Subtractive Reasoning and Fluency

Subtraction as Removal and Difference

Subtraction helps us in two important ways. We can use subtraction when something is taken away, and we can also use subtraction to find how many more or how many fewer one number is than another.

In this lesson, you will learn to tell the difference between these two kinds of subtraction stories:

• Removal (take-away): Start with an amount, then some are removed.
• Difference (comparison): Compare two amounts to find how far apart they are.

Both kinds use subtraction, but the story tells us why we subtract.

1. Subtraction as Removal

Subtraction as removal means you start with a group, and then some are taken away. The amount becomes smaller.

Words that often show removal are:

• took away
• gave away
• ate
• left
• removed
• spent

For example, if you have 9 apples and eat 3 apples, you have fewer apples now. You can write:

This means 3 apples were removed from 9 apples, so 6 apples are left.

You can think of removal like this:

1. Start with the whole amount.
2. Take some away.
3. Find what is left.

2. Subtraction as Difference

Subtraction as difference means you compare two amounts. Nothing has to be taken away. You are finding how much more, how much fewer, or how far apart the numbers are.

Words that often show difference are:

• how many more
• how many fewer
• how much greater
• how much less
• difference
• compare

For example, Mia has 8 stickers and Ben has 5 stickers. No one is taking stickers away. We are comparing their amounts.

Mia has 3 more stickers than Ben. The difference between 8 and 5 is 3.

You can think of difference like this:

1. Look at the two amounts.
2. Find which number is greater.
3. Subtract to find how far apart they are.

3. How to Tell Which Kind of Subtraction to Use

Ask yourself these questions when you read a problem:

• Did the story start with a number and then some were taken away? That is removal.
• Does the story compare two groups? That is difference.

Here is a helpful way to remember:

• Removal: “What is left?”
• Difference: “How many more or fewer?”

4. Worked Examples

Example 1: Removal

Sara had 12 crayons. She gave away 4 crayons. How many crayons does she have now?

We started with 12 crayons, and 4 were taken away. This is removal.

Answer: Sara has 8 crayons left.

Example 2: Difference

Jay has 14 toy cars. Leo has 9 toy cars. How many more toy cars does Jay have than Leo?

Nothing is being taken away. We are comparing 14 and 9. This is difference.

Answer: Jay has 5 more toy cars than Leo.

Example 3: Removal with a Bigger Number

A class had 25 cupcakes. The students ate 7 cupcakes. How many cupcakes are left?

We start with 25 cupcakes, and 7 are removed. This is removal.

Answer: 18 cupcakes are left.

Example 4: Difference with a Bigger Number

A tree is 17 feet tall. A bush is 9 feet tall. How much taller is the tree than the bush?

We are comparing two heights. Nothing is taken away. This is difference.

Answer: The tree is 8 feet taller than the bush.

5. Same Subtraction Sentence, Different Meaning

Sometimes the same subtraction sentence can match two different kinds of stories.

Look at this subtraction sentence:

It can mean:

• Removal: You had 15 cookies and ate 6. Now 9 are left.
• Difference: One child has 15 marbles and another has 6 marbles. The difference is 9 marbles.

This shows that subtraction can be used in more than one way.

6. Helpful Strategies

These strategies can help you solve subtraction problems:

• Use objects: Counters, blocks, or drawings can show what is taken away or compared.
• Use a number line: You can count back for removal or count the space between numbers for difference.
• Look for clue words: Words like “left” often mean removal. Words like “how many more” often mean difference.
• Check the story: Make sure your answer matches what the problem is asking.

7. Let’s Compare the Two Ideas

8. Quick Practice Thinking

Read each situation and think: removal or difference?

• There were 18 birds in a tree. 5 flew away. Removal
• Lena has 13 shells. Omar has 8 shells. Difference
• A store had 20 balloons. It sold 6 balloons. Removal
• A red ribbon is 11 inches long. A blue ribbon is 7 inches long. Difference

Summary

Subtraction can mean taking away or comparing two amounts. If something is removed from a group, use subtraction as removal. If you are finding how many more or fewer one amount is than another, use subtraction as difference.

When you read a word problem, think carefully about what is happening in the story. Ask, “Is something being taken away?” or “Am I comparing two amounts?” That will help you choose the right kind of subtraction.

Think-Addition to Subtract is a smart way to solve subtraction problems.

Sometimes subtraction can feel tricky. But if you know addition facts well, you can use addition to help you subtract.

When we think-addition to subtract, we ask a new question:

Instead of asking, "What is \(14 - 8\)?", we ask, "\(8\) plus what equals \(14\)?"

This works because subtraction and addition are related. They are part of the same fact family.

For example:

$$8 + 6 = 14$$

So:

$$14 - 8 = 6$$

And also:

$$14 - 6 = 8$$

Introduction: Why this works

Subtraction can mean finding how many are left. It can also mean finding the missing part or the difference between two numbers.

When we use think-addition, we are finding the missing part.

If you know one part and the whole, you can ask:

"What number do I add to this part to make the whole?"

That missing number is the answer to the subtraction problem.

Main Teaching Point 1: Turn subtraction into a missing-addend problem

Look at the subtraction problem:

$$13 - 9$$

We can turn it into an addition problem:

$$9 + \Box = 13$$

Now we ask, "What goes in the box?"

Since:

$$9 + 4 = 13$$

we know:

$$13 - 9 = 4$$

Main Teaching Point 2: Use what you know about addition facts

If you know your addition facts, this strategy becomes fast and easy.

Here are some related facts:

• \(7 + 3 = 10\), so \(10 - 7 = 3\)
• \(6 + 5 = 11\), so \(11 - 6 = 5\)
• \(4 + 8 = 12\), so \(12 - 4 = 8\)

Each time, subtraction asks for the number that is missing in the addition sentence.

Main Teaching Point 3: Count up to find the missing number

If you do not know the addition fact right away, you can count up.

For example, to solve \(15 - 12\), ask:

"\(12\) plus what equals \(15\)?"

Count up from \(12\):

• \(13\) is 1 more
• \(14\) is 2 more
• \(15\) is 3 more

So:

$$12 + 3 = 15$$

That means:

$$15 - 12 = 3$$

Main Teaching Point 4: This strategy is helpful when numbers are close together

Think-addition is especially helpful when the two numbers are close.

For example, in \(16 - 13\), the numbers are close together. It is easy to think:

"\(13\) plus what equals \(16\)?"

The answer is \(3\).

So:

$$16 - 13 = 3$$

Worked Example 1

Solve:

$$10 - 6$$

Step 1: Rewrite it as a missing-addition problem.

$$6 + \Box = 10$$

Step 2: Find the missing number.

$$6 + 4 = 10$$

Step 3: Write the subtraction answer.

$$10 - 6 = 4$$

Worked Example 2

Solve:

$$14 - 8$$

Step 1: Ask the addition question.

"\(8\) plus what equals \(14\)?"

Step 2: Use an addition fact.

$$8 + 6 = 14$$

Step 3: Write the answer.

$$14 - 8 = 6$$

Worked Example 3

Solve:

$$17 - 15$$

Step 1: Turn it into addition.

$$15 + \Box = 17$$

Step 2: Count up from \(15\).

• \(16\) is 1 more
• \(17\) is 2 more

So the missing number is \(2\).

Step 3: Write the subtraction sentence.

$$17 - 15 = 2$$

Worked Example 4

Solve:

$$21 - 18$$

Step 1: Ask:

"\(18\) plus what equals \(21\)?"

Step 2: Count up.

• \(19\) is 1 more
• \(20\) is 2 more
• \(21\) is 3 more

So:

$$18 + 3 = 21$$

That means:

$$21 - 18 = 3$$

How to do think-addition to subtract

1. Look at the subtraction problem.
2. Start with the smaller number.
3. Ask, "This number plus what equals the bigger number?"
4. Use an addition fact or count up.
5. The missing number is the subtraction answer.

Try to notice the pattern

In each problem, the answer tells how far apart the numbers are.

For example:

• \(11 - 9 = 2\) because \(9 + 2 = 11\)
• \(13 - 10 = 3\) because \(10 + 3 = 13\)
• \(18 - 14 = 4\) because \(14 + 4 = 18\)

The answer is the amount you need to add to get from the smaller number to the larger number.

When should I use this strategy?

Think-addition is a great choice when:

• you know the related addition fact
• the numbers are close together
• you want to solve the problem in your head

Be careful

Make sure you start with the smaller number when you think about the addition question.

For \(14 - 8\), ask:

"\(8\) plus what equals \(14\)?"

Do not ask:

"\(14\) plus what equals \(8\)?"

We add from the smaller part up to the whole.

Quick practice ideas

You can practice by turning subtraction into addition:

• \(12 - 7\) becomes \(7 + \Box = 12\)
• \(16 - 9\) becomes \(9 + \Box = 16\)
• \(19 - 17\) becomes \(17 + \Box = 19\)

Summary

Think-addition to subtract means using addition to find a subtraction answer.

You change a subtraction problem into a missing-addition problem. Then you find the number that makes the addition sentence true.

For example, if:

$$8 + 6 = 14$$

then:

$$14 - 8 = 6$$

This strategy helps you use what you already know about addition to become stronger at subtraction.

Constant Difference Strategy is a smart way to solve some subtraction problems more easily.

When we use this strategy, we change both numbers by the same amount. The difference stays the same, so the answer does not change.

This is helpful when one of the numbers is close to a friendly number like 10, 20, 50, 100, or 200.

For example, if you see \(400 - 199\), subtracting 199 can feel tricky. But 199 is very close to 200. So we can add 1 to both numbers:

Now the problem is easier, and the answer stays the same.

Why does this work?

Think about two kids standing on a number line. If both kids take the same number of steps to the right, the space between them stays the same.

Subtraction finds the distance between two numbers. If both numbers move the same amount, the distance does not change.

Here is the big idea:

• Add the same number to both numbers, and the difference stays the same.
• Subtract the same number from both numbers, and the difference stays the same.

We can show that idea like this:

and also

You do not need to memorize the letters. Just remember: move both numbers the same way.

When should you use the constant difference strategy?

• When the second number is close to a friendly number.
• When changing both numbers makes the subtraction easier to do in your head.
• When you want to turn a hard problem into an easier, equal problem.

How to use it

1. Look at the subtraction problem.
2. Find a number that is close to a friendly number.
3. Change both numbers by the same amount.
4. Solve the new, easier problem.

Let’s try some examples.

Example 1: \(52 - 19\)

The number 19 is close to 20. Add 1 to both numbers.

Now subtract:

So,

Why was this easier? Because subtracting 20 is simpler than subtracting 19.

Example 2: \(84 - 27\)

The number 27 is close to 30. Add 3 to both numbers.

Now solve:

So,

Example 3: \(300 - 98\)

The number 98 is close to 100. Add 2 to both numbers.

Now solve:

So,

Example 4: \(725 - 399\)

The number 399 is close to 400. Add 1 to both numbers.

Now solve:

So,

You can also subtract the same amount from both numbers.

Sometimes that makes the problem easier too.

Look at \(63 - 31\). We can subtract 1 from both numbers:

Now it is easy to solve:

So,

Helpful tip: Try to make the second number a friendly number. That usually makes subtraction faster.

Watch out!

• You must change both numbers.
• You must change them by the same amount.
• If you only change one number, the answer will change.

For example, this is not correct:

If you change 19 to 20, you must also change 52 to 53:

Let’s think on a number line.

If the distance from 19 to 52 is 33, then the distance from 20 to 53 is also 33.

Both numbers moved 1 step to the right, so the distance stayed the same.

Try these in your head:

• \(61 - 29 = 62 - 30 = 32\)
• \(150 - 49 = 151 - 50 = 101\)
• \(500 - 198 = 502 - 200 = 302\)

Summary

The constant difference strategy helps you solve subtraction problems by changing both numbers by the same amount.

This keeps the difference the same, but it can make the subtraction much easier.

When a number is close to a friendly number, try shifting both numbers and solving the new problem.

Counting Up on an Open Number Line is a smart way to subtract. Instead of taking away from a big number, we can start at the smaller number and count up to the bigger number.

This helps us find the difference between two numbers. The difference is how much more one number is than another number.

An open number line is a number line without every number written on it. We only write the numbers we need. Then we draw jumps to count up in parts.

When we count up, we usually make friendly jumps. Friendly jumps are easy jumps like:

• \(1\)s

• \(10\)s

• \(100\)s

These jumps make subtraction easier and faster.

Here is the big idea:

To solve $$45 - 32$$ by counting up, start at \(32\) and jump up to \(45\). Then add the jumps together.

How to count up on an open number line

1. Write the smaller number first. This is where you start.

2. Think of easy jumps to get closer to the bigger number.

3. Use jumps of \(10\) or \(100\) when you can.

4. Make a small jump at the end if needed.

5. Add all the jumps. The total is the difference.

Why this works

Subtraction can mean finding the distance between two numbers. On a number line, that distance is the amount you jump from the smaller number to the larger number.

So when we solve \(54 - 38\), we are really asking, “How far is it from \(38\) to \(54\)?”

Worked Example 1

Solve $$45 - 32$$

Start at \(32\). Count up to \(45\).

• Jump \(8\) to get from \(32\) to \(40\)

• Jump \(5\) to get from \(40\) to \(45\)

On the open number line, it looks like this:

\(32 \rightarrow 40 \rightarrow 45\)

Jumps: \(8\) and \(5\)

Add the jumps:

So, $$45 - 32 = 13$$

Worked Example 2

Solve $$73 - 58$$

Start at \(58\). Count up to \(73\).

• Jump \(2\) to get from \(58\) to \(60\)

• Jump \(10\) to get from \(60\) to \(70\)

• Jump \(3\) to get from \(70\) to \(73\)

Open number line:

\(58 \rightarrow 60 \rightarrow 70 \rightarrow 73\)

Jumps: \(2\), \(10\), \(3\)

Add the jumps:

So, $$73 - 58 = 15$$

Notice how we used a friendly number, \(60\), and then another friendly number, \(70\). That made the subtraction easier.

Worked Example 3

Solve $$156 - 128$$

Start at \(128\). Count up to \(156\).

• Jump \(2\) to get from \(128\) to \(130\)

• Jump \(20\) to get from \(130\) to \(150\)

• Jump \(6\) to get from \(150\) to \(156\)

Open number line:

\(128 \rightarrow 130 \rightarrow 150 \rightarrow 156\)

Jumps: \(2\), \(20\), \(6\)

Add the jumps:

So, $$156 - 128 = 28$$

Worked Example 4

Solve $$402 - 187$$

Start at \(187\). Count up to \(402\).

• Jump \(3\) to get from \(187\) to \(190\)

• Jump \(10\) to get from \(190\) to \(200\)

• Jump \(200\) to get from \(200\) to \(400\)

• Jump \(2\) to get from \(400\) to \(402\)

Open number line:

\(187 \rightarrow 190 \rightarrow 200 \rightarrow 400 \rightarrow 402\)

Jumps: \(3\), \(10\), \(200\), \(2\)

Add the jumps:

So, $$402 - 187 = 215$$

In this example, we used friendly jumps of \(10\) and \(100\). That is very helpful when numbers are bigger.

Tips for choosing jumps

• Try to jump to the next ten first, like from \(58\) to \(60\).

• Then make bigger jumps by tens or hundreds.

• Make one last small jump to land exactly on the bigger number.

• You do not need to use the same jumps every time.

For example, to solve \(45 - 32\), one student might jump \(8\) and then \(5\). Another student might jump \(10\) and then jump back in their thinking, but using friendly jumps to tens is usually easiest on an open number line.

Check your thinking

After you add your jumps, ask yourself: “Does my answer make sense?”

For example, in \(73 - 58\), the numbers are close together. So the difference should not be very large. Our answer, \(15\), makes sense.

Common mistake to avoid

Do not start at the bigger number when using this method. For counting up, start at the smaller number and move up to the bigger number.

Also, make sure you add all the jumps. If you forget one jump, your answer will be too small.

Lets remember

• Counting up is a subtraction strategy.

• Start at the smaller number.

• Jump up to the bigger number.

• Use friendly numbers like tens and hundreds.

• Add the jumps to find the difference.

Summary

Counting up on an open number line helps us subtract by finding the distance between two numbers. We start at the smaller number, make easy jumps to friendly numbers, and stop at the bigger number.

Then we add the jumps. That total is the answer to the subtraction problem.

Partial Differences Strategy is a way to subtract by working with one place value at a time.

Instead of doing all the subtraction at once, we break the numbers into hundreds, tens, and ones. Then we subtract each part and put the answers together.

This strategy helps us see what the numbers really mean. It helps us understand subtraction, not just memorize steps.

Let’s learn how it works.

What does partial differences mean?

A difference is the answer to a subtraction problem.

A partial difference is a smaller subtraction answer from just one place value. We find the hundreds difference, the tens difference, and the ones difference. Then we combine them.

For example, in \(58 - 24\):

• \(50 - 20 = 30\)
• \(8 - 4 = 4\)

Now add the partial differences:

$$30 + 4 = 34$$

So:

$$58 - 24 = 34$$

Steps for partial differences

1. Line up the numbers by place value.
2. Subtract the hundreds, tens, and ones.
3. Write each partial difference.
4. Add the partial differences together.

Example 1: A two-digit problem

Find \(67 - 25\).

First, break apart the numbers by place value:

• \(67 = 60 + 7\)
• \(25 = 20 + 5\)

Now subtract each part:

• Tens: \(60 - 20 = 40\)
• Ones: \(7 - 5 = 2\)

Add the partial differences:

$$40 + 2 = 42$$

So:

$$67 - 25 = 42$$

Example 2: A three-digit problem

Find \(348 - 123\).

Break apart by place value:

• Hundreds: \(300 - 100 = 200\)
• Tens: \(40 - 20 = 20\)
• Ones: \(8 - 3 = 5\)

Add the partial differences:

$$200 + 20 + 5 = 225$$

So:

$$348 - 123 = 225$$

What if one place value is tricky?

Sometimes one part is harder to subtract. We can still think carefully about the place values.

Let’s look at a problem where the tens are tricky.

Example 3: Using a negative partial difference

Find \(352 - 178\).

Subtract by place value:

• Hundreds: \(300 - 100 = 200\)
• Tens: \(50 - 70 = -20\)
• Ones: \(2 - 8 = -6\)

Now combine the partial differences:

$$200 + (-20) + (-6)$$

$$200 - 20 - 6 = 174$$

So:

$$352 - 178 = 174$$

This may look different, but it still works. One place value can take away from another place value, and the total answer is correct.

If negative numbers feel new, you can think of it like this:

• Start with \(200\)
• Take away \(20\), and you get \(180\)
• Take away \(6\), and you get \(174\)

Example 4: Another three-digit problem

Find \(604 - 251\).

Subtract by place value:

• Hundreds: \(600 - 200 = 400\)
• Tens: \(0 - 50 = -50\)
• Ones: \(4 - 1 = 3\)

Add the partial differences:

$$400 + (-50) + 3$$

$$400 - 50 + 3 = 353$$

So:

$$604 - 251 = 353$$

Why this strategy is helpful

• It helps you see the value of each digit.
• It helps you understand subtraction better.
• It is a good step toward faster subtraction later.
• It can make big numbers feel easier.

Tips to remember

• Always line up the ones, tens, and hundreds.
• Subtract each place value carefully.
• Add all the partial differences at the end.
• Check if your answer makes sense.

Quick check

Try these on your own:

• \(79 - 36\)
• \(456 - 214\)
• \(521 - 187\)

Summary

The partial differences strategy means subtracting by place value. We subtract the hundreds, tens, and ones separately, then combine the parts.

This strategy helps us understand what is happening in subtraction. When we work carefully with place value, we can solve both two-digit and three-digit subtraction problems.

Standard Subtraction Algorithm without Decomposing

Today we will learn how to subtract using the standard subtraction algorithm when we do not need to decompose.

This means we line up the numbers by place value, then subtract each column. We can do this when the digit on top is greater than or equal to the digit on the bottom in every column.

For example, in \(75 - 23\), the tens and ones are lined up like this:

The ones are lined up under the ones, and the tens are lined up under the tens. This helps us subtract correctly.

What does “without decomposing” mean?

Sometimes in subtraction, the top digit is smaller than the bottom digit in a column. Then we would need to regroup or decompose. But in this lesson, we are only working with problems where that does not happen.

That means:

• In the ones column, the top digit is the same as or bigger than the bottom digit.
• In the tens column, the top digit is the same as or bigger than the bottom digit.
• If there are hundreds, the same rule is true there too.

Steps for the standard subtraction algorithm

1. Write the larger number on top.
2. Write the smaller number under it.
3. Line up the digits by place value: ones under ones, tens under tens, hundreds under hundreds.
4. Start at the ones place.
5. Subtract each column.
6. Write each answer in the correct place.

Important idea: We always subtract straight down by place value.

We usually start with the ones, then move to the tens, then the hundreds.

Worked Example 1: Two-digit subtraction

Solve \(64 - 42\).

First, line up the digits by place value.

Now subtract the ones:

\(4 - 2 = 2\)

Next subtract the tens:

\(6 - 4 = 2\)

So the answer is:

So, \(64 - 42 = 22\).

Worked Example 2: A zero in the ones place

Solve \(90 - 50\).

Line up the digits:

Subtract the ones:

\(0 - 0 = 0\)

Subtract the tens:

\(9 - 5 = 4\)

Write the answer:

So, \(90 - 50 = 40\).

Worked Example 3: Three-digit subtraction

Solve \(876 - 253\).

Line up the hundreds, tens, and ones.

Start with the ones:

\(6 - 3 = 3\)

Next the tens:

\(7 - 5 = 2\)

Then the hundreds:

\(8 - 2 = 6\)

Write the answer:

So, \(876 - 253 = 623\).

Worked Example 4: When digits are equal in a column

Solve \(558 - 228\).

Line up the digits carefully:

Subtract the ones:

\(8 - 8 = 0\)

Subtract the tens:

\(5 - 2 = 3\)

Subtract the hundreds:

\(5 - 2 = 3\)

Write the answer:

So, \(558 - 228 = 330\).

How to know if you can use this method without decomposing

Look at each column before you start.

• If the top digit is bigger, you can subtract.
• If the top digit is the same, you can subtract.
• If the top digit is smaller, then that problem needs decomposing, so it is not the kind we are doing in this lesson.

For example:

• \(73 - 21\): yes, because \(3 \ge 1\) and \(7 \ge 2\)
• \(84 - 32\): yes, because \(4 \ge 2\) and \(8 \ge 3\)
• \(52 - 38\): no for this lesson, because in the ones place, \(2 < 8\)

Common mistakes to watch for

• Not lining up place values. Always put ones under ones and tens under tens.
• Subtracting the wrong way. In the standard algorithm, subtract the bottom digit from the top digit in each column.
• Forgetting a zero. If the answer in a column is \(0\), write it if it belongs in the answer, like in \(330\).

Helpful check

You can check your subtraction with addition. If your answer is correct, then:

difference + subtracted number = starting number

For Example 1, we found \(64 - 42 = 22\).

Check: \(22 + 42 = 64\)

That means the subtraction is correct.

Let’s remember

• Write the numbers vertically.
• Line up place values.
• Start with the ones.
• Subtract each column straight down.
• This works when the top digit is greater than or equal to the bottom digit in every column.

Summary

The standard subtraction algorithm without decomposing is a way to subtract numbers by lining them up by place value and subtracting each column. You can use this method when the top digit in every column is the same as or greater than the bottom digit. Start with the ones place, move left, and write each answer digit in the correct place.

Standard Subtraction Algorithm with Decomposing

Sometimes in subtraction, the top digit is too small to subtract the bottom digit. When that happens, we use decomposing.

Decomposing means breaking apart a larger place value into smaller parts. For example, 1 ten can be broken into 10 ones. Also, 1 hundred can be broken into 10 tens.

This is what some people call “borrowing,” but in math we can think of it as unbundling a ten or a hundred.

We use the standard subtraction algorithm to subtract numbers by lining up place values:

• ones under ones
• tens under tens
• hundreds under hundreds

Then we subtract from right to left: ones, tens, then hundreds.

Why do we decompose?

If there are not enough ones to subtract, we decompose 1 ten into 10 ones. If there are not enough tens to subtract, we decompose 1 hundred into 10 tens.

Here is the big idea:

• \(1\) ten = \(10\) ones
• \(1\) hundred = \(10\) tens

So the total amount does not change. We are only changing how the number is grouped.

For example:

\(52\) means 5 tens and 2 ones.

We can decompose it into 4 tens and 12 ones. It is still \(52\).

Steps for the standard subtraction algorithm with decomposing

1. Write the numbers in a column. Line up ones, tens, and hundreds.
2. Start with the ones place.
3. If the top digit is smaller than the bottom digit, decompose from the place to the left.
4. Subtract the ones.
5. Move to the tens place and do the same thing.
6. If needed, decompose a hundred into 10 tens.
7. Keep going until all place values are subtracted.

Worked Example 1: Two-digit subtraction with no decomposing

Find \(47 - 25\).

Start with the ones:

\(7 - 5 = 2\)

Now subtract the tens:

\(4 - 2 = 2\)

So, \(47 - 25 = 22\).

Worked Example 2: Two-digit subtraction with decomposing a ten

Find \(52 - 38\).

Start with the ones place.

We cannot do \(2 - 8\) because 2 ones is less than 8 ones.

So we decompose 1 ten from the 5 tens.

The 5 tens become 4 tens, and the 2 ones become 12 ones.

Now subtract the ones:

\(12 - 8 = 4\)

Now subtract the tens:

\(4 - 3 = 1\)

So, \(52 - 38 = 14\).

What happened?

We did not change 52 into a different number. We only regrouped it.

\(52 = 4\) tens and \(12\) ones, so the subtraction still works correctly.

Worked Example 3: Three-digit subtraction with decomposing a hundred

Find \(432 - 158\).

Start with the ones place.

We cannot do \(2 - 8\), so decompose 1 ten.

The 3 tens become 2 tens, and the 2 ones become 12 ones.

Now subtract the ones:

\(12 - 8 = 4\)

Next, subtract the tens.

We cannot do \(2 - 5\), so decompose 1 hundred.

The 4 hundreds become 3 hundreds, and the 2 tens become 12 tens.

Now subtract the tens:

\(12 - 5 = 7\)

Now subtract the hundreds:

\(3 - 1 = 2\)

So, \(432 - 158 = 274\).

Worked Example 4: Decomposing across a zero

Find \(403 - 186\).

Start with the ones place.

We cannot do \(3 - 6\), and there are 0 tens to decompose from.

So first decompose 1 hundred into 10 tens.

The 4 hundreds become 3 hundreds, and the 0 tens become 10 tens.

Now decompose 1 of those tens into 10 ones.

The 10 tens become 9 tens, and the 3 ones become 13 ones.

Now subtract the ones:

\(13 - 6 = 7\)

Subtract the tens:

\(9 - 8 = 1\)

Subtract the hundreds:

\(3 - 1 = 2\)

So, \(403 - 186 = 217\).

Helpful reminders

• Always line up place values correctly.
• Start subtracting from the right.
• If the top digit is smaller, decompose from the place to the left.
• Remember: 1 ten = 10 ones.
• Remember: 1 hundred = 10 tens.
• Decomposing changes the grouping, not the value.

Common mistakes to watch for

• Not lining up digits: ones must be under ones, tens under tens, hundreds under hundreds.
• Forgetting to reduce the digit on the left: if you take 1 ten, the tens digit becomes 1 less.
• Subtracting the smaller digit from the larger digit no matter what: always subtract top minus bottom after you decompose.
• Forgetting zeros matter: if there is a 0, you may need to decompose from the next place value over.

Let’s say it another way

In \(52 - 38\), you need to subtract 8 ones, but you only have 2 ones.

So you unbundle 1 ten. Now you have 12 ones. Then you can subtract 8 ones.

This is why decomposing helps the subtraction make sense.

Summary

The standard subtraction algorithm helps us subtract larger numbers in order by place value.

When the top digit is too small, we decompose by unbundling a ten into 10 ones or a hundred into 10 tens.

Then we subtract from right to left carefully. If you line up place values and decompose correctly, you can solve many subtraction problems.

Standard Algorithm for Three-Digit Subtraction

Today we will learn how to subtract 3-digit numbers using the standard algorithm. This means we line the numbers up by place value and subtract one place at a time.

When we subtract, we always start on the right. We subtract the ones first, then the tens, then the hundreds.

It is very important to line up the digits correctly:

• ones under ones
• tens under tens
• hundreds under hundreds

For example, in the number \(472\):

• \(4\) means 4 hundreds
• \(7\) means 7 tens
• \(2\) means 2 ones

When the top digit is smaller than the bottom digit, we need to regroup. Regrouping means taking 1 ten and turning it into 10 ones, or taking 1 hundred and turning it into 10 tens.

Let’s learn the steps.

1. Write the numbers vertically.
2. Start with the ones.
3. If the top digit is too small, regroup from the place to the left.
4. Subtract the tens.
5. Subtract the hundreds.
6. Check that your answer makes sense.

Example 1: No regrouping

Let’s solve \(584 - 231\).

Start with the ones: \(4 - 1 = 3\).

Now the tens: \(8 - 3 = 5\).

Now the hundreds: \(5 - 2 = 3\).

So, \(584 - 231 = 353\).

Example 2: Regroup in the ones place

Let’s solve \(452 - 178\).

Start with the ones: \(2 - 8\) does not work, because 2 is smaller than 8.

We regroup 1 ten from the 5 tens. The 5 tens become 4 tens. The 2 ones become 12 ones.

Now subtract the ones: \(12 - 8 = 4\).

Subtract the tens: \(4 - 7\) does not work, so we regroup again.

Take 1 hundred from the 4 hundreds. Now there are 3 hundreds, and the 4 tens become 14 tens.

Now subtract the tens: \(14 - 7 = 7\).

Subtract the hundreds: \(3 - 1 = 2\).

So, \(452 - 178 = 274\).

In this problem, we regrouped more than once. That is okay. We always regroup from the place just to the left.

Example 3: Regroup from hundreds to tens

Let’s solve \(603 - 245\).

Start with the ones: \(3 - 5\) does not work.

Look at the tens. There are 0 tens, so we cannot take 1 ten from there.

So first, regroup 1 hundred into 10 tens. The 6 hundreds become 5 hundreds, and the 0 tens become 10 tens.

Then regroup 1 ten into 10 ones. The 10 tens become 9 tens, and the 3 ones become 13 ones.

Now subtract the ones: \(13 - 5 = 8\).

Subtract the tens: \(9 - 4 = 5\).

Subtract the hundreds: \(5 - 2 = 3\).

So, \(603 - 245 = 358\).

This is an important kind of problem. If there are 0 tens, you may need to regroup from the hundreds first.

Example 4: Another subtraction with regrouping

Let’s solve \(700 - 463\).

Start with the ones: \(0 - 3\) does not work.

There are 0 tens too, so first regroup 1 hundred into 10 tens. The 7 hundreds become 6 hundreds, and the 0 tens become 10 tens.

Now regroup 1 ten into 10 ones. The 10 tens become 9 tens, and the 0 ones become 10 ones.

Subtract the ones: \(10 - 3 = 7\).

Subtract the tens: \(9 - 6 = 3\).

Subtract the hundreds: \(6 - 4 = 2\).

So, \(700 - 463 = 237\).

Helpful Tips

• Always line up hundreds, tens, and ones.
• Always start subtracting on the right.
• If the top digit is smaller, regroup from the place to the left.
• If there is a 0, you may need to regroup more than once.
• Work carefully, one place at a time.

How to Check Your Work

You can check subtraction with addition. Add your answer and the number you subtracted.

For example, in \(584 - 231 = 353\), check by adding:

If the sum matches the top number, your subtraction is correct.

Let’s Remember

The standard algorithm for 3-digit subtraction helps us subtract large numbers in an organized way. We line up place values, subtract from right to left, and regroup when needed.

With practice, you will get faster and more confident. Take your time, follow the steps, and remember that regrouping means trading 1 hundred for 10 tens or 1 ten for 10 ones.

Decomposing Across Zero is a subtraction skill that helps when you need to subtract from a number like 300, 500, or 402.

Sometimes, the digit you want to subtract from is too small. Then you decompose, which means you break apart a larger place value to make more ones or tens.

The tricky part is when the next digit is 0. A zero means there are no tens or no ones to borrow from right away. So we must move left until we find a place that has something to give.

Let’s remember place value:

• Ones are the last digit.
• Tens are the middle digit.
• Hundreds are the first digit in a 3-digit number.

For example, in 500:

• 5 hundreds
• 0 tens
• 0 ones

If you want to subtract ones from 500, you cannot take them from 0 ones. You also cannot borrow from 0 tens. So you go to the 5 hundreds first.

When you decompose across zero:

1. Start at the ones place.
2. If the top digit is too small, look to the left.
3. If that digit is 0, keep moving left until you find a digit greater than 0.
4. Take 1 from that place value.
5. Turn it into 10 of the next smaller place.
6. Keep going until you can subtract.

This may sound big, but let’s do it step by step.

Example 1: Subtracting from 300

Solve $$300 - 4$$

Look at the ones place. You need to do \(0 - 4\), but you cannot do that.

Look at the tens place. It is 0 tens, so there is nothing to borrow there.

Look at the hundreds place. There are 3 hundreds, so we can decompose 1 hundred.

Take 1 hundred from 3 hundreds. Now there are 2 hundreds left. That 1 hundred becomes 10 tens.

But we still need ones, and the ones place is still 0. So take 1 ten from the 10 tens. Now there are 9 tens, and that 1 ten becomes 10 ones.

Now the number is really:

$$300 = 2\text{ hundreds } 9\text{ tens } 10\text{ ones}$$

Now subtract:

• Ones: \(10 - 4 = 6\)
• Tens: \(9 - 0 = 9\)
• Hundreds: \(2 - 0 = 2\)

So, $$300 - 4 = 296$$

Example 2: A number with a zero in the tens place

Solve $$402 - 7$$

Look at the ones place. You need to do \(2 - 7\), but 2 is too small.

Look at the tens place. It is 0, so you cannot borrow from there.

Look at the hundreds place. There are 4 hundreds, so decompose 1 hundred.

4 hundreds becomes 3 hundreds, and 1 hundred becomes 10 tens.

Now use 1 of those tens to help the ones place. 10 tens becomes 9 tens, and 1 ten becomes 10 ones.

Add those 10 ones to the 2 ones already there. Now there are 12 ones.

The number is now:

$$402 = 3\text{ hundreds } 9\text{ tens } 12\text{ ones}$$

Now subtract:

• Ones: \(12 - 7 = 5\)
• Tens: \(9 - 0 = 9\)
• Hundreds: \(3 - 0 = 3\)

So, $$402 - 7 = 395$$

Example 3: Subtracting a 2-digit number across zero

Solve $$500 - 26$$

Start with the ones place: \(0 - 6\) does not work.

Look at the tens place: it is 0, so keep moving left.

Look at the hundreds place: there are 5 hundreds. Decompose 1 hundred.

5 hundreds becomes 4 hundreds, and 1 hundred becomes 10 tens.

Now we still need ones. Take 1 ten from the 10 tens. Then there are 9 tens left, and 1 ten becomes 10 ones.

Now the number is:

$$500 = 4\text{ hundreds } 9\text{ tens } 10\text{ ones}$$

Subtract by place value:

• Ones: \(10 - 6 = 4\)
• Tens: \(9 - 2 = 7\)
• Hundreds: \(4 - 0 = 4\)

So, $$500 - 26 = 474$$

Example 4: Watch the zeros carefully

Solve $$600 - 58$$

Ones: \(0 - 8\) does not work.

Tens: 0, so there is nothing to borrow.

Hundreds: 6, so we decompose 1 hundred.

6 hundreds becomes 5 hundreds, and 1 hundred becomes 10 tens.

Take 1 ten from the 10 tens to make ones. Then there are 9 tens, and 10 ones.

Now subtract:

• Ones: \(10 - 8 = 2\)
• Tens: \(9 - 5 = 4\)
• Hundreds: \(5 - 0 = 5\)

So, $$600 - 58 = 542$$

A helpful way to think about it

When you see a zero, think: “I need to go next door to the left.”

If that digit is also zero, think: “Keep going left until I find a digit that can help.”

Then trade:

• 1 hundred for 10 tens
• 1 ten for 10 ones

Common mistakes to avoid

• Do not borrow from 0. Zero has nothing to give.
• Do not forget to change every place value. If you take 1 hundred, the hundreds digit gets smaller by 1.
• Do not stop too soon. If you need ones, you may have to decompose from hundreds to tens, and then tens to ones.

Quick check questions

• For \(200 - 3\), do you borrow from the tens place first? No, because the tens digit is 0.
• In \(500 - 9\), where do you go to find a digit that can help? The hundreds place.
• If 1 hundred is decomposed, how many tens does it become? 10 tens.

Summary

Decomposing across zero means you move left past any zeros until you find a place value that has something to give.

Then you trade step by step until you have enough tens or ones to subtract. With careful place-value thinking, problems like \(300 - 4\) and \(500 - 26\) become much easier.

Inverse Relationship of Addition and Subtraction

Today we will learn how addition and subtraction are opposites. This is called an inverse relationship.

When we add, we put numbers together. When we subtract, we take away or find what is missing. Because these two operations are opposites, we can use addition to check subtraction, and we can use subtraction to check addition.

This is very helpful when you want to make sure your answer is correct.

What does inverse mean?

Inverse means opposite. Addition and subtraction work together like a team.

Look at these two number sentences:

These two equations are connected. They use the same three numbers: 7, 5, and 12.

If you know one fact, you can use it to find another fact. This is called a fact family.

Fact families

A fact family is a group of math facts that use the same three numbers.

For example, the numbers 4, 6, and 10 make this fact family:

• \(4 + 6 = 10\)
• \(6 + 4 = 10\)
• \(10 - 4 = 6\)
• \(10 - 6 = 4\)

The addition facts and subtraction facts belong together because addition and subtraction are inverses.

How to check subtraction with addition

When you subtract, you start with a whole amount. Then you take some away. The answer tells how many are left.

To check your subtraction answer, add the answer and the number you took away. If the sum matches the starting number, your subtraction is correct.

Here is the pattern:

Check it with addition:

Example pattern:

Check:

Since the addition gives us 15, the subtraction is correct.

Think of subtraction as a missing-addend problem

Sometimes subtraction asks, “What number do I add to get the total?”

For example:

You can think:

“\(9 + ? = 13\)”

The missing number is 4, so:

This shows again that subtraction and addition are opposites.

Worked Example 1

Solve:

Take 3 away from 9:

Now check with addition:

The check is correct, so the subtraction answer is correct.

Worked Example 2

Solve:

Think: “What number plus 8 equals 14?”

So:

Check:

The subtraction is correct.

Worked Example 3

Solve:

Subtract to find the difference:

Now check with inverse addition:

Since the sum is 25, the subtraction answer is correct.

Worked Example 4

A student solved this problem:

Is the answer correct? Let’s check with addition.

But the starting number was 31, not 29.

So \(31 - 12 = 17\) is not correct.

Let’s find the correct answer:

Check:

Now it is correct.

Steps to use inverse addition to check subtraction

1. Solve the subtraction problem.
2. Take your answer.
3. Add it to the number that was subtracted.
4. See if the sum matches the starting number.

Example:

1. \(18 - 7 = 11\)
2. Take the answer: 11
3. Add it to 7: \(11 + 7 = 18\)
4. The sum matches 18, so the subtraction is correct.

Why this matters

Using inverse operations helps you:

• check your work
• find mistakes
• understand how numbers are connected
• learn fact families

It also helps with bigger subtraction problems, because you do not have to guess if your answer is right. You can prove it by using addition.

Quick practice to think about

• \(16 - 9 = 7\) because \(7 + 9 = 16\)
• \(20 - 5 = 15\) because \(15 + 5 = 20\)
• \(42 - 18 = 24\) because \(24 + 18 = 42\)

Summary

Addition and subtraction are inverse operations, which means they are opposites. If you solve a subtraction problem, you can check it by adding the answer to the number you took away.

When the addition equation gives the starting number, your subtraction is correct. This is how fact families help us understand and prove our answers.

Take-From and Compare Word Problems are two common kinds of subtraction stories. In this lesson, you will learn how to tell them apart and how to solve them step by step.

Subtraction does not only mean taking away. Sometimes subtraction means finding how many are left, and sometimes it means finding the difference between two amounts.

That is why some word problems sound different, even though they both use subtraction.

There are 2 main kinds to learn:

• Take-From problems: something is taken away from a group.
• Compare problems: two groups are compared to find how many more or how many fewer.

Let’s learn each kind.

1. Take-From Problems

In a take-from problem, you start with some amount. Then some are taken away. You need to find how many are left.

These word clues often mean a take-from problem:

• left
• remain
• gave away
• lost
• spent
• ate
• flew away
• took away

The math model for a take-from problem is:

$$\text{start} - \text{taken away} = \text{left}$$

Example 1: Take-From

Lena had 14 stickers. She gave away 5 stickers. How many stickers does she have left?

Step 1: Find the starting amount. Lena had 14 stickers.

Step 2: Find how many were taken away. She gave away 5 stickers.

Step 3: Subtract.

$$14 - 5 = 9$$

Answer: Lena has 9 stickers left.

2. Compare Problems

In a compare problem, nothing has to be taken away from a group. Instead, you look at two amounts and find the difference between them.

These word clues often mean a compare problem:

• how many more
• how many fewer
• how many less
• what is the difference
• compare

The math model for a compare problem is:

$$\text{larger amount} - \text{smaller amount} = \text{difference}$$

Important: In compare problems, the answer tells how much bigger one amount is than the other.

Example 2: Compare with “how many more”

Ben has 12 marbles. Ava has 7 marbles. How many more marbles does Ben have than Ava?

Step 1: Find the two amounts: 12 and 7.

Step 2: Find the larger amount: 12.

Step 3: Subtract the smaller amount from the larger amount.

$$12 - 7 = 5$$

Answer: Ben has 5 more marbles than Ava.

Example 3: Compare with “how many fewer”

A class read 18 books in September and 11 books in October. How many fewer books did the class read in October than in September?

Step 1: Find the two amounts: 18 and 11.

Step 2: Find the larger amount: 18.

Step 3: Subtract.

$$18 - 11 = 7$$

Answer: The class read 7 fewer books in October than in September.

Notice something important: even when the question says how many fewer, you still subtract the smaller number from the larger number to find the difference.

How to Tell the Difference

Ask yourself these questions when you read a word problem:

1. Did the story start with one group and then some were taken away?
   If yes, it is probably a take-from problem.
2. Does the story tell about two groups and ask how many more or fewer?
   If yes, it is probably a compare problem.

Look at these examples:

• Take-From: Mia had 20 crayons. She lost 3 crayons. How many crayons are left?
• Compare: Mia has 20 crayons. Noah has 3 crayons. How many more crayons does Mia have than Noah?

Both problems use subtraction, but they tell different kinds of stories.

A Helpful Strategy: Circle the Important Words

When you solve a word problem, look for clue words and the numbers. You can ask:

• What amount do I start with?
• Is something taken away?
• Am I comparing two groups?
• What is the question asking me to find?

Example 4: A harder problem

There were 35 birds in a tree. Then 17 birds flew away. Later, Maya saw 9 birds in another tree. How many more birds were in the first tree at the start than in the other tree?

This problem has extra information, so let’s read carefully.

The question asks: How many more birds were in the first tree at the start than in the other tree?

So we compare:

• the first tree at the start: 35 birds
• the other tree: 9 birds

We do not use 17, because the question is about the first tree at the start.

Now subtract:

$$35 - 9 = 26$$

Answer: There were 26 more birds in the first tree at the start.

Watch Out for These Mistakes

• Mistake 1: Using addition when the question asks how many more or how many fewer.
  Remember: these questions usually ask for the difference, so use subtraction.
• Mistake 2: Subtracting in the wrong order.
  To find a difference, subtract the smaller number from the larger number.
• Mistake 3: Using numbers that do not answer the question.
  Sometimes a problem has extra information. Read the question again before solving.

Try These Steps Every Time

1. Read the whole problem slowly.
2. Find the numbers.
3. Look for clue words like left, gave away, how many more, or how many fewer.
4. Decide: take-from or compare?
5. Write a subtraction equation.
6. Solve.
7. Write the answer with words.

Let’s Compare the Two Types Side by Side

• Take-From: Start with one amount, subtract what is taken away, find what is left.
  Example: $$16 - 4 = 12$$
• Compare: Look at two amounts, subtract to find the difference.
  Example: $$16 - 4 = 12$$

Even if the equation looks the same, the story meaning is different.

Quick Practice Thinking

• “Sofia had 10 apples. She ate 2.”
  This is take-from because apples were taken away.
• “Sofia has 10 apples. Jay has 2 apples.”
  This is compare because we are comparing two groups.

Summary

Take-from problems happen when something is removed from a group. Compare problems happen when you look at two groups and find how many more or how many fewer.

When you see words like left or gave away, think take-from. When you see words like how many more or how many fewer, think compare.

Both kinds of problems often use subtraction. The most important job is to understand what the story is asking.