In order to be able to add whole numbers efficiently we need to be able to recall basic addition facts and use mental strategies. The mental strategies we use to solve addition problems are based on the commutative and associative laws. Together, these laws enable us to add numbers in any order and in flexible groupings.
Commutative Law: a + b = b + a for all real numbers
Associative Law: a + (b + c) = (a + b) + c for all real numbers
The commutative law means we only have to learn half the basic addition facts because we know that order does not matter for adding, e.g. 8 + 4 = 4 + 8. The associative law allows us to group numbers in convenient ways to add them, e.g. to add 8 and 5, we can split the 5 into 2 + 3 and use the known combination of 8 + 2 : 8 + 5 = 8 + (2 + 3) = (8 + 2) + 3 = 10 + 3 = 13.
In the example below we have used these laws to reorder and regroup numbers to quickly add 38 and 14. The steps written out seem long, but they are carried out very fast.
Example 1: 38 + 14 = 52
See Key Ideas, Associative and Commutative Laws for more detail.
The written algorithm for addition follows mental work with adding single digits and tens to two digit numbers. It is developed through a series of stages, beginning with illustration of the algorithm for two digit numbers with concrete materials and without the need for renaming and proceeding to larger numbers, renaming and more efficient execution.
Example 2: Adding two digit numbers without renaming, 13 + 26 = 39
Example 3: Adding two digit numbers with renaming, 19 + 34 = 53
Example 4: Expert performance of the addition algorithm,14 + 23 + 52 = 89
Example 5: Adding two and three digit numbers with renaming, 247 + 65 = 312
We need to be very careful when we add numbers that produce none of a particular grouping when added (eg, no ones, tens, thousands, etc.). When this happens a zero must be placed in the appropriate place value column.
The movies below demonstrate this.
Example 7: Full explanation with renaming, 328 + 476 = 804
Example 8: Expert performance with renaming, 459 + 543 = 1002
Example 9: Estimation and calculation, 97231 + 42859 + 65473 = 205563
Depending on where and when you went to school, you probably would have been taught one of the following methods of carrying.
You can choose to teach type 1, 2 or 3 as you prefer. In the past there were good reasons for not showing any carrying figures (type 4) but now that official calculations and bookkeeping records are done by computer it is not worthwhile training children to omit the carrying.
The quiz answers at the end of this section use any of types 1 to 3.
1. Farmer Jones has 256 horses and no other animals. He buys 134 sheep. How many animals does he now have altogether?
2. Sally has 8 dogs and Meg has 15dogs. How many dogs are there altogether?
3. Janeís mass is 38 kg and Jacobís mass is 49 kg. What is their total mass?
4. Sally rode 13 km further than Sam. If Sam rode 29 km, how far did Sally ride?
5. What number is 499 more than 1275?
6. Millie solved 192 maths problems this month. Last month she solved 29 more than this. How many maths problems did she solve in the two months?
7. The population of Pakistan is 131 000 000 and the population of Sri Lanka is 18 000 000. How many people live in Pakistan and Sri Lanka?
8. The areas of Western Australia, Queensland and South Australia are 1 700 000 km2, 1 400 000 km2 and 985 000 km2 respectively. What is the total area occupied by these three states?
To view the quiz answers, click here.
If you would like to do some more questions, click here to go to the mixed operations quiz at the end of the division section.
University of Melbourne