:
TU 9 44 23 11 28+ ---- 95 ---- Here, there's a 'carry' of 2 - that is, two lots of ten. Mistakes children make: One common mistake is to write down the whole of each part of the number. So:
TU
7
19 +
----
106
----
The six Units are straightforward enough - but instead of writing down a '1' in the Tens column, the child has written down a ten. You can sometimes spot this mistake because the 10 is squashed into the 'Tens' column. Another mistake I've seen is this:
TU
27
19 +
----
126
----
Here, the thinking seems to be that the 2 from the 27 goes into the Tens column, but the carried 1 gets shifted over into the Hundreds column. It can be tricky to spot this one. Sometimes the child writes the carried one in such a way that it looks like part of the original sum - a digit in the Hundreds column. Subtraction This is quite straightforward to begin with:
TU 15 4 - ----- 11 ----- Things soon get a little more complicated, though:
TU 37 19 - ---- 18 ---- Here, you cannot just take the nine from the seven - it's too big. There are two very different ways of handling this situation: 'borrow and pay back' and 'decomposition'. 'Borrow and pay back' is the older method. It tends not to be taught very much now.
TU 317 19 - ---- 18 ---- You've 'borrowed' one (one lot of Ten, that is) from the Tens column, so now you have 17 in the Units column and can take 9 from it, leaving 8 in the Units column.
TU
37
119 -
----
18
----
Since we borrowed a ten, you have to 'pay one back' - but you pay it back on the bottom line. So now you have 2 (two lots of ten, that is) to take away from the 3, leaving 1 in the Tens column. The problem with this method is that it really isn't very logical - there's no real reason for borrowing from the top and paying back at the bottom, for instance. For this reason children quite often find (or rather, found - since it's so rarely taught these days) it hard to understand. On the other hand, it is possible to learn 'borrowed one so pay one back' by rote, which some children manage to do if they can't understand the decomposition method. In contrast, the decomposition method is logical - but understanding it depends on having a firm grasp of the idea of bases and place value. This time you don't borrow anything - you just take a ten from the Tens column:
TU 317 19 - ---- 18 ---- That leaves 2 Tens, so before you do anything else, cross out the 3 and write in a 2:
TU
2
So, now you can take the 9 Units away from 17 Units; and the 1 ten away from the 2 Tens, giving the correct answer. One good trick to help children check their work is to add the result and the number subtracted together - the result should be the number they started with ( that is, 19+18=37). Mistakes children make Sometimes children don't understand that if they bring a digit into the Units column from the Tens column, it is worth 10 Units. Instead, they think it is worth only 1; generally, they then add the extra digit to what ever is in the Units column - and if that doesn't give them enough to do the subtraction, they bring across more digits till they have enough. So, in the sum above, they would start by doing this:
TU
2
But then they would get this:
TU
2
Since that still doesn't work, they would then do this:
TU
1
In other words, they'd end up with a result of zero! (This can also happen with 'borrow and payback' - they'd just end up paying more back. Or perhaps not, in which case the sum is wrong even within the terms of its own faulty logic.) Another fault is that children sometimes forget (or don't understand - or maybe just don't believe) that they can take a larger digit from a smaller one as long as it isn't in the position to furthest left of the number. So instead, they take the smallest number from the largest in each case:
TU
37
19 -
----
22
----
What's happened here is that the child has taken 7 from 9 and then 1 from 3. This is most likely to happen where 'you can't take a bigger number from a smaller one' has been drummed in by rote, without anyone making sure the child understands that 'the number' means 'the whole of the number' not 'each individual digit'. Multiplication It's very common for children to start learning their tables without understanding what multiplication is. Up to a point, this doesn't do much harm - it's very useful for children to know number facts off by heart. However, at some point they must come to understand the underlying concept. Multiplication is repeated addition. When we say 'six eights are forty-eight', we really mean
Multiplying numbers under 10 together is fairly straightforward, even before children have learned them by heart. Multiplying numbers over ten is a little bit more complicated. The trick is to break the sum down into separate stages.
HTU 23 41 x ----- ----- Rather than trying to multiply by 41, the trick is to multiply first by 40 and then by 1, and then add the results together. In fact, the 'multiply by 40' part gets broken down again, so that you multiply first by 10 and then by 4. The effect of multiplying by 10 is to put a zero in the Units column and push each digit left one place, so this is what you end up with when we multiply by 10:
HTU 23 41 x ----- 0 ----- Then multiply by 4, remembering to carry:
HTU 123 41 x ----- 920 ----- Notice that the result of multiplying 3 by 4 (actually 40) ends up in the Tens column, not the Units column. Next, multiply by 1 - this time, of course, you don't put down a 0, since you are multiplying by the digit in the Units column.
HTU 123 41 x ----- 920 23 ----- Finally, add the two results together:
HTU 123 41 x ----- 920 23 ----- 943 ----- Note that some teachers prefer to start by multiplying with the Units column, then the Tens. It's useful to show children that they can check their work by doing the multiplication the other way round - so our example would become:
HTU 141 23 x ----- 820 123 ----- 943 ---- Mistakes Children Make There are two common mistakes. One is to get confused about how many zeroes to put in when multiplying by anything other than Units. For instance, children sometimes think they need to put in two zeroes when multiplying by the Tens part of the sum 'because there are two numbers in the bottom line'. This is why it's useful to explain that what you're really doing is multiplying first by ten and then by whatever the actual digit in the number is. Second, and more commonly, children line the two parts of the result up wrong. This is especially true when multiplying large numbers, when there may be as many as five lines of partial results to add up. I've found that this problem is more common when the multiplication is started from the Units, which is why I advocate doing it the other way. However, it's probably best to go with whatever your child's teacher prefers, unless your child runs into real difficulty. One thing that can help is working on squared paper, so it's easy to line the columns up. Division Division means sharing a number into equal groups (with a remainder - the left overs - if necessary). Another way of thinking about it is to say that it's repeated subtraction - the reverse of multiplication, if you like. In fact, at its simplest, division just uses the number facts learned in the times tables:
____
9)45
----
5
Sometimes there is a remainder ____ 9)47 ---- 5 remainder 2 Long division, like long multiplication, is a bit more complicated: _____ 23)576 ----- Here the trick is to have an area on the page for working out - and to use it in an orderly way. Obviously, you can't divide 23 into 5, so you have to move the 5 (five lots of one hundred, remember) into the Tens column.
_____ 23) Now you can divide 57 by 23 (since 57 is larger than 23) - the question is, though, how many times it will go. With experience, the answer is clearly 2 - but a beginner needs a way to find that out. Here's how:
TU 57 23- ---- 34 ---- First, you take 23 away from 57. The answer, 34, is still bigger than 23. So repeat the operation, but this time use the new result as the starting figure:
TU 34 23- ---- 11 ---- Since 11 is smaller than 23, it's not possible to do another subtraction. So now you know that 23 goes into 57 twice with a remainder of 11. Now go back to the division sum, and write in 2 in the Tens column. Carry the remainder - 11 (really 11 lots of ten) - into the units column:
______ 23) Now repeat the subtraction procedure to find out how many times 23 will go into 116:
HTU
HTU 93 23- --- 70 ---
HTU 6
HTU 47 23- --- 24 ---
HTU 24 23- --- 1 --- So the answer is 5 times, with a remainder of 1, which you can now write into the sum.
______ 23) With a bit more experience, it's possible to short cut this process by estimating the answer and then checking by using multiplication. However, if your child uses that method to start with, they may not really understand what is going on. They may also make numerous mistakes, especially if they are still learning multiplication (particularly if long multiplication is involved). Mistakes Children Make One common mistake is to get confused over place value and what happens when a digit is moved from one column to the next. For instance, a child might do this:
_____
16)74
-----
_____
16)
Instead of taking the 7 into the Units column as single digit, the child has moved 70 along - so the sum really becomes 704 divided by 16. This quite often leads to confusion about which columns the answer should go into - the whole of the number may be squashed into the Units column, for instance. The other common problem is organising the working out (whether subtracting or multiplying) badly, and either making mistakes because of it, or transcribing the answers wrongly. |