Place Value and the Base-Ten System to 1,000

Grouping into Tens and Ones

Numbers can be made in different ways. One important way is by grouping objects into tens and ones.

When we count, we can count one by one. But when there are many objects, it is faster and easier to make groups of 10. In math, 10 ones make 1 ten.

This idea helps us understand place value. The ones tell how many single objects there are. The tens tell how many groups of 10 there are.

Here is the big idea:

$$10\text{ ones} = 1\text{ ten}$$

We can also think about it like this:

• If you have fewer than 10 ones, they stay as ones.
• If you get 10 ones, you can bundle them together to make 1 ten.
• A number can have some tens and some extra ones.

Why do we group into tens?

Grouping into tens helps us count bigger numbers more quickly. Instead of counting every object one at a time, we can count groups of 10 and then count the leftover ones.

For example, if you have 1 group of ten and 3 extra ones, you do not need to count all the way from 1. You can think:

$$10 + 3 = 13$$

So, 1 ten and 3 ones is 13.

How to find tens and ones in a number

1. Look for groups of 10 first.
2. Count how many tens there are.
3. Then count the extra ones left over.
4. Put them together to name the number.

You can also go the other way:

1. Start with a number.
2. Make as many groups of 10 as you can.
3. The objects left over are the ones.

Worked Example 1: A small number with only ones

Suppose you have 7 blocks.

Can you make a group of 10? No, because 7 is less than 10.

So the number 7 has:

• 0 tens
• 7 ones

We can write:

$$7 = 0\text{ tens } + 7\text{ ones}$$

Worked Example 2: Making one ten

Suppose you have 10 pennies.

Since 10 ones make 1 ten, we can bundle the 10 pennies into 1 group.

So the number 10 has:

• 1 ten
• 0 ones

We can write:

$$10 = 1\text{ ten } + 0\text{ ones}$$

Worked Example 3: Tens and extra ones

Suppose you have 14 crayons.

First, make a group of 10. That uses 10 crayons. Then count what is left.

There are 4 crayons left over.

So 14 has:

• 1 ten
• 4 ones

We can show it like this:

$$14 = 10 + 4$$

and also like this:

$$14 = 1\text{ ten } + 4\text{ ones}$$

Worked Example 4: More than one ten

Suppose you have 27 stickers.

Make groups of 10:

• 10 stickers = 1 ten
• another 10 stickers = 1 more ten

That makes 2 tens, which is 20 stickers. There are 7 stickers left over.

So 27 has:

• 2 tens
• 7 ones

We can write:

$$27 = 20 + 7$$

and

$$27 = 2\text{ tens } + 7\text{ ones}$$

Thinking about numbers with tens and ones

When you see a 2-digit number, the first digit tells the tens and the second digit tells the ones.

• In 13, the 1 means 1 ten, and the 3 means 3 ones.
• In 25, the 2 means 2 tens, and the 5 means 5 ones.
• In 40, the 4 means 4 tens, and the 0 means 0 ones.

This helps us read and understand numbers more easily.

Let’s practice thinking in groups

If a number has 3 tens and 2 ones, what number is it?

3 tens is 30. Add 2 ones.

$$30 + 2 = 32$$

So the number is 32.

If a number is 18, how many tens and ones does it have?

18 has 1 group of 10 and 8 extra ones.

So it has:

• 1 ten
• 8 ones

Helpful tips

• Always try to make groups of 10 first.
• Count the groups of 10 carefully.
• Then count the ones left over.
• Remember: you cannot have 10 ones left over, because 10 ones should be grouped into 1 ten.

Summary

Grouping into tens and ones is an important math idea. 10 ones make 1 ten. A number can be made from some tens and some ones.

When you group objects into tens, counting becomes easier. For example, 24 means 2 tens and 4 ones. The more you practice grouping, the better you will understand numbers.

Conceptualizing One Hundred

In math, numbers can be grouped in helpful ways. One important idea is that 10 tens make 1 hundred.

This helps us understand bigger numbers. When we count by ones, numbers grow slowly. When we group by tens, counting is faster. When we group 10 tens together, we get a new, bigger unit called one hundred.

Think about bundles of straws or sticks. One straw is 1 one. A bundle of 10 straws is 1 ten. Then 10 bundles of 10 straws make 1 hundred.

We can show this idea like this:

$$10\text{ tens} = 1\text{ hundred}$$

We can also write it with numbers:

$$10 \times 10 = 100$$

Why this matters

Understanding one hundred helps you read, build, and compare numbers. It shows how place value works in the base-ten system.

In base ten, each place is 10 times the value of the place to its right.

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

So a hundred is much bigger than a ten, and a ten is much bigger than a one.

Seeing one hundred

You can picture 100 in different ways:

• 100 single cubes
• 10 rods of ten
• 1 large square made of 100 small squares
• 1 number named one hundred

All of these show the same amount: 100.

If you had 10 groups of 10, you could count them by tens:

$$10,\ 20,\ 30,\ 40,\ 50,\ 60,\ 70,\ 80,\ 90,\ 100$$

When you say ten tens, you land on 100.

Place value chart

A place value chart helps us see where numbers belong.

The number 100 has:

• 1 in the hundreds place
• 0 in the tens place
• 0 in the ones place

That means 100 is made of 1 hundred, 0 tens, and 0 ones.

Making one hundred

Let’s build 100 step by step.

1. Start with 10 ones. That makes 1 ten.
2. Make 10 tens. That makes 1 hundred.

So we can say:

$$100 = 10\text{ tens}$$

And also:

$$100 = 100\text{ ones}$$

This means one hundred can be made from 100 ones or from 10 tens.

Worked Example 1

Question: How many tens are in 100?

Think: One hundred is made by bundling 10 tens together.

Answer: There are 10 tens in 100.

$$100 = 10\text{ tens}$$

Worked Example 2

Question: If you count by tens, what number do you say after 90?

Think: Count by tens:

$$10,\ 20,\ 30,\ 40,\ 50,\ 60,\ 70,\ 80,\ 90,\ 100$$

Answer: After 90 comes 100.

This shows that adding one more ten to 90 makes 100.

Worked Example 3

Question: Which is the same as 1 hundred?

• A. 5 tens
• B. 10 tens
• C. 9 tens

Think: We know:

$$1\text{ hundred} = 10\text{ tens}$$

Answer: B. 10 tens

Worked Example 4

Question: Maria has 8 tens. How many more tens does she need to make 1 hundred?

Think: One hundred is 10 tens. Maria has 8 tens.

She needs:

$$10 - 8 = 2$$

Answer: She needs 2 more tens.

And 2 tens is 20, so 80 and 20 make 100.

Helpful ways to remember

• 10 ones = 1 ten
• 10 tens = 1 hundred
• 100 is a bundle of 10 tens
• 100 has 1 in the hundreds place

Try thinking about it like this:

If you have 10 packs of crayons, and each pack has 10 crayons, then you have 100 crayons altogether.

That is because:

$$10\text{ groups of }10 = 100$$

Summary

One hundred is an important place value unit. It is made by grouping 10 tens together.

You can think of 100 as:

• 1 hundred
• 10 tens
• 100 ones

When you understand that 10 tens = 1 hundred, you are learning how numbers grow in the base-ten system. This helps you work with bigger numbers up to 1,000.

Counting by Ones, Tens, and Hundreds helps us understand how numbers grow and change in our base-ten number system.

When we count by ones, we add or take away 1 each time. When we count by tens, we add or take away 10 each time. When we count by hundreds, we add or take away 100 each time.

This is important because numbers are made of hundreds, tens, and ones. Knowing how to count in these ways helps us read, write, and understand numbers up to 1,000.

Let’s remember place value:

• The ones place tells how many ones.
• The tens place tells how many tens.
• The hundreds place tells how many hundreds.

For example, in the number \(356\):

• \(3\) means 3 hundreds
• \(5\) means 5 tens
• \(6\) means 6 ones

That means:

$$356 = 300 + 50 + 6$$

Now let’s learn how counting changes a number.

1. Counting by Ones

When we count by ones, we move to the very next number or the number right before it.

• Counting forward by ones means add 1.
• Counting backward by ones means subtract 1.

Examples:

• After \(27\) comes \(28\).
• After \(99\) comes \(100\).
• Before \(430\) comes \(429\).

Sometimes counting by ones crosses a boundary. A decade boundary is when we move from one group of ten to the next, like \(29\) to \(30\). A century boundary is when we move from one group of one hundred to the next, like \(199\) to \(200\).

Even when we cross a boundary, we are still just adding or subtracting 1.

2. Counting by Tens

When we count by tens, the tens change by 1 ten. The ones digit stays the same.

• Counting forward by tens means add 10.
• Counting backward by tens means subtract 10.

Examples:

• \(34, 44, 54, 64\)
• \(120, 130, 140, 150\)
• \(287, 277, 267, 257\)

Look closely at \(34, 44, 54, 64\). The ones digit is always \(4\). Only the tens change.

Sometimes counting by tens crosses a century boundary. For example:

$$180, 190, 200, 210$$

Here, after \(190\), adding 10 gives \(200\). We moved into a new hundred.

3. Counting by Hundreds

When we count by hundreds, the hundreds change by 1 hundred. The tens and ones digits stay the same.

• Counting forward by hundreds means add 100.
• Counting backward by hundreds means subtract 100.

Examples:

• \(245, 345, 445, 545\)
• \(600, 700, 800, 900\)
• \(932, 832, 732, 632\)

Look at \(245, 345, 445, 545\). The tens and ones stay \(45\). Only the hundreds digit changes.

4. What Stays the Same?

This is a helpful way to think:

• Count by ones: the number changes by 1.
• Count by tens: the ones digit stays the same.
• Count by hundreds: the tens and ones digits stay the same.

For example:

• By ones: \(58, 59, 60, 61\)
• By tens: \(58, 68, 78, 88\)
• By hundreds: \(58, 158, 258, 358\)

Worked Example 1: Counting forward by ones

Start at \(48\). Count the next 4 numbers by ones.

We add 1 each time:

$$48, 49, 50, 51, 52$$

The next 4 numbers are 49, 50, 51, 52.

Notice that we crossed a decade boundary when we went from \(49\) to \(50\).

Worked Example 2: Counting backward by tens

Start at \(163\). Count back 3 times by tens.

We subtract 10 each time:

$$163, 153, 143, 133$$

The numbers are 153, 143, 133.

The ones digit stayed \(3\) the whole time.

Worked Example 3: Counting forward by tens across a century boundary

Start at \(175\). Count forward 4 times by tens.

We add 10 each time:

$$175, 185, 195, 205, 215$$

The numbers are 185, 195, 205, 215.

Notice that after \(195\), the next number is \(205\). We crossed from the 100s into the 200s.

Worked Example 4: Counting backward by hundreds

Start at \(724\). Count back 3 times by hundreds.

We subtract 100 each time:

$$724, 624, 524, 424$$

The numbers are 624, 524, 424.

The tens and ones stayed \(24\). Only the hundreds changed.

Tips for Counting Carefully

• If you count by ones, check the next number in order.
• If you count by tens, keep the ones digit the same.
• If you count by hundreds, keep the last two digits the same.
• Go slowly when crossing numbers like \(29\) to \(30\), \(99\) to \(100\), or \(190\) to \(200\).

Try thinking about these patterns:

• \(67, 68, 69, 70\) is counting by ones.
• \(67, 77, 87, 97\) is counting by tens.
• \(67, 167, 267, 367\) is counting by hundreds.

The starting number can be almost any number. What changes is whether we add or subtract 1, 10, or 100.

Summary

Counting by ones means adding or subtracting \(1\). Counting by tens means adding or subtracting \(10\). Counting by hundreds means adding or subtracting \(100\).

When you count by tens, the ones digit stays the same. When you count by hundreds, the tens and ones digits stay the same. Be extra careful when your counting crosses a decade boundary like \(39\) to \(40\) or a century boundary like \(199\) to \(200\).

Skip Counting by 2s, 5s, and 10s

Skip counting means counting by numbers that are more than 1 each time. Instead of saying every number, we jump by the same amount again and again.

For this lesson, we will learn how to skip count by 2s, 5s, and 10s. This helps us see number patterns and helps us count faster.

Skip counting is also connected to the base-ten system. Our number system is built with ones, tens, and hundreds. When we count by 10s, we are counting groups of ten. When we count by 2s or 5s, we are counting equal groups too.

Why skip counting is useful

• It helps you count objects quickly.
• It helps you notice patterns in numbers.
• It helps you get ready for multiplication later.
• It helps with money and telling time.

Skip counting by 2s

When we skip count by 2s, we add 2 each time.

Start at 0:

$$0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20$$

You may notice something important. Numbers we say when we count by 2s have even numbers in the ones place: 0, 2, 4, 6, or 8.

If we start at another multiple of 2, we still add 2 each time.

For example:

$$6, 8, 10, 12, 14, 16$$

Skip counting by 5s

When we skip count by 5s, we add 5 each time.

Start at 0:

$$0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50$$

Look at the ones digits. They make a pattern:

• They end in 0 or 5.

• The ones digit goes back and forth: 5, 0, 5, 0, 5, 0.

If we start at a different multiple of 5, we still add 5 each time.

For example:

$$15, 20, 25, 30, 35, 40$$

Skip counting by 10s

When we skip count by 10s, we add 10 each time.

Start at 0:

$$0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100$$

Look at the pattern. The ones digit stays 0 every time. Only the tens digit changes as we count by 10s.

This is very important in place value. Counting by 10s means counting tens:

• \(10\) is 1 ten
• \(20\) is 2 tens
• \(30\) is 3 tens
• \(40\) is 4 tens

If we start at another multiple of 10, we still add 10 each time.

For example:

$$30, 40, 50, 60, 70, 80$$

How to skip count

1. Look at the starting number.

2. Decide if you are adding 2, 5, or 10.

3. Add the same amount each time.

4. Watch the number pattern to help you.

Patterns to remember

• By 2s: numbers end in 0, 2, 4, 6, 8.

• By 5s: numbers end in 0 or 5.

• By 10s: numbers end in 0.

Worked Example 1: Skip count by 2s

Start at \(4\). Count by 2s for 6 numbers.

We keep adding \(2\):

$$4, 6, 8, 10, 12, 14$$

Answer: \(4, 6, 8, 10, 12, 14\)

Worked Example 2: Skip count by 5s

Start at \(20\). Count by 5s for 5 numbers.

Add \(5\) each time:

$$20, 25, 30, 35, 40$$

Answer: \(20, 25, 30, 35, 40\)

Worked Example 3: Skip count by 10s

Start at \(50\). Count by 10s for 5 numbers.

Add \(10\) each time:

$$50, 60, 70, 80, 90$$

Answer: \(50, 60, 70, 80, 90\)

Worked Example 4: Find the missing numbers

Fill in the blanks:

$$10, \; 20, \; \underline{\hspace{1cm}}, \; 40, \; \underline{\hspace{1cm}}, \; 60$$

These numbers are going up by \(10\) each time.

After \(20\) comes \(30\).

After \(40\) comes \(50\).

Answer:

$$10, 20, 30, 40, 50, 60$$

Try thinking about groups

Skip counting can help you count objects in groups.

• 2, 4, 6, 8 means 4 groups of 2.

• 5, 10, 15, 20 means 4 groups of 5.

• 10, 20, 30, 40 means 4 groups of 10.

For example, if you have 3 groups of 10 blocks, you can skip count:

$$10, 20, 30$$

So there are \(30\) blocks.

Common mistakes to watch for

• Do not go back to counting by 1s.

• Make sure you add the same number each time.

• Check the ending digits for the pattern.

Quick check

• By 2s from \(8\): \(8, 10, 12, 14\)

• By 5s from \(25\): \(25, 30, 35, 40\)

• By 10s from \(70\): \(70, 80, 90, 100\)

Summary

Skip counting means adding the same number again and again.

When you skip count by 2s, add \(2\). When you skip count by 5s, add \(5\). When you skip count by 10s, add \(10\).

Look for patterns in the ones digit to help you. These patterns make counting faster and help you understand numbers better.

Base-Ten Block Representations help us see what a number is made of. We can build numbers with blocks and understand the value of each part.

Base-ten blocks are special math blocks. Each kind of block stands for a different amount.

• One small cube = 1 one
• One long rod = 1 ten = 10 ones
• One flat square = 1 hundred = 10 tens = 100 ones

That means:

When we build a number, we count how many hundreds, tens, and ones there are.

For example, in the number \(245\):

• 2 is in the hundreds place, so it means 2 hundreds
• 4 is in the tens place, so it means 4 tens
• 5 is in the ones place, so it means 5 ones

We can write that like this:

This is called breaking a number into parts. Base-ten blocks help us see those parts.

Let’s learn the three block types carefully.

Hundreds block: A big flat square stands for 100. It is the same as 10 tens or 100 ones.

Tens block: A long rod stands for 10. It is the same as 10 ones.

Ones block: A small cube stands for 1.

If you have 3 hundreds, 2 tens, and 6 ones, you have:

So the number is 326.

How to read base-ten blocks

1. Count the hundreds flats.
2. Count the tens rods.
3. Count the ones cubes.
4. Write the digits in order: hundreds, tens, ones.

This helps us make a 3-digit number.

How to build a number with base-ten blocks

1. Look at the number.
2. Read the hundreds digit and get that many hundreds flats.
3. Read the tens digit and get that many tens rods.
4. Read the ones digit and get that many ones cubes.

Now let’s try some examples together.

Example 1: Read a base-ten block model

A model shows:

• 1 hundred flat
• 3 tens rods
• 4 ones cubes

Step 1: Count the hundreds. There is 1 hundred, so that is \(100\).

Step 2: Count the tens. There are 3 tens, so that is \(30\).

Step 3: Count the ones. There are 4 ones, so that is \(4\).

The number is 134.

Example 2: Build a number with blocks

Build the number 257.

The digit 2 is in the hundreds place, so we need 2 hundreds flats.

The digit 5 is in the tens place, so we need 5 tens rods.

The digit 7 is in the ones place, so we need 7 ones cubes.

So we build:

• 2 hundreds
• 5 tens
• 7 ones

Example 3: Match blocks to a number

A model shows:

• 4 hundreds flats
• 0 tens rods
• 8 ones cubes

4 hundreds means \(400\).

0 tens means \(0\).

8 ones means \(8\).

The number is 408.

This example is important because sometimes a number has 0 tens or 0 ones. We still must show that place in the number.

Example 4: Think in parts

What number has 6 hundreds, 1 ten, and 2 ones?

6 hundreds = \(600\)

1 ten = \(10\)

2 ones = \(2\)

The number is 612.

Important idea: The same digit can have a different value depending on where it is.

Look at the digit 5 in these numbers:

• \(5\) means 5 ones
• \(50\) means 5 tens
• \(500\) means 5 hundreds

So place matters. A digit tells a different amount in the hundreds place, tens place, and ones place.

Base-ten blocks and expanded form

When we see blocks, we can write the number in expanded form. Expanded form shows the value of each place.

For example, 3 hundreds, 4 tens, and 9 ones is:

Expanded form helps us understand how the number is built.

Tips for success

• Count hundreds first, then tens, then ones.
• Do not mix up tens and ones.
• Remember: one ten is 10 ones.
• Remember: one hundred is 10 tens.
• If there are no blocks in one place, write 0 for that place.

Let’s practice thinking.

If you see 2 hundreds, 9 tens, and 3 ones, the number is:

If you need to build 471, use:

• 4 hundreds flats
• 7 tens rods
• 1 ones cube

If you see 5 hundreds, 6 tens, and 0 ones, the number is:

Summary

Base-ten blocks help us show numbers in a way we can see and touch. Hundreds flats stand for 100, tens rods stand for 10, and ones cubes stand for 1.

To read a model, count the hundreds, tens, and ones. To build a number, use the digits to choose the right number of hundreds, tens, and ones blocks.

When you understand base-ten blocks, it becomes easier to read, build, and break apart numbers up to 1,000.

Standard and Word Form to 1,000

Numbers can be written in different ways. In this lesson, you will learn how to write numbers in standard form and word form up to 1,000.

Standard form is the way we usually write a number using digits, like \(245\).

Word form is the way we write a number using words, like two hundred forty-five.

Both forms name the same number. We just show the number in different ways.

Let’s remember place value.

When we write a 3-digit number, each digit has a place and a job:

• Hundreds place
• Tens place
• Ones place

For example, in \(582\):

• \(5\) means 5 hundreds
• \(8\) means 8 tens
• \(2\) means 2 ones

So \(582\) is five hundred eighty-two.

How to read and write standard form to word form

1. Look at the hundreds digit.
2. Say that many hundreds.
3. Look at the tens and ones.
4. Say the tens and ones number.

Example: \(364\)

• \(3\) hundreds = three hundred
• \(64\) = sixty-four

So \(364\) in word form is three hundred sixty-four.

How to write word form to standard form

1. Listen or look for the hundreds word.
2. Write the hundreds digit.
3. Then write the tens digit and ones digit.

Example: seven hundred twelve

• seven hundred means \(7\) is in the hundreds place
• twelve means \(12\), so write \(1\) in the tens place and \(2\) in the ones place

So the standard form is \(712\).

Special things to remember

• If there are no tens, the tens digit is \(0\).
• If there are no ones, the ones digit is \(0\).
• Some numbers from 11 to 19 have special names, like eleven, twelve, and fifteen.
• Tens words are twenty, thirty, forty, fifty, sixty, seventy, eighty, and ninety.

Worked Example 1

Write \(126\) in word form.

Step 1: \(1\) in the hundreds place means one hundred.

Step 2: \(2\) in the tens place and \(6\) in the ones place make twenty-six.

Answer: one hundred twenty-six

Worked Example 2

Write four hundred five in standard form.

Step 1: four hundred means \(4\) in the hundreds place.

Step 2: There is no tens word, so the tens digit is \(0\).

Step 3: five means \(5\) in the ones place.

Answer:

Worked Example 3

Write \(890\) in word form.

Step 1: \(8\) hundreds means eight hundred.

Step 2: \(9\) tens means ninety.

Step 3: \(0\) ones means there are no ones to say.

Answer: eight hundred ninety

Worked Example 4

Write nine hundred thirty-two in standard form.

Step 1: nine hundred means \(9\) in the hundreds place.

Step 2: thirty means \(3\) in the tens place.

Step 3: two means \(2\) in the ones place.

Answer:

Try thinking about these number parts

The number \(708\) has:

• \(7\) hundreds
• \(0\) tens
• \(8\) ones

So its word form is seven hundred eight.

The number words six hundred seventy have:

• \(6\) in the hundreds place
• \(7\) in the tens place
• \(0\) in the ones place

So the standard form is \(670\).

Helpful tips

• Say the number slowly.
• Think: hundreds, tens, ones.
• If a place has nothing, use \(0\) in standard form.
• Check that the words and digits match the same number.

Summary

Standard form uses digits, like \(531\). Word form uses words, like five hundred thirty-one.

To change standard form to word form, read the hundreds, then the tens and ones.

To change word form to standard form, listen for the hundreds, tens, and ones, and write each digit in the correct place.

When you understand place value, it is easier to read, write, and understand numbers up to 1,000.

Expanded Form Representation helps us break a number into parts. These parts show the value of each digit.

In 2nd grade, we look at numbers using hundreds, tens, and ones. Expanded form shows how many hundreds, tens, and ones are in a number.

For example, the number \(345\) has:

• 3 hundreds
• 4 tens
• 5 ones

So its expanded form is:

$$345 = 300 + 40 + 5$$

This means:

• the 3 is really 300
• the 4 is really 40
• the 5 is really 5

Why do we use expanded form?

Expanded form helps us understand what a number is made of. It helps us read numbers, write numbers, and solve math problems.

Place value review

Every digit in a number has a place. The place tells its value.

• Hundreds place
• Tens place
• Ones place

Look at \(582\):

• 5 is in the hundreds place, so it means 500
• 8 is in the tens place, so it means 80
• 2 is in the ones place, so it means 2

So:

$$582 = 500 + 80 + 2$$

How to write a number in expanded form

1. Look at each digit.
2. Find its place: hundreds, tens, or ones.
3. Write the value of each digit.
4. Add the parts together.

Let’s practice this step by step.

Example 1

Write \(126\) in expanded form.

The digits are:

• 1 hundred = 100
• 2 tens = 20
• 6 ones = 6

So:

$$126 = 100 + 20 + 6$$

Example 2

Write \(470\) in expanded form.

The digits are:

• 4 hundreds = 400
• 7 tens = 70
• 0 ones = 0

So:

$$470 = 400 + 70 + 0$$

We can also write it as:

$$470 = 400 + 70$$

Since there are 0 ones, there is no ones part to add.

Example 3

Write \(908\) in expanded form.

The digits are:

• 9 hundreds = 900
• 0 tens = 0
• 8 ones = 8

So:

$$908 = 900 + 0 + 8$$

We can also write it as:

$$908 = 900 + 8$$

This shows that sometimes a number has 0 tens or 0 ones.

Example 4

What number is shown by this expanded form?

$$200 + 30 + 4$$

Let’s put the parts together:

• 200 means 2 hundreds
• 30 means 3 tens
• 4 means 4 ones

So the number is:

$$234$$

Helpful thinking

When you see a 3-digit number, ask yourself:

• How many hundreds?
• How many tens?
• How many ones?

Then write each part.

For \(651\):

• 6 hundreds = 600
• 5 tens = 50
• 1 one = 1

So:

$$651 = 600 + 50 + 1$$

Be careful!

• The digit tells how many, but the place tells the value.
• In \(352\), the 5 does not mean 5. It means 5 tens, or 50.
• In \(804\), the 0 means there are no tens.

Try to notice patterns

Numbers in expanded form always show place value parts added together.

For example:

• \(213 = 200 + 10 + 3\)
• \(560 = 500 + 60\)
• \(701 = 700 + 1\)

Summary

Expanded form breaks a number into hundreds, tens, and ones. It shows the value of each digit. To write expanded form, look at each digit, find its place, and add the values together.

Flexible Decomposition of Numbers means breaking a number apart in different correct ways.

We can use hundreds, tens, and ones to show a number. Sometimes a number can be split in more than one way.

For example, the number \(134\) can be shown as:

• \(1\) hundred, \(3\) tens, and \(4\) ones
• \(13\) tens and \(4\) ones

Both ways are correct because they both make \(134\).

In our number system:

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

This is why we can break numbers apart in flexible ways. We can trade:

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

Flexible means we can change how we group the number, but the total stays the same.

Let’s look at place value first.

• The hundreds place tells how many hundreds.
• The tens place tells how many tens.
• The ones place tells how many ones.

For \(247\):

• \(2\) hundreds = \(200\)
• \(4\) tens = \(40\)
• \(7\) ones = \(7\)

So:

$$247 = 200 + 40 + 7$$

That is the usual way to decompose the number. But we can also do it in other ways by trading hundreds for tens or tens for ones.

Here are some helpful ideas:

• If you trade \(1\) hundred, you get \(10\) tens.
• If you trade \(1\) ten, you get \(10\) ones.
• The number does not change when you trade. Only the way you show it changes.

Example 1: Decompose \(56\)

The usual way is:

$$56 = 5\text{ tens } + 6\text{ ones}$$

We can also trade one ten for ten ones.

If we take away \(1\) ten from \(5\) tens, we have \(4\) tens left.

Then we add \(10\) ones to the \(6\) ones:

$$6 + 10 = 16$$

So another way is:

$$56 = 4\text{ tens } + 16\text{ ones}$$

Both are correct:

• \(5\) tens and \(6\) ones
• \(4\) tens and \(16\) ones

Example 2: Decompose \(134\)

The usual way is:

$$134 = 1\text{ hundred } + 3\text{ tens } + 4\text{ ones}$$

Now trade the \(1\) hundred for \(10\) tens.

Then we have:

• \(10\) tens from the hundred
• plus \(3\) more tens

That makes \(13\) tens.

So:

$$134 = 13\text{ tens } + 4\text{ ones}$$

We can also write it as:

$$134 = 100 + 30 + 4$$

All of these show the same number.

Example 3: Decompose \(247\) in a new way

Start with:

$$247 = 2\text{ hundreds } + 4\text{ tens } + 7\text{ ones}$$

Trade \(1\) hundred for \(10\) tens.

• Now there is \(1\) hundred left.
• The \(4\) tens become \(14\) tens.
• The \(7\) ones stay the same.

So we can write:

$$247 = 1\text{ hundred } + 14\text{ tens } + 7\text{ ones}$$

We can even trade both hundreds for tens.

• \(2\) hundreds = \(20\) tens
• \(20\) tens + \(4\) tens = \(24\) tens

So another way is:

$$247 = 24\text{ tens } + 7\text{ ones}$$

Example 4: Decompose \(320\)

The usual way is:

$$320 = 3\text{ hundreds } + 2\text{ tens } + 0\text{ ones}$$

Trade the \(3\) hundreds for tens.

• \(3\) hundreds = \(30\) tens
• \(30\) tens + \(2\) tens = \(32\) tens

So:

$$320 = 32\text{ tens}$$

This is still correct, even though there are \(0\) ones.

How to decompose flexibly

1. Find the usual hundreds, tens, and ones.
2. Decide if you want to trade a hundred for tens or a ten for ones.
3. Make the trade carefully.
4. Check that the total number stays the same.

Let’s practice thinking

If a number is \(86\), the usual way is \(8\) tens and \(6\) ones.

If we trade one ten, we get:

• \(7\) tens
• \(16\) ones

So:

$$86 = 8\text{ tens } + 6\text{ ones} = 7\text{ tens } + 16\text{ ones}$$

If a number is \(150\), the usual way is \(1\) hundred, \(5\) tens, and \(0\) ones.

Trade the hundred for tens:

• \(1\) hundred = \(10\) tens
• \(10\) tens + \(5\) tens = \(15\) tens

So:

$$150 = 15\text{ tens}$$

Watch out for these mistakes

• Do not change the total value of the number.
• Remember: \(1\) hundred is not \(100\) tens. It is \(10\) tens.
• Remember: \(1\) ten is \(10\) ones.
• If you trade away a ten or a hundred, take it away from that place first.

Quick check

• \(72 = 7\) tens and \(2\) ones
• \(72 = 6\) tens and \(12\) ones
• \(180 = 1\) hundred, \(8\) tens, \(0\) ones
• \(180 = 18\) tens

Each pair shows the same number in different ways.

Summary

Flexible decomposition means breaking a number into parts in more than one correct way.

You can use hundreds, tens, and ones. You can trade:

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

This helps you see numbers in different ways. For example:

• \(134 = 1\) hundred, \(3\) tens, \(4\) ones
• \(134 = 13\) tens, \(4\) ones

When you decompose flexibly, the parts may look different, but the number stays the same.

Comparing Numbers Using Symbols

Numbers can be greater than, less than, or equal to each other. We use special math symbols to show this.

• e means greater than
• < means less than
• = means equal to

Here is what they mean:

$$8 > 5$$

This means 8 is greater than 5.

$$3 < 7$$

This means 3 is less than 7.

$$6 = 6$$

This means 6 is equal to 6.

How to Compare Numbers

When we compare numbers up to 1,000, we look at the digits from left to right.

Think about place value:

• Hundreds
• Tens
• Ones

We compare the biggest place first.

1. Look at the hundreds digit.
2. If the hundreds digits are the same, look at the tens digit.
3. If the tens digits are the same, look at the ones digit.

The first place that is different tells which number is greater.

Helpful Idea

You can think of the symbols like a hungry alligator. The alligator wants to eat the bigger number, so the open mouth points to the greater number.

For example:

$$9 > 4$$

The open mouth points to 9 because 9 is bigger.

Worked Example 1: Compare 45 and 51

Compare the tens first.

• 45 has 4 tens
• 51 has 5 tens

Since 4 tens is less than 5 tens, 45 is less than 51.

$$45 < 51$$

Worked Example 2: Compare 372 and 368

Look at the hundreds digit first.

• 372 has 3 hundreds
• 368 has 3 hundreds

The hundreds are the same, so look at the tens digit.

• 372 has 7 tens
• 368 has 6 tens

Since 7 tens is greater than 6 tens, 372 is greater than 368.

$$372 > 368$$

Worked Example 3: Compare 604 and 640

Look at the hundreds digit.

• 604 has 6 hundreds
• 640 has 6 hundreds

The hundreds are the same, so look at the tens digit.

• 604 has 0 tens
• 640 has 4 tens

Since 0 tens is less than 4 tens, 604 is less than 640.

$$604 < 640$$

Worked Example 4: Compare 888 and 888

Look at each place value.

• Hundreds: 8 and 8
• Tens: 8 and 8
• Ones: 8 and 8

All the digits are the same, so the numbers are equal.

$$888 = 888$$

Tips for Comparing Numbers

• Start at the left.
• Compare the hundreds first.
• If they are the same, compare the tens.
• If those are the same, compare the ones.
• Use >, <, or = to show your answer.

Lets Look at a Few More Quick Examples

$$200 > 199$$

200 has 2 hundreds. 199 has 1 hundred. So 200 is greater.

$$76 < 86$$

76 has 7 tens. 86 has 8 tens. So 76 is less.

$$540 = 540$$

Both numbers are exactly the same.

Summary

To compare numbers, look at the digits from left to right. Compare the hundreds, then the tens, then the ones.

Use these symbols:

• > means greater than
• < means less than
• = means equal to

Remember: the first place value that is different tells you which number is bigger.

With practice, comparing numbers gets easier and faster!

Ordering Numbers on an Open Number Line

An open number line is a number line with no numbers already marked on it, or only a few numbers marked. We use what we know about numbers to decide where other numbers should go.

When we order numbers on an open number line, we place them from least to greatest. That means the smallest number goes more to the left, and the greatest number goes more to the right.

This helps us build strong number sense. Number sense means understanding how big or small numbers are and how close numbers are to each other.

Important idea: On a number line, numbers get bigger as we move to the right.

Here is what to remember:

• Left means smaller numbers.
• Right means bigger numbers.
• Numbers that are close in value should be close together on the line.
• Numbers with a big difference should be farther apart.

Step 1: Compare the hundreds.

For 3-digit numbers, first look at the hundreds digit. A number with fewer hundreds is smaller.

For example, in \(245\) and \(678\), the number \(245\) is smaller because it has \(2\) hundreds and \(678\) has \(6\) hundreds.

Step 2: If the hundreds are the same, compare the tens.

For example, in \(412\) and \(437\), both have \(4\) hundreds. Then we look at the tens. \(1\) ten is less than \(3\) tens, so \(412\) is smaller.

Step 3: If the hundreds and tens are the same, compare the ones.

For example, in \(586\) and \(589\), both have \(5\) hundreds and \(8\) tens. Then we compare the ones. Since \(6<9\), we know \(586<589\).

How to place numbers on an open number line

1. Look at all the numbers.
2. Put them in order from least to greatest.
3. Draw a blank line with arrows.
4. Place the smallest number on the left.
5. Place the greatest number on the right.
6. Put the middle number where it belongs between them.
7. Make the spaces match the size of the differences as best you can.

You do not need every number marked. You only need to think carefully about where each number belongs.

Worked Example 1

Place \(124\), \(178\), and \(150\) on an open number line.

First, order the numbers.

All three numbers have \(1\) hundred. So we compare the tens:

• \(124\) has \(2\) tens
• \(150\) has \(5\) tens
• \(178\) has \(7\) tens

So the order is:

$$124 < 150 < 178$$

Now place them on the open number line.

\(124\) goes on the left. \(178\) goes on the right. \(150\) goes between them.

Since \(150\) is closer to \(124\) than to \(178\), it should be a little closer to \(124\).

Like this:

$$124 \qquad\qquad 150 \qquad\qquad\qquad 178$$

Worked Example 2

Place \(305\), \(290\), and \(318\) on an open number line.

First, order the numbers.

Look at the hundreds:

• \(290\) has \(2\) hundreds
• \(305\) has \(3\) hundreds
• \(318\) has \(3\) hundreds

So \(290\) is the smallest.

Now compare \(305\) and \(318\). They both have \(3\) hundreds, so compare the tens:

• \(305\) has \(0\) tens
• \(318\) has \(1\) ten

So:

$$290 < 305 < 318$$

Now place them on the open number line.

\(290\) goes left, \(318\) goes right, and \(305\) goes between them.

The jump from \(290\) to \(305\) is \(15\). The jump from \(305\) to \(318\) is \(13\). Those are close in size, so \(305\) should go about in the middle.

Like this:

$$290 \qquad\qquad 305 \qquad\qquad 318$$

Worked Example 3

Place \(460\), \(406\), and \(490\) on an open number line.

First, order the numbers.

All three numbers have \(4\) hundreds. So compare the tens:

• \(406\) has \(0\) tens
• \(460\) has \(6\) tens
• \(490\) has \(9\) tens

So:

$$406 < 460 < 490$$

Now place them on the open number line.

\(406\) goes on the left and \(490\) goes on the right.

Where does \(460\) go? It is much farther from \(406\) than from \(490\).

• From \(406\) to \(460\) is \(54\)
• From \(460\) to \(490\) is \(30\)

So \(460\) should be placed closer to \(490\) than to \(406\).

Like this:

$$406 \qquad\qquad\qquad 460 \qquad\qquad 490$$

Worked Example 4

Place \(721\), \(719\), and \(730\) on an open number line.

First, order the numbers.

Compare the hundreds. All have \(7\) hundreds.

Compare the tens:

• \(719\) has \(1\) ten
• \(721\) has \(2\) tens
• \(730\) has \(3\) tens

So:

$$719 < 721 < 730$$

Now place them on the open number line.

\(719\) and \(721\) are very close. They should be very close together on the line.

\(730\) is farther to the right.

Like this:

$$719 \quad 721 \qquad\qquad 730$$

Tips for success

• Always order the numbers first before drawing them.
• Start by comparing the hundreds digit.
• If needed, compare the tens digit next.
• If needed, compare the ones digit last.
• Think about how far apart the numbers are, not just which one is bigger.

Common mistakes to watch out for

• Mistake: Putting the greatest number on the left.
  Fix: Remember, numbers get bigger as you move right.
• Mistake: Looking only at the ones digit.
  Fix: Compare hundreds first, then tens, then ones.
• Mistake: Spacing numbers evenly when they are not evenly spaced.
  Fix: Numbers that are close should be close together.

Try this thinking:

If you need to place \(512\), \(540\), and \(518\), first put them in order:

$$512 < 518 < 540$$

Then think: \(512\) and \(518\) are only \(6\) apart, so they should be close together. \(540\) is much farther away, so it should be farther to the right.

Summary

An open number line is a blank number line that helps us show where numbers belong.

To order 3-digit numbers, compare the hundreds first, then tens, then ones. Place the smallest number on the left and the greatest number on the right.

Also think about distance. Numbers that are close in value should be close together on the number line.

Mental Math: 10 More and 10 Less

Numbers are made of hundreds, tens, and ones. When we find 10 more or 10 less, we are changing the tens part of the number.

The best part is this: the ones digit stays the same. Only the tens change by 1 ten.

For example, look at the number \(34\).

• \(34\) is 3 tens and 4 ones.
• 10 more means add 1 more ten.
• 10 less means take away 1 ten.

So:

$$34 + 10 = 44$$

$$34 - 10 = 24$$

Notice that the ones digit is still 4. Only the tens digit changed.

How to think about 10 more

• Keep the ones digit the same.
• Make the number 1 ten bigger.
• The tens digit goes up by 1.

How to think about 10 less

• Keep the ones digit the same.
• Make the number 1 ten smaller.
• The tens digit goes down by 1.

This works with bigger numbers too, even numbers with hundreds.

For example, in \(172\):

• 1 is in the hundreds place
• 7 is in the tens place
• 2 is in the ones place

If we add 10, we add 1 more ten.

$$172 + 10 = 182$$

If we subtract 10, we take away 1 ten.

$$172 - 10 = 162$$

Again, the ones digit stays the same: it is still 2.

Look for the pattern

When you move by tens, the numbers follow a pattern:

$$25,\ 35,\ 45,\ 55,\ 65$$

Every number has a 5 in the ones place. The tens are changing, but the ones stay the same.

This is why mental math with 10 more and 10 less can be quick and easy.

Worked Example 1

Find 10 more and 10 less than \(18\).

• \(18\) has 1 ten and 8 ones.
• 10 more means 2 tens and 8 ones: \(28\)
• 10 less means 0 tens and 8 ones: \(8\)

$$18 + 10 = 28$$

$$18 - 10 = 8$$

Worked Example 2

Find 10 more and 10 less than \(46\).

• The ones digit is 6, so it stays 6.
• For 10 more, the tens digit goes from 4 to 5.
• For 10 less, the tens digit goes from 4 to 3.

$$46 + 10 = 56$$

$$46 - 10 = 36$$

Worked Example 3

Find 10 more and 10 less than \(109\).

• \(109\) is 1 hundred, 0 tens, and 9 ones.
• 10 more gives 1 hundred, 1 ten, and 9 ones.
• 10 less gives 9 tens and 9 ones.

$$109 + 10 = 119$$

$$109 - 10 = 99$$

Even when a number has 0 tens, the rule still works.

Worked Example 4

Find 10 more and 10 less than \(253\).

• The ones digit is 3, so it stays 3.
• The tens digit is 5.
• 10 more changes 5 tens to 6 tens.
• 10 less changes 5 tens to 4 tens.

$$253 + 10 = 263$$

$$253 - 10 = 243$$

Tips to help you

• Do not count on by ones.
• Think: one more ten or one less ten.
• Check the ones digit. It should stay the same.
• If there is a hundreds digit, it usually stays the same too, unless the tens change past 9 tens or below 0 tens.

Try these in your head

• 10 more than \(61\) is \(71\)
• 10 less than \(61\) is \(51\)
• 10 more than \(340\) is \(350\)
• 10 less than \(340\) is \(330\)

What to remember

• 10 more means add 1 ten.
• 10 less means subtract 1 ten.
• The ones digit stays the same.
• Look at the tens place to help you answer quickly.

When you practice this, you will be able to solve it fast in your head!

Mental Math: 100 More and 100 Less

Today we will learn how to find 100 more and 100 less than a number.

This is a mental math skill. That means we can do it in our heads without writing a long addition or subtraction problem.

When we add or subtract 100, we look closely at the hundreds place.

In a 3-digit number, the digits have jobs:

• The first digit is the hundreds digit.
• The second digit is the tens digit.
• The third digit is the ones digit.

For example, in the number \(456\):

• \(4\) means 4 hundreds
• \(5\) means 5 tens
• \(6\) means 6 ones

We can write it like this:

$$456 = 400 + 50 + 6$$

If we add 100, we add 1 more hundred.

If we subtract 100, we take away 1 hundred.

Important idea: When finding 100 more or 100 less, the tens digit and ones digit stay the same. The hundreds digit changes.

Look at this pattern:

• \(356\) → 100 more is \(456\)
• \(356\) → 100 less is \(256\)

Notice that the tens digit \(5\) stayed the same, and the ones digit \(6\) stayed the same. Only the hundreds digit changed.

How to find 100 more:

1. Look at the hundreds digit.
2. Add 1 to that digit.
3. Keep the tens and ones digits the same.

How to find 100 less:

1. Look at the hundreds digit.
2. Subtract 1 from that digit.
3. Keep the tens and ones digits the same.

Here are some quick examples:

• 100 more than \(234\) is \(334\)
• 100 less than \(234\) is \(134\)
• 100 more than \(781\) is \(881\)
• 100 less than \(781\) is \(681\)

Let’s work through some examples together.

Example 1: Find 100 more than \(245\)

The hundreds digit in \(245\) is \(2\).

Add 1 to the hundreds digit: \(2 + 1 = 3\).

The tens digit \(4\) stays the same. The ones digit \(5\) stays the same.

So:

$$245 + 100 = 345$$

Example 2: Find 100 less than \(672\)

The hundreds digit in \(672\) is \(6\).

Subtract 1 from the hundreds digit: \(6 - 1 = 5\).

The tens digit \(7\) stays the same. The ones digit \(2\) stays the same.

So:

$$672 - 100 = 572$$

Example 3: Find 100 more than \(809\)

The hundreds digit in \(809\) is \(8\).

Add 1 to the hundreds digit: \(8 + 1 = 9\).

The tens digit is \(0\), and it stays the same. The ones digit \(9\) stays the same too.

So:

$$809 + 100 = 909$$

Example 4: Find 100 less than \(930\)

The hundreds digit in \(930\) is \(9\).

Subtract 1 from the hundreds digit: \(9 - 1 = 8\).

The tens digit \(3\) stays the same. The ones digit \(0\) stays the same.

So:

$$930 - 100 = 830$$

Let’s notice a pattern on a number line:

$$214,\ 314,\ 414,\ 514$$

Each number is 100 more than the number before it.

And here is a counting-back pattern:

$$648,\ 548,\ 448,\ 348$$

Each number is 100 less than the number before it.

Helpful tip: Say the number parts to yourself.

For \(523\), think: 5 hundreds, 2 tens, 3 ones.

100 more means 6 hundreds, 2 tens, 3 ones, so the answer is \(623\).

100 less means 4 hundreds, 2 tens, 3 ones, so the answer is \(423\).

Be careful! A common mistake is changing the tens or ones digit.

For example, 100 more than \(461\) is not \(462\).

Adding 100 changes the hundreds digit, not the ones digit.

The correct answer is:

$$461 + 100 = 561$$

Another common mistake is forgetting that the tens and ones stay the same.

For example, 100 less than \(387\) is 287, not \(277\).

Only the hundreds digit changes from \(3\) to \(2\).

Try these in your head:

• 100 more than \(154\) is \(254\)
• 100 less than \(154\) is \(54\)? No. Since we are working with hundreds, think of \(154\) as 1 hundred, 5 tens, 4 ones. Taking away 1 hundred gives \(54\), which is correct.
• 100 more than \(490\) is \(590\)
• 100 less than \(490\) is \(390\)

Sometimes the answer can become a 2-digit number after taking away 100. That is okay.

For example:

$$154 - 100 = 54$$

Summary

To find 100 more, add 1 to the hundreds digit.

To find 100 less, subtract 1 from the hundreds digit.

The tens and ones digits stay the same.

This helps us solve problems quickly using mental math.

Even and Odd Numbers

Numbers can be grouped in different ways. One important way is to decide if a number is even or odd.

Knowing if a number is even or odd helps us count, sort numbers, and understand how numbers work.

An even number can be split into 2 equal groups. When we pair up all the objects, none are left over.

An odd number cannot be split into 2 equal groups. When we pair up all the objects, 1 is left over.

We can think about it like this:

• Even numbers make perfect pairs.
• Odd numbers have 1 extra.

Here are some even numbers:

\(0, 2, 4, 6, 8, 10, 12, 14\)

Here are some odd numbers:

\(1, 3, 5, 7, 9, 11, 13, 15\)

A quick way to tell if a number is even or odd is to look at the ones digit.

• If the ones digit is \(0, 2, 4, 6,\) or \(8\), the number is even.
• If the ones digit is \(1, 3, 5, 7,\) or \(9\), the number is odd.

This works for small numbers and big numbers too, even all the way to \(1{,}000\).

For example:

• \(24\) is even because the ones digit is \(4\).
• \(57\) is odd because the ones digit is \(7\).
• \(130\) is even because the ones digit is \(0\).
• \(999\) is odd because the ones digit is \(9\).

Let’s connect this to place value. In a number like \(246\), there are:

• \(2\) hundreds
• \(4\) tens
• \(6\) ones

To tell if \(246\) is even or odd, we only need to look at the ones. The ones digit is \(6\), so \(246\) is even.

In a number like \(713\), there are:

• \(7\) hundreds
• \(1\) ten
• \(3\) ones

The ones digit is \(3\), so \(713\) is odd.

Worked Example 1

Is \(8\) even or odd?

Let’s pair 8 objects:

All 8 objects make pairs. There is nothing left over.

So, \(8\) is even.

Worked Example 2

Is \(11\) even or odd?

Let’s make pairs:

There is 1 left over.

So, \(11\) is odd.

Worked Example 3

Is \(36\) even or odd?

Look at the ones digit. The ones digit in \(36\) is \(6\).

Since \(6\) is one of \(0, 2, 4, 6, 8\), the number is even.

So, \(36\) is even.

Worked Example 4

Is \(145\) even or odd?

Look at the ones digit. The ones digit in \(145\) is \(5\).

Since \(5\) is one of \(1, 3, 5, 7, 9\), the number is odd.

So, \(145\) is odd.

Tips to Remember

• Even numbers can be shared into 2 equal groups.
• Odd numbers have 1 left over.
• Look at the ones digit to decide quickly.
• The hundreds and tens do not change whether a number is even or odd.

You can also notice a pattern when counting:

\(1\) odd, \(2\) even, \(3\) odd, \(4\) even, \(5\) odd, \(6\) even

The pattern keeps going: odd, even, odd, even.

Summary

An even number can be split into 2 equal groups with no leftovers. An odd number has 1 left over.

To tell if a number is even or odd, look at the ones digit.

• \(0, 2, 4, 6, 8\) means even
• \(1, 3, 5, 7, 9\) means odd

So whether the number is \(6\), \(42\), \(318\), or \(1{,}000\), the ones digit tells you if it is even or odd.

Benchmark Numbers and Rounding Concepts

Sometimes exact numbers are hard to use quickly. In math, we can use benchmark numbers to help us think. Benchmark numbers are friendly numbers like 10, 50, 100, and 200. They are easy to count by and easy to compare.

We also use benchmark numbers to help with rounding. Rounding means thinking about which friendly number a number is closest to. This helps us estimate, or make a close guess.

Estimating is helpful when we want to know about how many, about how much, or if an answer makes sense. We do not always need the exact number first. A close number can help us think faster.

What are benchmark numbers?

Benchmark numbers are numbers we know well and can use as helpers. In 2nd grade, common benchmark numbers are:

• 10
• 50
• 100
• 150
• 200
• and other tens or hundreds

These numbers are helpful because they are neat and easy to work with. For example, it is easier to think about 49 as being very close to 50 than to work with 49 exactly right away.

Why do we use benchmark numbers?

• To make counting and comparing easier
• To make a quick estimate
• To check if an answer is reasonable
• To think about numbers in groups of tens and hundreds

Thinking about what number is closest

When we round, we ask: Which benchmark number is this number closest to?

For example, think about 47. It is close to 50. It is not as close to 40. So if we use a benchmark number, 47 is about 50.

Think about 92. It is very close to 100. So we can say 92 is about 100.

Using a number line

A number line can help us see which benchmark number is closer.

Here is a number line from 40 to 50:

$$40 \qquad 41 \qquad 42 \qquad 43 \qquad 44 \qquad 45 \qquad 46 \qquad 47 \qquad 48 \qquad 49 \qquad 50$$

If the number is 47, it is only 3 away from 50, but 7 away from 40. That means 47 rounds to 50 when we use tens.

Rounding to the nearest ten

When we round to the nearest ten, we find the closest ten. The tens are:

$$10, 20, 30, 40, 50, 60, 70, 80, 90, 100$$

Examples:

• 21 is close to 20
• 38 is close to 40
• 54 is close to 50
• 68 is close to 70

A simple way to think is:

• If a number is just a little more than a ten, it may round down to that ten.
• If a number is close to the next ten, it may round up to the next ten.

For example:

• 32 is close to 30
• 39 is close to 40

Rounding to the nearest hundred

We can also use benchmark numbers that are hundreds. When we round to the nearest hundred, we find the closest hundred.

The hundreds are:

$$100, 200, 300, 400, 500, \dots$$

Examples:

• 112 is close to 100
• 189 is close to 200
• 260 is close to 300
• 341 is close to 300

We can think: Is the number closer to the hundred below or the hundred above?

Worked Example 1

Round 48 to a benchmark ten.

The tens around 48 are 40 and 50.

• 48 is 8 away from 40
• 48 is 2 away from 50

Since 48 is closer to 50, we say:

$$48 \approx 50$$

Worked Example 2

Round 73 to a benchmark ten.

The tens around 73 are 70 and 80.

• 73 is 3 away from 70
• 73 is 7 away from 80

Since 73 is closer to 70, we say:

$$73 \approx 70$$

Worked Example 3

Round 94 to a benchmark number.

94 is very close to 100. It is only 6 away from 100.

So we can estimate:

$$94 \approx 100$$

This helps us quickly think of 94 as about 100.

Worked Example 4

Round 176 to the nearest hundred.

The hundreds around 176 are 100 and 200.

• 176 is 76 away from 100
• 176 is 24 away from 200

Since 176 is closer to 200, we say:

$$176 \approx 200$$

Using benchmark numbers to estimate

Benchmark numbers help us make quick estimates in addition too.

Suppose there are 49 apples in one basket and 52 apples in another basket.

We can use benchmark numbers:

• 49 is about 50
• 52 is about 50

So the total is about:

$$50 + 50 = 100$$

The exact answer is:

$$49 + 52 = 101$$

Our estimate of 100 is very close, so it is reasonable.

Checking if an answer makes sense

Estimating can help us check our work.

If you add 61 + 37, you can estimate first:

• 61 is about 60
• 37 is about 40

Then:

$$60 + 40 = 100$$

Now find the exact answer:

$$61 + 37 = 98$$

Since 98 is close to 100, the answer makes sense.

Tips for students

• Look for the two friendly numbers around your number.
• Ask, “Which one is closer?”
• Use tens for smaller numbers like 34 or 67.
• Use hundreds for bigger numbers like 142 or 389.
• Remember: an estimate is close, not exact.

Let’s think together

• 36 is about 40
• 81 is about 80
• 98 is about 100
• 205 is about 200

Summary

Benchmark numbers are friendly numbers like 10, 50, and 100. We use them to estimate and to round numbers to a close, easy number. Rounding helps us decide which benchmark number is nearest. Estimating with benchmark numbers helps us count, compare, add, and check if an answer makes sense.