Where Did Adding & Subtracting Come From?
People have been adding and subtracting for a very, very long time! Long ago, people needed to count things like sheep, fish, and bags of grain. They had to figure out how to put groups together and take things away. Over time, smart thinkers found clever tricks to make math easier.
Here's the big question we will answer: Why do the strategies you use for adding and subtracting actually work? It's not magic — it's place value and some special math rules called properties of operations. Let's find out!
The Big Ideas
Before we dive in, let's learn four big ideas that make addition and subtraction strategies work. These ideas are like the building blocks that hold everything together.
Place Value
You Can Add in Any Order
You Can Group Numbers Any Way
Adding Zero Changes Nothing
See It With Pictures!
Let's look at why 36 + 27 works when you break numbers into tens and ones. This picture shows how place value lets you split numbers apart, add the pieces, and put the answer back together.
See what happened? We broke 36 into 30 + 6 and 27 into 20 + 7. Then we added the tens together (30 + 20 = 50) and the ones together (6 + 7 = 13). Finally, 50 + 13 = 63. It works because place value lets us split numbers into easier pieces, and the rules of addition let us rearrange and group them any way we want!
How the Math Rules Work
Let's look at the special math rules one at a time. Each rule is like a tool in a toolbox that helps you solve problems in different ways.
When you break a number into tens and ones, you are using place value. The number 45 is really just 40 and 5 sitting together. This lets you add or subtract the tens part and the ones part on their own. It's like taking apart a LEGO set — you can work with one section at a time!
This rule says you can flip the numbers around. Why does this help? Imagine you want to add 3 + 9. It might feel easier to start at 9 and count up 3 instead of starting at 3 and counting up 9. The answer is still 12 either way!
This is a big deal! When you see 4 + 6 + 3, you can add 4 + 6 first to get 10, then add 3 to get 13. Or you can add 6 + 3 first to get 9, then add 4 to get 13. Either way, you get the same answer. This rule is why the "make a ten" strategy works so well — you pick the numbers that are easy to add first!
Subtraction and addition are opposites. This means every subtraction fact has an addition fact hiding inside it. If 15 − 7 = 8, then 8 + 7 must equal 15. This is why "counting up" works for subtraction. To find 52 − 38, you can ask, "What do I add to 38 to get to 52?" That's 14!
Breaking Down Strategies
Now let's look at the most popular addition and subtraction strategies and see why each one works. Each strategy uses place value, the commutative property, or the associative property (or all three!).
Every strategy is just a different way of using the same rules! Whether you break apart numbers, make a ten, count up, or use friendly numbers, place value and the properties of operations are always working behind the scenes.
Worked Example
Let's solve a problem together, step by step, and see exactly why each step works.
When to Use Each Strategy
All strategies give you the right answer. But some feel easier for certain problems. Here's a handy chart to help you pick!
| Strategy | Best For | Why It Works |
|---|---|---|
| Break Apart | Any addition or subtraction | Place value lets you split tens and ones, then add or subtract each part. |
| Make a Ten | When ones add up close to 10 (like 8 + 5) | The associative property lets you move part of one number to the other to make 10. |
| Count Up | Subtraction when numbers are close together | Subtraction is the opposite of addition, so you can count up to find the difference. |
| Friendly Numbers | When one number is close to a ten (like 39 or 51) | You round to a ten, add or subtract, then fix the small extra bit. |
| Doubles / Near Doubles | When two numbers are the same or almost the same | If you know 6 + 6 = 12, then 6 + 7 is just one more — 13! The commutative and associative properties let you do this. |
What Comes Next?
The ideas you're learning now — place value, commutative property, and associative property — are the same ideas you'll use for bigger numbers later on!
| What You Know Now | What You'll Do Next |
|---|---|
| Breaking 2-digit numbers into tens and ones | Breaking 3-digit numbers into hundreds, tens, and ones (like 345 = 300 + 40 + 5) |
| Adding ones to make a ten | Adding tens to make a hundred |
| Checking subtraction with addition | Using the same check for multiplication and division |
| Commutative property for addition | Commutative property for multiplication too (3 × 4 = 4 × 3) |
So everything you learn right now is building a strong tower of math thinking. Each new year, you'll just add more floors to that tower. The foundation — place value and the properties of operations — stays the same all the way up!
Practice Problems
Try these problems on your own! Click "Show Answer" when you're ready to check.
What We Learned
Addition and subtraction strategies work because of two powerful ideas: place value and the properties of operations. Place value lets you break any number into tens and ones (like 47 = 40 + 7). You can add or subtract each part by itself and then put the answer back together. The commutative property says you can add numbers in any order. The associative property says you can group numbers any way you want. And since addition and subtraction are opposites, you can always check your subtraction by adding.
Whether you break apart, make a ten, count up, or use friendly numbers, every strategy follows the same math rules. That's why they all give you the right answer — and that's the really cool part! These same rules will help you with bigger numbers, multiplication, and so much more math to come.