Addition, Subtraction, Multiplication & Division
Understanding Addition & Subtraction
What is addition?
Addition means putting numbers together to find out how much they make in total.
+
=
4
+
=
2
4
2
Understanding Addition & Subtraction
What is addition?
Addition means putting numbers together to find out how much they make in total.
+
=
4
+
=
2
4
2
6
Other words we can use to describe addition are
add, sum, altogether, together, total, increase by
Understanding Addition & Subtraction
What is addition?
Addition means putting numbers together to find out how much they make in total.
+
=
4
+
=
2
4
2
6
What is subtraction?
Subtraction is the opposite: taking away a number from another.
Understanding Addition & Subtraction
What is addition?
Addition means putting numbers together to find out how much they make in total.
-
=
4
-
=
2
4
2
6
What is subtraction?
Subtraction is the opposite: taking away a number from another. It gives us the difference between two numbers.
Other words we can use to describe subtraction are
minus, difference, take away, deduct, remove
Adding Large Numbers
Column addition is a method we use to add large numbers by stacking them vertically and adding each place value (units, tens, hundreds, etc.) one by one. It helps us keep our numbers organized and ensures we don’t miss anything when adding big numbers together.
Adding Large Numbers
Example: Add 345 and 678.
Step 1: Write the numbers vertically
345
+ 678
Be sure to align the numbers by place value!
Step 2: Add the units place
345
+ 678
5 + 8 = 13
3
1
Write down the 3 from 13 in the units place...
...and carry the 1 over to the tens place
Adding Large Numbers
Example: Add 345 and 678.
Step 2: Add the units place
345
+ 678
23
1
1 + 4 + 7 = 12
...and again carry the 1 over to the hundreds place
1
Write down the 2 from 12
Adding Large Numbers
Example: Add 345 and 678.
Step 2: Add the units place
345
+ 678
1023
1
1
Write down the total of the column
1 + 3 + 6 = 10
Adding Large Numbers
Example: Add 345 and 678.
Step 2: Add the units place
345
+ 678
1023
Subtracting Large Numbers
Column subtraction works in a very similar way. The main difference is that instead of carrying a two digit over to the next column, we may need to borrow.
Subtracting Large Numbers
Example: 524 - 278
524
- 278
Write the bigger number on top and the smaller number underneath, making sure the digits are lined up (ones under ones, tens under tens, etc)
Subtracting Large Numbers
Example: 524 - 278
524
- 278
Start by subtracting from the right. Always begin with the ones (units) column. Subtract the bottom number from the top number.
In this case, that's 4 - 8, you can't do this because 4 is smaller than 8, this is where use 'borrowing'!
If the top digit is smaller than the bottom digit, borrow 1 from the next column to the left. Then, reduce the digit you borrowed from by one and add 10 to the current column.
We're going to 'borrow' one from here
'Borrowing' 1 from the tens column turns the 2 into a 1...
... and the 4 into 14.
Subtracting Large Numbers
Example: 524 - 278
524
- 278
Start by subtracting from the right. Always begin with the ones (units) column. Subtract the bottom number from the top number.
In this case, that's 4 - 8, you can't do this because 4 is smaller than 8, this is where use 'borrowing'!
If the top digit is smaller than the bottom digit, borrow 1 from the next column to the left. Then, reduce the digit you borrowed from by one and add 10 to the current column.
We're going to 'borrow' one from here
'Borrowing' 1 from the tens column turns the 2 into a 1...
... and the 4 into 14.
1
Subtracting Large Numbers
Example: 524 - 278
524
- 278
1
Now we can simply subtract 8 from 14. We're now ready to move on to the next column.
6
Subtracting Large Numbers
Example: 524 - 278
524
- 278
1
6
Because we have borrowed 1 from the 2 in the ten column, we only have 1 left
2
1
1 - 7 doesn't work, so again we need to borrow from the column to the left.
Subtracting Large Numbers
Example: 524 - 278
524
- 278
1
6
Because we have borrowed 1 from the 2 in the ten column, we only have 1 left
2
1
1 - 7 doesn't work, so again we need to borrow from the column to the left.
We now have 11 instead of 1 in the middle column.
1
Subtracting Large Numbers
Example: 524 - 278
524
- 278
1
46
2
1
1
11 - 7 = 4
Subtracting Large Numbers
Example: 524 - 278
524
- 278
1
246
2
4
1
In the left column we have 4 left after borrowing one.
5
4 - 2 = 2
Subtracting Large Numbers
Example: 524 - 278
524
- 278
246
Multiplication
Multiplication is a method of repeating addition. If you multiply 3 x 5, you are really taking 5 three times:
3 \times 5 = 15\\
5 + 5 + 5 = 15
Multiplication
Multiplication is a method of repeating addition. If you multiply 3 x 5, you are really taking 5 three times:
3 \times 5 = 15\\
5 + 5 + 5 = 15
You can think of multiplication as a 2-dimentional grid:
3
5
3 rows of 5 each
Multiplication
Multiplication is a method of repeating addition. If you multiply 3 x 5, you are really taking 5 three times:
3 \times 5 = 15\\
5 + 5 + 5 = 15
You can think of multiplication as a 2-dimentional grid:
3
5
3 rows of 5 each
= 3 \times 5 = 15
Multiplication
There are many methods you can use to multiply two numbers together, let's look at two of the most common methods, the column method and the grid method.
Multiplication
Column multiplication
Example: Multiply 47 x 36.
47
x 36
Write the numbers in column format. Start by multiplying each digit in the bottom number's units digit by each digit in the top number.
First, multiply 6 × 7= 42, so write the 2 and carry the 4 to the tens column
2
4
Then, multiply 6 x 4 = 24, and add the carried 4, 24 + 4 = 28.
28
Multiplication
Column multiplication
Example: Multiply 47 x 36.
47
x 36
For the next line, place a zero in the ones column since the next step is to multiply the bottom number's tens digit by each digit in the top number.
2
28
0
The bottom number's tens digit is 3, so multiply 3 x 7 = 21. Write 1 and carry 2.
1
2
Then we multiply 3 x 4 = 12, and add the 2 we carried, 12 + 2 = 14.
14
Multiplication
Column multiplication
Example: Multiply 47 x 36.
47
x 36
Finally, we add the results
2
28
0
1
14
+
1692
Final answer: 47 x 36 = 1692
Multiplication
Now we've seen how to multiply using the column method, let's look at multiplication using the grid method.
Multiplication
Grid multiplication
Example: Multiply 23 x 45.
Firstly, break each number into its place value components.
23 = 20 + 3
45 = 40 + 5
Multiply each part separately using a grid.
| 20 | 3 | |
|---|---|---|
| 40 | 800 | 120 |
| 5 | 100 | 15 |
20 x 40 = 800
40 x 3 = 120
20 x 5 = 100
3 x 5 = 15
Multiplication
Grid multiplication
Example: Multiply 23 x 45.
| 20 | 3 | |
|---|---|---|
| 40 | 800 | 120 |
| 5 | 100 | 15 |
Finally, add together the results in the table:
800
120
15
100
Multiplication
Grid multiplication
Example: Multiply 23 x 45.
| 20 | 3 | |
|---|---|---|
| 40 | 800 | 120 |
| 5 | 100 | 15 |
Finally, add together the results in the table:
800
120
15
100
+
+
+
=
1035
Division
The long division method is a step-by-step process for dividing large numbers by breaking them down into smaller parts. Let's look at an example.
Division
Example: Divide 432 ÷ 15.
432
15
Set up the division with the divisor, 15 outside of the division bar, and the dividend, 432, inside
We then work with the inside digits from left to right, dividing them by the divisor, 15.
15 does not go into 4, so include the next digit.
15 goes into 43 2 times so write 2 above the bar.
2
2 x 15 = 30, so subtract 43 - 30 = 13.
- 30
13
Bring down the next digit, 2.
2
15 goes into 132 8 times so write 8 above the bar.
8
8 x 15 = 120, so subtract 132 - 120 = 12
- 120
12
Final answer:
432 ÷ 15 = 28 remainder 12
Division
Example: Divide 432 ÷ 15.
432
15
Set up the division with the divisor, 15 outside of the division bar, and the dividend, 432, inside
We then work with the inside digits from left to right, dividing them by the divisor, 15.
15 does not go into 4, so include the next digit.
15 goes into 43 2 times so write 2 above the bar.
2
2 x 15 = 30, so subtract 43 - 30 = 13.
- 30
13
Bring down the next digit, 2.
2
15 goes into 132 8 times so write 8 above the bar.
8
8 x 15 = 120, so subtract 132 - 120 = 12
- 120
12
Final answer:
432 ÷ 15 = 28 remainder 12
Division
Example: Divide 432 ÷ 15.
432
15
2
- 30
13
2
8
- 120
12
Final answer:
432 ÷ 15 = 28 remainder 12
If you want to continue with decimals, you can continue to calculate your remainders:
Add a decimal point to your answer line after the last digit.
.
Since you now moved from units to tenths, you can add a zero to the last number.
0
Division
Example: Divide 432 ÷ 15.
432
15
2
- 30
13
2
8
- 120
12
If you want to continue with decimals, you can continue to calculate your remainders:
Add a decimal point to your answer line after the last digit.
.
Since you now moved from units to tenths, you can add a zero to the last number.
0
15 goes into 120 a total of 8 times
8
-120
Division
Example: Divide 432 ÷ 15.
432
15
2
- 30
13
2
8
- 120
12
If you want to continue with decimals, you can continue to calculate your remainders:
Add a decimal point to your answer line after the last digit.
.
Since you now moved from units to tenths, you can add a zero to the last number.
0
15 goes into 120 a total of 8 times
8
120 - 120 = 0. There are no more decimals or remainders. You have your final answer:
432 \div 15 = 28.8
-120
If you compare the two areas and you may notice that we doubled each side, but the area increased 4 times.
800m^2
200m^2
Thus, if you made each side 10 times longer, the area would increase by 100 times (10 x 10).
This is important to remember if you are converting from e.g. cm to m:
1m \times 1m = 1m^2
100cm \times 100cm = 10,000cm^2
1m^2
10,000cm^2
=
Well done! You have revised your addition, subtraction, multiplication and division. Remember: Your school may use slightly different methods of working out these, but the principals remain the same. You can use the ones you are most comfortable with.
With this information you should be one step closer to nailing the 11+ assessment. Good luck!
11+ Maths :: Addition, Subtraction, Multiplication & Division
By Pluspapers
