Adding Left to Right
Date: 05/20/2003 at 13:26:12 From: Tonya Subject: Addition in Math When I was in school, we were taught to add numbers from right to left. Now my daughter's teacher is teaching her to add left to right. How can I help her to understand both ways? The type of learning is different from when I was a kid. 25 25 15 15 -- -- 40 (my way) 30 10 --- 40 (new way)
Date: 05/20/2003 at 23:45:27 From: Doctor Peterson Subject: Re: Addition in Math Hi, Tonya. I'm not sure I agree with teaching this as a method of addition, but I do see a reason for doing it. It shows on one hand that there are different ways to do arithmetic - it isn't just a magic trick that you have to do the way you are taught or it won't work; and on the other hand, that it is possible to think about what you are doing and why, rather than just blindly following rules. The danger is that some teachers might just teach it as another set of rules, without showing the meaning behind it. Let's look behind the scenes at both methods. The traditional way (which was developed over centuries as the best method for paper and pencil, since it requires the least writing or erasing) starts at the right. We add the "ones" column, and get 5+5=10. We can't write this as the ones digit in the answer, because it is more than one digit; so we break it up as 1 ten and 0 ones, and write down the 0. The 1 ten is added to the tens column in the problem, giving us 1+2+1 = 4, which we write as the number of tens in the answer. The basic idea behind this doesn't really depend on the order of the columns: we add tens to tens and ones to ones, and if the "ones" go beyond 9, we move the tens into the tens column. The "new way" does exactly this; the only real difference is that it "carries" after adding the original numbers, rather than as part of the tens column when it is first added. That is, you add the tens to get 20+10 = 30; and you add the ones to get 5+5 = 10; and then you add these together to get 30+10 = 40. We could do the "new way" from right to left as well; the order doesn't matter. Let's compare that with the traditional way: new: old: 1 <-- tens from 5+5 25 25 + 15 + 15 ---- ---- 5+5 --> 10 0 <-- ones from 5+5 2+1 --> 3 4 <-- tens from 2+1, plus 1 ---- 40 I did two things differently from the "new" method you showed: First, I switched the order of the 10 and the 30, just to make it feel more familiar; second, I didn't write the 0 in the 30, since that results from adding tens only, and there will never be any ones in it. As you can see, the 10 you get in the old method, which is written in two separate places, is written all in one line this way, but still in the same columns. The 2+1 from the tens, plus the "carry" of 1, are written separately now, but still added in the tens column. So all the same work is being done, but it is perhaps a little clearer why we do it: we add tens, we add ones, and we add the results together. I think it's clear that this method is less efficient with pencil and paper, because there is more writing. And working from left to right in the traditional method would require erasing, because you would write 1+2 = 3, but then have to add 1 and change it to a 4 after you added 5+5. That was done in earlier days, because erasing is no trouble on a sand table or chalk board or abacus; it was only with the introduction of arithmetic on paper that the right-to-left method was needed. But the new method works well when I add in my head, and I suppose it could be argued that the new generation will have more reason to add in their heads than on paper. It's easier to work left to right in your head, because you remember numbers from left to right. I would say something like, 20 + 10 is 30, 5 + 5 is 10, 30 + 10 is 40. There are lots of other ways to add in your head; I might be more likely to think "25 + 5 is 30, plus 10 more is 40." With my own children, I like to have them solve a problem and then tell me how they did it, to emphasize that there are lots of ways to do it, and each problem might have its own special trick. If children get used to thinking, rather than just following rules, it can only help their understanding of numbers. My only concern is to make sure that they aren't just being taught more rules, and getting confused as to which they should follow. I hope this helps. Definitely do help your daughter not only to understand both methods, but to understand why they are the same, and what that teaches about addition. I hope the teacher is doing the same. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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