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| Using basic addition facts | Written algorithm for addition |
| Adding 3 or more digit numbers | Ways of recording carrying figures | Quick quiz |


Using basic addition facts

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

38 + 14

= 38 + (12 + 2)

= 38 + (2 + 12) (commutative law)

= (38 + 2) + 12 (associative law)

= 40 + 12

= 52

See Key Ideas, Associative and Commutative Laws for more detail.


The written algorithm for addition

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

Setting Out
Thinking

 

 

 

 

 

 

 

 

What do we add first? Ones. How many ones altogether? 9 ones.

tens
ones


 

 

What do we add next? Tens. How many tens altogether? 3 tens.

Tens
ones

The answer is 3 tens and 9 ones which is thirty nine.

Example 3: Adding two digit numbers with renaming, 19 + 34 = 53

Setting Out
Thinking

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

What do we add first? Ones. How many ones are there altogether? 13 ones. Are there enough ones to make a ten? Yes. 1 ten and 3 ones.

Tens
ones

 

 

 

 

   

 

Rename 10 ones as 1 ten.

Tens
ones


 

 

What do we add next? Tens. How many tens are there altogether? 5 tens. The answer is 5 tens and 3 ones which is fifty three.

Example 4: Expert performance of the addition algorithm,14 + 23 + 52 = 89



Adding 3 or more digit numbers

Example 5: Adding two and three digit numbers with renaming, 247 + 65 = 312

Setting Out
Thinking

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

hundreds
tens
ones

 

 

 


   

 

 

 

 

 

What do we add first? Ones. How many ones are there altogether? 12 ones. Are there enough ones to make a ten? Yes. 1 ten and 2 ones.

Rename 10 ones as 1 ten.

hundreds
tens
ones

 

 

 

 

 

 

 


 

 

 

 

 

 

 

What do we add next? Tens. How many tens are there altogether? 11 tens. Are there enough tens to make a hundred? Yes. 1 hundred and 1 ten.

Hundreds
tens
ones

 

 

 

 


 

 

 

 

 

 

 

Rename 10 tens as 1 hundred.

What do we add next? Hundreds. How many hundreds are there altogether? 3 hundreds. Are there enough hundreds to make a thousand? No.

The answer is 3 hundreds, 1 ten and 2 ones which we call 3 hundred and twelve.

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



Ways of recording carrying figures

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.

Type 1
Type 2
In this method the amounts carried across into the next column are recorded at the bottom of the column. In this method the amounts carried across into the next column are recorded at the top of the column.
   
Type 3
Type 4
In this method the amounts carried across into the next column are recorded underneath the line. In this method the traded amounts are not recorded at all - you tally them in your head. When bookkeeping was done by hand this ensured that there were no extra figures which could be confused with the original numbers.

The quiz answers at the end of this section use any of types 1 to 3.



Quick quiz

Perform the following additions:
1. 213 + 574
2. 693 + 27
3. 5908 + 439 + 6507
Estimate the answers to the following examples and then solve them:
4. 57 890 + 41 632
5. 102 337 + 8965 + 54 921

Further questions:

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
2003