Most of us are taught to do math on paper from right to left. And that's fine for doing math on paper. But if we want to do math in our head (even faster than we can on paper) there are many good reasons why it is better to work from left to right.After all, we read numbers from left to right, we pronounce numbers from left to right, and so it’s just more natural to think about (and calculate) numbers from left to right. When we compute the answer from right to left (as we probably do on paper), we generate the answer backward. That’s what makes it so hard to do math in our head.
Also, if we want to estimate our answer, it’s more important to know that our answer is “a little over 1200” than to know that the answer “ends in 8.” Thus, by working from left to right, we begin with the most significant digits of our problem. If we are used to working from right to left on paper, it may seem unnatural to work with numbers from left to right. In fact, our brain is already hard-wired for solving math problems from left to right; wehave spent years reading left to right, correct? But with practice you will find that it is the most natural and efficient way to do mental calculations.
With the first set of problems—two-digit addition—the left to- right method may not seem so advantageous. But be patient. If you stick to it, you will see that the only easy way to solve three-digit and larger addition problems, all subtraction problems, and most definitely all multiplication and division problems is from left to right. The sooner you get accustomed to computing this way, the better.
The easiest two-digit addition problems are those that do not require you to carry any numbers, when the first digits sum to 9 or below and the last digits sum to 9 or below. For example:
45
45
+ 34
?
Usually, you would first sum up 4 to 45, and then add 30 to the result. But by using the left to right approach, you first sum up 30 to 45, and then you add 4 to the result. After adding 30, you have the simpler problem 75 + 4, which equals 79.
Let’s illustrate this as follows:
45 + 34 = (45 + 30=>) 75 + 4 = 79
(first add 30) (then add 4)
Although this example is very simple, you’ll see the advantages of this method as you start to use it.
Now let’s try a calculation that requires you to carry a number:
84
+ 57 => 84 + 57 = (84 + 50=>) 134 + 7 = 141
? (first add 50) (then add 7)
Since, 57 = 50 + 7, you first sum up 84 to 50, and then you add 7 to the result.
If you’re working with three digits numbers, the process is the same. After each step, you arrive at a new (and simpler) addition problem. Let’s try the following:
759
+ 237
?
This example is a bit more complicated than the previous one, yet it’s very easy to solve using the left to right approach. Starting with 759, we add 200, then add 30, then add 7. After adding 200 (759 + 200 = 959), the problem becomes 959 + 37. After adding 30 (959 + 30 = 989), the problem simplifies to 989 + 7 = 996. This thought process can be diagrammed as follows:
759 + 237 = 959 + 37 = 989 + 7 = 996
(first add 200) (then add 30) (finally add 7)
All mental addition problems can be done by this method. The goal is to keep simplifying the problem until you are just adding a one-digit number. Notice that 759 + 237 requires you to hold on to six digits in your head, whereas 959 + 37 and 989 + 7 require only five and four digits, respectively. As you simplify the problem, the problem gets easier!
Another example:
858
+ 634
?
858 plus 634 is 1458 plus 34 is 1488 plus 4 is 1492.
You can split and solve the same this way as well:
858 + 634 = (800 + 600) + (50 + 30) + (8 + 4) = 1400 + 80 + 12 = 1492
2785
+ 634
?
2785 plus 634 is 3385 plus 34 is 3415 plus 4 is 3419.
Or, 2785 + 634 = (2700 + 600) + (80 + 30) + (5 + 4) = 3300 + 110 + 9 = 3419
Example for long addition:
326 + 678 + 245 + 567 = 900, 1100, 1600, 1620, 1690, 1730, 1790, 1804, & 1816
Look for opportunities to combine numbers to reduce the number of steps to the solution. This was done with 6+8 = 14 and 5+7 = 12 above. Look for opportunities to form 10, 100, 1000, and etc. between numbers that are not necessarily next to each other.
Practice some of your own if you like until you are comfortable doing left-to-right addition mentally.
My mother was an accountant forty years ago. She was noted to be able to add five digit numbers, left to right, with ease. She was German born and educated. I was born the US and went to decent public schools, but never learned this. Im glad to find this here. Im going to show my eight year old because easy math creates confident students.
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