Foundations of Multiplication and Division

Analyzing Equal Groups means looking at groups to see if they all have the same number of things.

This is an important math idea because multiplication is about equal groups. If every group has the same number, we can count faster using repeated addition.

If the groups do not have the same number, they are not equal groups. Then we should not use multiplication for them.

Let’s learn how to tell the difference.

What are equal groups?

Equal groups are groups that each have the same amount.

• 3 groups of 2 means each group has 2.
• 4 groups of 5 means each group has 5.
• If one group has more or fewer, the groups are not equal.

You can ask yourself these questions:

1. How many groups are there?
2. How many are in each group?
3. Does every group have the same number?

If the answer to the last question is yes, then they are equal groups.

Why equal groups matter

Equal groups help us move from adding one by one to adding in a pattern.

For example, if there are 3 groups with 4 stars in each group, we can add:

$$4 + 4 + 4 = 12$$

Because each group has the same number, this is an equal-groups situation.

How to analyze groups

When you look at groups of objects, do these steps:

1. Count the number of groups.
2. Count how many objects are in one group.
3. Check the other groups to make sure they match.
4. Decide: equal groups or not equal groups.

You are like a math detective. You are checking whether every group is the same size.

Example 1: Equal groups

There are 3 bowls. Each bowl has 2 apples.

• Bowl 1: 2 apples
• Bowl 2: 2 apples
• Bowl 3: 2 apples

Each bowl has 2 apples, so these are equal groups.

We can use repeated addition:

$$2 + 2 + 2 = 6$$

So there are 6 apples in all.

Example 2: Not equal groups

There are 3 bags of marbles.

• Bag 1: 4 marbles
• Bag 2: 4 marbles
• Bag 3: 3 marbles

Two bags have 4 marbles, but one bag has 3 marbles.

The groups are not equal because they do not all match.

We can still add to find the total:

$$4 + 4 + 3 = 11$$

But we should not call this equal groups.

Example 3: Finding the mistake

A student says, “I see 4 equal groups.”

The groups have:

• Group 1: 5 crayons
• Group 2: 5 crayons
• Group 3: 5 crayons
• Group 4: 4 crayons

Is the student correct?

No. One group has 4 crayons, not 5.

So the groups are not equal groups.

To be equal groups, all 4 groups must have the same number.

Example 4: Equal groups and total

There are 5 plates. Each plate has 3 cookies.

First, check if the groups are equal.

Each plate has 3 cookies, so yes, they are equal groups.

Now use repeated addition:

$$3 + 3 + 3 + 3 + 3 = 15$$

There are 15 cookies in all.

Equal groups and not equal groups

• Equal groups: every group has the same number.
• Not equal groups: at least one group has a different number.

Look carefully. Even if most groups match, all groups must match to be equal groups.

Try these thinking questions

1. There are 4 boxes. Each box has 2 pencils. Are they equal groups?

Yes, because each box has 2 pencils.

2. There are 3 jars with 6, 6, and 7 buttons. Are they equal groups?

No, because one jar has a different number.

3. There are 2 groups with 8 in each group. Are they equal groups?

Yes, because both groups have 8.

Helpful tips

• Count carefully.
• Check every group, not just one or two.
• Ask, “Does each group have the same number?”
• If yes, the groups are equal.
• If no, the groups are not equal.

Summary

Equal groups have the same number in every group.

Not equal groups have groups with different numbers.

When groups are equal, you can use repeated addition to find the total. This helps you get ready for multiplication.

Modeling Equal Groups means putting objects into groups that all have the same number in each group.

This is an important math idea because it helps us get ready for multiplication and division. When groups are equal, math is easier to see and understand.

For example, if 8 counters are put into 2 groups with 4 in each group, those are equal groups. Each group has the same amount.

If one group has 5 and another group has 3, those are not equal groups because the groups do not match.

What to look for in equal groups

• Each group has the same number of objects.
• You can count by groups instead of one by one.
• You can use repeated addition to find the total.

When we model equal groups, we can ask two helpful questions:

1. How many groups are there?
2. How many objects are in each group?

If we know those two things, we can find the total number of objects.

Equal groups and repeated addition

Repeated addition means adding the same number again and again.

If there are 3 groups with 2 stars in each group, we can add:

$$2 + 2 + 2 = 6$$

So, 3 equal groups of 2 make 6 altogether.

You can model equal groups with pictures or objects

You can use counters, cubes, buttons, drawings, or circles on paper. Put the objects into matching groups.

For example:

Group 1: ●●●

Group 2: ●●●

Group 3: ●●●

There are 3 groups, and each group has 3 objects. The total is:

$$3 + 3 + 3 = 9$$

Worked Example 1

There are 2 bowls. Each bowl has 4 apples. How many apples are there in all?

Step 1: Find the number of groups. There are 2 bowls, so there are 2 groups.

Step 2: Find how many are in each group. Each bowl has 4 apples.

Step 3: Add the equal groups.

$$4 + 4 = 8$$

Answer: There are 8 apples in all.

Worked Example 2

There are 4 bags. Each bag has 2 marbles. How many marbles are there in all?

Step 1: Number of groups = 4 bags.

Step 2: Number in each group = 2 marbles.

Step 3: Use repeated addition.

$$2 + 2 + 2 + 2 = 8$$

Answer: There are 8 marbles in all.

Worked Example 3

Sam made 3 equal groups of toy cars. There are 5 toy cars in each group. How many toy cars does Sam have?

Step 1: Count the groups. There are 3 groups.

Step 2: Count how many in each group. There are 5 in each group.

Step 3: Add the same number 3 times.

$$5 + 5 + 5 = 15$$

Answer: Sam has 15 toy cars.

Worked Example 4

Look at these groups:

●●●●
●●●●
●●●●
●●●●

How many groups are there, how many in each group, and how many objects are there in all?

Step 1: Count the groups. There are 4 groups.

Step 2: Count the objects in one group. There are 4 in each group.

Step 3: Add the equal groups.

$$4 + 4 + 4 + 4 = 16$$

Answer: There are 4 groups, 4 in each group, and 16 objects in all.

How to tell if groups are equal

Check each group carefully. Every group must have the same number.

Example of equal groups:

●●
●●
●●

Each group has 2, so the groups are equal.

Example of not equal groups:

●●●
●●
●●●

These are not equal because one group has 2 and the others have 3.

Tips for solving equal group problems

• First, count how many groups there are.
• Next, count how many objects are in each group.
• Make sure every group matches.
• Add the equal groups to find the total.

Let’s practice thinking

If there are 5 groups with 3 in each group, you can add:

$$3 + 3 + 3 + 3 + 3 = 15$$

If there are 2 groups with 6 in each group, you can add:

$$6 + 6 = 12$$

Equal groups help us see numbers in an organized way. Instead of counting every object one at a time, we can count group by group.

Summary

Equal groups are groups with the same number in each one. To model equal groups, count the number of groups and the number in each group. Then use repeated addition to find the total. This helps build strong number sense and prepares you for multiplication later.

Repeated Addition Equations

Sometimes we have equal groups. Equal groups means each group has the same number of things.

When we add the same number again and again, that is called repeated addition.

Repeated addition helps us count equal groups in an easy way.

For example, if there are 3 bowls and each bowl has 2 apples, we can add 2 three times:

$$2 + 2 + 2 = 6$$

This is a repeated addition equation.

How to make a repeated addition equation

1. Look for the number in each group.

2. Count how many groups there are.

3. Add the same number one time for each group.

4. Find the total.

Let’s practice thinking about groups.

• If there are 4 bags with 3 marbles in each bag, we add 3 four times.

• If there are 2 plates with 5 cookies on each plate, we add 5 two times.

• If there are 5 rows with 2 stars in each row, we add 2 five times.

Important idea: The number that repeats is the number in each group.

The number of times you write it is the number of groups.

Example 1

There are 2 boxes. Each box has 4 crayons. How many crayons are there in all?

Step 1: Number in each group = 4

Step 2: Number of groups = 2

Step 3: Write the repeated addition equation:

$$4 + 4 = 8$$

So, there are 8 crayons in all.

Example 2

There are 3 nests. Each nest has 2 eggs. How many eggs are there in all?

Step 1: Number in each group = 2

Step 2: Number of groups = 3

Step 3: Add 2 three times:

$$2 + 2 + 2 = 6$$

So, there are 6 eggs in all.

Example 3

There are 4 jars. Each jar has 3 buttons. How many buttons are there in all?

Step 1: Number in each group = 3

Step 2: Number of groups = 4

Step 3: Add 3 four times:

$$3 + 3 + 3 + 3 = 12$$

So, there are 12 buttons in all.

Example 4

There are 5 teams. Each team has 2 players. How many players are there in all?

Step 1: Number in each group = 2

Step 2: Number of groups = 5

Step 3: Add 2 five times:

$$2 + 2 + 2 + 2 + 2 = 10$$

So, there are 10 players in all.

Repeated addition can match pictures, rows, and groups

If you see equal rows or equal groups, you can write an addition equation.

• 3 rows of 4 dots means:

  $$4 + 4 + 4 = 12$$

• 2 groups of 6 shells means:

  $$6 + 6 = 12$$

• 4 groups of 1 toy means:

  $$1 + 1 + 1 + 1 = 4$$

Watch out for this!

Sometimes students mix up the number of groups and the number in each group.

Example: 3 bags with 2 apples in each bag.

The repeated addition equation is:

$$2 + 2 + 2 = 6$$

We write 2 because there are 2 apples in each group.

We write it 3 times because there are 3 groups.

Try thinking like this:

• What number is in each group?

• How many groups are there?

• How can I add the same number again and again?

Let’s do a few quick ones

1. 3 plates with 5 strawberries on each plate:

$$5 + 5 + 5 = 15$$

2. 4 cages with 2 birds in each cage:

$$2 + 2 + 2 + 2 = 8$$

3. 2 shelves with 7 books on each shelf:

$$7 + 7 = 14$$

Summary

Repeated addition means adding the same number more than one time.

It is a good way to show equal groups.

To write a repeated addition equation, use the number in each group, write it once for each group, and then find the total.

When you see equal groups, rows, or sets, you can use repeated addition to solve the problem.

Rectangular Arrays: Rows and Columns

Today we will learn about rectangular arrays. An array is a neat arrangement of objects in rows and columns. Arrays help us count objects more easily.

You may have seen arrays made with dots, tiles, chairs, or even muffins in a tray. When objects are lined up in a rectangle, we can quickly see how many there are.

What is a row?

A row goes across. It is horizontal, like words going across a page.

What is a column?

A column goes up and down. It is vertical, like a tall stack.

In an array, every row has the same number of objects. Every column also has the same number of objects. That is what makes the shape neat and easy to count.

Look at this array:

$$\begin{matrix} \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet \end{matrix}$$

This array has 2 rows because there are 2 lines going across.

It has 3 columns because there are 3 lines going up and down.

We can count all the dots:

$$3 + 3 = 6$$

So there are 6 dots in the array.

How rows and columns help us count

Instead of counting one by one, we can count by rows or by columns.

• Count by rows: add the number in each row.
• Count by columns: add the number in each column.

Both ways give the same total.

For example, if an array has 3 rows with 4 objects in each row, we can count:

$$4 + 4 + 4 = 12$$

If the same array has 4 columns with 3 objects in each column, we can count:

$$3 + 3 + 3 + 3 = 12$$

The total is still 12.

Worked Example 1

How many rows and columns are in this array?

$$\begin{matrix} \star & \star \\ \star & \star \\ \star & \star \end{matrix}$$

Step 1: Count the rows going across.

There are 3 rows.

Step 2: Count the columns going up and down.

There are 2 columns.

Step 3: Count all the stars.

We can count by rows:

$$2 + 2 + 2 = 6$$

So the array has 3 rows, 2 columns, and 6 stars.

Worked Example 2

How many objects are in this array?

$$\begin{matrix} \square & \square & \square & \square \\ \square & \square & \square & \square \end{matrix}$$

Step 1: Count the rows.

There are 2 rows.

Step 2: Count how many squares are in each row.

There are 4 squares in each row.

Step 3: Add the rows.

$$4 + 4 = 8$$

So there are 8 objects in the array.

We can also count by columns. There are 4 columns with 2 squares in each column.

$$2 + 2 + 2 + 2 = 8$$

Worked Example 3

Sara makes an array with 4 rows and 3 columns. How many objects does she use?

Let us picture it:

$$\begin{matrix} \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet \end{matrix}$$

There are 4 rows, and each row has 3 objects.

Add the rows:

$$3 + 3 + 3 + 3 = 12$$

So Sara uses 12 objects.

Worked Example 4

Tom says this array has 5 rows and 2 columns. Is he correct?

$$\begin{matrix} \circ & \circ & \circ & \circ & \circ \\ \circ & \circ & \circ & \circ & \circ \end{matrix}$$

Let us check carefully.

Rows go across. There are 2 rows.

Columns go up and down. There are 5 columns.

So Tom is not correct.

The array has 2 rows and 5 columns.

How many circles are there?

$$5 + 5 = 10$$

There are 10 circles.

Tips to remember

• Rows go across.
• Columns go up and down.
• An array is arranged in a neat rectangle.
• You can count by rows or by columns.
• Both ways give the same total.

Try it in real life

Look for arrays around you. You might see them in:

• an egg carton
• a muffin tray
• tiles on the floor
• windows in a building
• chairs lined up in a room

Ask yourself:

• How many rows are there?
• How many columns are there?
• How many objects are there all together?

Summary

A rectangular array is a group of objects arranged in rows and columns. Rows go across, and columns go up and down. Arrays help us count quickly by using equal groups in a neat rectangle.

Equations for Arrays

Today we will learn how to write equations for arrays.

An array is a set of objects lined up in rows and columns.

• A row goes across.
• A column goes up and down.

Arrays help us count objects that are arranged in neat lines. They also help us write addition equations.

When we look at an array, we can count by rows or by columns.

If each row has the same number, we can add the rows together. If each column has the same number, we can add the columns together.

Main Idea

To write an equation for an array:

1. Count how many rows there are.
2. Count how many objects are in each row.
3. Write a repeated addition equation.

You can also:

1. Count how many columns there are.
2. Count how many objects are in each column.
3. Write another repeated addition equation.

Both equations match the same array. They just count it in different ways.

Example 1: Count by Rows

Look at an array with 2 rows and 3 objects in each row.

It looks like this:

● ● ●
● ● ●

There are 2 rows. Each row has 3 objects.

So we add 3 two times:

$$3 + 3 = 6$$

This array has 6 objects in all.

Example 2: Count the Same Array by Columns

Now look at the same array again:

● ● ●
● ● ●

This time, count the columns.

There are 3 columns. Each column has 2 objects.

So we add 2 three times:

$$2 + 2 + 2 = 6$$

We got the same total, 6.

That means both equations match the array:

• $$3 + 3 = 6$$
• $$2 + 2 + 2 = 6$$

Example 3: A Bigger Array

Now look at an array with 4 rows and 2 objects in each row.

It looks like this:

● ●
● ●
● ●
● ●

Count by rows first.

There are 4 rows. Each row has 2 objects.

So the addition equation is:

$$2 + 2 + 2 + 2 = 8$$

Now count by columns.

There are 2 columns. Each column has 4 objects.

So the other equation is:

$$4 + 4 = 8$$

Both equations show 8 objects in all.

Example 4: Try Another One

Suppose an array has 3 rows and 4 objects in each row.

It looks like this:

● ● ● ●
● ● ● ●
● ● ● ●

Count by rows:

There are 3 rows of 4.

$$4 + 4 + 4 = 12$$

Count by columns:

There are 4 columns of 3.

$$3 + 3 + 3 + 3 = 12$$

Both equations match the same array.

How to Check Your Work

• Make sure the rows are equal. Each row should have the same number.
• Make sure the columns are equal. Each column should have the same number.
• Count carefully.
• Add the numbers to find the total.

Helpful Clues

• If you count by rows, write the number in each row again and again.
• If you count by columns, write the number in each column again and again.
• The total stays the same either way.

Let’s Think Together

If an array has 5 rows with 2 stars in each row, what equation can you write?

Count by rows:

$$2 + 2 + 2 + 2 + 2 = 10$$

If you count by columns, there are 2 columns with 5 stars in each column:

$$5 + 5 = 10$$

Summary

An array is a neat arrangement of objects in rows and columns.

We can write repeated addition equations for arrays by counting rows or counting columns.

For the same array, the equations may look different, but they have the same total.

When you see an array, ask yourself: How many rows? How many in each row? Then write the addition equation.

Skip Counting as Multiplicative Thinking

Sometimes we count objects one by one. But when objects are in equal groups, we can count faster.

Skip counting means counting by the same number again and again. This helps us find how many objects are in several equal groups.

For example, if there are 3 bags and each bag has 2 apples, we do not have to count every apple one at a time. We can skip count by 2: 2, 4, 6. There are 6 apples total.

This is called multiplicative thinking. That means we are thinking about equal groups instead of single objects.

What does skip counting show?

• Each jump shows one group.
• The number we count by shows how many are in each group.
• The last number we say is the total.

If there are 4 groups of 5, we skip count by 5 four times:

\(5, 10, 15, 20\)

So 4 groups of 5 make 20.

Skip counting is like repeated addition.

When groups are equal, we can add the same number again and again.

For 3 groups of 4:

$$4 + 4 + 4 = 12$$

We can also skip count:

\(4, 8, 12\)

Both ways show the same total: 12.

How to use skip counting

1. Find out how many are in each group.
2. Find out how many groups there are.
3. Skip count by the group size one time for each group.
4. The last number is the total.

Worked Example 1

There are 2 plates. Each plate has 3 cookies. How many cookies are there?

Step 1: Each group has 3 cookies.

Step 2: There are 2 groups.

Step 3: Skip count by 3 two times: 3, 6.

Step 4: The total is 6.

We can also show it with addition:

$$3 + 3 = 6$$

Answer: There are 6 cookies.

Worked Example 2

There are 5 boxes. Each box has 2 crayons. How many crayons are there?

Each group has 2. There are 5 groups.

Skip count by 2 five times:

\(2, 4, 6, 8, 10\)

The last number is 10.

We can check with repeated addition:

$$2 + 2 + 2 + 2 + 2 = 10$$

Answer: There are 10 crayons.

Worked Example 3

There are 4 rows of chairs. Each row has 3 chairs. How many chairs are there?

This is 4 equal groups of 3.

Skip count by 3 four times:

\(3, 6, 9, 12\)

So there are 12 chairs.

Repeated addition says:

$$3 + 3 + 3 + 3 = 12$$

Answer: There are 12 chairs.

Worked Example 4

A toy shelf has 6 rows. Each row has 5 toy cars. How many toy cars are on the shelf?

Each group has 5. There are 6 groups.

Skip count by 5 six times:

\(5, 10, 15, 20, 25, 30\)

The total is 30.

We can show it with repeated addition too:

$$5 + 5 + 5 + 5 + 5 + 5 = 30$$

Answer: There are 30 toy cars.

Helpful idea: Use a number line in your mind

When you skip count, you can imagine jumps on a number line.

• If each group has 2, make jumps of 2.
• If each group has 5, make jumps of 5.
• The number of jumps tells how many groups there are.

Example: 3 groups of 4 means 3 jumps of 4:

\(0 \to 4 \to 8 \to 12\)

So the total is 12.

Things to remember

• Skip counting works best when groups are equal.
• Count by the number in each group.
• Say one skip-count number for each group.
• The last number you say is the total amount.

Watch out!

• Do not count groups and objects together.
• Make sure all groups have the same number.
• Do not stop counting too soon. Count once for every group.

Let’s compare

Suppose there are 3 baskets with 4 oranges in each basket.

• Counting by ones: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
• Skip counting by 4: 4, 8, 12

Both ways get the same answer, but skip counting is faster because the oranges are in equal groups.

Summary

Skip counting helps us find totals in equal groups. We count by the number in each group, one time for each group. Skip counting shows repeated addition and helps us think about multiplication in a simple way.

Fair Sharing Foundations means taking a group of things and splitting them into equal groups. When we share fairly, each group gets the same number.

This is an important math idea because it helps us understand division. Division is about sharing or splitting into equal groups.

For example, if 8 cookies are shared fairly between 2 children, each child gets the same number of cookies. We can find out how many by making 2 equal groups.

Fair sharing is not random sharing. It must be equal. If one group has more and another has less, the sharing is not fair.

When we solve fair sharing problems, we can ask:

• How many things are there altogether?
• How many groups do we need to make?
• How many will be in each group if the groups are equal?

We can use objects, drawings, circles, counters, or repeated moves of one item at a time to show fair sharing.

How to share fairly:

1. Count the total number of objects.
2. Know how many groups you need.
3. Give one object to each group.
4. Keep going, one at a time, until all objects are used.
5. Check that every group has the same number.

If all groups have the same number, the sharing is fair.

Here is a simple way to think about it:

$$\text{total objects} \div \text{number of groups} = \text{objects in each group}$$

You do not have to remember the division sign yet to understand the idea. Just remember: split into equal groups.

Worked Example 1: Share 6 apples into 2 equal groups

We have 6 apples and want 2 equal groups.

Give one apple to Group 1 and one apple to Group 2. Keep going until all 6 apples are shared.

• First round: 1 apple in each group
• Second round: 2 apples in each group
• Third round: 3 apples in each group

Now all 6 apples are used, and both groups have 3 apples.

So, $$6 \div 2 = 3$$

Each group gets 3 apples.

Worked Example 2: Share 12 crayons among 3 children

We have 12 crayons and 3 children. We want each child to get the same number.

Share one crayon at a time to each child.

• After 1 round, each child has 1
• After 2 rounds, each child has 2
• After 3 rounds, each child has 3
• After 4 rounds, each child has 4

All 12 crayons are shared fairly.

So, $$12 \div 3 = 4$$

Each child gets 4 crayons.

Worked Example 3: Share 10 toy cars into 5 equal groups

We need to make 5 equal groups from 10 toy cars.

Place one toy car in each of the 5 groups. That uses 5 cars.

Place one more toy car in each group. That uses the other 5 cars.

Now every group has 2 toy cars.

So, $$10 \div 5 = 2$$

Each group has 2 toy cars.

Worked Example 4: Is this fair sharing?

There are 9 strawberries shared into 3 groups.

Suppose the groups look like this:

• Group 1: 4 strawberries
• Group 2: 3 strawberries
• Group 3: 2 strawberries

These groups are not equal, so this is not fair sharing.

To share 9 strawberries fairly into 3 groups, each group should have 3 strawberries.

So, $$9 \div 3 = 3$$

Using drawings can help

You can draw circles to show groups. Then put marks or dots into each circle one at a time.

For example, to share 8 stars into 4 groups, draw 4 circles and place 1 star in each circle. Then place 1 more star in each circle.

Now each circle has 2 stars, so $$8 \div 4 = 2$$.

Fair sharing and repeated addition

After sharing, you can check your answer by adding the equal groups.

If 12 objects are shared into 3 equal groups and each group has 4, then:

$$4 + 4 + 4 = 12$$

This shows the sharing is correct.

Things to remember

• Fair sharing means equal groups.
• Every group must have the same number.
• You can share one at a time to keep it fair.
• You can check by counting each group.
• You can also check with repeated addition.

Let's think together

If 15 blocks are shared into 3 equal groups, how many go in each group?

Share one block at a time into 3 groups until all 15 are used. Each group gets 5.

So, $$15 \div 3 = 5$$

If 14 stickers are shared into 2 equal groups, each group gets 7 stickers.

So, $$14 \div 2 = 7$$

Brief Summary

Fair sharing means splitting a group of objects into equal groups. To solve these problems, count the total, make the right number of groups, and share one at a time until all objects are used. If each group has the same number, the sharing is fair.

Division as Repeated Subtraction

Sometimes we have a total number of things, and we want to split them into equal groups. Division helps us find out how many equal groups we can make.

One way to understand division is to use repeated subtraction. This means we keep subtracting the same number again and again until there is nothing left.

Each time we subtract, we are taking away one group. Then we count how many groups we took away.

For example, if we have 12 apples and we take away groups of 3 apples, we can subtract 3 again and again:

We subtracted 3 4 times, so:

This means 12 apples can be made into 4 equal groups of 3.

How repeated subtraction works

1. Start with the total number.
2. Subtract the same group size each time.
3. Keep going until you get to 0.
4. Count how many times you subtracted.

The number of times you subtract is the answer to the division problem.

We can think about division questions like this:

• How many groups of 2 are in 8?
• How many groups of 5 are in 15?
• How many groups of 4 are in 16?

Lets practice with some examples.

Worked Example 1

Find: \(8 \div 2\)

Start at 8. Subtract 2 each time:

We subtracted 2 4 times.

So, 8 things can be split into 4 equal groups of 2.

Worked Example 2

Find: \(15 \div 5\)

Start at 15. Subtract 5 each time:

We subtracted 5 3 times.

So, 15 things make 3 equal groups of 5.

Worked Example 3

Find: \(18 \div 6\)

Start at 18. Subtract 6 each time:

We subtracted 6 3 times.

So, 18 things can be put into 3 equal groups of 6.

Worked Example 4

Find: \(20 \div 4\)

Start at 20. Subtract 4 each time:

We subtracted 4 5 times.

So, 20 things make 5 equal groups of 4.

A helpful way to think

When you see a division problem like \(12 \div 3\), ask yourself:

How many times can I subtract 3 from 12 until I get to 0?

If you can subtract 3 four times, then the answer is 4.

Lets look at one more idea.

Division and subtraction work together. Repeated subtraction shows us the number of equal groups.

For \(10 \div 2\):

There are 5 subtractions, so:

Tips to remember

• Start with the total number.
• Subtract the same number each time.
• Stop when you reach 0.
• Count the number of subtractions.
• That count is the division answer.

Summary

Division as repeated subtraction means taking away the same amount again and again.

When you reach 0, count how many times you subtracted. That tells you how many equal groups there are.

So if you know how to subtract the same number many times, you can use that to solve division problems.