Table of Contents
Understanding Addition
Addition is the operation of combining quantities to find “how many in all” or “how much altogether.” In this chapter, we focus on how addition works, how to write it, and how to think about it in different ways.
We will work mostly with whole numbers (0, 1, 2, 3, …) in this chapter. Later chapters will extend these ideas to other kinds of numbers.
The Addition Symbol and Basic Language
The symbol for addition is the plus sign $+$.
An addition statement (also called an addition expression) looks like:
$$
3 + 4
$$
The full sentence
$$
3 + 4 = 7
$$
is called an equation.
For an addition equation $a + b = c$:
- $a$ and $b$ are called addends
- $c$ is called the sum
You can read $3 + 4 = 7$ aloud as:
- “Three plus four equals seven”
- “Three and four is seven”
- “The sum of three and four is seven”
Ways to Picture Addition
Addition is one idea, but you can picture it in several ways. These different pictures help you understand what is really happening when you add.
1. Putting Groups Together
Think of $2 + 3$ as putting two groups together:
- A group of 2 objects
- A group of 3 objects
Count all the objects together:
$$
2 + 3 = 5
$$
This interpretation is often called combining or joining.
2. Moving Forward on a Number Line
Imagine a number line, with 0, 1, 2, 3, and so on in order from left to right.
To compute $2 + 3$ on a number line:
- Start at $2$
- Move $3$ steps to the right (forward)
- You land on $5$
So:
$$
2 + 3 = 5
$$
Addition can be thought of as moving to the right on the number line.
3. Adding as “How Many More”
Sometimes you know how many you have and how many more you get.
Example:
You have 4 apples. Your friend gives you 2 more.
“How many apples do you have now?”
You started with 4, added 2 more:
$$
4 + 2 = 6
$$
So addition often answers questions like “How many now?” or “How many in all?”
Basic Addition Facts
Very young learners usually memorize basic addition facts with small numbers, such as all sums with addends from 0 to 10.
Some key ideas:
- Adding zero
If you add zero, the number does not change:
$$
a + 0 = a \quad\text{and}\quad 0 + a = a
$$
Example: + 0 = 7$ - Adding one
Adding one gives the next number in the counting sequence:
$$
a + 1 = \text{the next whole number after } a
$$
Example: + 1 = 6$ - Doubles
A “double” is when both addends are the same:
$$
2 + 2 = 4,\quad 3 + 3 = 6,\quad 7 + 7 = 14
$$
Doubles are helpful facts that can make other additions easier. - Near doubles
If you know + 5 = 10$, then:
$$
5 + 6 = 5 + 5 + 1 = 11 \
6 + 5 = 5 + 5 + 1 = 11
$$
These facts are the building blocks for adding larger numbers.
Properties of Addition
Certain patterns are always true in addition with numbers. These are called properties. They are rules that work for all numbers and help us add more easily.
Commutative Property
The order of the addends does not change the sum.
For any numbers $a$ and $b$:
$$
a + b = b + a
$$
Examples:
- $2 + 5 = 7$ and $5 + 2 = 7$
- $10 + 3 = 13$ and $3 + 10 = 13$
This means you can rearrange addends in any order without changing the total.
Associative Property
When adding three or more numbers, you can group them in different ways and the sum stays the same.
For any numbers $a, b,$ and $c$:
$$
(a + b) + c = a + (b + c)
$$
Examples:
- $(2 + 3) + 4 = 5 + 4 = 9$
+ (3 + 4) = 2 + 7 = 9$ - $(1 + 5) + 2 = 6 + 2 = 8$
+ (5 + 2) = 1 + 7 = 8$
The parentheses just show which numbers you add first. With addition, changing the grouping does not change the result.
Identity Property
The number $0$ is the additive identity because adding $0$ does not change a number.
For any number $a$:
$$
a + 0 = a \quad\text{and}\quad 0 + a = a
$$
Example:
- $9 + 0 = 9$
- $0 + 12 = 12$
Strategies for Mental Addition
You do not always need to line numbers up and write them down. Many additions can be done in your head using simple strategies.
Making a Ten
Numbers that add up to $10$ are especially useful. For example:
- $3 + 7 = 10$
- $4 + 6 = 10$
- $8 + 2 = 10$
You can use these to help with other sums.
Example: $8 + 5$
Think:
- $8$ needs $2$ more to make $10$
- Break $5$ into $2 + 3$
- Add $8 + 2 = 10$, then $10 + 3 = 13$
So $8 + 5 = 13$.
Breaking Numbers Apart (Decomposing)
You can break one addend into easier parts.
Example: $27 + 5$
Think:
- $27 + 3 = 30$
- $5 = 3 + 2$, so after adding 3, you still have 2 left
- $30 + 2 = 32$
So $27 + 5 = 32$.
This uses the idea that:
$$
27 + 5 = 27 + (3 + 2) = (27 + 3) + 2
$$
which is an application of the associative property.
Adding in Steps
For larger sums, it often helps to add in steps.
Example: $34 + 29$
One way:
- Add tens: $30 + 20 = 50$
- Add ones: $4 + 9 = 13$
- Combine: $50 + 13 = 63$
So $34 + 29 = 63$.
Another way (using a “nice” number):
- Notice $29$ is close to $30$
- $34 + 30 = 64$
- But you added $1$ too much, so subtract $1$:
- 1 = 63$
So again $34 + 29 = 63$.
Written Addition: Lining Up Place Values
For larger whole numbers, you usually write them in columns and add them by place value (ones, tens, hundreds, …).
Example: $237 + 58$
Step 1: Line up the digits by place value.
237
+ 58Think of $58$ as $058$:
237
+ 058
Step 2: Add the ones (rightmost) column: $7 + 8 = 15$
Write down 5 in the ones place, and carry 1 ten to the tens column.
1 ← carried
237
+ 058
----
5Step 3: Add the tens column: $3 + 5 = 8$, plus the carried 1 makes $9$.
1
237
+ 058
----
95Step 4: Add the hundreds column: $2 + 0 = 2$.
1
237
+ 058
----
295
So:
$$
237 + 58 = 295
$$
Here you used:
- Place value (ones, tens, hundreds)
- Carrying (also called regrouping) when a column sum is 10 or more
Word Problems with Addition
Addition word problems usually involve combining amounts, joining groups, or finding a total.
Typical phrases that suggest addition:
- “in all”
- “altogether”
- “total”
- “combined”
- “how many now”
Example:
Maria has 8 red pencils and 5 blue pencils.
How many pencils does she have in all?
You are combining two amounts:
- Red: 8 pencils
- Blue: 5 pencils
So:
$$
8 + 5 = 13
$$
Maria has 13 pencils altogether.
Checking an Addition
You can check your result in a few simple ways:
- Reverse the addends
Since $a + b = b + a$, check that both give the same sum.
Example: If + 9 = 16$, also think + 7 = 16$. - Estimate
Round the numbers and see if your answer is close.
Example: For 8 + 305$, estimate 0 + 300 = 500$.
Your exact answer 3$ is close to 0$, so it seems reasonable. - Use subtraction (to be learned in detail later)
If $a + b = c$, then $c - a$ should give $b$.
For now, it is enough to know this idea; the details of subtraction are treated in the next chapter.
Common Mistakes to Avoid
- Not lining up place values when doing written addition
Always align ones under ones, tens under tens, and so on. - Forgetting to carry when a column sum is 10 or more
Example: In + 8$, + 8 = 15$; write 5 and carry 1 to the tens place. - Misreading a word problem
Make sure the situation really involves combining amounts before using addition. Some problems with similar wording may require other operations, which are treated in their own chapters.
Summary
- Addition combines quantities to find a total.
- The plus sign $+$ is used to show addition.
- The numbers added are called addends; the result is the sum.
- You can picture addition by joining groups, moving right on a number line, or adding “how many more.”
- Key properties of addition: commutative, associative, and identity ($0$).
- Mental strategies (making tens, breaking apart numbers) and place-value methods help with both small and large sums.
- Word problems often use phrases like “in all,” “altogether,” or “total” to indicate addition.