Trade-first subtraction is the traditional borrowing algorithm with the work put in a different order. Instead of switching between trading and subtracting column by column, a student first scans the whole problem and makes every trade that will be needed. Only then does any subtracting happen.
The reason for splitting it in two is simple. Most borrowing errors come from switching tasks mid-problem: a child trades, subtracts, trades again, and loses track of which column has already been reduced. Doing all the trades first means the second stage is nothing but easy subtraction.
How the method works
- Line the numbers up by place value.
- Look at each column, right to left. If the top digit is smaller than the bottom digit, trade one unit from the column to its left before doing anything else.
- Record every trade as you go: the column you took from goes down by one, the column you gave to goes up by ten.
- When no column has a smaller digit on top, subtract every column in one pass.
A worked example
Take 471 - 293.
Stage one, all the trading
4 7 1 - 2 9 3 ones: 1 is less than 3, so trade a ten 4 6 11 - 2 9 3 tens: 6 is less than 9, so trade a hundred 3 16 11 - 2 9 3
Stage two, all the subtracting
3 16 11 - 2 9 3 ------------ 1 7 8 so 471 - 293 = 178
Reading the trades. When the tens column went from 6 to 16, the hundreds column dropped from 4 to 3. Nothing was created. One hundred simply moved across and became ten tens.
An example with zeros
Zeros are the case that separates a student who understands trading from one who is copying steps. In 600 - 247 the ones column needs a ten, but the tens column has nothing to give, so the trade has to travel.
6 0 0 - 2 4 7 trade a hundred into the tens 5 10 0 now trade a ten into the ones 5 9 10 - 2 4 7 ------------ 3 5 3 so 600 - 247 = 353
Why it works
Every trade rewrites the top number without changing its value. In the first example, the top number begins as 400 + 70 + 1 and ends as 300 + 160 + 11. Add those up and both come to 471. Because the value never changed, subtracting from the rewritten form gives the same difference as subtracting from the original.
Common mistakes
Adding ten without reducing the neighbour
A student turns the ones into 11 but leaves the tens digit as 7. The top number has silently grown by ten and the answer is ten too large. Every trade changes two columns, never one.
Trading a column that did not need it
Trading when the top digit is already larger is not wrong mathematically, but it adds work and invites a slip. Scan first, trade only where the top digit is smaller.
Starting to subtract before the trading is finished
This is the habit the method exists to break. If a column is subtracted and then a later trade reaches back into it, the recorded answer for that column is already stale.
Check yourself
842 - 367
ones: 2 < 7, trade -> 8 3 12 tens: 3 < 6, trade -> 7 13 12 subtract: 7 13 12 - 3 6 7 4 7 5 475
703 - 458
ones need a ten, tens have none, so trade a hundred first -> 6 10 3 then a ten into the ones -> 6 9 13 subtract: 6 9 13 - 4 5 8 2 4 5 245
1,000 - 649
the trade travels across three columns 1 0 0 0 -> 0 9 9 10 minus 0 6 4 9 3 5 1 351
Why does trading never change the answer?
Because a trade only rewrites the top number into a different but equal form. One hundred becomes ten tens, one ten becomes ten ones. The total value on top stays exactly the same, so the difference stays the same too.
See also
Sources
- Trade-first subtraction appears in the Everyday Mathematics curriculum as a staged form of the standard regrouping algorithm. Examples and diagrams on this page are our own.