Counting On and Counting Back to Add and Subtract

Section 1

Where Did Adding and Subtracting Come From?

People have been counting for a very, very long time! Long before there were schools or books, people needed to count things like sheep, fish, and stones. Let's look at how people learned to count and add.

A Long, Long Time Ago

Cave people used their fingers to count things. They held up one finger for each thing they saw. This is just like what you do!

Tally Marks

People started making scratches on bones and rocks to keep track of how many things they had. Each scratch stood for one thing.

Numbers Were Born

People made up number symbols like 1, 2, 3, 4, 5 so they could write down how many things they counted.

Adding and Subtracting

Once people had numbers, they figured out they could put groups together (add) or take some away (subtract) by counting!

Here is the big question we will answer today: How can counting help us add and subtract? You already know how to count forward (1, 2, 3, 4…) and maybe even count backward (10, 9, 8, 7…). Those same skills will help you add and subtract!

Section 2

The Big Ideas

There are four important ideas to know. Let's look at each one!

Count On to Add

When you add, you start at one number and count forward. If you add 3, you count on 3 more.

Count Back to Subtract

When you subtract, you start at one number and count backward. If you subtract 2, you count back 2.

Use a Number Line

A number line is a line with numbers on it. You can hop forward to add and hop backward to subtract.

Adding and Subtracting Are Connected

Adding is the opposite of subtracting. If you count on 2 to add 2, you can count back 2 to subtract 2!

✦ Key Takeaway

Think of a number line like a sidewalk. Each square on the sidewalk is a number. When you add, you walk forward. When you subtract, you walk backward. The number of steps you take is the number you are adding or subtracting!

Section 3

See It on a Number Line!

Let's see how counting on helps us add. We will solve 5 + 3 by starting at 5 and counting on 3 more.

Start at 5. Count on 3 hops: 6, 7, 8. You land on 8!

See how it works? You start at 5 and then hop forward 3 times. Each hop is counting one more number. You say: "6… 7… 8!" So 5 + 3 = 8. You don't have to count all the way from 1. You just count on from the number you start with!

Now let's see counting back to subtract. We will solve 8 − 3.

Start at 8. Count back 3 hops: 7, 6, 5. You land on 5!

This time you start at 8 and hop backward 3 times. You say: "7… 6… 5!" So 8 − 3 = 5. Counting back is just counting in the other direction!

Section 4

How Does It Work?

Let's learn the steps for counting on to add and counting back to subtract.

Counting On to Add

4 + 2 = ?

Start at 4. Count on 2: say "5, 6." The answer is 6!

Here is how to count on:

Step 1: Look at the first number. That is where you start.
Step 2: Look at the second number. That tells you how many to count forward.
Step 3: Count forward that many times. Say each number out loud!
Step 4: The last number you say is the answer.

Let's try it with objects! Imagine you have 4 blue stars and you get 2 more yellow stars.

⭐⭐⭐⭐ + 🌟🌟 — You don't need to count all 4 blue stars again. Just start at 4 and count on: "5, 6!" → 4 + 2 = 6

Counting Back to Subtract

7 − 2 = ?

Start at 7. Count back 2: say "6, 5." The answer is 5!

Here is how to count back:

Step 1: Look at the first number. That is where you start.
Step 2: Look at the second number. That tells you how many to count backward.
Step 3: Count backward that many times.
Step 4: The last number you say is the answer.

✦ Key Takeaway

Think of it like going up and down stairs. When you add, you go up the stairs — count one step for each number you add. When you subtract, you go down the stairs — count one step for each number you subtract. The step you land on is your answer!

Section 5

More Ways to See It

Let's look at a chart that shows how counting on and counting back work for different problems.

Problem	What You Do	What You Say	Answer
3 + 2	Start at 3, count on 2	"4, 5"	5
6 + 1	Start at 6, count on 1	"7"	7
4 + 3	Start at 4, count on 3	"5, 6, 7"	7
5 − 2	Start at 5, count back 2	"4, 3"	3
9 − 1	Start at 9, count back 1	"8"	8
7 − 3	Start at 7, count back 3	"6, 5, 4"	4

Do you notice a pattern? When we add, the answer is bigger than where we started. When we subtract, the answer is smaller. That makes sense! Adding means more. Subtracting means less.

Let's also look at using your fingers! Your fingers are a great tool for counting on and counting back.

💡 Using Your Fingers to Solve 6 + 3

Put the number 6 in your head. Now hold up 3 fingers. ✊ ← 6 is in your head. 👆👆👆 ← hold up 3 fingers. Touch each finger and count: "7… 8… 9!" → 6 + 3 = 9

Section 6

Let's Solve One Together!

Let's solve this problem step by step: You have 7 apples. You eat 2 apples. How many apples do you have left?

7 − 2 = ?

Step 1 — What Do We Know?You start with 7 apples. You eat 2 apples. Eating means taking away, so we need to subtract. 7 − 2 = ?

Step 2 — Start at 7Put the number 7 in your head. That is our starting number.

Step 3 — Count Back 2We need to subtract 2, so we count back 2 times: First count back: 7 → 6. Second count back: 6 → 5.

Step 4 — We Landed On Our Answer!The last number we said was 5.

7 − 2 = 5 — You have 5 apples left! 🍎🍎🍎🍎🍎

Section 7

Counting On vs. Counting Back

Let's compare these two ways of using counting. They are like two sides of the same coin!

	Counting On (Add)	Counting Back (Subtract)
Direction	Forward → → →	Backward ← ← ←
Numbers get	Bigger	Smaller
Means	Put together / more	Take away / less
Example	3 + 2 → say "4, 5" → 5	5 − 2 → say "4, 3" → 3
Symbol	+ (plus sign)	− (minus sign)

Counting on is great when you add a small number like 1, 2, or 3. If you need to add a big number like 8, it might take a lot of hops! Later, you will learn faster tricks. But for now, counting on and counting back are wonderful tools.

✦ Key Takeaway

Adding and subtracting are opposites. They undo each other! If you count on 2 to go from 5 to 7, you can count back 2 to go from 7 to 5. It's like taking two steps forward and then two steps back — you end up where you started!

Section 8

What's Coming Next?

Right now, you are learning to count on and count back with small numbers. This is a super important skill! Here is what you will learn next as you grow as a math learner.

What You Know Now	What You'll Learn Next
Count on 1, 2, or 3	Add bigger numbers like 5 + 7
Count back 1, 2, or 3	Subtract bigger numbers like 14 − 6
Use fingers to count	Use ten frames and blocks
Adding and subtracting are opposites	Fact families (like 3 + 4 = 7, 7 − 4 = 3)

Everything you learn in math builds on what you already know. Counting on and counting back are the first big building blocks. You are doing great by learning them now! Soon you will be adding and subtracting so fast that you won't even need to count on your fingers.

Section 9

Practice Time!

Try these problems on your own! Use counting on or counting back. Click "Show Answer" when you are ready to check.

PROBLEM 1 — TELL ME WHAT YOU KNOW

When you see the problem 6 + 2, do you count forward or backward? Why?

PROBLEM 2 — COUNT ON TO ADD

Solve this problem by counting on: 5 + 2 = ?

PROBLEM 3 — COUNT BACK TO SUBTRACT

Solve this problem by counting back: 9 − 3 = ?

PROBLEM 4 — STORY PROBLEM

You have 4 crayons. Your friend gives you 3 more crayons. How many crayons do you have now? Use counting on to find the answer.

PROBLEM 5 — THINK ABOUT IT

Sam solved 6 + 2 and got 8. Then Sam solved 8 − 2 and got 6. Why did Sam end up back at 6? Can you explain?

Summary

What We Learned

Today you learned that you can use counting to help you add and subtract! When you see a plus sign (+), you count on — that means counting forward from the first number. When you see a minus sign (−), you count back — that means counting backward from the first number. You can use a number line, your fingers, or even count in your head!

The most important thing to remember is that adding and subtracting are opposites. Adding makes numbers bigger by counting forward. Subtracting makes numbers smaller by counting backward. You are already a great counter, and now you are a great adder and subtracter too! Keep practicing, and it will get easier and easier.