Addition and Subtraction Strategies Within 1,000

Adding and Subtracting Multiples of 100

In this lesson, we will learn how to add and subtract multiples of 100.

A multiple of 100 is a number like 100, 200, 300, 400, and so on. These numbers have 0 tens and 0 ones.

When we add or subtract multiples of 100, we can focus on the hundreds. This makes big numbers easier to work with.

Think of it like this: if you know that \(2 + 3 = 5\), then you can also know that \(200 + 300 = 500\).

The hundreds digit works a lot like a basic fact. We add or subtract the hundreds, and the tens and ones stay 0.

What does a multiple of 100 look like?

• \(100 = 1\) hundred
• \(200 = 2\) hundreds
• \(500 = 5\) hundreds
• \(900 = 9\) hundreds

So when you see a number like \(700\), you can think, “That is 7 hundreds.”

Adding multiples of 100

To add multiples of 100, count the hundreds.

For example, if you add \(300 + 200\), you are adding 3 hundreds + 2 hundreds.

That equals 5 hundreds, which is \(500\).

We can write it like this:

Subtracting multiples of 100

To subtract multiples of 100, take away the hundreds.

For example, if you subtract \(800 - 300\), you are finding 8 hundreds - 3 hundreds.

That equals 5 hundreds, which is \(500\).

We can write it like this:

A helpful strategy

You can cover up the two zeros and work with the first digit.

Then put the two zeros back at the end.

• \(400 + 300\) becomes \(4 + 3 = 7\), so \(700\)
• \(900 - 200\) becomes \(9 - 2 = 7\), so \(700\)

This works because the numbers are all made of whole hundreds.

Worked Example 1

Solve: \(100 + 400\)

Step 1: Think about the hundreds.

\(100\) is 1 hundred.

\(400\) is 4 hundreds.

Step 2: Add the hundreds.

\(1 + 4 = 5\)

Step 3: Write the answer in hundreds.

Worked Example 2

Solve: \(600 + 300\)

Step 1: Think about the hundreds.

\(600\) is 6 hundreds.

\(300\) is 3 hundreds.

Step 2: Add the hundreds.

\(6 + 3 = 9\)

Step 3: Write the answer in hundreds.

Worked Example 3

Solve: \(700 - 200\)

Step 1: Think about the hundreds.

\(700\) is 7 hundreds.

\(200\) is 2 hundreds.

Step 2: Subtract the hundreds.

\(7 - 2 = 5\)

Step 3: Write the answer in hundreds.

Worked Example 4

Solve: \(900 - 500\)

Step 1: Think about the hundreds.

\(900\) is 9 hundreds.

\(500\) is 5 hundreds.

Step 2: Subtract the hundreds.

\(9 - 5 = 4\)

Step 3: Write the answer in hundreds.

How to check your thinking

• Ask: Are both numbers multiples of 100?
• Look for two zeros at the end.
• Add or subtract the hundreds digits.
• Write the two zeros back in the answer.

Be careful!

• \(200 + 300\) is not \(50\). It is \(500\).
• \(800 - 100\) is 7 hundreds, so it is \(700\).
• Always make sure your answer is still a multiple of 100.

Try thinking with hundreds

These number sentences can be read in an easy way:

• \(500 + 200\) means 5 hundreds + 2 hundreds
• \(600 - 400\) means 6 hundreds - 4 hundreds

When you think this way, big numbers feel smaller and easier.

Summary

Multiples of 100 are numbers like \(100\), \(200\), and \(900\).

To add or subtract them, focus on the hundreds. Add or subtract the first digit, then keep the two zeros.

If you know small facts like \(3 + 4 = 7\) or \(8 - 2 = 6\), you can use them to solve problems like \(300 + 400 = 700\) and \(800 - 200 = 600\).

Base-Ten Models for Three-Digit Operations

When we work with numbers up to 1,000, it helps to see the number. Base-ten models let us build numbers with blocks.

We use:

• Hundred flats to show hundreds
• Ten rods to show tens
• Unit blocks to show ones

These models help us add and subtract three-digit numbers. They also help us understand regrouping.

What each block means

• 1 unit block = \(1\)
• 1 ten rod = \(10\)
• 1 hundred flat = \(100\)

And these blocks can trade with each other:

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

This is called regrouping. We regroup when we need to make adding or subtracting easier.

Step 1: Build a number

To build the number \(243\), use:

• 2 hundred flats
• 4 ten rods
• 3 unit blocks

We can write it like this:

$$243 = 200 + 40 + 3$$

This shows that every three-digit number has hundreds, tens, and ones.

Using base-ten models to add

When we add, we put more blocks together. Then we check if we have enough ones to trade for a ten, or enough tens to trade for a hundred.

Worked Example 1: Add without regrouping

Find \(123 + 145\).

First, build each number.

• \(123\) = 1 hundred, 2 tens, 3 ones
• \(145\) = 1 hundred, 4 tens, 5 ones

Now put the blocks together.

• Hundreds: \(1 + 1 = 2\) hundreds
• Tens: \(2 + 4 = 6\) tens
• Ones: \(3 + 5 = 8\) ones

No group has 10 or more, so we do not need to trade.

So the sum is:

$$123 + 145 = 268$$

Worked Example 2: Add with regrouping

Find \(278 + 145\).

Build the numbers.

• \(278\) = 2 hundreds, 7 tens, 8 ones
• \(145\) = 1 hundred, 4 tens, 5 ones

Put the blocks together.

• Hundreds: \(2 + 1 = 3\) hundreds
• Tens: \(7 + 4 = 11\) tens
• Ones: \(8 + 5 = 13\) ones

Now regroup.

• \(13\) ones means 1 ten and 3 ones
• Add that 1 ten to the 11 tens: now there are 12 tens
• \(12\) tens means 1 hundred and 2 tens
• Add that 1 hundred to the 3 hundreds: now there are 4 hundreds

Now we have:

• 4 hundreds
• 2 tens
• 3 ones

So the sum is:

$$278 + 145 = 423$$

Using base-ten models to subtract

When we subtract, we take blocks away. Sometimes there are not enough ones or tens. Then we regroup by trading a larger block for smaller blocks.

Worked Example 3: Subtract with one regroup

Find \(352 - 126\).

Build \(352\):

• 3 hundreds
• 5 tens
• 2 ones

We need to take away \(126\):

• 1 hundred
• 2 tens
• 6 ones

But there are only 2 ones, and we need to take away 6 ones. So we regroup.

Trade 1 ten for 10 ones.

• 5 tens becomes 4 tens
• 2 ones becomes 12 ones

Now subtract:

• Ones: \(12 - 6 = 6\)
• Tens: \(4 - 2 = 2\)
• Hundreds: \(3 - 1 = 2\)

So the difference is:

$$352 - 126 = 226$$

Worked Example 4: Subtract with more regrouping

Find \(401 - 178\).

Build \(401\):

• 4 hundreds
• 0 tens
• 1 one

We need to take away \(178\):

• 1 hundred
• 7 tens
• 8 ones

First look at the ones. We have 1 one, but we need to take away 8 ones. We cannot do that yet.

There are 0 tens, so we cannot trade a ten for ones. We need to trade 1 hundred for 10 tens.

• 4 hundreds becomes 3 hundreds
• 0 tens becomes 10 tens

Now trade 1 of those tens for 10 ones.

• 10 tens becomes 9 tens
• 1 one becomes 11 ones

Now subtract:

• Ones: \(11 - 8 = 3\)
• Tens: \(9 - 7 = 2\)
• Hundreds: \(3 - 1 = 2\)

So the difference is:

$$401 - 178 = 223$$

How base-ten models help

• They help you see what each digit means.
• They help you understand why we regroup.
• They help you add and subtract carefully.
• They make big numbers easier to work with.

Tips to remember

1. Build the number with hundreds, tens, and ones.
2. For addition, put blocks together.
3. If you have 10 or more ones, trade for a ten.
4. If you have 10 or more tens, trade for a hundred.
5. For subtraction, take blocks away.
6. If you do not have enough ones, trade 1 ten for 10 ones.
7. If you do not have enough tens, trade 1 hundred for 10 tens.

Quick check questions

• How would you build \(364\)?
• What can 10 ones trade for?
• What can 1 hundred trade for?
• In \(256 + 178\), do you need to regroup the ones? the tens?
• In \(320 - 145\), do you need to regroup? Why?

Summary

Base-ten models use hundred flats, ten rods, and unit blocks to show numbers. We can add by joining blocks and regrouping when we make a group of 10. We can subtract by taking blocks away and regrouping when we need more tens or ones. These models help us understand three-digit addition and subtraction in a clear, visual way.

Partial sums is a smart way to add big numbers by breaking them into parts.

When we use partial sums with three-digit numbers, we split each number into hundreds, tens, and ones. Then we add the hundreds, add the tens, add the ones, and finally put all the sums together.

This helps us stay organized and see the value of each digit clearly.

Let’s remember place value first:

• In the number \(347\), the \(3\) means 3 hundreds.
• The \(4\) means 4 tens.
• The \(7\) means 7 ones.

So we can break apart \(347\) like this:

We can do the same with any three-digit number. For example:

Now let’s learn the steps for partial sums.

1. Break apart each number into hundreds, tens, and ones.
2. Add the hundreds.
3. Add the tens.
4. Add the ones.
5. Add those partial sums together to get the total.

This is called partial sums because we first find smaller sums, or parts of the total.

Here is our first example.

Example 1: Add \(123 + 245\)

First, break apart each number:

Now add like parts:

• Hundreds: \(100 + 200 = 300\)
• Tens: \(20 + 40 = 60\)
• Ones: \(3 + 5 = 8\)

Now add the partial sums:

So,

That was simple because none of the parts made a new ten or a new hundred.

Example 2: Add \(356 + 423\)

Break apart the numbers:

Add the same place values:

• Hundreds: \(300 + 400 = 700\)
• Tens: \(50 + 20 = 70\)
• Ones: \(6 + 3 = 9\)

Add the partial sums:

So,

Now let’s try one where a part is bigger than 10.

Example 3: Add \(278 + 145\)

Break apart the numbers:

Add the parts:

• Hundreds: \(200 + 100 = 300\)
• Tens: \(70 + 40 = 110\)
• Ones: \(8 + 5 = 13\)

Now we have these partial sums:

Add them together:

So,

It is okay if a partial sum is more than \(9\) ones or more than \(9\) tens. We can still add the parts together at the end.

Let’s look closely at why this works.

In example 3, \(70 + 40 = 110\). That means 11 tens. And \(8 + 5 = 13\). That means 13 ones.

When we combine everything, the tens and ones make the correct total. Partial sums lets us add in parts first, then put all the parts together.

Example 4: Add \(487 + 276\)

Break apart the numbers:

Add the hundreds, tens, and ones:

• Hundreds: \(400 + 200 = 600\)
• Tens: \(80 + 70 = 150\)
• Ones: \(7 + 6 = 13\)

Now add the partial sums:

Add step by step:

So,

Let’s talk about what to remember when you use partial sums.

• Always line up place values in your thinking. Hundreds go with hundreds, tens go with tens, and ones go with ones.
• Break numbers apart carefully. For example, \(562\) is \(500 + 60 + 2\), not \(500 + 6 + 2\).
• Add all the partial sums at the end. Do not forget one of the parts.
• It is okay if you get sums like \(130\) or \(15\). Just add them in at the end.

Here is a helpful way to write partial sums:

For \(364 + 218\):

So,

You can also think of it like sorting blocks:

• Put all the hundreds together.
• Put all the tens together.
• Put all the ones together.
• Then count the whole amount.

Common mistakes to watch out for:

• Adding a ten to a hundred by mistake.
• Forgetting to break apart a number correctly.
• Stopping after finding the partial sums and forgetting to add them all together.

Let’s check one more problem together.

Add \(632 + 157\).

Break apart:

Add like parts:

• Hundreds: \(600 + 100 = 700\)
• Tens: \(30 + 50 = 80\)
• Ones: \(2 + 7 = 9\)

Add the partial sums:

So,

Summary

Partial sums means breaking numbers into hundreds, tens, and ones. Then you add each place value and combine the parts to find the total.

This strategy is helpful because it makes big addition problems easier to see and solve. When you stay organized and add like parts together, you can add three-digit numbers with confidence.

Open Number Lines with Hundreds, Tens, and Ones

An open number line is a line we use to show jumps. We do not have to write every number. We only write the numbers we need.

Open number lines help us add and subtract bigger numbers. We can make big jumps of hundreds, medium jumps of tens, and small jumps of ones.

This is a great way to work with numbers within 1,000 because it helps us see how numbers are built from hundreds, tens, and ones.

Think about place value:

• 0 = 1 hundred
• 10 = 1 ten
• 1 = 1 one

When we add on an open number line, we usually move to the right.

When we subtract on an open number line, we usually move to the left.

How to use an open number line

1. Write the starting number.
2. Break the other number into hundreds, tens, and ones.
3. Make jumps for the hundreds, then tens, then ones.
4. Land on the answer.

For example, the number \(235\) has:

• 2 hundreds
• 3 tens
• 5 ones

So we can think of it as:

$$235 = 200 + 30 + 5$$

This helps us know what jumps to make on the number line.

Adding with an open number line

Suppose we want to find \(324 + 156\).

First, break apart \(156\):

$$156 = 100 + 50 + 6$$

Start at \(324\). Then jump:

• +100 to get to \(424\)
• +50 to get to \(474\)
• +6 to get to \(480\)

So:

$$324 + 156 = 480$$

We can show the jumps like this:

$$324 \rightarrow 424 \rightarrow 474 \rightarrow 480$$

Each jump matches a place value part of \(156\).

Worked Example 1

Find \(241 + 123\).

Break apart \(123\):

$$123 = 100 + 20 + 3$$

Start at \(241\).

• Jump +100: \(241 \rightarrow 341\)
• Jump +20: \(341 \rightarrow 361\)
• Jump +3: \(361 \rightarrow 364\)

Answer:

$$241 + 123 = 364$$

Worked Example 2

Find \(478 + 215\).

Break apart \(215\):

$$215 = 200 + 10 + 5$$

Start at \(478\).

• Jump +200: \(478 \rightarrow 678\)
• Jump +10: \(678 \rightarrow 688\)
• Jump +5: \(688 \rightarrow 693\)

Answer:

$$478 + 215 = 693$$

Notice how the jumps can cross into a new hundred. That is okay. The number line helps us see where we land.

Subtracting with an open number line

Now let us subtract. Suppose we want to find \(562 - 134\).

Break apart \(134\):

$$134 = 100 + 30 + 4$$

Start at \(562\). Then jump left:

• -100 to get to \(462\)
• -30 to get to \(432\)
• -4 to get to \(428\)

So:

$$562 - 134 = 428$$

We can show the jumps like this:

$$562 \rightarrow 462 \rightarrow 432 \rightarrow 428$$

Worked Example 3

Find \(705 - 243\).

Break apart \(243\):

$$243 = 200 + 40 + 3$$

Start at \(705\).

• Jump -200: \(705 \rightarrow 505\)
• Jump -40: \(505 \rightarrow 465\)
• Jump -3: \(465 \rightarrow 462\)

Answer:

$$705 - 243 = 462$$

Worked Example 4

Find \(430 - 178\).

Break apart \(178\):

$$178 = 100 + 70 + 8$$

Start at \(430\).

• Jump -100: \(430 \rightarrow 330\)
• Jump -70: \(330 \rightarrow 260\)
• Jump -8: \(260 \rightarrow 252\)

Answer:

$$430 - 178 = 252$$

You can choose smart jumps too

Sometimes students break tens or ones into smaller parts to make an easier landing. That is okay on an open number line.

For example, to add \(356 + 27\), you might do:

• +20 to get to \(376\)
• +4 to get to \(380\)
• +3 to get to \(383\)

That still works because:

$$27 = 20 + 4 + 3$$

Open number lines are called open because there is more than one good way to jump.

Tips to remember

• Start at the first number.
• Break apart the second number.
• Use hundreds, tens, and ones.
• Add by jumping right.
• Subtract by jumping left.
• Check that your jumps match the number you are adding or subtracting.

Watch out for these mistakes

• Do not forget a jump. If you are adding \(146\), you need \(+100\), \(+40\), and \(+6\).
• Make sure subtraction jumps go left, not right.
• Be careful when counting tens and ones after a jump.

Let us review

An open number line helps us add and subtract by using place value. We break numbers into hundreds, tens, and ones, and then make jumps.

Big jumps are hundreds. Medium jumps are tens. Small jumps are ones. This makes solving three-digit problems easier to see and understand.

When you use an open number line, you are showing your thinking step by step. That is a smart math strategy.

Subtracting Across Zeros means subtracting when there is a zero in the top number, and we need to regroup to subtract.

This can look tricky at first. But if we go slowly, one place at a time, we can do it.

Remember that in a 3-digit number, the digits stand for hundreds, tens, and ones.

For example, in \(402\):

• \(4\) means 4 hundreds
• \(0\) means 0 tens
• \(2\) means 2 ones

When we subtract, we start at the ones place. If the top digit is too small, we regroup.

But what if there is a zero? Then that place has nothing to give. We may need to go to the next place value and regroup step by step.

Big idea: If a zero cannot give, look to the left until you find a digit that can give. Then move one value over at a time.

Main Teaching Points

Here is how to subtract across zeros:

1. Start at the ones place.
2. If the top digit is smaller than the bottom digit, you need to regroup.
3. If the tens digit is 0, it cannot give 1 ten.
4. Go to the hundreds place and take 1 hundred.
5. That 1 hundred becomes 10 tens.
6. If you still need ones, take 1 ten from those tens. That 1 ten becomes 10 ones.
7. Now subtract.

It helps to think of regrouping like trading:

• 1 hundred = 10 tens
• 1 ten = 10 ones

We are not changing how much the number is. We are only changing how it is grouped.

Worked Example 1

Let’s solve:

$$302 - 1$$

Start with the ones place.

In the ones place, \(2 - 1 = 1\). That is easy.

In the tens place, \(0 - 0 = 0\).

In the hundreds place, \(3 - 0 = 3\).

So the answer is:

$$302 - 1 = 301$$

This problem has a zero, but we did not need to regroup across it.

Worked Example 2

Now let’s try a problem where we do need to subtract across a zero:

$$401 - 3$$

Start at the ones place. We need to do \(1 - 3\), but 1 is too small.

Can the tens place help? No. The tens digit is \(0\), so there are 0 tens.

So we look to the hundreds place. There are 4 hundreds. We take 1 hundred from the 4 hundreds.

Now the 4 hundreds become 3 hundreds, and that 1 hundred becomes 10 tens.

But we still need ones. So we take 1 ten from the 10 tens.

Now the 10 tens become 9 tens, and that 1 ten becomes 10 ones.

Now the number is regrouped like this:

• 3 hundreds
• 9 tens
• 11 ones

Now subtract:

• Ones: \(11 - 3 = 8\)
• Tens: \(9 - 0 = 9\)
• Hundreds: \(3 - 0 = 3\)

So:

$$401 - 3 = 398$$

Worked Example 3

Let’s try another one:

$$500 - 8$$

Start at the ones place. We need to do \(0 - 8\), but 0 is too small.

Can the tens place help? No. The tens digit is also \(0\).

So go to the hundreds place. There are 5 hundreds. Take 1 hundred away.

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

Next, take 1 ten from the 10 tens to make ones.

Now 10 tens becomes 9 tens, and 1 ten becomes 10 ones.

Now subtract:

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

So:

$$500 - 8 = 492$$

Worked Example 4

Now let’s subtract a 2-digit number:

$$602 - 27$$

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

Can the tens place help? No. The tens digit is \(0\).

So go to the hundreds place. Take 1 hundred from 6 hundreds.

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

Then take 1 ten from the 10 tens to help the ones place.

Now the 10 tens become 9 tens, and the 2 ones become 12 ones.

Now subtract:

• Ones: \(12 - 7 = 5\)
• Tens: \(9 - 2 = 7\)
• Hundreds: \(5 - 0 = 5\)

So:

$$602 - 27 = 575$$

Tips to Remember

• Always start at the ones place.
• If the top digit is too small, regroup.
• If there is a zero, look to the left for a digit that can give.
• Trade 1 hundred for 10 tens.
• Trade 1 ten for 10 ones.
• Take your time and move one place at a time.

A Simple Way to Think About It

Imagine you have 1 big box of 100 blocks, but no tens and not enough ones.

You can open the big box and make 10 groups of ten. Then you can open 1 group of ten and make 10 ones.

This is what regrouping across zeros does. It helps you get the place value pieces you need to subtract.

Common Mistakes

• Forgetting that 0 cannot give. If the tens digit is 0, you must look left.
• Not changing both places. If you take 1 hundred, the hundreds go down by 1 and the tens go up by 10.
• Rushing. Go slowly and check each place.

Brief Summary

Subtracting across zeros means regrouping when a zero is in the way.

If a place has 0, it cannot give. Look to the left, take from a bigger place, and trade step by step.

Then subtract ones, tens, and hundreds carefully. With practice, subtracting across zeros gets easier.

Standard Algorithm for Three-Digit Addition

Today we will learn how to add three-digit numbers using the standard algorithm. This means we line the numbers up in columns and add one place at a time.

When we add, we work from right to left:

• ones
• tens
• hundreds

Sometimes a column makes a number that is 10 or more. When that happens, we regroup. We write the extra value above the next column.

Let’s remember the place values in a three-digit number:

• The right digit is the ones place.
• The middle digit is the tens place.
• The left digit is the hundreds place.

For example, in the number \(372\):

• \(3\) means 3 hundreds
• \(7\) means 7 tens
• \(2\) means 2 ones

When we use the standard algorithm, it is very important to line up the digits by place value.

Like this:

The ones are under the ones, the tens are under the tens, and the hundreds are under the hundreds.

Steps for the standard algorithm

1. Write the numbers one above the other.
2. Line up hundreds, tens, and ones.
3. Start with the ones column.
4. If the sum is 10 or more, regroup the extra ten to the tens column.
5. Add the tens column next.
6. If the sum is 10 or more, regroup the extra hundred to the hundreds column.
7. Add the hundreds column.
8. Write the final answer.

Let’s look at some examples.

Example 1: No regrouping

Add \(243 + 125\).

Step 1: Add the ones. \(3 + 5 = 8\). Write 8 in the ones place.

Step 2: Add the tens. \(4 + 2 = 6\). Write 6 in the tens place.

Step 3: Add the hundreds. \(2 + 1 = 3\). Write 3 in the hundreds place.

So, \(243 + 125 = 368\).

Example 2: Regroup in the ones column

Add \(256 + 127\).

Step 1: Add the ones. \(6 + 7 = 13\).

13 ones is 1 ten and 3 ones. Write the 3 in the ones place and regroup the 1 ten above the tens column.

Step 2: Add the tens. \(1 + 5 + 2 = 8\). Write 8 in the tens place.

Step 3: Add the hundreds. \(2 + 1 = 3\). Write 3 in the hundreds place.

So, \(256 + 127 = 383\).

Example 3: Regroup in the ones and tens columns

Add \(468 + 275\).

Step 1: Add the ones. \(8 + 5 = 13\).

Write 3 in the ones place. Regroup 1 ten above the tens column.

Step 2: Add the tens. \(1 + 6 + 7 = 14\).

14 tens is 1 hundred and 4 tens. Write 4 in the tens place and regroup 1 hundred above the hundreds column.

Step 3: Add the hundreds. \(1 + 4 + 2 = 7\). Write 7 in the hundreds place.

So, \(468 + 275 = 743\).

Example 4: Another regrouping example

Add \(389 + 246\).

Step 1: Add the ones. \(9 + 6 = 15\).

Write 5 in the ones place. Regroup 1 ten above the tens column.

Step 2: Add the tens. \(1 + 8 + 4 = 13\).

Write 3 in the tens place. Regroup 1 hundred above the hundreds column.

Step 3: Add the hundreds. \(1 + 3 + 2 = 6\).

Write 6 in the hundreds place.

So, \(389 + 246 = 635\).

Tips to help you

• Always line up the digits carefully.
• Start with the ones column.
• If a column equals 10 or more, regroup.
• Do not forget to add the regrouped number in the next column.
• Check that your answer makes sense.

Common mistakes to watch for

• Putting numbers in the wrong columns
• Starting with the hundreds instead of the ones
• Forgetting to regroup
• Writing the regrouped number but not adding it

Here is a quick check with place values:

If you add \(358 + 241\), you can think:

• ones: \(8 + 1 = 9\)
• tens: \(5 + 4 = 9\)
• hundreds: \(3 + 2 = 5\)

This helps us see that each digit belongs in the correct place.

Summary

The standard algorithm for three-digit addition helps us add big numbers in an organized way. We line up the hundreds, tens, and ones. Then we add from right to left: ones, tens, hundreds.

If a column makes 10 or more, we regroup and move the extra value to the next column. With careful lining up and regrouping, you can solve three-digit addition problems correctly.

Using Addition to Check Subtraction

When we subtract, we take one number away from another number.

Sometimes we want to make sure our subtraction answer is correct. A smart way to check is to use addition.

Why does this work? Because addition and subtraction are a fact family. They work together.

If we know that $$15 - 7 = 8$$ then we can check by adding:

$$7 + 8 = 15$$

If the addition sentence is true, then the subtraction answer is correct.

The Big Idea

In subtraction, we have:

• whole: the starting number
• part: the number taken away
• difference: the answer

To check subtraction with addition, add the difference and the part taken away.

If their sum equals the whole, your subtraction is correct.

Here is the rule:

$$\text{subtrahend} + \text{difference} = \text{whole}$$

In second grade words, that means:

number taken away + answer = starting number

How to Check a Subtraction Problem

1. Solve the subtraction problem.
2. Take your answer, called the difference.
3. Add it to the number that was taken away.
4. See if you get the starting number.

If you do, the subtraction is correct.

If you do not, go back and check your subtraction work.

Worked Example 1: Two-Digit Numbers

Solve:

$$14 - 6 = 8$$

Now check with addition.

Add the answer to the number taken away:

$$8 + 6 = 14$$

The sum is 14, which matches the starting number.

So, $$14 - 6 = 8$$ is correct.

Worked Example 2: A Bigger Two-Digit Problem

Solve:

$$52 - 19 = 33$$

Check with addition:

$$33 + 19 = 52$$

Let’s see it by parts:

$$33 + 10 = 43$$

$$43 + 9 = 52$$

We got back to 52, so the subtraction answer is correct.

Worked Example 3: Three-Digit Numbers

Solve:

$$326 - 104 = 222$$

Check with addition:

$$222 + 104 = 326$$

Add by place value:

• Ones: \(2 + 4 = 6\)
• Tens: \(2 + 0 = 2\)
• Hundreds: \(2 + 1 = 3\)

So:

$$222 + 104 = 326$$

The subtraction answer is correct.

Worked Example 4: Three-Digit Numbers with Regrouping

Solve:

$$500 - 278 = 222$$

Now check with addition:

$$278 + 222 = 500$$

Let’s add carefully:

• Ones: \(8 + 2 = 10\). Write 0 ones and make 1 extra ten.
• Tens: \(7 + 2 = 9\), and 1 more ten makes 10 tens. Write 0 tens and make 1 extra hundred.
• Hundreds: \(2 + 2 = 4\), and 1 more hundred makes 5 hundreds.

So the sum is:

$$278 + 222 = 500$$

That means $$500 - 278 = 222$$ is correct.

What If the Check Does Not Match?

Sometimes the addition check does not give the starting number. That means something is wrong.

Look at this example:

$$63 - 28 = 45$$

Now check it:

$$45 + 28 = 73$$

But the starting number was 63, not 73.

So the subtraction answer is not correct.

We need to try the subtraction again.

The correct subtraction is:

$$63 - 28 = 35$$

Now check:

$$35 + 28 = 63$$

Now it matches, so 35 is the correct answer.

Tips to Remember

• Start with a subtraction problem.
• Use addition to check your answer.
• Add the answer and the number taken away.
• The sum should equal the starting number.
• If it does not match, check your subtraction again.

Let’s Practice Thinking

If you solve $$81 - 20 = 61$$, how do you check it?

Add the difference and the number taken away:

$$61 + 20 = 81$$

It matches, so the subtraction is correct.

If you solve $$407 - 105 = 302$$, how do you check it?

$$302 + 105 = 407$$

It matches, so the subtraction is correct.

Summary

Subtraction and addition work together. You can check a subtraction answer by adding the answer back to the number that was taken away.

If the sum equals the starting number, your subtraction is correct. This is a great way to check your work, especially with bigger numbers.

Estimating Sums and Differences means finding a number that is close to the real answer. We do not need the exact answer. We just want a quick, smart guess.

Estimating helps us check our work. If we add or subtract and get an answer that seems too big or too small, an estimate can help us notice that.

When we estimate, we often use rounding. Rounding means changing a number to a nearby number that is easier to work with.

For 2nd grade, a helpful way to round is to think about the tens.

• If the ones digit is 0, 1, 2, 3, or 4, round down to the lower ten.
• If the ones digit is 5, 6, 7, 8, or 9, round up to the next ten.

Here are some quick rounding examples:

• (23 \approx 20\)
• (47 \approx 50\)
• (81 \approx 80\)
• (196 \approx 200\)
• (342 \approx 340\)

Notice that we are choosing numbers that are easier to add or subtract in our heads.

To estimate a sum, round each addend first, then add the rounded numbers.

To estimate a difference, round each number first, then subtract the rounded numbers.

Lets look at some examples step by step.

Example 1: Estimate a sum

Estimate: \(34 + 22\)

First, round each number to the nearest ten.

• \(34 \approx 30\)
• \(22 \approx 20\)

Now add the rounded numbers.

So, the estimate for \(34 + 22\) is 50.

The exact answer is \(56\), and 50 is close. That means the estimate makes sense.

Example 2: Estimate a difference

Estimate: \(78 - 31\)

Round each number to the nearest ten.

• \(78 \approx 80\)
• \(31 \approx 30\)

Now subtract.

So, the estimate for \(78 - 31\) is 50.

The exact answer is \(47\), which is close to 50.

Example 3: Estimate with 3-digit numbers

Estimate: \(146 + 233\)

We can still round to the nearest ten.

• \(146 \approx 150\)
• \(233 \approx 230\)

Add the rounded numbers.

So, the estimate is 380.

This helps us know the exact answer should be close to 380.

Example 4: Estimate a 3-digit difference

Estimate: \(592 - 218\)

Round each number to the nearest ten.

• \(592 \approx 590\)
• \(218 \approx 220\)

Now subtract.

So, the estimate is 370.

The exact answer is close to 370, so this estimate is useful.

How estimation helps you check your work

Suppose you solve \(146 + 233\) and get 619. Does that make sense?

Lets estimate:

Since 619 is not close to 380, that answer does not make sense. The estimate helps us catch a mistake.

Tips for estimating

• Look at the ones digit to decide whether to round up or down.
• Round both numbers in the problem the same way: usually to the nearest ten.
• Remember, an estimate is close, not exact.
• Use estimation to check if your exact answer is reasonable.

Lets practice thinking

If you see \(61 + 19\), you can think:

• \(61 \approx 60\)
• \(19 \approx 20\)

So the sum should be about 80.

If you see \(405 - 182\), you can think:

• \(405 \approx 410\)
• \(182 \approx 180\)

So the difference should be about 230.

Summary

Estimating sums and differences means finding an answer that is close to the exact answer.

We estimate by rounding numbers, usually to the nearest ten, and then adding or subtracting.

Estimation helps us solve problems quickly and check whether an exact answer makes sense.