Subtraction Concepts and Strategies

Subtraction as Taking Away

Today we will learn about subtraction. Subtraction means taking away from a group.

When we subtract, we start with some things. Then some things are taken away. We look at how many are left.

You can think:

• Start with a group.
• Take away some of the group.
• Count what is left.

We use the minus sign to show subtraction: \(-\)

For example, if you have 5 apples and 2 are taken away, you can write:

$$5 - 2 = 3$$

This means: start with 5, take away 2, and 3 are left.

How to subtract by taking away

1. Look at the first number. This is how many you start with.
2. Look at the second number. This is how many are taken away.
3. Count how many are left.

You can use real objects to help, like blocks, crayons, or fingers.

You can also draw pictures. Draw the starting group, cross out the ones taken away, and count the ones still there.

Worked Example 1

Sara has 4 balloons. 1 balloon flies away. How many balloons are left?

Start with 4. Take away 1.

$$4 - 1 = 3$$

Answer: 3 balloons are left.

Worked Example 2

There are 7 cookies on a plate. 3 cookies are eaten. How many cookies are left?

Start with 7. Take away 3.

You can count back: 6, 5, 4.

$$7 - 3 = 4$$

Answer: 4 cookies are left.

Worked Example 3

8 birds are in a tree. 5 birds fly away. How many birds are left?

Start with 8. Take away 5.

You can use fingers or draw 8 birds and cross out 5.

$$8 - 5 = 3$$

Answer: 3 birds are left.

Worked Example 4

Liam has 10 toy cars. He gives away 6 toy cars. How many toy cars does he have now?

Start with 10. Take away 6.

Count back 6 numbers: 9, 8, 7, 6, 5, 4.

$$10 - 6 = 4$$

Answer: 4 toy cars are left.

Ways to help yourself subtract

• Use objects: Move some away and count what is left.
• Use drawings: Cross out the ones taken away.
• Use fingers: Put some fingers down to show taking away.
• Use counting back: Start at the first number and count back the second number.

Let’s look carefully at one more problem

Suppose you see:

$$6 - 2$$

This means start with 6. Take away 2.

If you count back 2 numbers, you say: 5, 4.

So:

$$6 - 2 = 4$$

Important idea

In subtraction, the answer tells how many are left. The answer to a subtraction problem is called the difference.

For example:

$$9 - 4 = 5$$

The difference is 5.

Watch out!

• The first number tells how many you have at the start.
• The second number tells how many are taken away.
• Do not count the crossed-out objects. Count only the ones left.

Try thinking about subtraction like a story

If 12 ducks are in a pond and 2 swim away, there are fewer ducks now.

We can write:

$$12 - 2 = 10$$

Taking away makes the group smaller.

Summary

Subtraction means taking away. You start with a group, take some away, and count how many are left.

You can subtract with objects, pictures, fingers, or by counting back. The answer tells how many are left after some are taken away.

Subtraction as Comparison (Difference)

Sometimes subtraction means take away. But subtraction can also mean compare.

When we compare, we look at two groups and ask, How many more? or How many fewer?

The answer tells us the difference. The difference is the gap between the two numbers.

For example, if one child has 7 stickers and another child has 4 stickers, we can compare 7 and 4. We ask, How many more is 7 than 4?

We use subtraction to find that gap:

$$7 - 4 = 3$$

This means the difference between 7 and 4 is 3.

How to compare with subtraction

1. Look at the two numbers.

2. Find the bigger number.

3. Take away the smaller number.

4. The answer is the difference.

You can think, How far apart are the numbers?

You can also use objects, drawings, or your fingers to help.

Compare with pictures in your mind

Imagine 2 rows of cubes. One row has 8 cubes. The other row has 5 cubes.

Match the cubes one by one. After matching, 3 cubes are extra in the longer row. So the difference is 3.

That is why:

$$8 - 5 = 3$$

Words that tell us to compare

• How many more?

• How many fewer?

• How many less?

• What is the difference?

• How many are left to match?

These words often mean we should compare two amounts.

Worked Example 1

Ava has 6 apples. Ben has 2 apples. How many more apples does Ava have than Ben?

First, find the bigger number: 6.

Then subtract the smaller number: 2.

$$6 - 2 = 4$$

Ava has 4 more apples than Ben.

Worked Example 2

There are 9 birds on one tree and 7 birds on another tree. What is the difference?

We compare 9 and 7.

$$9 - 7 = 2$$

The difference is 2.

This also means one tree has 2 more birds than the other tree.

Worked Example 3

Liam has 10 toy cars. Mia has 6 toy cars. How many fewer toy cars does Mia have than Liam?

"How many fewer" means compare.

Find the bigger number: 10.

Subtract the smaller number: 6.

$$10 - 6 = 4$$

Mia has 4 fewer toy cars than Liam.

Worked Example 4

One basket has 13 oranges. Another basket has 8 oranges. How many more oranges are in the first basket?

We compare 13 and 8.

$$13 - 8 = 5$$

The first basket has 5 more oranges.

A helpful way to think

If you know the subtraction fact, you know the difference.

For example:

$$12 - 9 = 3$$

This means 12 is 3 more than 9.

It also means 9 is 3 fewer than 12.

The difference stays the same: 3.

Compare and match

You can line up objects to compare them.

• If 7 blocks match with 7 blocks, they are the same.

• If 2 extra blocks are left in one group, the difference is 2.

This matching idea helps us see the gap clearly.

Be careful

• Use the bigger number first when finding the difference.

• Read the question words carefully.

• "How many more" and "how many fewer" both ask for the difference.

Try these ideas when solving

• Circle the two numbers.

• Ask: Which number is bigger?

• Subtract to find the gap.

• Say the answer with words like more, fewer, or difference.

Summary

Subtraction as comparison means finding the difference between two groups.

We ask questions like How many more? or How many fewer?

To solve, take the smaller number away from the bigger number.

The answer tells us the gap between the two amounts.

Subtraction as Finding the Missing Part

Sometimes subtraction means taking away. But subtraction can also mean finding the missing part.

When we know the whole and we know one part, we can use subtraction to find the missing part.

Think about a puzzle. If one piece is missing, we can figure out what piece we still need. That is what we do in this kind of subtraction.

Whole, part, part

• The whole is the total amount.
• A part is one piece of the whole.
• The other part may be missing.

If we know the whole and one part, we can find the missing part with subtraction.

We can think:

• What goes with this part to make the whole?
• How many more do I need?

Here is the big idea:

$$\text{whole} - \text{known part} = \text{missing part}$$

For example, if there are 8 apples in all, and 5 are red, then some are green. We can find the green apples by subtracting:

$$8 - 5 = 3$$

So, 3 apples are green. The missing part is 3.

You can also think about addition.

If we know one part is 5 and the whole is 8, we can ask, 5 and what number make 8?

$$5 + 3 = 8$$

So, the missing part is still 3.

This is why subtraction and addition work together. Subtraction can help us find a missing part, and addition can help us check the answer.

How to find the missing part

1. Find the whole.
2. Find the part you know.
3. Subtract to find the part that is missing.
4. Check with addition.

Worked Example 1

There are 7 balloons in all. 4 balloons are blue. How many balloons are not blue?

Step 1: Find the whole. The whole is 7.

Step 2: Find the known part. The known part is 4 blue balloons.

Step 3: Subtract.

$$7 - 4 = 3$$

Answer: 3 balloons are not blue.

Check:

$$4 + 3 = 7$$

The parts make the whole, so the answer is correct.

Worked Example 2

There are 10 fish in a tank. 6 fish are orange. How many fish are not orange?

Whole: 10

Known part: 6

Missing part: ?

$$10 - 6 = 4$$

Answer: 4 fish are not orange.

Check:

$$6 + 4 = 10$$

Worked Example 3

Ella has 12 crayons. 9 crayons are in her box. The rest are on the table. How many crayons are on the table?

We know the whole is 12 crayons.

We know one part is 9 crayons in the box.

The missing part is the crayons on the table.

$$12 - 9 = 3$$

Answer: 3 crayons are on the table.

Check:

$$9 + 3 = 12$$

Worked Example 4

There are 15 children on the playground. 8 children are on the swings. The rest are on the slide side. How many children are on the slide side?

Whole: 15

Known part: 8

Missing part: ?

$$15 - 8 = 7$$

Answer: 7 children are on the slide side.

Check:

$$8 + 7 = 15$$

Helpful ways to think

• Ask: What part is missing?
• Ask: How many more to make the whole?
• Use subtraction: whole minus part.
• Use addition to check: part plus missing part equals whole.

Look for clue words

Sometimes story problems use words that help us know we are finding a missing part.

• in all
• the rest
• how many are not
• how many more
• missing

If you see these words, think: Do I know the whole and one part? If yes, subtraction can help.

Try this thinking

If there are 14 birds in all and 10 are in the tree, how many are on the ground?

We know:

• Whole = 14
• Known part = 10

So we subtract:

$$14 - 10 = 4$$

The missing part is 4.

A quick picture in your mind

You can imagine a number bond:

• The big circle is the whole.
• The two small circles are the parts.
• If one small circle is empty, subtraction helps you fill it in.

Example:

Whole = 9, one part = 2, missing part = ?

$$9 - 2 = 7$$

So the missing part is 7.

Important idea to remember

When you find a missing part, you are not taking something away that disappears. You are figuring out the part you do not know yet.

Subtraction helps answer the question: What part is missing from the whole?

Summary

Subtraction can mean finding the missing part. When you know the whole and one part, subtract to find the other part.

Then check your answer with addition. If the two parts make the whole, your answer is correct.

The Minus Symbol and Equations

Today we will learn about the minus symbol and how to write subtraction equations.

Subtraction means taking away. It can also mean how many are left or how many more.

When we subtract, we use the minus symbol: \(-\).

We also use the equals symbol: \(=\).

The equals symbol means is the same as or the answer is.

A subtraction equation has numbers and symbols that tell a subtraction story.

Here is the shape of a subtraction equation:

$$8 - 3 = 5$$

This equation says: Start with 8. Take away 3. There are 5 left.

Parts of a subtraction equation

• The first number tells how many we start with.
• The minus symbol \(-\) tells us to take away.
• The second number tells how many we take away.
• The equals symbol \(=\) tells us the answer comes next.
• The last number tells how many are left.

How to write a subtraction equation

1. Find how many there are at the start.
2. Find how many are taken away.
3. Write the minus symbol \(-\).
4. Write the equals symbol \(=\).
5. Find how many are left.

We can also use subtraction to find a missing part.

For example, if there are 9 apples and 4 are red, then the rest are not red. We can write:

$$9 - 4 = 5$$

This means 5 apples are not red.

We can also use subtraction to find a difference.

If one child has 7 blocks and another child has 5 blocks, we can ask: How many more blocks does the first child have?

We write:

$$7 - 5 = 2$$

The difference is 2.

Worked Examples

Example 1: Taking away

Liam has 6 balloons. 2 balloons fly away. How many balloons are left?

Start with 6. Take away 2.

Write the equation:

$$6 - 2 = 4$$

Liam has 4 balloons left.

Example 2: Taking away with more objects

There are 10 cookies on a plate. 3 cookies are eaten. How many cookies are left?

Start with 10. Take away 3.

Write the equation:

$$10 - 3 = 7$$

There are 7 cookies left.

Example 3: Finding a missing part

There are 8 birds in a tree. 5 birds are blue. The rest are yellow. How many birds are yellow?

Start with 8 birds. Take away the 5 blue birds.

Write the equation:

$$8 - 5 = 3$$

There are 3 yellow birds.

Example 4: Finding a difference

Emma has 9 crayons. Noah has 6 crayons. How many more crayons does Emma have?

We compare 9 and 6. Subtract the smaller number from the bigger number.

Write the equation:

$$9 - 6 = 3$$

Emma has 3 more crayons than Noah.

Helpful tips

• The number before the minus sign is the number you start with.
• The number after the minus sign is the number you take away.
• The number after the equals sign is the answer.
• You can use fingers, drawings, or objects to help subtract.

Watch out!

• Do not mix up \(-\) and \(=\).
• Make sure the first number matches how many you start with.
• Make sure the second number matches how many are taken away.
• Count carefully to find how many are left.

Summary

The minus symbol \(-\) means take away.

The equals symbol \(=\) means the answer is the same as.

A subtraction equation tells a subtraction story with numbers and symbols.

We can write subtraction equations to show:

• taking away,
• how many are left,
• a missing part,
• or how many more.

When you see a story problem, ask:

1. How many do I start with?
2. How many are taken away?
3. How many are left?

Then write the subtraction equation with \(-\) and \(=\).

Subtracting Zero and Subtracting All

Today we will learn two easy subtraction ideas:

• Subtracting zero
• Subtracting all

Subtraction means taking away. When we subtract, we start with a number and take some away.

Let’s learn what happens when we take away nothing and what happens when we take away everything.

1. Subtracting Zero

Zero means none or nothing.

If you subtract zero, you are taking away nothing. So the number stays the same.

We can say:

$$a - 0 = a$$

That means any number minus zero is still the same number.

For example, if you have 5 apples and take away 0 apples, you still have 5 apples.

$$5 - 0 = 5$$

2. Subtracting All

Subtracting all means subtracting the whole number you started with.

If you take away everything, there is nothing left. Nothing left means zero.

We can say:

$$a - a = 0$$

That means any number minus itself equals zero.

For example, if you have 4 blocks and take away all 4 blocks, you have 0 blocks left.

$$4 - 4 = 0$$

How to Think About It

• If you take away 0, nothing changes.
• If you take away all, you have 0 left.

You can remember it like this:

• Minus 0 = stays the same
• Minus the same number = 0

Worked Examples

Example 1: \(7 - 0\)

Start with 7. Take away 0. You are not taking anything away.

So 7 stays 7.

$$7 - 0 = 7$$

Example 2: \(3 - 3\)

Start with 3. Take away all 3.

Now there is nothing left.

$$3 - 3 = 0$$

Example 3: \(10 - 0\)

Start with 10. Take away nothing.

The number does not change.

$$10 - 0 = 10$$

Example 4: \(12 - 12\)

Start with 12. Take away all 12.

Everything is gone, so 0 is left.

$$12 - 12 = 0$$

Try to Notice the Pattern

• \(1 - 0 = 1\)
• \(2 - 0 = 2\)
• \(6 - 0 = 6\)

When we subtract zero, the answer is the number we started with.

• \(1 - 1 = 0\)
• \(2 - 2 = 0\)
• \(6 - 6 = 0\)

When we subtract all, the answer is zero.

Helpful Tips

• Look for a 0 after the minus sign. If you see it, the answer stays the same.
• Look to see if both numbers are the same. If they are the same, the answer is 0.

Let’s Say It Together

• Subtract 0, the number stays the same.
• Subtract all, the answer is 0.

Summary

Subtracting zero means taking away nothing, so the number stays the same. Subtracting all means taking away the whole amount, so zero is left. These are important subtraction patterns to remember and use when solving math problems.

Strategy: Counting Back helps us subtract.

When we subtract, we start with a number and take some away. One way to do this is called counting back.

To count back, start at the first number. Then move backward the number of times you are taking away.

For example, in \(9 - 3\), start at 9. Then count back 3 numbers: 8, 7, 6. So the answer is \(6\).

How to count back

1. Look at the subtraction problem.
2. Start at the first number.
3. Count backward one number at a time.
4. Stop when you have counted back the second number.
5. The number you land on is the answer.

You can use your fingers to help. You can also picture a number line in your head.

When you count back, say one number for each step. This helps you not count too many or too few.

Example 1

Solve \(7 - 2\).

Start at 7.

Count back 2 numbers:

• 6 (1)
• 5 (2)

So, $$7 - 2 = 5$$

Example 2

Solve \(10 - 4\).

Start at 10.

Count back 4 numbers:

• 9 (1)
• 8 (2)
• 7 (3)
• 6 (4)

So, $$10 - 4 = 6$$

Example 3

Solve \(13 - 5\).

Start at 13.

Count back 5 numbers:

• 12 (1)
• 11 (2)
• 10 (3)
• 9 (4)
• 8 (5)

So, $$13 - 5 = 8$$

Example 4

Solve \(18 - 6\).

Start at 18.

Count back 6 numbers:

• 17 (1)
• 16 (2)
• 15 (3)
• 14 (4)
• 13 (5)
• 12 (6)

So, $$18 - 6 = 12$$

Tips for counting back

• Start at the bigger number, the first number in the subtraction problem.
• Count backward slowly.
• Use fingers to keep track of how many steps you took.
• If the number you take away is small, counting back is a great strategy.

Be careful!

Sometimes children make a small mistake. They start counting the first number as a step. Do not do that.

In \(8 - 3\), start at 8, but do not say 8 as step 1. Count back from 8:

• 7 (1)
• 6 (2)
• 5 (3)

So, $$8 - 3 = 5$$

Let’s think about what subtraction means

Subtraction means taking away. If you have 12 blocks and take away 2 blocks, you have fewer blocks left.

You can count back to find how many are left: 12, then back to 11, then back to 10. So \(12 - 2 = 10\).

Summary

Counting back is a subtraction strategy. Start at the first number and move backward. Count back the second number of steps. The number you land on is your answer.

Strategy: Think Addition

Sometimes subtraction can feel tricky. A smart way to solve a subtraction problem is to think addition.

When we use think addition, we ask: “What number can I add to the smaller number to make the bigger number?”

For example, in the subtraction problem \(12 - 8\), we can think: “\(8\) plus what equals \(12\)?”

If we know that \(8 + 4 = 12\), then we know that:

$$12 - 8 = 4$$

So subtraction and addition can help each other.

Why does this work?

Subtraction can mean finding a missing part. If the whole is \(12\) and one part is \(8\), we can find the missing part by thinking about addition.

We can think of it like this:

$$8 + \Box = 12$$

The missing number is \(4\), so:

$$12 - 8 = 4$$

How to use the Think Addition strategy

1. Look at the subtraction problem.

2. Find the smaller number and the bigger number.

3. Ask: “Smaller number plus what equals bigger number?”

4. Use what you know about addition facts.

5. The number you add is the answer to the subtraction problem.

Worked Example 1

Solve \(7 - 5\).

Think: “\(5\) plus what equals \(7\)?”

We know:

$$5 + 2 = 7$$

So:

$$7 - 5 = 2$$

Worked Example 2

Solve \(10 - 6\).

Think: “\(6\) plus what equals \(10\)?”

Count on from \(6\): \(7, 8, 9, 10\).

That is \(4\) more.

So:

$$6 + 4 = 10$$

and

$$10 - 6 = 4$$

Worked Example 3

Solve \(14 - 9\).

Think: “\(9\) plus what equals \(14\)?”

You can count on: \(10, 11, 12, 13, 14\).

That is \(5\) numbers more.

So:

$$9 + 5 = 14$$

Therefore:

$$14 - 9 = 5$$

Worked Example 4

Solve \(16 - 7\).

Think: “\(7\) plus what equals \(16\)?”

You might know the fact:

$$7 + 9 = 16$$

So:

$$16 - 7 = 9$$

Tips to help you

• Start with the smaller number in the subtraction problem.

• Think about addition facts you already know.

• You can count on to find the missing number.

• If you know \(3 + 5 = 8\), then you also know \(8 - 3 = 5\) and \(8 - 5 = 3\).

Let’s look at one more way

For \(13 - 11\), ask: “\(11\) plus what equals \(13\)?”

We know:

$$11 + 2 = 13$$

So:

$$13 - 11 = 2$$

When is Think Addition a good strategy?

• It is helpful when the numbers are close together.

• It is helpful when you know your addition facts.

• It helps you find the missing part.

Summary

To use Think Addition, change a subtraction problem into an addition question.

Ask: “What can I add to the smaller number to make the bigger number?”

The number you add is the answer to the subtraction problem.

So if \(12 - 8\), think \(8 + 4 = 12\). That means \(12 - 8 = 4\).

Using a Number Line to Subtract

Subtraction means taking away. It can also mean finding how many are left or finding the space between two numbers.

A number line helps us see numbers in order. We can use it to subtract by making jumps backward.

When we subtract on a number line, we start at the first number. Then we jump back the second number. The number where we land is the answer.

For example, in \(8 - 3\), we start at 8. Then we jump back 3 spaces. We land on 5. So \(8 - 3 = 5\).

Main Idea 1: Start at the first number

In a subtraction problem, the first number tells you where to begin on the number line.

If the problem is \(9 - 2\), start at 9.

Main Idea 2: Jump backward

The second number tells you how many jumps to make backward.

Backward means moving to the left on the number line.

For \(9 - 2\), make 2 jumps back:

9 to 8 is 1 jump.
8 to 7 is 2 jumps.

So \(9 - 2 = 7\).

Main Idea 3: The number you land on is the answer

After you make all the backward jumps, look at the number where you stop. That number is the difference.

The difference is the answer to a subtraction problem.

Main Idea 4: A number line can also show the distance between numbers

Sometimes subtraction means finding how far apart two numbers are.

If you want to solve \(9 - 6\), you can start at 6 and count up to 9.

6 to 7 is 1.
7 to 8 is 2.
8 to 9 is 3.

So the distance between 6 and 9 is 3, and \(9 - 6 = 3\).

How to Subtract on a Number Line

1. Look at the subtraction problem.

2. Start at the first number.

3. Jump backward the number of times shown by the second number.

4. Say the number where you land.

You can also find the distance between two numbers by counting the jumps from the smaller number to the bigger number.

Worked Example 1

Solve \(6 - 1\).

Start at 6. Jump back 1 space.

$$6 \rightarrow 5$$

You land on 5.

So, \(6 - 1 = 5\).

Worked Example 2

Solve \(10 - 4\).

Start at 10. Jump back 4 spaces.

$$10 \rightarrow 9 \rightarrow 8 \rightarrow 7 \rightarrow 6$$

You land on 6.

So, \(10 - 4 = 6\).

Worked Example 3

Solve \(13 - 5\).

Start at 13. Jump back 5 spaces.

$$13 \rightarrow 12 \rightarrow 11 \rightarrow 10 \rightarrow 9 \rightarrow 8$$

You land on 8.

So, \(13 - 5 = 8\).

Worked Example 4

Solve \(12 - 9\) by finding the distance between the numbers.

Start at 9 and count up to 12.

$$9 \rightarrow 10 \rightarrow 11 \rightarrow 12$$

That is 3 jumps.

So, \(12 - 9 = 3\).

Helpful Tips

• Start on the correct number.

• Move left to subtract.

• Count each jump carefully.

• The last number is your answer.

• If the numbers are close together, you can count the space between them.

Let’s Think

If you solve \(7 - 3\), start at 7 and jump back 3 times.

$$7 \rightarrow 6 \rightarrow 5 \rightarrow 4$$

So \(7 - 3 = 4\).

If you solve \(15 - 2\), start at 15 and jump back 2 times.

$$15 \rightarrow 14 \rightarrow 13$$

So \(15 - 2 = 13\).

Summary

A number line is a great tool for subtraction.

• Start at the first number.

• Jump backward by the second number.

• The number where you land is the answer.

• You can also use a number line to find the distance between two numbers.

When you use a number line, subtraction becomes easy to see. Backward jumps show taking away, and counting the space between numbers shows how far apart they are.

Fact Families and Inverse Relationship

Today we will learn about fact families. A fact family is a group of math facts that use the same 3 numbers.

We will also learn about the inverse relationship. That means addition and subtraction work together. If you know an addition fact, you can use it to help with subtraction. If you know a subtraction fact, you can use it to help with addition.

Fact families help us see number patterns. They help us check our work, too.

What is a fact family?

A fact family uses 3 numbers. With those 3 numbers, we can write:

• 2 addition facts
• 2 subtraction facts

Let’s use the numbers 3, 4, and 7.

These numbers make one fact family:

All 4 math facts use the same numbers: 3, 4, and 7.

How addition and subtraction are related

Addition puts parts together. Subtraction takes a part away or finds a missing part.

If we know that

then we also know:

Why? Because 3 and 4 are the parts, and 7 is the whole. If we take one part away from the whole, the other part is left.

This is why addition and subtraction are called inverse operations. They are opposite, but they help each other.

Think of a number bond

You can think about a fact family like this:

• The whole is the biggest number.
• The parts are the two smaller numbers.

For 3, 4, and 7:

• Whole: 7
• Parts: 3 and 4

The addition facts put the parts together to make the whole.

The subtraction facts start with the whole and take away one part to find the other part.

How to make a fact family

1. Find the 3 numbers.
2. Find the biggest number. That is the whole.
3. Write 2 addition facts with the two smaller numbers first.
4. Write 2 subtraction facts starting with the biggest number.

Worked Example 1

Use the numbers 2, 5, and 7.

Step 1: The biggest number is 7, so 7 is the whole.

Step 2: Write the addition facts.

Step 3: Write the subtraction facts.

These 4 equations are one fact family.

Worked Example 2

Use the numbers 6, 1, and 7.

The biggest number is 7.

Addition facts:

Subtraction facts:

Notice that we use the same 3 numbers each time.

Worked Example 3

Suppose you know this addition fact:

Can you find the other facts in the family?

Yes. The 3 numbers are 8, 2, and 10.

The second addition fact is:

The subtraction facts are:

So one addition fact can help us find the subtraction facts.

Worked Example 4

Suppose you know this subtraction fact:

What fact family does it belong to?

The 3 numbers are 9, 4, and 5.

The biggest number is 9, so 9 is the whole.

The full fact family is:

Tips to help you

• Look for the biggest number. It is the whole.
• The two smaller numbers are the parts.
• Addition facts end with the biggest number.
• Subtraction facts begin with the biggest number.
• Always check that all 4 facts use the same 3 numbers.

What stays the same and what changes?

• In the addition facts, the two smaller numbers can switch places.
• In the subtraction facts, the biggest number stays first.
• The answer in one fact can help you write another fact.

For example, if

then

and

Let’s check our thinking

Do the numbers 4, 4, and 8 make a fact family?

Yes. They do. The facts are:

Sometimes the facts repeat when the two parts are the same. That is okay.

Why fact families are useful

• They help you learn addition and subtraction facts.
• They help you see how numbers are connected.
• They help you solve missing number problems.
• They help you check your answers.

For example, if you solve

you can check with addition:

If the addition fact is true, your subtraction answer makes sense.

Summary

A fact family is a group of 4 facts that use the same 3 numbers. It has 2 addition facts and 2 subtraction facts.

Addition and subtraction are inverse, or opposite, operations. They work together. When you know one fact, you can often find the others.

Remember: find the biggest number, use the same 3 numbers, and write both addition and subtraction facts.

Decomposing to Subtract means breaking a number apart to make subtraction easier.

Sometimes, when we subtract, it helps to go to 10 first. The number 10 is a friendly number because it is easy to work with.

For example, in \(14 - 6\), we can break apart the 6 into \(4\) and \(2\). Why? Because \(14 - 4 = 10\). Then we subtract the extra 2.

$$14 - 6 = 14 - 4 - 2 = 10 - 2 = 8$$

This is called decomposing. Decomposing means taking one number and splitting it into smaller parts.

How to Decompose to Subtract

1. Look at the first number.

2. Think: How much do I subtract to get to 10?

3. Break apart the number you are subtracting.

4. Subtract to 10 first.

5. Subtract the rest.

Let’s practice that idea.

If we have \(13 - 5\), first ask: how much do we subtract from 13 to get to 10?

The answer is \(3\), because \(13 - 3 = 10\).

Now break apart the 5 into \(3\) and \(2\).

$$13 - 5 = 13 - 3 - 2 = 10 - 2 = 8$$

So, \(13 - 5 = 8\).

Why This Helps

Subtracting to 10 first makes the problem simpler.

Instead of trying to subtract the whole number all at once, we do it in two easy steps.

• Step 1: Get to 10

• Step 2: Subtract what is left

Worked Examples

Example 1: \(12 - 3\)

How much do we subtract from 12 to get to 10? We subtract \(2\).

So we break apart 3 into \(2\) and \(1\).

$$12 - 3 = 12 - 2 - 1 = 10 - 1 = 9$$

The answer is 9.

Example 2: \(15 - 7\)

How much do we subtract from 15 to get to 10? We subtract \(5\).

So we break apart 7 into \(5\) and \(2\).

$$15 - 7 = 15 - 5 - 2 = 10 - 2 = 8$$

The answer is 8.

Example 3: \(16 - 8\)

How much do we subtract from 16 to get to 10? We subtract \(6\).

So we break apart 8 into \(6\) and \(2\).

$$16 - 8 = 16 - 6 - 2 = 10 - 2 = 8$$

The answer is 8.

Example 4: \(18 - 9\)

How much do we subtract from 18 to get to 10? We subtract \(8\).

So we break apart 9 into \(8\) and \(1\).

$$18 - 9 = 18 - 8 - 1 = 10 - 1 = 9$$

The answer is 9.

Let’s Notice a Pattern

When the first number is bigger than 10, we can often make subtraction easier by going to 10.

Here are some ways to get to 10:

• From 11, subtract 1

• From 12, subtract 2

• From 13, subtract 3

• From 14, subtract 4

• From 15, subtract 5

• From 16, subtract 6

• From 17, subtract 7

• From 18, subtract 8

• From 19, subtract 9

If you know how far a number is from 10, you can decompose the number you are subtracting.

Try Thinking Through One More Problem

Let’s solve \(17 - 9\).

First, ask: how much from 17 to get to 10? The answer is \(7\).

Now break apart 9 into \(7\) and \(2\).

$$17 - 9 = 17 - 7 - 2 = 10 - 2 = 8$$

So, \(17 - 9 = 8\).

Tips to Remember

• Break apart the number you subtract.

• Try to get to 10 first.

• Then subtract the rest.

• Take it one small step at a time.

Summary

Decomposing to subtract means breaking apart a number to make subtraction easier.

A smart way is to subtract enough to get to 10, then subtract what is left.

For example:

$$14 - 6 = 14 - 4 - 2 = 10 - 2 = 8$$

When you use 10 as a helper number, subtraction can feel quick and easy.