About the Operations of Addition and Subtraction

• Only like things can be added or subtracted
• Cycles for addition and subtraction algorithms
• Two different subtraction algorithms
• References

Only like things can be added or subtracted

In their first year of schooling, students meet this apparently simple idea; that only like things can be added or subtracted. For example:

2 apples and 3 apples make a total of 5 apples

5 apples with 3 apples removed, leaves 2 apples

If we are trying to combine things which are not ‘like’ we may need to rename first, e.g. 2 apples and 3 bananas need to be renamed with their ‘common denominator’ to be seen as one group.

2 apples and 3 bananas? (THINK: these are all fruit)

2 pieces of fruit and 3 pieces of fruit makes a total of 5 pieces of fruit

2 apples and 3 bananas makes a total of 5 pieces of fruit

This powerful idea is at the heart of all addition and subtraction situations and needs to be made explicit for students as they move into more complex situations (see table below for various examples of addition).

In contrast, it is rare for ‘like things’ to be multiplied or divided – for example we multiply speed by time, not speed by speed. An exception is to multiply lengths (e.g. height and width of rectangle) but then the answer is not a length but an area.

In summary:

• Multiplication and division change the type of quantity,
• Addition and subtraction keep them the same.

Types of numbers	Examples of how we only add and subtract like things
Whole Numbers (natural numbers)	2 ones + 3 ones = 5 ones 2 tens + 3 tens = 5 tens 2 hundreds + 3 hundreds = 5 hundreds 2 tens + 3 ones = 20 ones + 3 ones = 23 ones
Fractions	2 fourths + 3 fourths = 5 fourths 2 ninths + 3 ninths = 5 ninths
Decimals	2 tenths + 3 tenths = 5 tenths 2 hundredths + 3 hundredths = 5 hundredths
Measurement	2 metres + 3 metres = 5 metres 2 kg + 3 kg = 5 kg 20 m + 3 m = 23 m 2 m + 0.5 m = 2.5 m 0.2 m + 0.45 m = 2 tenths m + 45 hundredths m = 20 hundredths m + 45 hundredths m = 65 hundredths m = 0.65 m
Other, including algebra and trigonometry	2 (groups of 8) + 3 (groups of 8) = 5 (groups of 8) 2 × 8 + 3 × 8 = 5 × 8 2y + 3y = 5y 2 t2+ 3 t2= 5 t2 2sint + 3sint = 5sint

Cycles for addition and subtraction algorithms

Addition and subtraction algorithms follow repeated cycles of combining or taking away and trading. These processes can be illustrated by the following cycles (the diagram has been adapted from Booker et al, 2004) These are powerful visual images as they emphasise the link between addition and subtraction (inverse operations).

Furthermore, by replacing ‘place value’ by ‘like-sized pieces’ we can adapt the vertical algorithm and these cycles of 3 steps for mixed numbers as well. Hence, we can use the cycle for addition and subtraction for various types of numbers.

The cycles above show what happens in the symbolic world.

In the world of concrete materials, the same cycles work. We select pieces of the same size (e.g. tens, or tenths, or halves), move the pieces to combine two groups for addition or to take some pieces away for subtraction, and we exchange groups of ten pieces for another size.

Two different subtraction algorithms

There are two formal written subtraction algorithms in common use in the community:

Decomposition Algorithm

This method is messier BUT its advantage is that it does record exactly what is done with materials in a ‘take-away’ context (trade 1 hundred for 10 tens to make 12 tens, leaving 5 hundreds).

This method can be complicated in some cases, e.g. 1000 − 24 requires a series of 3 trades (1 cube for 10 flats, 1 flat for 10 longs and 1 long for 10 minis) and recording this can be messy. In this case it is actually simpler to solve 100-24 mentally and then add 900 or work out 999 – 24 and then add 1.

 

Equal additions Algorithm

This method is tidier BUT it is not so easy to explain because it relies on ‘difference between’ rather than ‘take away’ idea of subtraction. That is, the difference between the two numbers remains constant if you add 100 to each number. The 100 added to each is made as 10 tens in the top number but just as 1 hundred in the bottom number.

References

G Booker et al, Teaching Primary Mathematics. (3rd edn), Addison Wesley Longman, 2004.