Mental math and estimation are two topics that have been ignored or minimized in schools. If you graduated from high school before 1990, your math education was mostly computation and other processes (mathematicians call these algorithms). That left little time for other math topics. If you are still in school or graduated after 1990, your math education was expanded to include probability, statistics, geometry and problem solving. Even with the general use of calculators to speed up our learning of all the new material, mental math and estimation were again put on the back burner and left there.
Suppose your second-grader is having trouble remembering some addition facts. She comes to you with the problem 5+7 because she cannot remember the fact. Suppose that you don’t tell her the answer, but reply with “What is 5+5?” She readily answers that 10 is the correct answer. Then you say, “How much more is 5+7 than 5+5”? If she can tell you that it is two more, she now can figure out the answer on her own: 5+7= 5+5+2, which is 10+2, which is 12. You and your second-grader have just discovered the strategy of substitution.
Let’s try another one. Let’s use our new strategy to add 17+7. We think of 17 as 10+7, so we have 10+7+7. The last 7 is really 3+4 (because we had to add something to 7 to get 10). We now have 17+7 is really 10+7+3+4. 7+3 is 10, and 10 added to 10 is 20. Now add the 4 to get 24. Notice that we never really had to “carry” anything.
If you can add 10s to any number, then you can add 9s, too. Since 9 is really one less than 10, we can add a 9 by adding 10, then backing up one. For example, 23+9 can be thought of as 23+10-1, or 33–1, which is 32.
One last example: At the dinner table, your middle school student asks you what the result of 27+48 is. With your hands occupied with the knife and fork, you can’t do the problem “in the air” and, with the calculator in the next room, you may have to admit to your student that you don’t know the answer. But wait! You now have the strategy of substitution! Think of 27 as 20+7 and 48 as 40+8. Add the tens first: 20+40 is 60. So now you only need to deal with 7+8. You know that 7+3 is 10, so you think of 7+8 as 7+3+5 which is 15. So 60+15 is 60+10+5=75. Again you never had to “carry” anything!
See if you can do these “in your head”. Then check the answers at the bottom of the column.
1) 37 + 17 2) 19 + 57 3) 286 + 428
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Answers:
(1) 37 + 17 = 30 + 10 + 7 + 7 = 40 + 7 + 3 + 4 = 50 + 4 = 54
(2) 19 + 57 = 57 + 19 = 57 + 10 + 9 = 67 + 9 = 67 + 10 – 1 = 77 – 1 = 76
(3) 286 + 428 = 200 + 400 + 80 + 20 + 6 + 8 = 600 + 100 + 6 + 8 = 700 + 6 + 4 + 4 = 714
