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How do you teach addition? Play.Be sure to visit my
BLOG
to see how I use algebra to teach addition, multiplication and more to even very young students. There are actual lessons from my tutoring sessions and other things that may be of interest to you. This page, Addition and Subtraction And Problem Solving Using Manipulatives Gets into the nitty-gritty of using the blocks and reveals some secrets most are not aware of for best use of manipulatives.
Teaching addition is fun. Fun does not mean endless drills and worksheets. Some Asian learning centers add fun by doing endless drills and worksheets with timed trials and beat the clock games which can be an effective way to teach addition but not exactly the "funnest" way. Same with multiplication, repetition is the mother of skill but if you wait to long it becomes tedium. A lot of parents ask me how to teach addition without making it totally boring for their youngster. Using the top tray to build numbers puts the child in a situation where they cannot fail. And it's fun to boot. Easy and fun go together in the beginning. Older students can build their own tens using two hands. For very young students you start by putting a nine in the tens box and letting them pick the piece that makes a ten. Then an eight, then a seven etc. It is visually obvious what piece they need to put there. Blind students can feel what will go there. Next set up nine, eight, seven, six all the way down to one. These exercises are important to do for all students all the way through high school. You will not have them building walls and towers but you will have them building the numbers using two hands and going for speed. See who can build out a top tray the fastest. Racing is fun for older students. Even them teen-agers. I need to add a section on the degrees of difficulty...you keep it easy and pretty soon they are doing "difficult" problems...without difficulty never even knowing it's supposed to be hard or that other students think what they are doing is hard.
Let the student fill in the missing blocks. They love this. It makes them happy. Laugh along with them. Have fun. Give them encouragement. They will try to put a block in that is either to big or two small. Say “too big” or “too small”. DON NOT SAY, “NO, that's the wrong piece.” Instead say “try another one.” REMOVE THE “NO” FROM THE LESSON. If you insist on saying “no” I will hire math thugs to hunt you down and beat you with sticks. Some people are so used to saying “no” they have a hard time at first. Cut it out. This is important. 
I have found that for very young students (2 to 3) with limited motor skills, making a ten out of a six and a four is easier than making a ten out of a nine and a one, because the four is easier for their little hands to handle, even though the one fitting into the place left by the nine is supposed to be more visually obvious and therefore a lower degree of difficulty. Note that this child is not even two and a half years old, just having fun playing blocks. He has the two and the five together but it's too short. He eventually put the two with the eight and got a five out to go with the other five. Lastly he put the one in with the nine. There was trial and error, and he had fun doing it. Put the child in a math rich environment and they will learn math.
Skyscrapers: 
For students with more advanced motor skills build the tens vertically. Students really enjoy this variation the first couple of times they see it. Don't put them in order either it's more fun to fit the pieces in. 
The tens completed vertically. Any random order is fine and if they want to stair step it that's fine too.
You can also do this with nines and eights and sevens...then it gets a little too short and isn't as much fun. More on
how to teach addition past ten using walls and making buildings.
Building towers: 
Practice the smaller numbers by building towers. Here are the simple rules for building towers. The teacher is the only one who can use a six block, the students have to use combinations for seven, six, and five and so on. Students will always try to use the same blocks over and over again. Gently encourage them to use different combinations. Build up high without knocking it over, if it falls over start again...you can go a lot higher than three stroies! 
Three stories is fun, more is more fun. Students can race, building towers the fastest and the highest...keep it fun. Looking at this static webpage doesn't do it justice. Here is a video that captures the excitement:
And here is a link to the blog post that talks about these addends towers.
This is a Mortensen Math Activity. You can even get a kit from them specifically about this. The numbers like to party. Here is an example of how to teach addition using the having a party story with sevens. Empty the sevens out of a top tray. Put a one seven in the sevens place and say “Seven is having a party. Only sevens can come to seven's party. But six wants to come. Who can six bring to make seven?” 
At this point you can either put a six in there and let the student find the one or you can let them get both the six and the one and build the seven, depending on how young the student is or their skill level. You can also leave all the tens in the tray and replace each ten with a combination...the numbers are swimming in the tens pool nine wants to swim in the tens pool who must nine bring? ETC. Use your imagination to teach addition, make up stories and have fun. Teach addition in degrees, first get these combinations mastered then move on to two digits and three digits. Then it will be a matter of bigger is funner, rather than drudgery, pain and tears. 
The completed seven party. Note there are only three combinations for seven using two blocks!
Practice building the 25 combinations until the students don't have to think about them at all. Parents, when you teach addition to your children, they should MASTER addition facts, all 45 addends. Don't count on a public school teacher to do it for you. They are overworked underpaid and are often placed in a position where it s difficult for them to succeed. They are there to assist you in the education of your child, not to do it all for you. Note: we have not taken out paper and pencil yet, students who can't make numbers yet or need practice making their numbers can still learn how to add and do algebra and multiplication and...you get the point. Do not make the mistake of spending all of your time on addition. During an hour lesson you will count, sing songs, practice some addition, do some algebra, pattern multiplication, count by tens and more. You might also spend time learning to make the symbols 0, 1, 2, 3...with a paper and pencil. Or on a white board with dry erase markers. Check out
my blog
for more real life teaching examples, five minutes lessons and more. There are also more examples of how I teach addition and what I do in an hour. Here is a short vid showing the basic idea of "Having a Party":
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