When I was in school (like many of you) I learned to write the equations vertically and begin adding in the one's place, carrying the one anytime the answer was bigger than 9. Basically this method is a great written strategy or algorithm. If you've ever tried to "carry the one" in your head, it can be challenging, especially as the numbers get bigger. There are just so many parts and pieces to keep straight.
Most adults have adopted other strategies for computing big numbers mentally. Sometimes I add the tens first, then the ones and finish by adding the tens and ones together. Other times I adjust one of the numbers to make it easier to do mentally and make a change at the end to compensate for that adjustment. And finally, there are times when I simply start with the bigger number and skip count up, first by 10s and then by 1s. What's more interesting is that children who have not been introduced to any particular method naturally discover these strategies on their own. In the long run these methods promote the understanding of number and the mental capacity to compute large numbers in our heads.
We are also using estimation to consider the reasonability of our answer. Estimation is a great way to check our work to see if it is close.
Below is a video modeling and explaining these strategies.