#1 is adding by place value. Some of the kids call it decomposing the number, which is perfectly fabulous. This is basically just adding a place at a time. What makes this way so great, is that it focuses on adding the value of each place, instead of just the number in the place. For example, 235 + 316 could be broken up into 200 + 300, 30 + 10, and 5 + 6. Once you add those parts together, you add their sums for your answer. I know this looks a little long to those of us who were taught the traditional algorithm. While it does involve more steps, it really helps the kids see the value of what they're adding. When carrying, they understand it's a group of 10, not a 1.
The second way is by adding-on using a number line. You begin with your first number, then add on the second number in whatever number chunks you wish. Begin at 235, then add 300 to get you to 535. Then add 10 to get you to 545. Finally, add 6 to get you to 551. We're basically just breaking up that 316 into smaller, easier-to-add-mentally, chunks. I love this way of adding because the kids can customize it for their own skill level. If they need smaller chunks, they can add small numbers. If they see those bigger jumps, they can do that as well. This way really helps the kids make their numbers work for them, which will lead to stronger mental math.
The last way is the typical vertical algorithm. I've warned my kids that if they choose vertical, which is fine, they'd better be able to explain what all the numbers they've carried mean! No carrying a 1! It's a group of 10!
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