Models For Addition Alogrithms

On Tuesday in class, we learned about Addition Alogrithms...

Alogrithm: step-by-step procedure for computing
1. adding digits
2. regrouping or "carrying"

Examples
Partial Sums
In this method, the digits for each place value are added, and the partial sums are recorded before there is any regrouping. This method is easy for kids, because it allows them to look at the problem in a much simpler way. It is less overwhelming for them to work with smaller numbers.


1.   +345              2.               +345    = 3 hundreds + 4 tens + 5
        278                                278    = 2 hundreds + 7 tens + 8
          13                                             5 hundreds + 11 tens + 13
        11                          Regrouping:   6 hundreds + 2 tens + 3
        5                                               = 623
        623

I was never taught this way of adding in Elementary School, Junior High, or High School. I can't believe that over all those years I never saw a teacher add this way. Not even once! It would have been really helpful in my early years of math, because I would have been able to actually see what I was doing. In the partial sums method, it is obvious exactly what is being added. You can clearly see that there is three hundereds, four tens, and five ones in 345. If I become an Elementary teacher, I will make sure my students learn this method.



Left to Right Addition

First Step                  Second Step                   Third Step
  +897                           +897                             +897
    537                              537                               537
  13                                132                                1324
                                        4                                    43

To add 897 and 537 from left to right, first the 8 and 5 are added in the hundreds colum. In the second step, 9 and 3 are added in the tens colum. Because regrouping (carrying) is necessary, 3 in the hundreds colum is scratched out and replaced by the 4. In the third step, the units digits are added. Again regrouping is necessary, so 2 in the tens column is scratced out and replaced by 3.

This method is comfortable for kids, because they learn to read from left to right, some find it natural to add in this direction as well. I thought this was interesting. This is another method that I have never seen before. The first time doing this in class was kind of awkward. It was weird for me to add this way. It felt backwards. However, I think that if I had learned this when I was younger it would have worked very well, with little or no confusion.


Associative Property for Addition
For any whole numbers a,b, and c,
a+(b+c) = (a+b) +c

In any sume of three numbers, the middle number may be added to either of the two end numbers.

6 + 7 = 6 + (4 + 3) = (6 + 4) + 3 =10 + 3 = 13


Commutative Property for Addition
a+b = b+a

When two numbers are added the numbers may be interchanged without affecting the sum.

26 + 37 + 4 = 26 + 4 + 37 = (26 + 4) = 37 = 30 + 37

The numbers 26, 37,and 4 are arranged more conveniently on the right side of the following equation than on the left, because 26 + 4 = 30 and it is easy to compute 30 +37.
Breaking down problems helps kids understand and see how the problem is actually being solved.

video from xoax.net

This video is a great visual explanation of addition. It specifically shows kids how adding works step by step. The different colored markers help to see what is changing in each problem.

So that's all for addition... Can't wait to see the new things we will learn about subtraction. I'm sure there will be something I've never heard of!