Place Value and Base-Ten Architecture

Unitizing Tens and Hundreds means grouping small units into bigger units. In our number system, 10 ones make 1 ten, and 10 tens make 1 hundred. This helps us read, write, and understand numbers more easily.

Think of single cubes. One cube is 1 one. If you snap together 10 cubes, you make a ten. If you put together 10 tens, you make a hundred. We are still counting the same amount, but now we are counting it in bigger groups.

This is called unitizing. Unitizing means we treat a group as one new unit. So 10 ones can be thought of as 1 ten. And 10 tens can be thought of as 1 hundred.

Here is the big idea:

• 10 ones = 1 ten
• 10 tens = 1 hundred
• 1 hundred = 100 ones

We can write these ideas with numbers:

\(10 \times 1 = 10\)

\(10 \text{ ones} = 1 \text{ ten}\)

\(10 \times 10 = 100\)

\(10 \text{ tens} = 1 \text{ hundred}\)

When we group by tens, counting gets faster. Instead of counting 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, we can say, “That is 1 ten.” Then 20 is 2 tens, 30 is 3 tens, and so on.

When we group by hundreds, counting gets even faster. Instead of thinking about 100 single ones, we can say, “That is 1 hundred.”

Place value shows the value of a digit by where it is. In a 3-digit number:

• the right digit tells the number of ones
• the middle digit tells the number of tens
• the left digit tells the number of hundreds

For example, in the number \(243\):

• \(2\) means 2 hundreds
• \(4\) means 4 tens
• \(3\) means 3 ones

So \(243\) is:

$$243 = 2\text{ hundreds} + 4\text{ tens} + 3\text{ ones}$$

This also means:

$$243 = 200 + 40 + 3$$

Notice how each place is 10 times bigger than the place to its right.

• 1 ten is 10 times as much as 1 one.
• 1 hundred is 10 times as much as 1 ten.

This is why our base-ten system is called base ten. We keep making new units by groups of 10.

Worked Example 1: Make a ten from ones

If you have 10 ones, how many tens is that?

Step 1: Count the ones: \(10\).

Step 2: Group them into one group of 10.

Step 3: One group of 10 ones is 1 ten.

Answer: \(10\) ones = \(1\) ten.

Worked Example 2: Count tens

If you have 6 tens, how many ones is that?

Step 1: Each ten is 10 ones.

Step 2: Multiply or skip-count by 10 six times.

$$6 \text{ tens} = 6 \times 10 = 60 \text{ ones}$$

Answer: \(6\) tens = \(60\) ones.

Worked Example 3: Make a hundred from tens

If you have 10 tens, how many hundreds is that?

Step 1: Count the tens: \(10\) tens.

Step 2: Group 10 tens together.

Step 3: 10 tens make 1 hundred.

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

Answer: \(10\) tens = \(1\) hundred.

Worked Example 4: Read a number using hundreds, tens, and ones

What does \(372\) mean?

Step 1: Look at each digit and its place.

• \(3\) is in the hundreds place, so it means 3 hundreds.
• \(7\) is in the tens place, so it means 7 tens.
• \(2\) is in the ones place, so it means 2 ones.

Step 2: Write it in expanded form.

$$372 = 300 + 70 + 2$$

Step 3: Think about the units.

$$372 = 3\text{ hundreds} + 7\text{ tens} + 2\text{ ones}$$

Answer: \(372\) means 3 hundreds, 7 tens, and 2 ones.

Helpful ways to think about unitizing

• 10 pennies can be thought of as 1 group of ten pennies.
• 10 bundles of 10 sticks can be thought of as 1 bundle of one hundred sticks.
• A number can be made of hundreds, tens, and ones at the same time.

Common mistakes to avoid

• Do not think that 10 tens is 10. 10 tens = 100.
• Do not forget the unit. A 4 in the tens place means 4 tens, not 4 ones.
• Remember that the same digit can have different values in different places.

For example, compare these numbers:

• In \(4\), the 4 means 4 ones.
• In \(40\), the 4 means 4 tens.
• In \(400\), the 4 means 4 hundreds.

Even though the digit is the same, its value changes because its place changes.

Quick check

1. How many tens are in \(20\)? 2 tens
2. How many ones are in \(5\) tens? 50 ones
3. How many hundreds are in \(10\) tens? 1 hundred
4. What does \(185\) mean? 1 hundred, 8 tens, 5 ones

Summary

Unitizing helps us group numbers into bigger units. 10 ones make 1 ten, and 10 tens make 1 hundred. This helps us understand place value and read numbers up to 1,000 more easily.

Base-Ten Block Representations

Numbers are made of parts. In our base-ten number system, those parts are ones, tens, and hundreds.

Base-ten blocks help us see these parts. They are special blocks we use to build numbers.

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

When we use base-ten blocks, we are showing the value of each digit in a number.

For example, in the number \(243\):

• The 2 means 2 hundreds
• The 4 means 4 tens
• The 3 means 3 ones

So \(243\) can be modeled with:

• 2 flats
• 4 longs
• 3 cubes

This is called a base-ten block representation.

Why do base-ten blocks help?

They help us understand that bigger places are made from smaller places.

• 10 ones make 1 ten
• 10 tens make 1 hundred

We can write that as:

\(10 \text{ ones} = 1 \text{ ten}\)

\(10 \text{ tens} = 1 \text{ hundred}\)

This is why the system is called base ten. Every time we get 10 of something, we can trade it for 1 of the next bigger unit.

How to model a number with base-ten blocks

1. Look at the hundreds digit.
2. Use that many flats.
3. Look at the tens digit.
4. Use that many longs.
5. Look at the ones digit.
6. Use that many cubes.

Let’s practice with some worked examples.

Example 1: Model \(16\)

The number \(16\) has:

• 1 ten
• 6 ones

So we use:

• 1 long
• 6 cubes

We can also write:

$$16 = 10 + 6$$

This shows that \(16\) is made of 1 group of ten and 6 extra ones.

Example 2: Model \(34\)

The number \(34\) has:

• 3 tens
• 4 ones

So we use:

• 3 longs
• 4 cubes

We can write:

$$34 = 30 + 4$$

If we counted all the blocks, we would get \(34\).

Example 3: Model \(152\)

The number \(152\) has:

• 1 hundred
• 5 tens
• 2 ones

So we use:

• 1 flat
• 5 longs
• 2 cubes

We can write:

$$152 = 100 + 50 + 2$$

This helps us see that the digit 5 in \(152\) does not mean 5 ones. It means 5 tens, or \(50\).

Example 4: What number do these blocks show?

Suppose you see:

• 3 flats
• 2 longs
• 7 cubes

Let’s find the number:

• 3 flats = 3 hundreds = 300
• 2 longs = 2 tens = 20
• 7 cubes = 7 ones = 7

Add them together:

$$300 + 20 + 7 = 327$$

So the blocks show 327.

Important idea: A digit’s place changes its value

Look at the digit 4 in these numbers:

• \(4\) means 4 ones
• \(40\) means 4 tens
• \(400\) means 4 hundreds

Base-ten blocks help us see this clearly:

• \(4\) = 4 cubes
• \(40\) = 4 longs
• \(400\) = 4 flats

Even though the digit is the same, its place changes its value.

Trading blocks

Sometimes we can trade smaller blocks for a bigger block.

• 10 cubes can be traded for 1 long
• 10 longs can be traded for 1 flat

For example, if you have 10 ones, you can trade them for 1 ten.

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

If you have 2 tens and 10 ones, you can trade the 10 ones for 1 more ten. Then you have 3 tens.

This helps when building numbers and when adding.

Reading base-ten block representations

When you look at a picture or set of blocks, ask yourself:

1. How many hundreds are there?
2. How many tens are there?
3. How many ones are there?

Then put the digits together in that order: hundreds, tens, ones.

For example:

• 4 flats
• 0 longs
• 6 cubes

This is \(406\).

The 0 in the tens place is important. It tells us there are no tens.

Common mistakes to avoid

• Do not count a long as 1. A long is 10.
• Do not count a flat as 1. A flat is 100.
• Always check the place: hundreds, tens, then ones.
• Remember that 0 can be a place holder, like in \(406\).

Try thinking like this

If you see a number such as \(281\), you can say:

• 2 hundreds
• 8 tens
• 1 one

And if you want to build it, you would use:

• 2 flats
• 8 longs
• 1 cube

Summary

Base-ten blocks help us model numbers by showing ones, tens, and hundreds. A cube stands for 1, a long stands for 10, and a flat stands for 100.

To model a number, match each digit to its place value. To read blocks, count the hundreds, tens, and ones, then write the number.

When you understand base-ten blocks, you can better understand how numbers are built.

Standard, Word, and Expanded Forms

Numbers can be written in different ways. In 3rd Grade, it is important to know how to read, write, and understand numbers up to 1,000.

There are 3 main ways to write a number:

• Standard form – writing the number with digits, like \(456\)
• Word form – writing the number with words, like four hundred fifty-six
• Expanded form – showing the value of each digit, like $$456 = 400 + 50 + 6$$

These 3 forms all name the same number. They just show it in different ways.

Why is this important?

When you know all 3 forms, you understand what each digit in a number means. This helps you read numbers, compare numbers, add and subtract, and solve word problems.

Place Value Review

To understand expanded form, you need to know place value. In a 3-digit number, each digit has a place:

• Hundreds
• Tens
• Ones

Look at the number \(456\).

• The \(4\) is in the hundreds place, so it means \(400\).
• The \(5\) is in the tens place, so it means \(50\).
• The \(6\) is in the ones place, so it means \(6\).

So:

$$456 = 400 + 50 + 6$$

1. Standard Form

Standard form is the usual way we write a number using digits.

Examples:

• \(23\)
• \(109\)
• \(782\)

When you see standard form, think: What is each digit worth?

2. Word Form

Word form means writing the number in words.

Examples:

• \(23\) is twenty-three
• \(109\) is one hundred nine
• \(782\) is seven hundred eighty-two

When writing word form:

• Say the number out loud.
• Write the hundreds, tens, and ones in order.
• For numbers like twenty-one or fifty-six, use a hyphen.

3. Expanded Form

Expanded form shows the value of each digit by adding them together.

Examples:

• \(23 = 20 + 3\)
• \(109 = 100 + 9\)
• \(782 = 700 + 80 + 2\)

Expanded form helps you see how a number is built from hundreds, tens, and ones.

How to Change a Number into Expanded Form

1. Look at each digit.
2. Find its place value.
3. Write its value.
4. Add the values together.

Example with \(364\):

• \(3\) hundreds = \(300\)
• \(6\) tens = \(60\)
• \(4\) ones = \(4\)

So:

$$364 = 300 + 60 + 4$$

What About Zero?

Sometimes a number has a zero in one place. Zero means there are none in that place.

Look at \(407\):

• \(4\) hundreds = \(400\)
• \(0\) tens = \(0\)
• \(7\) ones = \(7\)

So the expanded form can be written as:

$$407 = 400 + 0 + 7$$

Many times, we leave out the zero part and write:

$$407 = 400 + 7$$

The word form is four hundred seven.

Worked Examples

Example 1: Write \(245\) in word form and expanded form.

Step 1: Look at each digit.

• \(2\) hundreds = \(200\)
• \(4\) tens = \(40\)
• \(5\) ones = \(5\)

Expanded form:

$$245 = 200 + 40 + 5$$

Word form: two hundred forty-five

Example 2: Write six hundred thirty-two in standard form and expanded form.

Step 1: Think about the place values.

• six hundred = \(600\)
• thirty = \(30\)
• two = \(2\)

Put them together in standard form:

$$632$$

Expanded form:

$$632 = 600 + 30 + 2$$

Example 3: Write \(508\) in word form and expanded form.

Step 1: Find the value of each digit.

• \(5\) hundreds = \(500\)
• \(0\) tens = \(0\)
• \(8\) ones = \(8\)

Expanded form:

$$508 = 500 + 8$$

Word form: five hundred eight

Notice that there are no tens.

Example 4: Write the number shown in expanded form, $$700 + 20 + 9$$, in standard form and word form.

Step 1: Add the place values together.

• \(700\) means 7 hundreds
• \(20\) means 2 tens
• \(9\) means 9 ones

Standard form:

$$729$$

Word form: seven hundred twenty-nine

Tips to Remember

• Standard form uses digits.
• Word form uses words.
• Expanded form shows place value by adding.
• Always look at the hundreds, tens, and ones places.
• If a digit is \(0\), that place has no value to add.

Try Thinking About These

• What is the word form of \(361\)?
• What is the expanded form of \(914\)?
• What is the standard form of eight hundred six?
• What number is $$300 + 50 + 1$$?

Brief Summary

A number can be written in standard form, word form, and expanded form. Standard form uses digits, word form uses words, and expanded form shows the value of each digit. When you understand place value, you can change a number from one form to another.

Flexible Decomposition of Numbers

Numbers can be broken apart in different ways. This is called decomposing a number.

Sometimes we break a number into hundreds, tens, and ones in the usual way. For example, \(456\) can be written as \(400 + 50 + 6\).

But numbers can also be broken apart in flexible ways. That means there is more than one correct way to split the same number.

For example, \(456\) is not only \(400 + 50 + 6\). It can also be:

• \(400 + 40 + 16\)
• \(300 + 150 + 6\)
• \(200 + 200 + 50 + 6\)

All of these parts still add up to \(456\). That is why all of them are correct.

Why is this important?

Flexible decomposition helps you understand numbers better. It also helps when you add, subtract, compare numbers, and solve word problems.

When you decompose flexibly, you are taking value from one place and moving it to another place without changing the total number.

For example, 1 ten is the same as 10 ones. So in \(456\), you can take one ten from \(50\) and turn it into 10 ones:

$$456 = 400 + 40 + 16$$

The total stays the same. Only the parts change.

Remember these place value facts:

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

These facts help us make new decompositions.

Standard decomposition means breaking a number into hundreds, tens, and ones in the usual way.

For example:

$$372 = 300 + 70 + 2$$

Non-standard decomposition means breaking a number into different parts that still make the same total.

For example:

$$372 = 300 + 60 + 12$$

Here, one ten was changed into 10 ones. So \(70 + 2\) became \(60 + 12\).

You can also change one hundred into 10 tens. That helps make even more decompositions.

For example, in \(372\), the \(300\) can become \(200 + 100\), and that \(100\) can be thought of as \(10\) tens. So one correct decomposition is:

$$372 = 200 + 170 + 2$$

It still equals \(372\).

How to decompose a number flexibly

1. Write the number in standard form.
2. Choose one part to break apart.
3. Trade a hundred for 10 tens, or a ten for 10 ones.
4. Check that the parts still add to the original number.

Let’s look at some examples.

Example 1: Decompose \(245\)

First, write it in standard form:

$$245 = 200 + 40 + 5$$

Now trade one ten for 10 ones.

Then \(40 + 5\) becomes \(30 + 15\).

$$245 = 200 + 30 + 15$$

This is a flexible decomposition.

Check:

$$200 + 30 + 15 = 245$$

Example 2: Decompose \(618\) in a different way

Standard form:

$$618 = 600 + 10 + 8$$

Trade one hundred for 10 tens.

Then \(600\) can become \(500 + 100\), and \(100\) is the same as \(10\) tens.

So \(500 + 100 + 10 + 8\) is the same as \(500 + 110 + 8\).

$$618 = 500 + 110 + 8$$

Check:

$$500 + 110 + 8 = 618$$

Example 3: Find two ways to decompose \(456\)

Standard form:

$$456 = 400 + 50 + 6$$

Way 1: Trade one ten for 10 ones.

$$456 = 400 + 40 + 16$$

Way 2: Trade one hundred for 10 tens.

Then \(400 + 50\) can be thought of as \(300 + 150\).

$$456 = 300 + 150 + 6$$

Both are correct because both equal \(456\).

Example 4: Is this decomposition correct?

Maria says:

$$582 = 400 + 180 + 2$$

Let’s check by adding the parts.

$$400 + 180 + 2 = 582$$

Yes, it is correct.

How did this happen? The standard form is:

$$582 = 500 + 80 + 2$$

Maria traded one hundred for 10 tens. So \(500 + 80\) became \(400 + 180\).

Helpful thinking tips

• If you take away 1 ten, add 10 ones.
• If you take away 1 hundred, add 10 tens.
• The number does not change if the total stays the same.
• Always add the parts to check your work.

Watch out for mistakes

• Do not forget that 1 ten = 10 ones, not 1 one.
• Do not forget that 1 hundred = 10 tens.
• Make sure the parts add back to the original number.

For example, \(456 = 400 + 45 + 6\) is not correct because:

$$400 + 45 + 6 = 451$$

The total changed, so the decomposition is wrong.

Let’s practice thinking

If a number is \(734\), its standard form is:

$$734 = 700 + 30 + 4$$

If you trade one ten for 10 ones, you get:

$$734 = 700 + 20 + 14$$

If you trade one hundred for 10 tens, you get:

$$734 = 600 + 130 + 4$$

Both are correct flexible decompositions.

Summary

Flexible decomposition means breaking a number into different parts without changing its value.

You can use place value to do this:

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

A number can have many correct decompositions. The most important rule is that all the parts must add up to the original number.

Lesson: The Role of Zero as a Placeholder

When we write numbers, each digit has a place. In 3rd Grade, we often work with the ones place, tens place, and hundreds place.

Zero is a very important digit because it can act as a placeholder. A placeholder shows that a place is empty, while keeping the other digits in the correct places.

For example, in the number \(405\), the \(0\) means there are no tens. But it also helps the \(4\) stay in the hundreds place and the \(5\) stay in the ones place.

Without the zero, the number would look different. If we wrote \(45\), that would mean 4 tens and 5 ones. That is not the same as \(405\), which means 4 hundreds, 0 tens, and 5 ones.

Why place value matters

Every digit has a value based on its place:

• Ones place: groups of 1
• Tens place: groups of 10
• Hundreds place: groups of 100

Look at this number:

$$405 = 4 \text{ hundreds} + 0 \text{ tens} + 5 \text{ ones}$$

That means:

$$405 = 400 + 0 + 5$$

The zero does not mean “nothing important.” It means there are no tens, and that information is important.

Zero holds a place

Think of place value like labeled boxes:

• Hundreds box
• Tens box
• Ones box

If one box is empty, we still need to show that it is empty. We use a zero to show the empty place.

For the number “four hundred five”:

• 4 goes in the hundreds place
• 0 goes in the tens place
• 5 goes in the ones place

So we write:

$$405$$

We do not write \(4005\), because that would mean 4 thousands, 0 hundreds, 0 tens, and 5 ones. That is a much bigger number.

Reading and writing numbers with zero

When you hear a number word, listen for the hundreds, tens, and ones.

• “Three hundred two” means 3 hundreds, 0 tens, and 2 ones: \(302\)
• “Six hundred forty” means 6 hundreds, 4 tens, and 0 ones: \(640\)
• “Nine hundred nine” means 9 hundreds, 0 tens, and 9 ones: \(909\)

If a place has no value, use zero as the placeholder.

Worked Example 1

Write “two hundred five” as a number.

Step 1: Find the hundreds. There are 2 hundreds, so put \(2\) in the hundreds place.

Step 2: Is there a tens number named? No. So put \(0\) in the tens place.

Step 3: There are 5 ones, so put \(5\) in the ones place.

$$205$$

So, “two hundred five” is written as 205.

Worked Example 2

What does \(470\) mean?

Look at each digit by place:

• \(4\) in the hundreds place means 4 hundreds
• \(7\) in the tens place means 7 tens
• \(0\) in the ones place means 0 ones

$$470 = 400 + 70 + 0$$

So, \(470\) means four hundred seventy.

The zero shows there are no ones.

Worked Example 3

Write “seven hundred three” as a number.

We have:

• 7 hundreds
• 0 tens
• 3 ones

So we write:

$$703$$

Be careful: do not write \(73\), because that means 7 tens and 3 ones. Do not write \(7003\), because that means 7 thousands and 3 ones.

Worked Example 4

What is the difference between \(405\) and \(450\)?

Let’s break them apart:

$$405 = 400 + 0 + 5$$

$$450 = 400 + 50 + 0$$

In \(405\), the zero is in the tens place, so there are no tens.

In \(450\), the zero is in the ones place, so there are no ones.

That is why the numbers are different. The place of the zero matters.

Common mistakes to watch for

• Writing \(45\) instead of \(405\). This leaves out the hundreds place.
• Writing \(4005\) instead of \(405\). This adds a thousands place that does not belong.
• Forgetting that zero can be in the tens place or the ones place.

Helpful way to check

Ask yourself:

1. How many hundreds are there?
2. How many tens are there?
3. How many ones are there?

If one answer is “none,” write a zero in that place.

For example, “eight hundred one” means:

• 8 hundreds
• 0 tens
• 1 one

So the number is:

$$801$$

Summary

Zero is a placeholder in our base-ten number system. It shows that a place has no value, but it keeps the other digits in the correct places.

In numbers like \(405\), the zero tells us there are no tens. This helps us know the number is 4 hundreds and 5 ones, not 45 and not 4005.

When writing numbers, always think about the hundreds, tens, and ones places. If one place is empty, use zero to hold that place.

Skip Counting Sequences and Multiples

Skip counting means counting forward by the same amount each time instead of by 1.

For example, if you count by 2s, you say numbers like 2, 4, 6, 8, and 10. Each number is 2 more than the number before it.

Skip counting helps us notice number patterns. It also helps us get ready for multiplication, because skip counting by the same number again and again is like adding that number over and over.

What are multiples?

A multiple of a number is a number you get when you keep adding that same number.

• Multiples of 2: 2, 4, 6, 8, 10, 12, ...
• Multiples of 5: 5, 10, 15, 20, 25, 30, ...
• Multiples of 10: 10, 20, 30, 40, 50, ...
• Multiples of 100: 100, 200, 300, 400, ...

When we skip count, we are saying the multiples of that number.

Skip counting by 2s

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

Example:

$$2,\ 4,\ 6,\ 8,\ 10,\ 12$$

We can show the pattern like this:

$$2 \rightarrow 4 \rightarrow 6 \rightarrow 8$$

Each jump is +2.

Numbers you say when skip counting by 2s are often called even numbers.

Skip counting by 5s

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

Example:

$$5,\ 10,\ 15,\ 20,\ 25,\ 30$$

Look at the ones digits: 5, 0, 5, 0, 5, 0. That pattern repeats.

This is a helpful clue. Multiples of 5 end in 0 or 5.

Skip counting by 10s

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

Example:

$$10,\ 20,\ 30,\ 40,\ 50$$

When counting by 10s, the tens digit changes by 1 each time, and the ones digit stays 0.

All multiples of 10 end in 0.

Skip counting by 100s

When you skip count by 100s, you add 100 each time.

Example:

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

When counting by 100s, the hundreds digit changes by 1 each time. The tens and ones digits stay the same if you start on a full hundred.

You can start from any number

You do not have to start at 0, 2, 5, 10, or 100. You can start at any given number and keep adding the skip-counting amount.

For example, if you start at 7 and count by 2s, you get:

$$7,\ 9,\ 11,\ 13,\ 15$$

If you start at 23 and count by 5s, you get:

$$23,\ 28,\ 33,\ 38,\ 43$$

We are still adding the same amount each time. That is what makes it skip counting.

How to find the next number in a skip-counting pattern

1. Look at how much the numbers are increasing.
2. Keep adding that same amount.
3. Check that the pattern stays the same.

Example: In the pattern 30, 40, 50, 60, the numbers go up by 10 each time. So the next number is 70.

How skip counting connects to repeated addition

Skip counting by 5s means adding 5 again and again.

For example:

$$5,\ 10,\ 15,\ 20$$

This is the same as:

$$5$$

$$5+5=10$$

$$5+5+5=15$$

$$5+5+5+5=20$$

So skip counting helps us see groups of the same size.

Worked Example 1: Count by 2s

Fill in the missing numbers:

$$8,\ 10,\ \square,\ \square,\ 16$$

Step 1: Find the pattern. From 8 to 10 is +2.

Step 2: Keep adding 2.

$$8,\ 10,\ 12,\ 14,\ 16$$

Answer: The missing numbers are 12 and 14.

Worked Example 2: Start from any number and count by 5s

Start at 12. Write the next 4 numbers when skip counting by 5s.

Step 1: Start at 12.

Step 2: Add 5 each time.

$$12+5=17$$

$$17+5=22$$

$$22+5=27$$

$$27+5=32$$

Answer: $$12,\ 17,\ 22,\ 27,\ 32$$

Worked Example 3: Count by 10s

What are the next 3 numbers?

$$46,\ 56,\ 66,\ \square,\ \square,\ \square$$

Step 1: Find the change. From 46 to 56 is +10. From 56 to 66 is +10.

Step 2: Keep adding 10.

$$66+10=76$$

$$76+10=86$$

$$86+10=96$$

Answer: The next 3 numbers are 76, 86, 96.

Notice that the ones digit stayed the same. It stayed 6 the whole time.

Worked Example 4: Count by 100s

Start at 350. Count by 100s for 4 more numbers.

Step 1: Add 100 each time.

$$350+100=450$$

$$450+100=550$$

$$550+100=650$$

$$650+100=750$$

Answer: $$350,\ 450,\ 550,\ 650,\ 750$$

Notice that the last two digits stayed 50. Only the hundreds changed.

Helpful patterns to remember

• Count by 2s: add 2 each time.
• Count by 5s: add 5 each time.
• Count by 10s: add 10 each time.
• Count by 100s: add 100 each time.

• Multiples of 2 are even numbers.
• Multiples of 5 end in 0 or 5.
• Multiples of 10 end in 0.
• When counting by 100s, the hundreds digit changes by 1 each time.

Tips for success

• Say the numbers out loud to hear the pattern.
• Use a number line if you need to see the jumps.
• Check that the amount added each time stays the same.
• If you get stuck, start again and add step by step.

Summary

Skip counting means adding the same number again and again. We can skip count by 2s, 5s, 10s, and 100s, and we can start from any number. The numbers we say in skip counting are called multiples. Looking for patterns in the digits can help us extend and create skip-counting sequences.

Relative Magnitude and Number Lines helps us understand where numbers belong, which numbers are greater or smaller, and how far apart numbers are.

A number line is a straight line that shows numbers in order. As you move to the right, the numbers get bigger. As you move to the left, the numbers get smaller.

When we talk about relative magnitude, we are comparing the size of numbers. We can ask questions like:

• Which number is greater?
• Which number is smaller?
• Which number is between two other numbers?
• How far apart are two numbers?

Number lines are a great tool because they let us see numbers in order and compare them easily.

Main Idea 1: Numbers go in order on a number line.

Here is a simple number line:

$$0 \qquad 1 \qquad 2 \qquad 3 \qquad 4 \qquad 5$$

Each step to the right adds 1. So after 3 comes 4, and after 4 comes 5.

Each step to the left subtracts 1. So before 5 comes 4, and before 4 comes 3.

This helps us know the order of numbers.

Main Idea 2: Bigger numbers are farther to the right.

On a number line, the number farther to the right is always greater.

• 8 is greater than 5 because 8 is to the right of 5.
• 12 is less than 15 because 12 is to the left of 15.

We can write these comparisons like this:

\(8 > 5\)

\(12 < 15\)

Main Idea 3: The spaces between tick marks must be equal.

On a number line, the marks are called tick marks. The distance between each tick mark should be the same. This is important because each step should stand for the same amount.

For example, if one step means 1, then every step means 1. If one step means 10, then every step means 10.

Main Idea 4: Number lines can count by 1s, 10s, 100s, or other equal amounts.

Sometimes a number line shows every number. Sometimes it skips count by equal steps.

• Counting by 1s: \(24, 25, 26, 27\)
• Counting by 10s: \(40, 50, 60, 70\)
• Counting by 100s: \(300, 400, 500, 600\)

To place a number correctly, first figure out what each jump is worth.

Main Idea 5: A number can be placed on empty, partially labeled, or fully labeled number lines.

A fully labeled number line shows many or all of the numbers.

A partially populated number line shows only some numbers, so we must figure out the missing ones.

An empty number line may show no numbers except a start and end, so we use what we know about order and equal spacing.

How to place a number on a number line

1. Look at the numbers that are already shown.
2. Figure out the pattern or step size.
3. Count from one labeled mark to the next.
4. Place the number where it belongs.
5. Check: Is the number in the right order? Is it the right distance from the other numbers?

Worked Example 1: Fully labeled number line

A number line shows:

$$10 \qquad 11 \qquad 12 \qquad 13 \qquad 14 \qquad 15$$

Where should 13 go?

Step 1: Find 13 in the counting order.

Step 2: 13 comes after 12 and before 14.

Answer: 13 goes on the tick mark between 12 and 14.

This shows that 13 is greater than 12 but less than 14.

Worked Example 2: Partially populated number line

A number line shows 20, 25, and 30 on equally spaced tick marks. What number is halfway between 20 and 30?

Step 1: We see that 25 is in the middle.

Step 2: The distance from 20 to 25 is 5, and from 25 to 30 is also 5.

Answer: The number halfway between 20 and 30 is \(25\).

This helps us see that numbers in the middle are equally far from both ends.

Worked Example 3: Number line counting by 10s

A number line has equally spaced tick marks labeled 40, 50, 60, 70. Where should 65 go?

Step 1: The number line counts by 10s.

Step 2: 65 is more than 60 but less than 70.

Step 3: 65 is halfway between 60 and 70 because:

$$60 + 5 = 65 \qquad \text{and} \qquad 65 + 5 = 70$$

Answer: 65 goes halfway between 60 and 70.

This shows that even if a number is not labeled, we can still place it by using what we know about distance.

Worked Example 4: Empty number line

An empty number line has one tick mark labeled 100 and another farther right labeled 200. Where could 150 go?

Step 1: 150 is between 100 and 200.

Step 2: 150 is halfway between them because:

$$100 + 50 = 150 \qquad \text{and} \qquad 150 + 50 = 200$$

Answer: 150 should be placed halfway between 100 and 200.

This helps us understand the size of 150. It is bigger than 100, smaller than 200, and right in the middle.

Comparing numbers on a number line

We can use a number line to compare numbers quickly.

• If a number is to the right, it is greater.
• If a number is to the left, it is smaller.
• If a number is between two numbers, it is greater than the one on the left and less than the one on the right.

For example, if we see 345, 350, and 355 on a number line, then:

• \(355 > 350\)
• \(345 < 350\)
• 350 is between 345 and 355

Using place value to help

Place value helps us know where numbers belong.

For example, think about 472 and 527.

• 472 has 4 hundreds.
• 527 has 5 hundreds.

Since 5 hundreds is more than 4 hundreds, 527 is greater than 472. On a number line, 527 would be to the right of 472.

Now think about 348 and 352.

• Both have 3 hundreds.
• 348 has 4 tens.
• 352 has 5 tens.

So 352 is greater and goes to the right of 348.

Common mistakes to watch for

• Forgetting equal spaces: Tick marks must be evenly spaced.
• Ignoring the pattern: Always check if the line counts by 1s, 10s, or another amount.
• Placing a number in the wrong order: Make sure the number is between the correct numbers.
• Guessing without checking distance: A number like 90 should be much closer to 100 than to 50.

Try thinking about these questions:

• Is 78 closer to 70 or 80?
• Which number is greater: 601 or 610?
• What number is halfway between 300 and 400?
• Where would 999 go on a number line from 990 to 1,000?

Summary

A number line shows numbers in order from least to greatest. Numbers to the right are greater, and numbers to the left are smaller. We can use number lines to place numbers, compare numbers, and see how far apart numbers are.

When using a number line, always look for equal spacing and the counting pattern. Then use order and distance to decide where each number belongs.

Comparing Numbers Using Place Value

Numbers can be compared to find out which one is greater, which one is less, or if they are equal.

When we compare numbers, we use what we know about place value. Place value tells us the value of each digit based on where it is in the number.

In 3rd grade, we often compare numbers up to 1,000. That means we look at the hundreds, tens, and ones places.

For example, in the number \(452\):

• \(4\) is in the hundreds place, so it means \(400\)
• \(5\) is in the tens place, so it means \(50\)
• \(2\) is in the ones place, so it means \(2\)

So, \(452 = 400 + 50 + 2\).

Symbols we use when comparing:

• \(>\) means greater than
• \(<\) means less than
• \(=\) means equal to

Example:

• \(8 > 5\) means 8 is greater than 5
• \(3 < 7\) means 3 is less than 7
• \(6 = 6\) means 6 is equal to 6

How to compare numbers using place value:

1. Look at the digit in the greatest place value first.
2. If those digits are different, the number with the bigger digit is the greater number.
3. If those digits are the same, move to the next place value to the right.
4. Keep going until you find digits that are different, or until all digits match.

For 3-digit numbers, start with the hundreds place. If the hundreds are the same, compare the tens. If the tens are the same, compare the ones.

Worked Example 1: Compare 324 and 421

Start with the hundreds place.

• \(324\) has \(3\) hundreds.
• \(421\) has \(4\) hundreds.

Since \(4\) hundreds is more than \(3\) hundreds, \(421\) is greater.

So we write:

$$324 < 421$$

Worked Example 2: Compare 587 and 563

First compare the hundreds place.

• Both numbers have \(5\) hundreds.

Since the hundreds are the same, compare the tens place.

• \(587\) has \(8\) tens.
• \(563\) has \(6\) tens.

Because \(8\) tens is more than \(6\) tens, \(587\) is greater.

So we write:

$$587 > 563$$

Worked Example 3: Compare 746 and 749

Compare the hundreds place first.

• Both numbers have \(7\) hundreds.

Now compare the tens place.

• Both numbers have \(4\) tens.

The hundreds and tens are the same, so compare the ones place.

• \(746\) has \(6\) ones.
• \(749\) has \(9\) ones.

Since \(9\) ones is more than \(6\) ones, \(749\) is greater.

So we write:

$$746 < 749$$

Worked Example 4: Compare 608 and 608

Compare each place value.

• Hundreds: both have \(6\)
• Tens: both have \(0\)
• Ones: both have \(8\)

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

So we write:

$$608 = 608$$

Important idea: A digit can have different values depending on its place.

For example, the digit \(5\) in \(523\) means \(500\), but the digit \(5\) in \(356\) means \(50\).

That is why place value helps us compare numbers correctly.

Let’s look at a place value comparison:

Compare \(392\) and \(389\).

• Hundreds: both have \(3\)
• Tens: \(392\) has \(9\), and \(389\) has \(8\)

We stop at the tens place because the digits are different there. Since \(9\) tens is greater than \(8\) tens, \(392\) is greater than \(389\).

$$392 > 389$$

Tips to remember:

• Always start with the greatest place value.
• For 3-digit numbers, compare hundreds first.
• If the hundreds are the same, compare tens.
• If the tens are the same, compare ones.
• If every digit matches, use \(=\).

A quick way to think about it:

• More hundreds means the number is greater.
• If hundreds match, more tens means the number is greater.
• If hundreds and tens match, more ones means the number is greater.

Summary

To compare numbers using place value, look at the biggest place first. Compare hundreds, then tens, then ones. Use \(>\), \(<\), or \(=\) to show whether one number is greater than, less than, or equal to another number.

Rounding to the Nearest Ten and Hundred

Sometimes a number has a lot of digits, and we want a number that is close to it but easier to use. This is called rounding.

When we round, we find which ten or hundred a number is closest to.

For example, 47 is close to 50. So, when we round 47 to the nearest ten, it becomes 50.

Rounding helps us:

• estimate answers,
• understand how big numbers are,
• use friendly numbers that are easier to think about.

Part 1: Rounding to the Nearest Ten

When rounding to the nearest ten, we ask: Which ten is the number closest to?

The tens are numbers like 0, 10, 20, 30, 40, 50, and so on.

Every number between two tens belongs somewhere in between. For example, 34 is between 30 and 40.

A helpful way to round is to look at the ones digit.

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

Why does 5 round up? Because 5 is the middle point between two tens. When a number is exactly in the middle, we round up.

Here is the idea on a number line for 34:

$$30 \qquad 31 \quad 32 \quad 33 \quad 34 \quad 35 \quad 36 \quad 37 \quad 38 \quad 39 \qquad 40$$

Since 34 is less than 35, it is closer to 30 than to 40.

So, $$34 \approx 30$$ when rounding to the nearest ten.

Part 2: Rounding to the Nearest Hundred

When rounding to the nearest hundred, we ask: Which hundred is the number closest to?

The hundreds are numbers like 0, 100, 200, 300, 400, and so on.

For example, 172 is between 100 and 200.

A helpful way to round to the nearest hundred is to look at the tens digit and the number after the hundred.

• If the number is from 0 to 49 past a hundred, round down.
• If the number is from 50 to 99 past a hundred, round up.

The midpoint is 50. That means 150 is exactly halfway between 100 and 200, so it rounds up to 200.

Here is the idea on a number line for 172:

$$100 \qquad \qquad 150 \qquad \qquad 200$$

Since 172 is more than 150, it is closer to 200 than to 100.

So, $$172 \approx 200$$ when rounding to the nearest hundred.

How to Round to the Nearest Ten

1. Find the two tens your number is between.
2. Look at the ones digit.
3. If it is 0 to 4, round down.
4. If it is 5 to 9, round up.

How to Round to the Nearest Hundred

1. Find the two hundreds your number is between.
2. Look at whether the number is below or above the midpoint, 50.
3. If it is 0 to 49 past the hundred, round down.
4. If it is 50 to 99 past the hundred, round up.

Worked Example 1: Round 23 to the nearest ten

23 is between 20 and 30.

The ones digit is 3.

Since 3 is less than 5, we round down.

$$23 \approx 20$$

Worked Example 2: Round 68 to the nearest ten

68 is between 60 and 70.

The ones digit is 8.

Since 8 is 5 or more, we round up.

$$68 \approx 70$$

Worked Example 3: Round 145 to the nearest hundred

145 is between 100 and 200.

The midpoint is 150.

145 is less than 150, so it is closer to 100.

$$145 \approx 100$$

Worked Example 4: Round 650 to the nearest hundred

650 is between 600 and 700.

The midpoint is 650.

Because 650 is exactly in the middle, we round up.

$$650 \approx 700$$

Tips to Help You Remember

• To round to the nearest ten, check the ones digit.
• To round to the nearest hundred, think about the middle number ending in 50.
• 0, 1, 2, 3, 4 means round down.
• 5, 6, 7, 8, 9 means round up.

Common Mistakes to Watch For

• Do not just pick the bigger number. Always choose the closest ten or hundred.
• Do not forget that 5 rounds up.
• When rounding to the nearest hundred, do not look only at the ones digit. Think about where the number is between two hundreds.

Try Thinking About These

• 42 rounded to the nearest ten is 40 because 42 is closer to 40 than 50.
• 97 rounded to the nearest ten is 100 because 97 is closer to 100 than 90.
• 320 rounded to the nearest hundred is 300 because 320 is less than 350.
• 389 rounded to the nearest hundred is 400 because 389 is more than 350.

Summary

Rounding means finding the nearest ten or hundred.

To round to the nearest ten, look at the ones digit. To round to the nearest hundred, look at where the number is between two hundreds and use the midpoint of 50.

If the number is less than 5 ones or less than 50 past a hundred, round down. If it is 5 or more ones, or 50 or more past a hundred, round up.

Rounding helps make numbers easier to understand and use.

Parity means finding out if a number is even or odd.

This is an important idea in 3rd Grade Maths because it helps us understand how numbers are built. We can tell whether a number can be split into two equal groups or if one is left over.

An even number can be grouped into pairs with no leftovers.

An odd number can be grouped into pairs with one leftover.

Let’s look at small numbers first.

• 2 dots can make 1 pair, so 2 is even.
• 4 dots can make 2 pairs, so 4 is even.
• 5 dots can make 2 pairs and 1 leftover, so 5 is odd.
• 7 dots can make 3 pairs and 1 leftover, so 7 is odd.

A quick way to check parity is to look at the ones digit.

If the ones digit is one of these numbers, the whole number is even:

\(0, 2, 4, 6, 8\)

If the ones digit is one of these numbers, the whole number is odd:

\(1, 3, 5, 7, 9\)

This works for numbers up to 1,000 and beyond. The ones digit tells us whether the number can be paired evenly.

For example:

• \(18\) ends in \(8\), so it is even.
• \(23\) ends in \(3\), so it is odd.
• \(140\) ends in \(0\), so it is even.
• \(999\) ends in \(9\), so it is odd.

Even numbers can also be written as a sum of two equal addends. Equal addends are two parts that are the same size.

For example:

• \(8 = 4 + 4\)
• \(12 = 6 + 6\)
• \(20 = 10 + 10\)

These are all even because the number splits into two equal parts.

Odd numbers cannot be split into two equal whole-number parts without a leftover one.

For example, \(9\) cannot be written as two equal whole numbers. We can get close with \(4 + 5\), but those addends are not equal.

So:

• If a number can be made into pairs with no leftover, it is even.
• If a number has one leftover when making pairs, it is odd.
• If a number can be split into two equal addends, it is even.

Worked Example 1: Use pairing

Is \(6\) even or odd?

Make pairs: \((2, 2, 2)\). There are 3 pairs and no leftover.

So \(6\) is even.

We can also write:

$$6 = 3 + 3$$

Worked Example 2: Use pairing with a leftover

Is \(11\) even or odd?

Make pairs from 11 objects. We can make 5 pairs, and 1 object is left over.

So \(11\) is odd.

Worked Example 3: Use the ones digit

Is \(42\) even or odd?

Look at the ones digit. The ones digit is \(2\).

Since \(2\) is in the even list \((0,2,4,6,8)\), \(42\) is even.

It can also be split into two equal addends:

$$42 = 21 + 21$$

Worked Example 4: A larger number

Is \(357\) even or odd?

Look only at the ones digit. The ones digit is \(7\).

Since \(7\) is odd, \(357\) is odd.

It cannot be split into two equal whole-number addends without a leftover one.

Tips to remember

• Even means pairs fit perfectly.
• Odd means one is left over.
• Check the ones digit for a quick answer.
• Even numbers can be written as two equal addends.

Try thinking about these:

• Does \(30\) end in an even digit or an odd digit?
• Can \(14\) be split into two equal addends?
• If \(19\) is grouped into pairs, is there a leftover?

Summary

Numbers are either even or odd. Even numbers can be paired with no leftovers, and odd numbers have one leftover. You can quickly tell by looking at the ones digit. Even numbers can also be written as a sum of two equal addends.