Mental methods for subtraction
Children do not merely absorb information passively but interpret selectively and construct their own meanings from it (Labinowicz, 1985, see references). Hence they will devise variations of the algorithms taught at school and sometimes invent their own methods to suit the question. Children will use their understanding of place value and their number sense to work mentally with numbers. Many children can work out answers to arithmetic questions before they are formally taught how to do the algorithms.
This section shows some subtractions carried out mentally by children. They have been loosely classified according to the mathematical principles involved. These questions have been done mentally by children; the writing in the movies and the adult voice are only used to help explain the thinking concisely.
Classifications
- Complementary
Addition
- Subtraction in
stages
- Rounding
- Equal additions
principle
- Renaming
principle
- Other methods
Complementary Addition
Students very often use addition to build up from the subtrahend to the minuend. For example to subtract 8 from 11, see how many need to be added to 8 to get 11. In this case, counting 8, 9, 10, 11 shows that three have to be added; 11 - 8 must be 3.
The movies below show some examples of mental computation, which uses the principle of complementary addition. Note that:
- There will be more than one possible way of 'adding
up' to the larger number.
- These mental computations displayed in the written form may be presented in various ways.
Movies:
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Subtraction in stages
In this method of mental computation, the subtraction is carried out in two or more steps. For example, taking away 34 can be done by taking away 30 and then taking away 4.
Here are some examples of mental computation using subtraction in stages.
Movies:
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Rounding
Many students round off either the subtrahend or the minuend or both, before carrying out subtraction.
Here are some examples where students round off numbers for subtracting.
Movies:
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Equal additions principle
The equal additions principle is often observed in children's mental computation. This principle is that adding equal quantities to the subtrahend and the minuend leaves the answer unaffected. For example, 23 - 16 = 7, so that 123 - 116 = 7 and 1023 - 1016 = 7. This principle is the basis of the equal additions written algorithm for subtraction. (see equal additions).
Here are some examples of mental computation using the equal additions principle.
Movies:
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Renaming principle
Students use the renaming principle when subtracting mentally almost automatically. This principle is also the basis of the written subtraction method called decomposition (see decomposition).
Here is an example of mental computation using the renaming principle.
Janet:
Janet wanted to work out one thousand four hundred subtract eight hundred.
Immediately she thought of one thousand four hundred as fourteen hundred.
Fourteen minus eight is six, so the answer is six hundred.
Other methods
These are individual innovative methods devised by children to suit specific questions. Mental computation often takes advantage of particular properties of the actual numbers involved.
The movies below show some mixed, unclassified examples of mental computation.
Movies:
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