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Subtraction

For subtraction, Everyday Mathematics offers five methods. The "trade first method", the "left to right subtraction method", and the "counting up method" are already present in the 3rd grade student reference book, and the "partial differences method" and the "same change rule" make their first appearance in the 4th grade student reference. Notice that the traditional right to left subtraction method is not part of the Everyday Mathematics curriculum.

Trade First

Closest to the traditional standard is a method that Everyday Mathematics calls "trade first". It is the Everyday Math focus algorithm for subtraction. It is a two stage process, first working right to left to do all the borrowing (recording the intermediate results above the top number) and then a second pass, in any order, doing the subtractions. The intermediate results are two-digit numbers, so one needs to use wide columns, and it is recommended to separate the columns with clear vertical lines. In the example at right, 325 - 58, we first recognize that the ones column needs borrowing, so we replace 2*10+5 by 1*10+15. Then we recognize that the tens column also needs borrowing, so we replace 3*100+1*10 by 2*100+11*10. Then we do the subtraction in each column.


     2|11|
| 1|15
3| 2| 5
- | 5| 8
--|--|--
2| 6| 7

Left to Right Subtraction

The second standard method of EM is left to right subtraction, the way one might well do the problem mentally, but carried out with paper and pencil. Here "left to right" refers to the decomposition of the second number. In the example at right, 325-58, the 58 is decomposed as 50+8. The individual subtractions are done mentally.


      325
- 50
---
275
- 8
---
267

Counting Up

The third standard method in EM is the "counting up method". It is carried out in two stages. In the first pass we count up from the smaller to the larger number, first by ones, then tens, and so on, and then the odd remainder, and then in a second pass we add up the addends. The example at right displays the process on our familiar example, 325 - 58.


      58      and then
+ 2
60 2
+ 40 + 40
100 + 200
+ 200 + 25
300 ---
+ 25 267
325

Partial Differences

The fourth standard method of EM, the "partial differences" method, is again a two-stage method. We operate first on each column individually, keeping track of the sign if a borrow would be needed, and then we combine the results using mental arithmetic. With this method, for 325 - 58 the second stage involves the mental arithmetic 300 - 30 - 3 = 267.


       325
- 58
---
+300
- 30
- 3
---
267

Same Change Rule

The fifth method is the "same change" rule. It is based on the recognition that a subtraction problem is easier if the smaller number ends in one or more zeroes. We may bring that about without changing the answer by changing both terms by the same amount. So, 325 - 58 may be changed into 327 - 60, which we do mentally to obtain our 267. If the second problem is still difficult then we might apply the same change rule again, but this is not discussed in the Everyday Math student reference book.


    325   ->   327
- 58 - 60
--- ---
267

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Counting Up Subtraction  11k v. 1 Aug 9, 2010, 11:45 AM Jeffrey Rainaldi
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Left to Right Subtraction  11k v. 1 Aug 9, 2010, 11:44 AM Jeffrey Rainaldi
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Partial Differences Subtraction  18k v. 1 Aug 9, 2010, 11:43 AM Jeffrey Rainaldi
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Trade First Subtraction  12k v. 1 Aug 9, 2010, 11:47 AM Jeffrey Rainaldi
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