Section 1
Where Did Multiplication Come From?
People have been multiplying for thousands of years! Long ago, farmers needed to count big groups of things, like sheep or bags of grain. Instead of counting one by one, they found faster ways. Let's look at some cool moments in the history of multiplication.
About 4,000 years ago
About 3,000 years ago
About 2,300 years ago
About 500 years ago
All these people figured out the same thing you are learning right now: multiplication is just a fast way to add equal groups, and division is a fast way to split things into equal groups. You are part of a long history of math thinkers!
Section 2
The Big Ideas
Before we start solving problems, let's learn four important ideas. These are the building blocks you'll use again and again.
Multiplication = Equal Groups
Division = Sharing Equally
They Are Opposites!
Fact Families Stick Together
KEY TAKEAWAY
Section 3
See It: Arrays Show Both!
An array is a picture that shows objects in rows and columns. Arrays are amazing because they let you see multiplication AND division at the same time! Look at the array below. It shows 4 rows of 6 stars.
See how one picture gives you two facts? The array shows 4 × 6 = 24 (4 rows of 6). It also shows 24 ÷ 4 = 6 (24 stars split into 4 rows gives 6 in each row). You can even flip it: 6 × 4 = 24 and 24 ÷ 6 = 4. One array, four facts!
Section 4
How It Works: The Relationship
Here's the biggest secret in 3rd-grade math: multiplication and division are connected. If you know one fact, you can figure out the other. Let's see how.
MULTIPLICATION
This means "8 groups of 7 equals 56."
Since 8 groups of 7 makes 56, we can "go backwards." If we start with 56 and make 8 equal groups, each group will have 7.
DIVISION (GOING BACKWARDS)
Start with 56. Split into 8 groups. Each group has 7.
And we can also ask: if we start with 56 and put 7 in each group, how many groups do we get?
ANOTHER DIVISION FACT
Start with 56. Put 7 in each group. You get 8 groups.
And don't forget — you can swap the order in multiplication! 7 × 8 = 56 is also true. This is called the commutative property, which is a big word that just means "you can swap the numbers and the answer is the same."
FUN FACT
Section 5
Fact Families: Numbers That Belong Together
A fact family is a group of three numbers that are connected by multiplication and division. Every fact family makes exactly four math facts — two multiplication and two division. Let's look at one!
The triangle shape helps you remember: the product (the big answer) goes at the top, and the two factors (the numbers you multiply) go at the bottom. To divide, start at the top and go down to find the missing factor!
Here's a table with more fact families. See if you can spot the pattern!
| Factors | Multiply #1 | Multiply #2 | Divide #1 | Divide #2 |
|---|---|---|---|---|
| 3, 7 | 3 × 7 = 21 | 7 × 3 = 21 | 21 ÷ 3 = 7 | 21 ÷ 7 = 3 |
| 5, 8 | 5 × 8 = 40 | 8 × 5 = 40 | 40 ÷ 5 = 8 | 40 ÷ 8 = 5 |
| 6, 9 | 6 × 9 = 54 | 9 × 6 = 54 | 54 ÷ 6 = 9 | 54 ÷ 9 = 6 |
| 8, 7 | 8 × 7 = 56 | 7 × 8 = 56 | 56 ÷ 8 = 7 | 56 ÷ 7 = 8 |
| 9, 9 | 9 × 9 = 81 | 9 × 9 = 81 | 81 ÷ 9 = 9 | 81 ÷ 9 = 9 |
DID YOU NOTICE?
Section 6
Worked Example: Solve Step by Step
Let's solve a problem together. Read each step carefully!
Problem: Mia has 72 stickers. She wants to put them into 8 equal piles. How many stickers are in each pile?
- 1
Step 1 — What are we asked?
We need to find how many stickers go in each pile when 72 stickers are shared equally into 8 piles. This is a division problem!72 ÷ 8 = ? - 2
Step 2 — Think of a multiplication fact
Ask yourself: "8 times what number equals 72?" Think about the 8 times table:8 × 7 = 56… not enough. 8 × 8 = 64… still not 72. 8 × 9 = 72… Yes! That's it! - 3
Step 3 — Write the answer
Since 8 × 9 = 72, we know that 72 ÷ 8 = 9. - 4
Step 4 — Check it!
Does 9 × 8 really equal 72? Let's see: 9 × 8 = 72. ✅ It works!Answer: Mia puts 9 stickers in each pile.
KEY TAKEAWAY
Section 7
Helpful Strategies
There's more than one way to multiply and divide! Here are some strategies that make these problems easier. Try them all and pick the ones you like best.
| Strategy | How It Works | Example |
|---|---|---|
| Use a Fact Family | Turn division into multiplication (or the other way around) to use what you already know. | Don't know 42 ÷ 6? Think: 6 × ? = 42. Answer: 7! |
| Skip Counting | Count by the same number over and over: 3, 6, 9, 12… | For 4 × 3, count by 3 four times: 3, 6, 9, 12. |
| Break Apart (Distributive) | Split a hard problem into two easy problems and add the answers. | 7 × 6 = (5 × 6) + (2 × 6) = 30 + 12 = 42 |
| Doubles | Use doubling for ×2, ×4, and ×8 facts. | For 4 × 6, double 6 to get 12, then double 12 to get 24. |
| Use 10s and Subtract | For ×9, multiply by 10 and subtract one group. | 9 × 7 = (10 × 7) − 7 = 70 − 7 = 63 |
The break apart strategy is especially powerful. Let's look at it more closely. Say you need to find 8 × 7. That's a tough one! But you can break the 8 into 5 + 3:
BREAK APART STRATEGY
Split the 8 into 5 + 3, multiply each part by 7, then add!
KEY TAKEAWAY
Section 8
What Comes Next?
Great news — everything you're learning now will help you with bigger math later! Here's a peek at what's ahead.
| What You Know Now | What You'll Learn Later |
|---|---|
| Multiply within 100 (like 8 × 7 = 56) | Multiply bigger numbers (like 28 × 37) in 4th and 5th grade |
| Divide within 100 (like 56 ÷ 8 = 7) | Long division with larger numbers and remainders |
| Fact families connect × and ÷ | Algebra! In 6th grade, you'll use letters like x and solve equations like 8 × x = 56 |
| Break apart strategy | The distributive property — a rule used all through high school math! |
By memorizing your multiplication and division facts now, you're building a strong math foundation. It's like learning to read — once you know words well, you can read whole books! Once you know these facts well, you can do any math problem that comes your way.
LOOK AHEAD
Section 9
Practice Problems
Time to try some on your own! Start with the easier ones and work your way up. Click "Show Answer" when you're ready to check.
Question 1: PROBLEM 1 — WHAT DO YOU KNOW?
Show answer
Question 2: PROBLEM 2 — BASIC CALCULATION
Show answer
Question 3: PROBLEM 3 — INTERMEDIATE
Show answer
Question 4: PROBLEM 4 — WORD PROBLEM
Show answer
Question 5: PROBLEM 5 — CHALLENGE
Show answer
Summary
Lesson Review
Summary
In this lesson, you learned that multiplication means counting equal groups, and division means splitting into equal groups. The most important idea is that they are connected — if you know one fact, like 8 × 7 = 56, you can figure out the related division facts: 56 ÷ 8 = 7 and 56 ÷ 7 = 8. Three numbers that go together this way form a fact family with four facts.
You also learned powerful strategies to help you solve problems faster: using the fact family relationship, skip counting, the break apart strategy, doubling, and using 10s and subtracting for 9s facts. You can use arrays to see these facts as pictures. Keep practicing your facts every day, and soon you'll know them all by heart — and that will make every math topic that comes next much easier!