What are some important thinking strategies for subtraction?

Edward C. Rathmell
University of Northern Iowa

Children initially show the whole, take away a part, and count what is left to get the answer. This is commonly called take-away and count what is left or count all. Initially, this strategy is appropriate, but children should be encouraged to learn more efficient thinking for larger numbers.

There are two generalizations, that students learn about subtraction, to help them solve basic facts. One involves subtracting zero, n - 0 = n. The other involves subtracting a number from itself, n - n = 0.

After counting all, students learn some other counting strategies. If students are subtracting a small number, counting back is efficient. For example, for 9 - 2, you can just think, 8, 7. Counting back is not as easy as counting on, but most students can count back 1 or 2. Some students count back more, but this strategy becomes inefficient to count back many more than two.

Similarly, if the part you are subtracting is nearly the same size as the whole, counting up is efficient. For example, for 9 - 7, you can just start at 7 and count up to 9. By keeping track of how many counts you make, you know the difference, ...8, 9. That's 2 more. This strategy can be easily used for problems where the difference is no more than 2. Some children use it for problems with greater differences, but again, it becomes inefficient for greater differences.

Sometimes students use doubles. For example, for 11 - 6, you can think, since 5 and 5 make 10, the other part must be 1 more or 6. This can be used as long as the part is about half of the whole. The other part can be found by adjusting or compensating after checking to see if doubling the part makes the whole.

Similarly, some students use ten to help them solve subtraction facts with the whole greater than 10. Ten is used as a bridge or stepping stone to get from the known part to the whole or from the whole to the known part. For example, for 13 - 9, you can start at 9 and add 1 to make ten, then add 3 more to get 13. That's 4 more. Or you can think, start at 13 and subtract 3 to get to 10, then subtract 1 more to get to 9. That's 4 less.

If the part being subtracted is just a little more than the one's digit of the whole, you can use ten by subtracting that amount and using ten as a stepping stone. For 14 - 6, think 14 minus 4 is ten, then subtract 2 more to get to 8.

Ten can also be used in a other ways. For example, for 13 - 9, you can think 13 - 10 is 3, so 13 - 9 is 1 more or 4. This is particularly effective when you are subtracting nine. Also, for problems with a difference of 8 or 9, you can use ten as a bridge or stepping-stone when you take away the part. For example, for 16 -7, 16 - 6 is 10, then take away 1 more to get 9.

All of these strategies are overridden by the one main strategy that you hope students will eventually begin to use. That is using additionfacts that are already known. Since students know addition facts and each addition fact simply tells you the parts and the whole, they can use that information to answer related subtraction facts. For example, if you know that 3 + 5 is 8, then you also know 8 - 3 and 8 - 5 since the parts and the whole are the same for all of these basic fact problems.

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