M2N2e: ways to compensate

M2N2. Students will build fluency with multi-digit addition and subtraction.
e) Use basic properties of addition (commutative, associative, and identity) to simplify problems (e.g. 98 + 17 by taking two from 17 and adding it to the 98 to make 100 and replacing the original problem by the sum 100 + 15).
Students, and adults, often use different mental computation strategies. The one that is discussed in this standard is often explained by using the associative property of addition: 98 + 17 = 98 + (2 + 15) = (98 + 2) + 15

However, we can also explain it slightly differently. "98 + 17" means we are putting together 98 and 17. If we pretended 98 were 100, that means we actually have 2 more than we are supposed to. So, if we don't want to change the final answer, we have to make 17 smaller by 2. In other words, 98 + 7 = (98 + 2) + (17 - 2). In general, if we added a number to one of the addends, we have to subtract the same number from the other addend to compensate.

What about subtraction? Let's think about 83 - 18. Subtracting 20 mentally is much easier. But if we subtract 20 instead of 18, we will be taking away 2 more than we are supposed to. So, to compensate for that, we must make the starting number bigger by 2, too. That is, 83 - 18 = (83 + 2) - (18 + 2). Alternately, you might think if we make 83 into 89, then there will be no re-grouping needed. But, in that case, you are starting with 6 more. So, if we want to keep the answer the same, we must take away 6 more than 18 as well. Thus, 83 - 18 = (83 + 6) - (18 + 6). As it turns out, for subtraction, if we add (or subtract) the same number to both the minuend and the subtrahend, the difference stays the same. This idea is sometimes called the equal addition principle of subtraction.

What about multiplication? How do we compensate? Let's think about 35 x 16. If we had 70, it might be easier to multiply mentally. But if we realize that 35 x 16 means 16 groups of 35 [I'm using the Japanese convention of writing the number in a group first]. So, if we make 35 into 70, you are actually putting 2 of those 35's together, and there will be only 8 groups. Or 70 x 8. Thus, we see that 35 x 16 = (35 x 2) x (16 ÷ 2). In general, if we multiply a factor by a number, then we must divide the other factor by the same number to keep the product the same.

For division, let's think about 112 ÷ 14. One way to interpret 112 ÷ 14 is to figure out how many in each group if we split 112 into 14 equal groups. The answer should be the same if we only consider 7 groups with a half as many total. So, 112 ÷ 14 = 56 ÷ 7. In general, if we multiply (or divide) both the dividend and the divisor by the same number, the quotient does not change. In the GPS, this particular idea is actually explicitly mentioned in M4N3(d). I sometime call this relationship the equal multiplication principle of division. Probably the most common place where we see the use of this principle is with problems like 2400 ÷ 400.

When you look at these four ways of making compensations, you notice that there are parallels between addition/multiplication and subtraction/division. With addition and multiplication, we do "opposite" to the two numbers to keep the result the same. However, with subtraction and division, we do the same to both numbers. Although only the division situation is mentioned explicitly in the GPS, looking at these compensation strategies may be useful in helping students develop a deeper understanding of the four arithmetic operations and how they may relate to each other.