Survey of mental methods for subtraction The following section shows some subtractions carried out mentally by children. They have been loosely classified according to the mathematical principles involved. The writing in the movies and the adult voice are only used to help explain the thinking concisely. Classifications Terminology Complementary addition Subtraction in stages Rounding Equal additions principle Renaming principle Other methods 74   - 32   —— 42   minuend subtrahend difference Complementary addition Students very often use addition to build up from the subtrahend to the minuend. For example, to subtract 8 from 11, first see what needs to be added to 8 to get 11. In this case, counting 8, 9, 10, 11 shows that three have to be added. Therefore, 11 - 8 must be 3. Alternatively add 2 to get 10 and then another one to get 11. The movies below show some examples of mental computation, which uses the principle of complementary addition. There will be more than one possible way of 'adding up' to the minuend (larger number). The idea of mental computation is to select something easy. Mark Veena These movies display the mental computation in a written form, to help the explanation. Mostly, people would do this without any writing. There is no correct way to write these methods. Subtraction in stages For subtraction in stages, the subtraction is carried out in two or more steps. For example, with the question 53 - 34, when taking away the 34, 30 can be taken away first and then 4. Or it could be done by taking away 4, then 30, or even 3 then 31. The steps are selected to make the steps of subtraction as easy as possible. Vineeth Clare Rounding Many students round off either the subtrahend or the minuend or both, before carrying out subtraction. There are many possible ways of doing this. In the first movie, the numbers are rounded to 570 and 370, but they could have been rounded to 560 and 360, or anything else easy. A right choice is an easy choice for the user. Maggie Hendy Equal additions principle The equal additions principle is often observed in children's mental computation. According to this principle, adding equal quantities to the subtrahend and the minuend leaves the answer unaffected. For example, 23 - 16 = 7 .......... so that if I add 100 to both numbers, I know that 123 - 116 = 7 ........ and if I add a thousand to both numbers, I know that 1023 - 1016 = 7. This principle is the basis of the equal additions algorithm for subtraction. (Click here to go to the Teaching Algorithms for Subtraction page, equal additions.) Andy Peliwe When using the equal additions principle for mental computation, a good choice of the quantity added will make either the subtrahend or the minuend a round number (e.g. to the nearest ten). As shown in Andy's movie, the subtrahend (27) is rounded to the nearest ten (30) by adding 3 to both. The subtraction then becomes much easier to solve mentally. Renaming principle Students use the renaming principle when subtracting mentally almost automatically. The renaming uses place value equivalencies (e.g. ten tens are one hundred). This principle is the basis of the written subtraction algorithm called decomposition. (Click here to go to the Teaching Algorithms for Subtraction page, decomposition.) Here are two examples of mental computation using the renaming principle. Janet wanted to work out 1400 - 800 Immediately she thought of one thousand four hundred as fourteen hundred. Fourteen minus eight is six, so the answer is six hundred. James wanted to work out 85 - 73. He thought of 85 as "seventy and fifteen" (the renaming step) and then only had to subtract 3 from 15. Answer 12. Other methods There are many other individual innovative methods devised by children and adults to suit specific questions and many combinations of methods. Mental computation usually takes advantage of particular properties of the actual numbers involved. These movies show two other common methods, which do not fit into the classifications above. John John starts by subtracting 8 ones from 4 ones and records an answer of negative four ones. He then subtracts 3 tens from 7 tens giving 4 tens (40). The answer of 36 is found by adding -4 and 40. Cathy Since the numbers are close together, Cathy counts back from 103 to 98, recording the difference as she counts.