Welcome to the World of Division!
Hello, Math Explorer! Get ready for an exciting adventure into division. Have you ever shared sweets with your friends to make sure everyone gets the same number? Or have you ever put your toys into equal groups? If you have, you've already used division!
In this chapter, we're going to learn all about how to divide things up fairly. It's a super useful skill that helps us solve problems every day. Don't worry if it seems tricky at first, we'll go step-by-step and have fun along the way. You can do it!
Section 1: What is Division?
Division is simply about sharing equally or making equal groups.
Imagine you have 12 yummy cookies to share equally among 3 friends. How many cookies does each friend get?
You can give one cookie to each friend, then another, and another, until they are all gone. You will find that each friend gets 4 cookies!
We can write this as a math problem:
$$12 \div 3 = 4$$
We read this as "Twelve divided by three equals four."
Meet the Division Family!
Every division problem has three main parts. Let's learn their special names:
- Dividend: This is the total number you start with. (In our example, it was the 12 cookies).
- Divisor: This is the number you are dividing by. (It was the 3 friends).
- Quotient: This is the answer! (It was the 4 cookies each friend got).
Dividend $$ \div $$ Divisor $$ = $$ Quotient
Key Takeaway
Division is all about sharing fairly. The number you start with is the dividend, the number you divide by is the divisor, and the answer is the quotient.
Section 2: The Super Link Between Multiplication and Division
Guess what? If you know your multiplication facts, you are already a division star! Multiplication and division are opposites, like a secret team. They are part of something called a "Fact Family".
Let's look at our cookie example again.
We know that $$12 \div 3 = 4$$.
This is because $$3 \times 4 = 12$$. See the connection? The same three numbers are used!
So, a fact family for 3, 4, and 12 would be:
$$3 \times 4 = 12$$
$$4 \times 3 = 12$$
$$12 \div 3 = 4$$
$$12 \div 4 = 3$$
Quick Review Box
How to check your division answer:
To check if your division is correct, you can use multiplication!
Is $$20 \div 5 = 4$$ correct?
Check: Does $$5 \times 4 = 20$$? Yes, it does! So the answer is correct. Hooray!
Key Takeaway
Multiplication and division are best friends! You can always use one to check the other. Knowing your times tables will make division much, much easier.
Section 3: Leftovers are Okay! Meet the Remainder
What if you have 13 cookies to share among 3 friends? Let's see... each friend gets 4 cookies, but wait! There is 1 cookie left over. You can't split the last cookie without breaking it.
This leftover number is called the remainder.
We write the answer like this:
$$13 \div 3 = 4 \text{ R } 1$$
This is read as "Thirteen divided by three is four, with a remainder of one."
Very Important Rule: The remainder must ALWAYS be smaller than the divisor. If it's bigger, it means you could have shared one more time!
Example: In $$13 \div 3$$, the remainder is 1, which is smaller than the divisor (3). So it's correct!
Did you know?
The division symbol $$ \div $$ is called an obelus! It was first used a very long time ago, in 1659. It looks like a fraction with a dot for the numerator and a dot for the denominator.
Key Takeaway
When a number cannot be divided perfectly, the amount left over is called the remainder. Remember, the remainder is always less than the divisor.
Section 4: Tackling Bigger Numbers with Long Division
What if you need to solve $$68 \div 2$$? That's too big to do in our heads! For this, we use a method called column form or long division. It looks a bit like a house!
To make it easy, we can remember a family to help us with the steps:
Dad - Mum - Sister - Brother
D = Divide
M = Multiply
S = Subtract
B = Bring Down
Let's try it! Step-by-Step Example 1: $$68 \div 2$$
Step 1: Set it up
Write the dividend (68) inside the "house" and the divisor (2) outside.
Step 2: DAD - Divide
Look at the first digit inside, which is 6. Ask: "How many times does 2 go into 6?"
$$6 \div 2 = 3$$. Write the 3 on top, right above the 6.
Step 3: MUM - Multiply
Multiply the number you just wrote on top (3) by the divisor (2).
$$3 \times 2 = 6$$. Write the 6 under the 6 inside the house.
Step 4: SISTER - Subtract
Subtract the number you just wrote from the number above it.
$$6 - 6 = 0$$. Write the 0 below.
Step 5: BROTHER - Bring Down
Bring down the next digit from the dividend (8) next to the 0.
Step 6: Repeat the steps!
Now we look at the new number, 08 (or just 8).
Divide: $$8 \div 2 = 4$$. Write the 4 on top, next to the 3.
Multiply: $$4 \times 2 = 8$$. Write the 8 under the 8.
Subtract: $$8 - 8 = 0$$.
Bring Down: There are no more numbers to bring down! We are finished.
The number on top is our answer! So, $$68 \div 2 = 34$$.
Another Example! With a Remainder: $$137 \div 4$$
Let's use our D-M-S-B family again!
1. Divide: Can 4 go into 1? No. So we look at the first TWO digits: 13. How many 4s in 13? (Think: $$4 \times 1=4, 4 \times 2=8, 4 \times 3=12, 4 \times 4=16$$... too big! So, it's 3). Write 3 on top.
2. Multiply: $$3 \times 4 = 12$$. Write 12 under the 13.
3. Subtract: $$13 - 12 = 1$$. Write 1 below.
4. Bring Down: Bring down the 7 to make the new number 17.
-- REPEAT --
1. Divide: How many 4s in 17? (Think: $$4 \times 4 = 16$$). Write 4 on top.
2. Multiply: $$4 \times 4 = 16$$. Write 16 under the 17.
3. Subtract: $$17 - 16 = 1$$.
4. Bring Down: Nothing left to bring down. The 1 at the bottom is our remainder!
So, $$137 \div 4 = 34 \text{ R } 1$$.
Common Mistakes to Avoid
- Forgetting to subtract: Always remember the "Sister" step!
- Remainder is too big: If your remainder is bigger than the divisor, go back and check your division step. You can probably make a bigger group.
- Messy columns: Keep your numbers lined up neatly. It helps you avoid mistakes!
Key Takeaway
Long division helps us solve big problems one small step at a time. Just remember the family: Divide, Multiply, Subtract, Bring Down!
Section 5: Solving Division Word Problems
Division is all around us! Word problems are just division stories. Look for clue words like:
- share equally
- split between
- how much for each
- groups of
Example 1: A baker made 45 cupcakes. He wants to put them into boxes with 5 cupcakes in each box. How many boxes will he need?
Thinking: We have a total (45) and we are making equal groups (of 5). This is a division problem!
Calculation: $$45 \div 5 = 9$$
Answer: The baker will need 9 boxes.
Example 2: 25 students are going on a trip. Each car can hold 4 students. How many cars are needed?
Thinking: We need to split 25 students into groups of 4.
Calculation: $$25 \div 4 = 6 \text{ R } 1$$
Answer: The answer is 6 with a remainder of 1. This means 6 cars will be full, but there is 1 student left over. We can't leave that student behind! So, they will need one more car. They need 7 cars altogether.
Sometimes the remainder is very important in the story!
Key Takeaway
Read word problems carefully to find the total amount and the size of the groups. Draw pictures if it helps! Think about what the remainder means for the final answer.
Section 6: Check Your Work with Estimation!
Estimation is like making a smart guess. It helps you check if your answer is sensible. Before you solve a big division problem, you can round the numbers to make it easier.
Problem: $$123 \div 4$$
Thinking: Let's estimate first. 123 is very close to 120. And I know that $$12 \div 4 = 3$$. So, $$120 \div 4$$ should be 30. My real answer should be close to 30.
Let's calculate the real answer: $$123 \div 4 = 30 \text{ R } 3$$.
Wow! 30 R 3 is very close to our estimate of 30. Our answer is very likely correct!
Key Takeaway
Before you divide, try to estimate the answer by using numbers that are easier to work with. This is a great way to catch silly mistakes!