# Primary Mathematics/Subtracting numbers

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## Subtracting numbers

Subtraction is when we ask the question, if I have this many, and take away that many, how many are left? Like I may have 7 apples, but I eat 2, so I have 7-2 = 5 left.

We are not allowed to subtract a larger natural number from a smaller one because we can not have less than zero of anything.

Subtraction is just addition reversed. You can see this in the following way. If I have an addition, like 3+5 = 8, I can write it in a triangle like so:

```   8
/ \
3 + 5
```

Now the left side and the right side of the triangle are two subtractions: 8 - 3 = 5, and 8 - 5 = 3. This nicely shows that when you subtract, say, 8 - 5, you want to know what the other leg of the triangle is, given that the top is 8 and one leg is 5. To say this in another way, you want to know what must be added to 5 to get 8.

Below is a subtraction table for small numbers. To use it, find your first number in the left column, your second number in the top row. Then the difference is found where that row and column intersect. These differences follow a very regular pattern and are easy to remember with a little practice.

Subtraction table
- 0 1 2 3 4 5 6 7 8 9
0 0
1 1 0
2 2 1 0
3 3 2 1 0
4 4 3 2 1 0
5 5 4 3 2 1 0
6 6 5 4 3 2 1 0
7 7 6 5 4 3 2 1 0
8 8 7 6 5 4 3 2 1 0
9 9 8 7 6 5 4 3 2 1 0
10 10 9 8 7 6 5 4 3 2 1
11 11 10 9 8 7 6 5 4 3 2
12 12 11 10 9 8 7 6 5 4 3
13 13 12 11 10 9 8 7 6 5 4
14 14 13 12 11 10 9 8 7 6 5
15 15 14 13 12 11 10 9 8 7 6
16 16 15 14 13 12 11 10 9 8 7
17 17 16 15 14 13 12 11 10 9 8
18 18 17 16 15 14 13 12 11 10 9

## Subtraction with large numbers

When larger numbers are involved, use the following procedure. Subtraction is more complicated than addition, so pen and paper are usually required.

• If the answer of subtracting these digits will not be less than zero, subtract these digits. The answer is the right-most digit of the answer.
• In this case, move one digit left in each number and repeat the process.
• If it was not possible to perform the subtraction because the result would be less than zero, add 10 to the digit of the first number to make the subtraction possible. Do the subtraction, the answer is the right-most digit of the answer.
• Reduce the next digit to the left in the first number by 1. This will be possible if that digit is 1 or more.
• If reducing that digit was possible, the borrowing procedure is complete. Continue to the next round, moving one digit to the left in each number. You will now use the reduced digit in the next round.
• If reducing that digit was not possible because it was 0, add 10 to it and then reduce it by one. Now one must borrow again from next left digit. Reduce the next digit to the left by 1, borrowing again if it is 0. Continue this process until you reduce by 1 a digit that is 1 or more. Then the borrowing procedure is complete. Repeat the process with the next digit of each number. You will now be using the first of the reduced digits in the next round.

To make this clearer, see the examples below.

Example 1, no borrowing required.

```  36    36
-  5, -  5         ∴36-5=31
----  ----
1    31
```

Example 2, 1 round of borrowing required.

```      3    3
40   40   40
- 2, - 2, - 2
---  ---  ---　　　　　∴40-2=38
8    8   38
```

Example 3, 2 rounds of borrowing required.

```       09    09
100   100   100
- 15, - 15, - 15
---   ---   ---     　∴100-15=85
5     5    85
```

## Teaching subtraction

Subtracting numbers and adding numbers should be taught together, the process of teaching subtraction is the same as teaching addition. You should not teach children negative numbers at a young age as there is no way to physically show them a negative number.

As with addition it is important for children to remember the simple combinations of single digit numbers. Moving to 2 digit addition (having learnt place values), start with simple problems:

``` 45
-32
===

```

first 5-2=3 second 40-30=10

``` 45
-32
===
3
10
```

``` 45
-32
===
3
10
===
13
```

When this is grasped move on to harder problems:

``` 52
-35
====
```

There are 2 methods that I would like to illustrate, the first uses carrying:

Start by saying that 5 is bigger than 2 so we take 10 from the 50 and add it to the 2 to get 12

``` 5412
- 3 5
======
```

Is the usual way of writing this.

Now subtract 5 from 12 to get 7. This is another 'pair' that needs to be taught - all the numbers up to 10 subtracted from all larger numbers up to 20 are 'required', or it is possible to say that 12-2 is 10 and that 5-2 is 3 and 10-3 is 7, which is how I do it, but is not necesseraly the 'easiest' way. Different children will want to do this in different ways and will often develop their own methods for mental subtraction (ie when you don't write it down). For instance I do this: 56-38 becomes 50-40+6+2 : note that I have used both the fact that 56=50+6 and that 38=40-2 (this is why I personally feel that every child needs to know the pairs that sum to 10) and that a+b-(c-d) = a-c+b+d allthough you can 'see' this as the 'big chunk' from 40 to 50 and then the two small end bits added on (6 and 2) without expaining using algebra, again teachers should remember this general truth (that addition can be re-ordered and that minus minus = plus) in order to answer questions and with a view to the child learning it in this form.

To continue:

``` 5412
- 3 5
======
7
```

Then we take 30 from 40 (remember that it is now 40 since we gave the other 10 to the units) to get 10

``` 5412
- 3 5
======
7
10
```

And finally add these two together

``` 5412
- 3 5
======
7
10
======
17
```

An alternate method of subtraction is to perform subtraction starting on the left and heading to the right. This second method was used by some young children, which shows that young children can understand negative numbers if told about them:

The same sum:

``` 52
-35
====
```

This time we will take 30 from 50 first (working left to right here - much more 'obvious') to get 20

``` 52
-35
====
20
```

Then we will take 5 from 2 to get -3

``` 52
-35
====
20
-3
```

Then we take the 3 from the 20

``` 52
-35
====
20
-3
====
17
```

To finish the sum. Here students need to know what 20-3 is: This is easiest to look at as 10+10-3 and then know that 10-3 is 7 (again, numbers adding to 10).

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