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Borrow and pay back

Q: I teach at an FE college and a student recently started doing subtraction using "borrow one pay one back". I haven't seen this method before and would be interested in knowing how it works. Can you help?

A: I went to primary school in Scotland and this is the formal method of subtraction I was taught. This method is also called "Equal addition". At college when the decomposition method was introduced it seemed to start a video in my head of my primary teacher showing us the hundreds, tens and units columns. When you are in a column where the number on the top line is smaller than the number to be subtracted, you "borrow one" from the next column to the left.

The traditional formal method of decomposition for subtraction is shown on the left of the first diagram (below); the same sum is shown as "borrow one, pay one back" on the right. For instance, for 6 - 7 in the tens column, one lump of 100 has to be borrowed from the hundreds column. You then have to pay one back to the hundreds column, by adding one to the bottom number (thus taking away an extra 100 from the top number). In this case, the 8 in the hundreds column to the left becomes 9 (8 + 1).

The easiest way for me to explain how the "borrow one, pay one back" method works is by writing the script of my thought processes as I go through the sum. Although I talk about 1 in each case in the picture, we are in fact borrowing 10, 100 and so on.

Take the example 4000 - 324. Although this subtraction can be performed using mental methods, some will find a paper and pencil method easier. The method can be applied to any sum. Begin on the extreme right-hand (units) column:

* The first column is "zero take away 4". This cannot be done, so I borrow 1 from the top row of the column to the left and pay 1 back to the bottom row of that column (the reason for this is that it takes away the lump of 10 that I borrowed from that column). The sum becomes 10-4=6, and I write the answer under the units column.

* The next column is now "0 take away 3" (the 3 is from the 1 I added to the 2 in the previous step). Again, this cannot be done, so I borrow another 1, this time from the thousands column, on the left, and pay 1 back to the bottom of that column. This now reads 10 - 3 = 7 and so I write 7 under the original 2.

* The next column is now "0 take away 4" (3 + 1). This cannot be done, so I borrow 1 from the top of the next column and then pay 1 back to the bottom of that column. We now have 10 - 4 = 6. Write 6 under the original 3.

* The last column is now 4 take away 1 (0 + 1), and this can be done: 4 - 1 = 3. Write this in the answer line beneath the unwritten 0. This gives the answer to 4000 - 324 as 3676.

The framework for mathematics suggests flexibility of methods used by pupils to carry out mathematical operations.

I am currently trying to update my knowledge of the NC levels for key stages 3 and 4. Our department is poorly stocked on such documents. Please could you tell me where to obtain lists of NC level descriptors and what pupils should know for each level and for each year (Years 7-9); NC GCSE grade descriptors and what pupils should know for each grade and for each of the four main areas (number; using and applying maths; shape and space; data handling); and key skills topics for Years 7-9.

Visit the National Curriculum website at;Subject@id=22amp;Session@id=D_ yis3e4CTrLs7ag596PwI

Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) Email your questions to Mathagony Aunt at Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX

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