Division Concepts and Applications

Partitive Division means fair sharing.

In fair sharing, we know how many groups there are. We need to find out how many items go in each group.

For example, if 12 apples are shared equally into 3 baskets, partitive division helps us find how many apples are in each basket.

We can write that as \(12 \div 3 = 4\).

This means 12 items shared into 3 equal groups gives 4 items in each group.

Think of it like this:

• The total number tells how many items there are altogether.
• The number of groups tells how many equal groups we are making.
• The answer tells how many are in each group.

In partitive division, we ask:

“If I share these items fairly into this many groups, how many will be in each group?”

How to solve fair sharing problems

1. Read the total number of items.
2. Find the number of groups.
3. Share the items equally.
4. Count how many items are in each group.

You can solve these problems by:

• drawing pictures,
• using counters or blocks,
• skip counting,
• or using multiplication you already know.

Important idea: Every group must have the same number of items. That is what makes it fair.

Worked Example 1

There are 8 cookies. They are shared equally among 2 children. How many cookies does each child get?

We know:

• Total items: 8 cookies
• Number of groups: 2 children

Share 8 cookies into 2 equal groups:

Child 1: 4 cookies

Child 2: 4 cookies

So, each child gets 4 cookies.

$$8 \div 2 = 4$$

Check: \(2 \times 4 = 8\). That matches the total.

Worked Example 2

15 crayons are shared equally into 3 boxes. How many crayons go in each box?

We know:

• Total items: 15 crayons
• Number of groups: 3 boxes

Share fairly:

Put 1 crayon in each box again and again until all 15 crayons are used.

Each box gets 5 crayons.

$$15 \div 3 = 5$$

Check: \(3 \times 5 = 15\).

Worked Example 3

18 stickers are shared equally among 6 students. How many stickers does each student get?

We know:

• Total items: 18 stickers
• Number of groups: 6 students

We can use multiplication to help:

Ask: What number times 6 equals 18?

Since \(6 \times 3 = 18\), each student gets 3 stickers.

$$18 \div 6 = 3$$

Check: \(6 \times 3 = 18\).

Worked Example 4

20 toy cars are shared equally into 4 bags. How many toy cars are in each bag?

We know:

• Total items: 20 toy cars
• Number of groups: 4 bags

Share equally into 4 groups.

Each bag gets 5 toy cars.

$$20 \div 4 = 5$$

Check: \(4 \times 5 = 20\).

Using pictures in your mind

Imagine drawing circles for the groups. Then place one item in each circle until all the items are gone.

If the sharing is fair, every circle will have the same number.

For \(12 \div 3\), draw 3 circles. Then place 12 dots into the circles one at a time. You will end with 4 dots in each circle.

How partitive division is different from another kind of division

In this lesson, we know the number of groups first.

We are finding the number in each group.

That is why it is called fair sharing.

Clue words that may help you notice a fair sharing problem are:

• shared equally
• shared among
• put into groups
• each group gets the same

Be careful!

• Do not just guess.
• Make sure the groups are equal.
• Check your answer with multiplication.

Try these questions in your head:

• \(10 \div 2 = 5\) because 10 shared into 2 equal groups gives 5 in each group.
• \(16 \div 4 = 4\) because 16 shared into 4 equal groups gives 4 in each group.
• \(21 \div 7 = 3\) because 21 shared into 7 equal groups gives 3 in each group.

Summary

Partitive division means sharing a total amount into a known number of equal groups.

You already know how many groups there are, and you need to find how many are in each group.

To solve, share equally, count what is in each group, and check with multiplication.

Quotitive Division (Measurement Grouping) means we know how many are in each group, and we need to find how many groups we can make.

This kind of division is called measurement grouping. We are measuring how many equal groups fit into a total.

For example, if you have 12 cookies and put 3 cookies in each bag, the question is: How many bags can you fill?

That is a division problem:

$$12 \div 3 = 4$$

This means 12 things can be split into groups of 3, and that makes 4 groups.

In quotitive division, we ask:

• How many groups?
• How many times does this group size fit into the total?

We are not asking how many are in each group. We already know the group size.

Look for clue words like these:

• in each group
• in each bag
• in each box
• groups of
• how many groups
• how many bags, boxes, teams, or rows

How to solve quotitive division:

1. Find the total number.
2. Find the size of each group.
3. Divide the total by the group size.
4. The answer tells how many groups you can make.

You can also think about skip counting or repeated subtraction.

For example, to solve \(15 \div 5\), you can ask: how many times can I count by 5 until I get to 15?

$$5,\ 10,\ 15$$

That is 3 jumps, so:

$$15 \div 5 = 3$$

You can also subtract 5 again and again:

$$15 - 5 = 10$$

$$10 - 5 = 5$$

$$5 - 5 = 0$$

You subtracted 5 three times, so there are 3 groups.

Division and multiplication are connected.

If:

$$4 \times 3 = 12$$

then:

$$12 \div 3 = 4$$

This helps us check our work. If 4 groups have 3 in each group, then there are 12 in all.

Worked Example 1

Sara has 10 crayons. She puts 2 crayons in each cup. How many cups can she fill?

Step 1: Total = 10 crayons

Step 2: Group size = 2 crayons in each cup

Step 3: Divide

$$10 \div 2 = 5$$

Answer: Sara can fill 5 cups.

You can check with multiplication:

$$5 \times 2 = 10$$

Worked Example 2

There are 18 apples. Each basket holds 6 apples. How many baskets are needed?

Step 1: Total = 18 apples

Step 2: Group size = 6 apples in each basket

Step 3: Divide

$$18 \div 6 = 3$$

Answer: You need 3 baskets.

Think of it as jumps of 6:

$$6,\ 12,\ 18$$

There are 3 jumps, so there are 3 groups.

Worked Example 3

A teacher has 20 stickers. She gives out stickers in groups of 4. How many groups can she make?

Step 1: Total = 20 stickers

Step 2: Group size = 4 stickers

Step 3: Divide

$$20 \div 4 = 5$$

Answer: She can make 5 groups.

Check:

$$5 \times 4 = 20$$

Worked Example 4

There are 14 toy cars. Each shelf can hold 3 toy cars. How many full shelves can be made?

Step 1: Total = 14 toy cars

Step 2: Group size = 3 cars on each shelf

Step 3: Divide

$$14 \div 3 = 4\text{ remainder }2$$

This means 3 fits into 14 4 full times, with 2 left over.

Answer: You can make 4 full shelves, and 2 cars are left over.

We can see it with subtraction:

$$14 - 3 = 11$$

$$11 - 3 = 8$$

$$8 - 3 = 5$$

$$5 - 3 = 2$$

We made 4 groups of 3, and 2 are left.

Important idea: In quotitive division, the answer tells the number of groups, not the number in each group.

Let’s compare:

• Quotitive division: 12 cookies, 3 in each bag. How many bags?
• The answer is \(12 \div 3 = 4\) bags.

So when the group size is known, division helps us find how many groups can be made.

Tips to remember:

• Find the total.
• Find how many are in each group.
• Use division to find the number of groups.
• Use multiplication to check.
• If not all items fit equally, there may be a remainder.

Quick practice questions to think about:

• \(16 \div 4\): How many groups of 4 are in 16?
• \(21 \div 7\): How many groups of 7 are in 21?
• \(13 \div 5\): How many full groups of 5 are in 13? How many are left over?

Summary: Quotitive division means the size of each group is known, and we are finding how many groups can be made. We can solve by dividing, skip counting, or repeated subtraction. We can check our answer with multiplication.

Division as Repeated Subtraction means we solve a division problem by taking away the same number again and again until there is nothing left, or until we cannot subtract anymore.

Division asks, "How many equal groups?" Repeated subtraction helps us find the answer by counting how many times we subtract the divisor.

For example, in \(12 \div 3\), we keep subtracting 3 from 12:

\(12 - 3 = 9\)
\(9 - 3 = 6\)
\(6 - 3 = 3\)
\(3 - 3 = 0\)

We subtracted 3 a total of 4 times, so:

$$12 \div 3 = 4$$

This works because division and subtraction can work together. Each subtraction removes one equal group.

Important words:

• Dividend: the number we start with
• Divisor: the number we subtract each time
• Quotient: how many times we subtract

In \(12 \div 3 = 4\):

• 12 is the dividend
• 3 is the divisor
• 4 is the quotient

How to use repeated subtraction:

1. Start with the first number, called the dividend.
2. Subtract the second number, called the divisor.
3. Keep subtracting the same divisor.
4. Count how many times you subtract.
5. The number of subtractions is the quotient.

Let’s look at some examples.

Example 1: \(10 \div 2\)

Start with 10 and subtract 2 each time.

\(10 - 2 = 8\)
\(8 - 2 = 6\)
\(6 - 2 = 4\)
\(4 - 2 = 2\)
\(2 - 2 = 0\)

We subtracted 2 5 times.

$$10 \div 2 = 5$$

Example 2: \(15 \div 5\)

Start with 15 and subtract 5 each time.

\(15 - 5 = 10\)
\(10 - 5 = 5\)
\(5 - 5 = 0\)

We subtracted 5 3 times.

$$15 \div 5 = 3$$

Example 3: \(14 \div 4\)

Now let’s try a problem where we do not reach 0 exactly.

\(14 - 4 = 10\)
\(10 - 4 = 6\)
\(6 - 4 = 2\)

We cannot subtract 4 again from 2, because 2 is smaller than 4.

We subtracted 4 3 times, with 2 left over.

So:

$$14 \div 4 = 3 \text{ remainder } 2$$

The 2 left over is called the remainder.

Example 4: A word problem

Lina has 18 stickers. She puts them into groups of 3. How many groups can she make?

We use repeated subtraction:

\(18 - 3 = 15\)
\(15 - 3 = 12\)
\(12 - 3 = 9\)
\(9 - 3 = 6\)
\(6 - 3 = 3\)
\(3 - 3 = 0\)

She subtracted 3 6 times.

$$18 \div 3 = 6$$

Lina can make 6 groups.

How repeated subtraction connects to multiplication

If \(12 \div 3 = 4\), then \(4 \times 3 = 12\).

If \(15 \div 5 = 3\), then \(3 \times 5 = 15\).

This shows that division is the opposite of multiplication.

Tips to remember:

• Subtract the same number each time.
• Count each subtraction carefully.
• If you reach 0, the division is exact.
• If a small number is left, that number is the remainder.

Let’s compare two problems:

For \(16 \div 4\):

\(16 - 4 = 12\)
\(12 - 4 = 8\)
\(8 - 4 = 4\)
\(4 - 4 = 0\)

That is 4 subtractions, so:

$$16 \div 4 = 4$$

For \(17 \div 4\):

\(17 - 4 = 13\)
\(13 - 4 = 9\)
\(9 - 4 = 5\)
\(5 - 4 = 1\)

That is 4 subtractions, with 1 left over, so:

$$17 \div 4 = 4 \text{ remainder } 1$$

Why this strategy helps

Repeated subtraction is a great way to understand what division means. It helps you see each equal group being taken away one at a time.

It is especially helpful when you are learning division for the first time, because it shows exactly what is happening in the problem.

Summary

Division as repeated subtraction means subtracting the divisor again and again from the dividend.

The number of times you subtract is the quotient.

If you cannot subtract anymore and a small amount is left, that amount is the remainder.

When you use repeated subtraction, you can better understand how division works and how it connects to multiplication.

Inverse Relationship with Multiplication

Sometimes division can feel tricky. A helpful way to solve division problems is to remember that division and multiplication are connected.

They are called inverse operations. That means they can help undo each other.

For example, if you know that \(4 \times 3 = 12\), then you also know that \(12 \div 4 = 3\) and \(12 \div 3 = 4\).

This is an important idea: when you divide, you can think, “What number times the divisor equals the dividend?”

So instead of only thinking about division, you can turn the problem into a multiplication problem with a missing number.

For example:

can be thought of as:

If you know that \(3 \times 6 = 18\), then \(18 \div 3 = 6\).

Words to know:

• Dividend: the number being divided
• Divisor: the number you divide by
• Quotient: the answer to a division problem

In \(18 \div 3 = 6\):

• 18 is the dividend
• 3 is the divisor
• 6 is the quotient

You do not always need to say these big words, but it is good to know what they mean.

How to use multiplication to solve division

1. Look at the division problem.
2. Change it into a multiplication problem with a missing number.
3. Ask: What number times the divisor gives the dividend?
4. Use what you know about multiplication facts to find the answer.

Here is the pattern:

This means division is like finding a missing factor.

Fact families can help too. A fact family is a group of multiplication and division facts that use the same numbers.

For example, the numbers 5, 7, and 35 make this fact family:

• \(5 \times 7 = 35\)
• \(7 \times 5 = 35\)
• \(35 \div 5 = 7\)
• \(35 \div 7 = 5\)

If you know one multiplication fact, you can use it to help with two division facts.

Worked Example 1

Solve \(12 \div 4\).

Think: \(4 \times ? = 12\)

We know:

So:

The quotient is 3.

Worked Example 2

Solve \(20 \div 5\).

Turn it into a missing-factor multiplication problem:

We know:

So:

This means 20 split into groups of 5 makes 4 groups.

Worked Example 3

Solve \(42 \div 6\).

Think:

Now use your multiplication facts. Since:

then:

The missing factor is 7, so the quotient is 7.

Worked Example 4

There are 32 crayons. They are put into 8 equal groups. How many crayons are in each group?

Write the division problem:

Now think of the matching multiplication problem:

We know:

So:

There are 4 crayons in each group.

Tips for solving division with multiplication

• Look for a multiplication fact you already know.
• Ask yourself, “What number times this divisor makes the dividend?”
• Use skip-counting if you are not sure. For example, for \(24 \div 6\), count by 6: 6, 12, 18, 24. That is 4 counts, so \(24 \div 6 = 4\).
• Check your answer by multiplying.

Check your answer

After you solve a division problem, multiply to see if it is correct.

Example: If you think \(27 \div 3 = 9\), check it:

It matches, so the answer is correct.

What to remember

• Division and multiplication are inverse operations.
• You can solve division by finding a missing factor.
• \(a \div b = ?\) can be rewritten as \(b \times ? = a\).
• Multiplication facts help you solve division facts.

Let’s try a few more in your head:

• \(15 \div 3\) means \(3 \times ? = 15\), so the answer is 5.
• \(28 \div 4\) means \(4 \times ? = 28\), so the answer is 7.
• \(54 \div 9\) means \(9 \times ? = 54\), so the answer is 6.

Summary

When you see a division problem, you can rewrite it as a multiplication problem with a missing number. This works because multiplication and division are opposites that help each other. If you know your multiplication facts, you can use them to solve division problems more easily.

Division with 0 and 1 Edge Cases

Division helps us split things into equal groups or find how many groups we can make. Sometimes division problems have 0 or 1 in them. These are special cases, and they follow important rules.

In this lesson, we will learn three big ideas:

• Any number divided by 1 stays the same.
• 0 divided by any nonzero number is 0.
• Division by 0 is not allowed.

Let’s look at each rule step by step.

1. Any number divided by 1 is itself

When we divide by 1, we are asking, “If I put all the items into 1 group, how many items are in that group?” Since there is only one group, it gets all the items.

For example, if you have 7 apples and put them into 1 group, that group has 7 apples.

So:

$$7 \div 1 = 7$$

This is true for any number:

• $$4 \div 1 = 4$$
• $$12 \div 1 = 12$$
• $$100 \div 1 = 100$$

You can also think about multiplication. Since division and multiplication are connected, we can ask: “What number times 1 equals the starting number?”

For example:

$$9 \div 1 = 9$$

because

$$9 \times 1 = 9$$

Rule: Dividing by 1 does not change the number.

2. Zero divided by a nonzero number is 0

Now let’s think about dividing 0. If you have 0 cookies and want to share them equally into groups, there is still nothing to share.

If there are 3 groups, each group gets 0 cookies.

So:

$$0 \div 3 = 0$$

This works with any nonzero number:

• $$0 \div 2 = 0$$
• $$0 \div 5 = 0$$
• $$0 \div 100 = 0$$

We can check with multiplication. We ask, “What number times 3 equals 0?” The answer is 0, because:

$$0 \times 3 = 0$$

So:

$$0 \div 3 = 0$$

Rule: If the first number is 0 and the second number is not 0, the answer is 0.

3. Division by 0 is not allowed

Now for the trickiest case: dividing by 0.

If you see a problem like:

$$6 \div 0$$

this means, “How many groups of 0 are in 6?” But groups of 0 do not help us make 6 in a normal way. There is no number you can multiply by 0 to get 6, because:

• $$1 \times 0 = 0$$
• $$5 \times 0 = 0$$
• $$100 \times 0 = 0$$

Any number times 0 is always 0, never 6.

So:

$$6 \div 0$$

has no answer. We say it is undefined, which means it is not a number answer we can use.

The same is true for any number divided by 0:

• $$3 \div 0$$ is undefined
• $$10 \div 0$$ is undefined
• $$100 \div 0$$ is undefined

Important: Division by 0 is never allowed.

What about $$0 \div 0$$?

This one is also not allowed. It may look like the answer should be 0, but it is still a division-by-0 problem.

So:

$$0 \div 0$$

is also undefined.

Helpful ways to remember

• Divide by 1: Keep the number the same.
• 0 divided by a number: The answer is 0, as long as the number is not 0.
• Divide by 0: Not allowed.

You can remember it like this:

• $$a \div 1 = a$$
• $$0 \div a = 0$$, if \(a \ne 0\)
• $$a \div 0$$ is undefined

Worked Example 1

Find:

$$8 \div 1$$

Step 1: Think about the rule for dividing by 1.

Any number divided by 1 stays the same.

Answer:

$$8 \div 1 = 8$$

Worked Example 2

Find:

$$0 \div 4$$

Step 1: Think about sharing 0 things into 4 groups.

There is nothing to share, so each group gets 0.

Answer:

$$0 \div 4 = 0$$

Worked Example 3

Find:

$$15 \div 0$$

Step 1: Check the divisor, the number we are dividing by.

The divisor is 0.

Step 2: Use the rule.

Division by 0 is not allowed.

Answer:

$$15 \div 0$$ is undefined.

Worked Example 4

Decide if each statement is true or false:

1. $$11 \div 1 = 11$$
2. $$0 \div 9 = 9$$
3. $$7 \div 0 = 0$$

Step 1: Check each one with the rules.

• $$11 \div 1 = 11$$ is true because dividing by 1 keeps the number the same.
• $$0 \div 9 = 9$$ is false because 0 divided by a nonzero number is 0.
• $$7 \div 0 = 0$$ is false because division by 0 is undefined.

Common mistakes to watch for

• Thinking $$0 \div 5 = 5$$. This is wrong. The answer is 0.
• Thinking $$6 \div 0 = 0$$. This is wrong. Division by 0 is undefined.
• Forgetting that dividing by 1 keeps the number unchanged.

Quick check

• $$13 \div 1 = 13$$
• $$0 \div 7 = 0$$
• $$9 \div 0$$ is undefined

Summary

Special division rules help us with 0 and 1. Any number divided by 1 stays the same. Zero divided by any nonzero number equals 0. But dividing by 0 is undefined, which means it does not have a number answer.

Conceptualizing Remainders means understanding what happens when we divide and there are some pieces left over.

Sometimes a number can be divided into equal groups with nothing left. Other times, after making equal groups, a few items remain. Those extra items are called a remainder.

For example, if 10 cookies are shared equally among 3 children, each child gets 3 cookies. That uses 9 cookies. There is 1 cookie left over, so the answer is 3 remainder 1.

We can write that like this:

$$10 \div 3 = 3 \text{ remainder } 1$$

A remainder is always smaller than the divisor. The divisor is the number we are dividing by.

In the example above, the divisor is 3. The remainder is 1. Since 1 is smaller than 3, the answer makes sense.

If the remainder were 3 or more, we could make another full group. That means it would not really be a remainder yet.

Here is the big idea:

• Dividend: the total number we start with
• Divisor: the number in each group, or the number of groups
• Quotient: the answer showing how many equal groups or how many in each group
• Remainder: the amount left over

When you divide with remainders, you are asking:

1. How many equal groups can I make?
2. How many are left after that?

One way to find a remainder is to use multiplication. Multiply the divisor by the quotient. Then subtract from the dividend.

It looks like this:

$$\text{remainder} = \text{dividend} - (\text{divisor} \times \text{quotient})$$

Let's learn with examples.

Worked Example 1

Divide 14 by 4.

We want to find how many groups of 4 we can make from 14.

4 + 4 + 4 = 12, so we can make 3 full groups of 4.

We started with 14 and used 12.

$$14 - 12 = 2$$

So 2 are left over.

The answer is:

$$14 \div 4 = 3 \text{ remainder } 2$$

Check: Is the remainder smaller than the divisor?

Yes. The remainder is 2, and the divisor is 4. Since \(2 < 4\), the answer is correct.

Worked Example 2

Divide 17 by 5.

We make groups of 5.

5 + 5 + 5 = 15, so we can make 3 full groups.

Now subtract:

$$17 - 15 = 2$$

So the answer is:

$$17 \div 5 = 3 \text{ remainder } 2$$

This means 17 items can make 3 equal groups of 5, with 2 left over.

Worked Example 3

Divide 22 by 6.

Let's use multiplication facts.

\(6 \times 3 = 18\)

\(6 \times 4 = 24\), but 24 is too big because it is more than 22.

So the quotient is 3.

Now find the leftover part:

$$22 - 18 = 4$$

So:

$$22 \div 6 = 3 \text{ remainder } 4$$

Check the remainder: \(4 < 6\), so it is correct.

Worked Example 4

There are 19 pencils. They are put into boxes with 4 pencils in each box. How many full boxes can be made, and how many pencils are left?

We are dividing 19 by 4.

\(4 \times 4 = 16\)

\(4 \times 5 = 20\), and that is too many.

So we can make 4 full boxes.

Now subtract to find the remainder:

$$19 - 16 = 3$$

So 4 full boxes can be made, and 3 pencils are left.

We write:

$$19 \div 4 = 4 \text{ remainder } 3$$

In a word problem, the remainder tells us the leftover amount that does not fit into a full group.

How to Tell if a Remainder Makes Sense

• The remainder must be less than the divisor.
• If the remainder is 0, there is nothing left over.
• If the remainder is the same as the divisor or bigger, make another group.

For example, suppose someone says:

$$15 \div 4 = 2 \text{ remainder } 7$$

That cannot be right because the remainder 7 is bigger than the divisor 4.

Since 7 is enough to make another group of 4, we must keep dividing.

Helpful Strategy

1. Find the biggest multiplication fact you know that is close to the dividend.
2. Make sure it does not go over the dividend.
3. Subtract to find the remainder.
4. Check that the remainder is smaller than the divisor.

Let's Think About Leftovers

Remainders happen in real life all the time.

• 13 apples shared into groups of 5 gives 2 full groups and 3 apples left.
• 21 students lining up in rows of 4 gives 5 full rows and 1 student left.
• 11 toy cars packed in bags of 2 gives 5 full bags and 1 toy car left.

These leftovers are remainders.

Important Reminder

A remainder is not a mistake. It is part of the answer when things do not divide equally.

When you see a division problem, ask yourself:

How many full groups can I make, and what is left over?

Summary

When we divide, sometimes there are items left after making equal groups. Those leftover items are called the remainder.

The remainder must always be smaller than the divisor. If it is not, we can make another full group.

To solve a division problem with a remainder, make as many equal groups as possible, then count what is left over.

Tape Diagrams for Division

Division helps us split a whole into equal parts. A tape diagram is a picture that looks like a long bar. We can break the bar into equal boxes to show how division works.

Tape diagrams help us see 3 important parts of a division problem:

• Dividend: the whole amount we start with
• Divisor: the number of equal groups or the number in each group
• Quotient: the answer to the division problem

When we use a tape diagram, we draw one bar for the whole. Then we split the bar into equal parts. The equal parts help us find the missing number.

For example, in the problem \(12 \div 3\), the whole is 12. We split 12 into 3 equal parts. Then we find how much is in each part.

How to Draw a Tape Diagram for Division

1. Read the division problem carefully.
2. Find the whole amount.
3. Draw one long bar to show the whole.
4. Split the bar into equal parts.
5. Use what you know to find the missing number.
6. Write the division equation.

Sometimes we know how many groups there are. Then we split the tape into that many parts.

Sometimes we know how many are in each group. Then we keep making equal parts until the whole is used up. The number of parts tells us the answer.

Division as Sharing

Sharing means we split a whole into equal groups. A tape diagram makes equal sharing easy to see.

Example 1: Sharing equally

There are 12 apples. 3 children share them equally. How many apples does each child get?

We know:

• Whole amount: 12
• Number of groups: 3

Draw one bar for 12. Split it into 3 equal parts.

Each part must be:

So each child gets 4 apples.

We can also check with multiplication:

That means our division answer is correct.

Example 2: Another sharing problem

15 stickers are shared equally among 5 friends. How many stickers does each friend get?

We know:

• Whole amount: 15
• Number of groups: 5

Split a tape diagram into 5 equal parts. The whole bar is 15.

Now find the value of each part:

Each friend gets 3 stickers.

Check:

Division as Finding the Number of Groups

Sometimes we know the whole and the size of each group. We need to find how many groups there are.

Example 3: Finding the number of groups

18 crayons are put into boxes. Each box holds 6 crayons. How many boxes are needed?

We know:

• Whole amount: 18
• Each group has: 6

Draw a tape for 18. Make equal parts of 6 until the whole is filled.

The parts are:

• 6
• 6
• 6

There are 3 equal parts, so there are 3 boxes.

The equation is:

Check:

Example 4: A bigger one

20 cookies are packed into bags. Each bag holds 4 cookies. How many bags are needed?

We know:

• Whole amount: 20
• Each group has: 4

Draw one tape for 20. Mark off equal parts of 4.

Count the parts:

• 4
• 4
• 4
• 4
• 4

There are 5 parts, so:

So 5 bags are needed.

How Tape Diagrams Help

• They show the whole clearly.
• They show that the parts must be equal.
• They help us decide if we are finding the size of each group or the number of groups.
• They connect pictures to equations.

What to Ask Yourself

• What is the whole amount?
• Do I know the number of groups?
• Or do I know how many are in each group?
• What does each part of the tape diagram mean?

Common Mistakes to Watch For

• Do not make parts that are not equal.
• Do not forget that the whole bar is the total amount.
• Read the problem carefully so you know what you are finding.
• Always check with multiplication when you can.

Try to Think About These

If a problem says \(16 \div 4\), ask: Am I splitting 16 into 4 equal groups? Or am I making groups of 4 from 16? A tape diagram can help with both. The picture helps you see the same equation in a clear way.

Summary

A tape diagram is a bar picture that helps us understand division. We use one bar for the whole and split it into equal parts. Sometimes we know the number of groups, and sometimes we know the size of each group. Tape diagrams help us find the missing number and connect division to multiplication.