Addition is one of four basic arithmetic operations, with the other 3 being subtraction, multiplication, division. Addition is one of the next steps after counting, and the result of addition is a sum.

There are a number of ways to approach addition.

### Combining sets

One of the most common ways to teach/learn addition is by combining sets. To combine sets, we first need to know how to count. Once we know how to count, combining sets (at least smaller ones) becomes relatively simple. Start with small numbers of objects that we can easily count to; this can be objects drawn on a page, or any object in real life such as a marble, peanut, or shapes on a page. To add the numbers in both sets, we just need to count the number of objects in each set, then combine them.

Example

Below is an example of adding two sets of objects.

In this example, there is 1 triangle in each set. We add 2 sets of 1 triangle each to get a total of 2 triangles. Using this method works for smaller numbers, but will get more difficult with larger numbers of objects.

We can also use number lines to add. This also involves counting, but rather than counting objects, we just count along the number line. Generally, we start at one point on the number line, then add the other number(s) by counting spaces equivalent to the number that we are adding.

Example

Add 3 + 4 on a number line.

The above figure shows two ways we could've approached the addition problem. Above the number line in green, we started at 3, then counted 4 spaces to result in 7. Below the line in orange, we started at 4 and counted 3 spaces, again to result in 7. This shows us that the order in which we add does not matter. This is always true, and is referred to as the commutative property of addition.

Adding using a number line is very intuitive, but also tedious, especially as the numbers being added get larger.

One of the most commonly taught algorithms for addition involves lining up the addends based on place value then adding the digits vertically starting from the ones place (right-most digit). This algorithm is sometimes referred to as the standard addition algorithm. To use this algorithm, it is important to be relatively comfortable with all the addition facts. While understanding place value is not absolutely necessary in order to be able to use the algorithm, it helps us understand why the algorithm works. Below are the steps for performing the standard addition algorithm, as well as an example of how to use it.

1. Line up the numbers being added based on place value and size; the largest number should be on top, and their ones places should be aligned. Draw a line below the addends.
2. Add the columns vertically starting from the ones place (right to left). Write the result below the line. If the sum of a column is larger than 9, write the ones place digit below the line, and write a 1 in the following column (to the left). This is referred to as "carrying."
3. Continue adding from right to left until all columns have been added; when adding columns, add any carried 1s as well. Treat empty spaces in a column as adding a 0, and just bring any values in the column down below the line.

Example

1. In the first step, we lined up 247 and 37 based on their ones places (in this case both have a 7 in the ones place), placed 247 above because it is larger, and drew a line below.
2. We then added 7 + 7 to get 14, wrote the 4 below the line in the same column, and carried the 1 to the second column (the tens place) since the sum was larger than 9.
3. We continued adding each column from right to left, including carried 1s. The second column was 4 + 3 along with a carried 1, so 4 + 3 + 1 = 8. The final column was simply 2 because 37 does not have a non-zero digit in the hundreds place, so we just bring down the 2.

The reason that we carry a 1 when the sum is larger than 9 is because of how the decimal system works. The number 247 can be broken up as 200 + 40 + 7. The 2 is in the hundreds place, the 4 is in the tens place, and the 7 is in the ones place. Similarly, the 3 in 37 is in the tens place, and the 7 is in the ones place, so it can be broken up as 30 + 7. When we add columns, and the sum exceeds 9, we carry a 1 to indicate that we've "filled" the place, so we need to add 1 to the following place to show this. In the above example, 7 + 7 = 14. 14 is made up of one 10, and four 1s, represented by the carried 1 and the 4 in the ones place.

There are many different ways to approach addition that can make it easier to perform the operation. Below are just a few examples.

### Rounding numbers

Rounding numbers to the closest multiple of 10 (as well as 5), can make them easier to add. For example, if we are adding 17 and 8, we can round 17 to 20 (+3), and 8 to 10 (+2), and add them to get 30, keeping track of how much we added. We then subtract the difference between our rounded values from our total value, so:

30 - 3 - 2 = 25

### Counting up

When the numbers being added are relatively small, counting up from the larger number is one of the simpler ways to add. When adding 8 + 3, we can just start at 8, and count 3 numbers past 8. This is helpful when starting out learning addition, which usually happens after we are fairly comfortable with counting. Starting at 8, we can count 3 more numbers past 8, to get 11. We can use our fingers as we count to keep track of how many numbers we are counting.

### Grouping/rearranging numbers

Because of the associative and commutative properties of addition, we can move numbers around when we are adding, and group them in whatever way is simplest for us to perform the addition.

Example

Add 3 + 12 + 7 + 8.

One way to simplify this addition problem is to group 3 and 7, and 12 and 8. This is because 3 + 7 = 10, and 12 + 8 = 20, so 10 + 20 = 30. If we recognize addition facts that complement each other well, we can make it easier to perform the addition by grouping them rather than just adding whatever numbers we have from left to right.