Number Sense and Base-Ten Structure

Base-Ten Positional Structure means that the value of a digit depends on where it is placed in a number.

In our number system, each place is worth 10 times as much as the place to its right. This is called the base-ten system.

For example, in the number \(352\):

• The \(3\) is in the hundreds place, so it means \(300\).
• The \(5\) is in the tens place, so it means \(50\).
• The \(2\) is in the ones place, so it means \(2\).

Even though \(3\), \(5\), and \(2\) are single digits, their values change because of their positions.

Let’s look at the places in a number:

• Ones
• Tens
• Hundreds
• Thousands

Moving left in a number makes the value of a digit 10 times greater.

Moving right in a number makes the value of a digit 10 times smaller.

Here is a place value chart:

Thousands | Hundreds | Tens | Ones

For example, in \(4,582\):

• \(4\) means \(4,000\)
• \(5\) means \(500\)
• \(8\) means \(80\)
• \(2\) means \(2\)

This number can be written in expanded form:

$$4,582 = 4,000 + 500 + 80 + 2$$

Expanded form helps us see the value of each digit clearly.

Important idea: A digit in one place is worth 10 times what it would be worth in the place to its right.

Look at the digit \(7\):

• In the ones place, \(7\) means \(7\).
• In the tens place, \(7\) means \(70\).
• In the hundreds place, \(7\) means \(700\).
• In the thousands place, \(7\) means \(7,000\).

Each time the \(7\) moves one place left, its value becomes 10 times greater.

We can show this with multiplication:

$$7 \times 10 = 70$$

$$70 \times 10 = 700$$

$$700 \times 10 = 7,000$$

This pattern is what makes the base-ten system work.

Worked Example 1

What is the value of each digit in \(246\)?

Step 1: Look at each place.

• \(2\) is in the hundreds place, so it is \(200\).
• \(4\) is in the tens place, so it is \(40\).
• \(6\) is in the ones place, so it is \(6\).

Answer:

$$246 = 200 + 40 + 6$$

Worked Example 2

In the number \(5,353\), what does each \(5\) mean?

Step 1: Find the place of each \(5\).

• The first \(5\) is in the thousands place, so it means \(5,000\).
• The second \(5\) is in the tens place, so it means \(50\).

Step 2: Compare them.

The \(5\) in the thousands place is worth much more than the \(5\) in the tens place because it is farther to the left.

Answer:

$$5,353 = 5,000 + 300 + 50 + 3$$

Worked Example 3

The digit \(8\) is in the tens place in one number and in the hundreds place in another number. How do the values compare?

If \(8\) is in the tens place, it means \(80\).

If \(8\) is in the hundreds place, it means \(800\).

Now compare:

$$800 = 10 \times 80$$

So, the \(8\) in the hundreds place is 10 times the value of the \(8\) in the tens place.

Worked Example 4

Write \(6,104\) in expanded form and tell the value of the digit \(1\).

Step 1: Name each digit’s value.

• \(6\) is in the thousands place, so it means \(6,000\).
• \(1\) is in the hundreds place, so it means \(100\).
• \(0\) is in the tens place, so it means \(0\) tens.
• \(4\) is in the ones place, so it means \(4\).

Step 2: Write the expanded form.

$$6,104 = 6,000 + 100 + 4$$

Answer: The digit \(1\) has a value of \(100\).

Things to remember

• A digit’s value depends on its place.
• Each place to the left is 10 times greater.
• Each place to the right is 10 times smaller.
• Expanded form shows the value of each digit.

Quick check

1. What is the value of the \(9\) in \(394\)?
2. In \(7,281\), what digit is in the hundreds place?
3. Is the \(6\) in \(60\) worth 10 times or 100 times the \(6\) in \(6\)?
4. Write \(8,430\) in expanded form.

Answers:

1. The \(9\) is in the tens place, so its value is \(90\).
2. The digit in the hundreds place is \(2\).
3. The \(6\) in \(60\) is worth 10 times the \(6\) in \(6\).
4. $$8,430 = 8,000 + 400 + 30$$

Summary

In the base-ten system, each digit has a value based on its place. Ones, tens, hundreds, and thousands are all connected because each place is 10 times the place to its right. When you understand place value, you can read, write, compare, and break apart numbers more easily.

Reading and Writing Large Numbers helps us understand how big numbers are built. In 4th grade, you will read, say, and write whole numbers all the way up to one million.

Big numbers may look tricky at first, but they become much easier when you break them into parts. The secret is to use place value and to read numbers in periods.

In our base-ten number system, each place is worth 10 times as much as the place to its right. That means a digit can have different values depending on where it is in the number.

Here are the place values you will use:

• Ones
• Tens
• Hundreds
• Thousands
• Ten thousands
• Hundred thousands
• Millions

We can organize these places into periods. A period is a group of three digits.

• Ones period: ones, tens, hundreds
• Thousands period: thousands, ten thousands, hundred thousands
• Millions period: millions

For example, in the number \(456{,}321\), we can separate the digits into periods like this:

$$456\,|\,321$$

The left group, \(456\), is the thousands period. The right group, \(321\), is the ones period.

To read a large number, follow these steps:

1. Start at the left.
2. Read the first period as a 3-digit number.
3. Say the period name, such as thousand or million.
4. Read the next period.
5. Do not say the period name for the ones period.

Important: When reading whole numbers, we usually do not say “and” in the middle of the number. For example, \(245\) is read as two hundred forty-five, not “two hundred and forty-five.”

It also helps to remember that commas separate periods. Each comma shows where one period ends and the next begins.

Here is how some numbers are grouped:

• \(7{,}532\) has 2 periods: \(7\,|\,532\)
• \(84{,}190\) has 2 periods: \(84\,|\,190\)
• \(305{,}817\) has 2 periods: \(305\,|\,817\)
• \(1{,}000{,}000\) has 3 periods: \(1\,|\,000\,|\,000\)

When a period has zeros, you still read the number carefully. Some zeros are silent, but they still hold places so the other digits have the correct value.

For example:

• \(4{,}008\) is four thousand eight
• \(40{,}050\) is forty thousand fifty
• \(308{,}002\) is three hundred eight thousand two

Now let’s learn how to write large numbers in different ways.

You may write a number in these forms:

• Standard form: using digits, like \(72{,}419\)
• Word form: using words, like seventy-two thousand four hundred nineteen
• Expanded form: showing the value of each digit, like \(70{,}000 + 2{,}000 + 400 + 10 + 9\)

Expanded form is helpful because it shows how the number is built from place values.

For example, the number \(583{,}204\) means:

$$500{,}000 + 80{,}000 + 3{,}000 + 200 + 4$$

The zero in the tens place means there are no tens, so we do not need to write \(+0\) unless a teacher asks for it.

Worked Example 1: Read a 4-digit number

Read the number \(6{,}241\).

Step 1: Separate into periods.

$$6\,|\,241$$

Step 2: Read the thousands period: \(6\) is six thousand.

Step 3: Read the ones period: \(241\) is two hundred forty-one.

So \(6{,}241\) is read as six thousand two hundred forty-one.

Worked Example 2: Read a number with zeros

Read the number \(70{,}305\).

Step 1: Separate into periods.

$$70\,|\,305$$

Step 2: Read \(70\) in the thousands period: seventy thousand.

Step 3: Read \(305\) in the ones period: three hundred five.

So \(70{,}305\) is read as seventy thousand three hundred five.

Notice that the zero in the tens place does not get read, but it is still important.

Worked Example 3: Write a number in standard form

Write four hundred twenty-three thousand eighteen in standard form.

Step 1: Find the thousands period: four hundred twenty-three thousand means \(423\,|\,\_\_\_\).

Step 2: Find the ones period: eighteen means \(018\).

Step 3: Put the periods together.

$$423{,}018$$

So the standard form is \(423{,}018\).

Worked Example 4: Read, write, and expand a 6-digit number

Look at the number \(905{,}146\).

First, read it by periods:

$$905\,|\,146$$

\(905\) is nine hundred five thousand.

\(146\) is one hundred forty-six.

So the number is nine hundred five thousand one hundred forty-six.

Now write it in expanded form:

$$900{,}000 + 5{,}000 + 100 + 40 + 6$$

Notice that there are no ten thousands and no tens hundreds? Actually, there are no ten thousands because that digit is \(0\). Each nonzero digit shows its value by its place.

Here are some helpful tips for reading and writing large numbers:

• Use commas to break numbers into periods of three digits.
• Read one period at a time.
• Say the period name after each group except the last group.
• Watch for zeros. They hold a place even when you do not say them.
• Check each digit’s place value if you are unsure.

Let’s look at a few more quick examples:

• \(12{,}000\) = twelve thousand
• \(120{,}000\) = one hundred twenty thousand
• \(12{,}345\) = twelve thousand three hundred forty-five
• \(500{,}060\) = five hundred thousand sixty
• \(1{,}000{,}000\) = one million

Be careful with numbers that have missing digits in the middle. For example, \(600{,}007\) is not read as “six hundred thousand seven thousand.” It is read as six hundred thousand seven.

When writing word form, use the place value words correctly:

• Use thousand for the thousands period.
• Use million for the millions period.
• Do not add extra place value words.

For example:

• \(34{,}210\) = thirty-four thousand two hundred ten
• \(700{,}001\) = seven hundred thousand one
• \(1{,}002{,}015\) = one million two thousand fifteen

Summary

Large numbers are easier to read and write when you break them into periods of three digits. Read each period from left to right and say the period name, such as thousand or million. You can write numbers in standard form, word form, and expanded form by using place value carefully.

Standard and Expanded Form

Numbers can be written in different ways. Two important ways are standard form and expanded form.

Standard form means writing a number the usual way, using digits. For example, \(4,582\) is in standard form.

Expanded form means writing a number to show the value of each digit. For example, \(4,582\) in expanded form is:

Expanded form helps us see how much each digit is really worth. This is called place value.

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

• Ones = \(1\)
• Tens = \(10\)
• Hundreds = \(100\)
• Thousands = \(1{,}000\)
• Ten thousands = \(10{,}000\)

When you look at a number, each digit has a job based on where it is.

For example, in \(3,416\):

• The \(3\) is in the thousands place, so it means \(3{,}000\).
• The \(4\) is in the hundreds place, so it means \(400\).
• The \(1\) is in the tens place, so it means \(10\).
• The \(6\) is in the ones place, so it means \(6\).

So the expanded form of \(3,416\) is:

To write a number in expanded form, follow these steps:

1. Look at each digit.
2. Find the place value of each digit.
3. Write the value of each digit as an addition sentence.

To write a number in standard form from expanded form, follow these steps:

1. Look at each value in the expanded form.
2. Place each value in the correct place.
3. Put the digits together to make one number.

Worked Example 1

Write \(2,345\) in expanded form.

First, name each digit's value:

• \(2\) thousands = \(2{,}000\)
• \(3\) hundreds = \(300\)
• \(4\) tens = \(40\)
• \(5\) ones = \(5\)

So:

Worked Example 2

Write \(7,090\) in expanded form.

Be careful: the digit \(0\) still has a place, but its value is \(0\).

• \(7\) thousands = \(7{,}000\)
• \(0\) hundreds = \(0\)
• \(9\) tens = \(90\)
• \(0\) ones = \(0\)

So the expanded form is:

We usually do not need to write the zero-value parts.

Worked Example 3

Write the expanded form \(5,000 + 600 + 20 + 8\) in standard form.

Put each value in the correct place:

• \(5,000\) means \(5\) in the thousands place
• \(600\) means \(6\) in the hundreds place
• \(20\) means \(2\) in the tens place
• \(8\) means \(8\) in the ones place

So the number is:

Worked Example 4

Write the expanded form \(8,000 + 300 + 4\) in standard form.

Think about the places:

• \(8,000\) means \(8\) in the thousands place
• \(300\) means \(3\) in the hundreds place
• There is no tens value, so the tens digit is \(0\)
• \(4\) means \(4\) in the ones place

So the standard form is:

This example is important because sometimes a place has no value. When that happens, we use a zero to hold the place.

Helpful Tips

• Always check the place of each digit.
• Read the number from left to right: thousands, hundreds, tens, ones.
• In expanded form, add the value of each digit.
• If a place has no value, use \(0\) in standard form.

Common Mistakes to Avoid

• Do not use the digit alone. For example, in \(4,582\), the \(5\) means \(500\), not just \(5\).
• Do not forget zeros. For example, \(6,042\) has a \(0\) in the hundreds place.
• Make sure each part of expanded form matches the correct place value.

Let’s look at one more number together: \(6,042\).

• \(6\) thousands = \(6{,}000\)
• \(0\) hundreds = \(0\)
• \(4\) tens = \(40\)
• \(2\) ones = \(2\)

Its expanded form is:

Its standard form is:

Summary

Standard form is the regular way we write a number using digits, like \(4,582\). Expanded form shows the value of each digit, like \(4,000 + 500 + 80 + 2\).

When changing between forms, use place value to help you. Ask yourself what each digit is worth, then write the number the correct way.

Expanded Notation with Multipliers helps us show the value of each digit in a number.

Every digit has a place value. A digit can mean ones, tens, hundreds, or thousands depending on where it is in the number.

When we use expanded notation with multipliers, we write a number as the sum of each digit multiplied by its place value.

For example, in the number \(3,482\):

• \(3\) is in the thousands place, so it means \(3 \times 1{,}000\)
• \(4\) is in the hundreds place, so it means \(4 \times 100\)
• \(8\) is in the tens place, so it means \(8 \times 10\)
• \(2\) is in the ones place, so it means \(2 \times 1\)

So we can write:

This is like regular expanded form, but instead of only adding the values, we also show the multiplication that makes each value.

Regular expanded form for \(3,482\) is:

Expanded notation with multipliers is:

Both forms show the same number. The multiplier form helps us see why each digit has its value.

Main idea: each place is 10 times greater than the place to its right.

• ones: \(1\)
• tens: \(10\)
• hundreds: \(100\)
• thousands: \(1{,}000\)

That means a digit in the tens place is multiplied by \(10\), a digit in the hundreds place is multiplied by \(100\), and so on.

Here is a simple way to do it:

1. Look at each digit.
2. Name its place value.
3. Write the digit times that place value.
4. Add all the parts together.

Worked Example 1

Write \(56\) in expanded notation with multipliers.

The digit \(5\) is in the tens place, so it means \(5 \times 10\).

The digit \(6\) is in the ones place, so it means \(6 \times 1\).

So:

If we multiply, we get:

Worked Example 2

Write \(304\) in expanded notation with multipliers.

The digit \(3\) is in the hundreds place, so it means \(3 \times 100\).

The digit \(0\) is in the tens place, so it means \(0 \times 10\).

The digit \(4\) is in the ones place, so it means \(4 \times 1\).

So:

This can also be written as:

Notice the zero in the tens place. It is important because it shows there are no tens.

Worked Example 3

Write \(7,219\) in expanded notation with multipliers.

• \(7\) thousands \(= 7 \times 1{,}000\)
• \(2\) hundreds \(= 2 \times 100\)
• \(1\) ten \(= 1 \times 10\)
• \(9\) ones \(= 9 \times 1\)

So:

And in regular expanded form:

Worked Example 4

What number is shown by:

First, multiply each part:

• \(4 \times 1{,}000 = 4{,}000\)
• \(6 \times 100 = 600\)
• \(0 \times 10 = 0\)
• \(3 \times 1 = 3\)

Now add:

So the number is 4,603.

Helpful Tips

• Start from the left and name each place: thousands, hundreds, tens, ones.
• Each digit is multiplied by the value of its place.
• Do not forget zeros. A zero still tells us about the place.
• You can check your work by multiplying and adding the parts.

Watch out for these mistakes:

• Mixing up place values. For example, in \(482\), the \(4\) means \(4 \times 100\), not \(4 \times 10\).
• Forgetting the ones place. The last digit is always multiplied by \(1\).
• Leaving out a zero place when it matters, like in \(304\).

Let’s compare

For the number \(2,540\):

• Standard form: \(2,540\)
• Expanded form: \(2,000 + 500 + 40 + 0\)
• Expanded notation with multipliers: \((2 \times 1{,}000) + (5 \times 100) + (4 \times 10) + (0 \times 1)\)

All three forms show the same number in different ways.

Summary

Expanded notation with multipliers shows how much each digit is worth by multiplying the digit by its place value.

To write a number this way, break it into digits, match each digit to its place, and write each part as multiplication.

This helps you understand how our base-ten number system works and why digits change value depending on their place.

Composing and Decomposing Numbers means putting numbers together and breaking numbers apart. This helps us understand how numbers are built in our base-ten system.

In base ten, we group by tens. That means:

•  ones make 1 ten
•  tens make 1 hundred
•  hundreds make 1 thousand

When we compose a number, we join parts together to make one whole number.

When we decompose a number, we split it into parts in different ways.

For example, the number \(347\) can be composed from:

• 3 hundreds
• 4 tens
• 7 ones

We can write that as:

$$347 = 300 + 40 + 7$$

This is called expanded form. It shows the value of each digit.

But numbers can also be decomposed in more than one way. That is very important.

For example, \(347\) can also be broken apart like this:

• 2 hundreds, 14 tens, and 7 ones
• 3 hundreds, 3 tens, and 17 ones
• 34 tens and 7 ones

All of these still equal \(347\). We are just regrouping the same amount.

Here is why this works:

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

So if we take 1 hundred away, we can replace it with 10 tens. If we take 1 ten away, we can replace it with 10 ones.

Main Idea: The number does not change when we regroup it. Only the way we describe it changes.

Lets look at some examples.

Worked Example 1: Standard decomposition

Decompose \(582\) into hundreds, tens, and ones.

The digits tell us the place value:

• 5 is in the hundreds place  \(500\)
• 8 is in the tens place  \(80\)
• 2 is in the ones place  \(2\)

So:

$$582 = 500 + 80 + 2$$

That means \(582\) is 5 hundreds, 8 tens, and 2 ones.

Worked Example 2: Compose a number from parts

Compose a number with 4 hundreds, 6 tens, and 9 ones.

Write the value of each part:

• 4 hundreds = \(400\)
• 6 tens = \(60\)
• 9 ones = \(9\)

Add them together:

$$400 + 60 + 9 = 469$$

So the composed number is 469.

Worked Example 3: Decompose in a different way

Decompose \(463\) using tens and ones, with fewer hundreds.

Start with the standard form:

$$463 = 4 hundreds + 6 tens + 3 ones$$

Now change 1 hundred into 10 tens.

If we take away 1 hundred from 4 hundreds, we have 3 hundreds left.

Then add 10 tens to the 6 tens:

$$6 tens + 10 tens = 16 tens$$

So now the number is:

$$463 = 3 hundreds + 16 tens + 3 ones$$

This is another correct way to decompose \(463\).

Worked Example 4: Decompose using more ones

Decompose \(215\) into hundreds, tens, and ones, but use no tens.

Start with:

$$215 = 2 hundreds + 1 ten + 5 ones$$

We want no tens, so change 1 ten into 10 ones.

Then:

• 2 hundreds stay the same
• 1 ten becomes 10 ones
• 10 ones + 5 ones = 15 ones

So:

$$215 = 2 hundreds + 15 ones$$

This is a correct decomposition too.

How to Decompose a Number

1. Read the number carefully.
2. Find the value of each digit by its place.
3. Write the number in standard expanded form.
4. If needed, regroup one place value into the next smaller unit.
5. Check that the total stays the same.

Helpful Checks

• If you decompose a number in a new way, add the parts to make sure they equal the original number.
• Remember: 1 hundred can become 10 tens.
• Remember: 1 ten can become 10 ones.

Lets check one together:

Is \(528 = 4 hundreds + 12 tens + 8 ones\)?

First, find the value:

• 4 hundreds = \(400\)
• 12 tens = \(120\)
• 8 ones = \(8\)

Add them:

$$400 + 120 + 8 = 528$$

Yes! This is correct.

Why This Matters

Composing and decomposing numbers helps with adding, subtracting, and understanding large numbers.

For example, when you regroup in subtraction, you are really decomposing one ten into 10 ones. When you regroup in addition, you may compose 10 ones into 1 ten.

Summary

Numbers can be built and broken apart in many ways. The usual way uses hundreds, tens, and ones, such as $$347 = 300 + 40 + 7$$.

But we can also regroup, like changing 1 hundred into 10 tens or 1 ten into 10 ones. This helps us see that the same number can have different correct decompositions.

Comparing Whole Numbers means deciding which number is greater, which number is smaller, or whether the numbers are equal.

When we compare numbers, we often use these symbols:

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

For example, \(9 > 4\) means 9 is greater than 4. Also, \(3 < 8\) means 3 is less than 8. And \(6 = 6\) means both numbers are the same.

To compare whole numbers well, we use place value. Place value tells us the value of each digit based on where it is in the number.

In a number like \(4{,}582\):

• 4 is in the thousands place
• 5 is in the hundreds place
• 8 is in the tens place
• 2 is in the ones place

This helps us compare numbers one place at a time, starting with the greatest place value.

Step 1: Look at the number of digits.

A number with more digits is greater than a number with fewer digits.

For example, compare \(523\) and \(4{,}018\).

• \(523\) has 3 digits.
• \(4{,}018\) has 4 digits.

So, $$523 < 4{,}018$$

Step 2: If the numbers have the same number of digits, compare from left to right.

Start with the greatest place value. If those digits are the same, move to the next place.

For example, compare \(6{,}214\) and \(6{,}193\).

• Both numbers have 4 digits.
• Compare the thousands digits: 6 and 6. They are the same.
• Compare the hundreds digits: 2 and 1.
• Since \(2 > 1\), \(6{,}214\) is greater.

So, $$6{,}214 > 6{,}193$$

Step 3: Keep going place by place until you find a difference.

Sometimes the first few digits match. That is okay. Just keep comparing the next place.

For example, compare \(7{,}405\) and \(7{,}450\).

• Thousands: 7 and 7, same
• Hundreds: 4 and 4, same
• Tens: 0 and 5
• Since \(0 < 5\), \(7{,}405\) is less than \(7{,}450\).

So, $$7{,}405 < 7{,}450$$

Step 4: If every digit matches, the numbers are equal.

For example, compare \(3{,}333\) and \(3{,}333\).

Each digit is the same, so $$3{,}333 = 3{,}333$$

A helpful way to remember the symbols is to think of the open side of the symbol as a hungry mouth. The hungry mouth opens toward the greater number.

For example:

• In \(8 > 2\), the open side points to 8 because 8 is greater.
• In \(5 < 9\), the open side points to 9 because 9 is greater.

Worked Example 1

Compare \(345\) and \(354\).

1. Both numbers have 3 digits.
2. Compare the hundreds digits: 3 and 3. They are the same.
3. Compare the tens digits: 4 and 5.
4. Since \(4 < 5\), the first number is smaller.

Answer: $$345 < 354$$

Worked Example 2

Compare \(8{,}901\) and \(8{,}876\).

1. Both numbers have 4 digits.
2. Compare the thousands digits: 8 and 8. They are the same.
3. Compare the hundreds digits: 9 and 8.
4. Since \(9 > 8\), the first number is greater.

Answer: $$8{,}901 > 8{,}876$$

Worked Example 3

Compare \(4{,}209\) and \(4{,}209\).

1. Both numbers have 4 digits.
2. Thousands digits match.
3. Hundreds digits match.
4. Tens digits match.
5. Ones digits match.

Answer: $$4{,}209 = 4{,}209$$

Worked Example 4

Compare \(9{,}999\) and \(10{,}000\).

1. \(9{,}999\) has 4 digits.
2. \(10{,}000\) has 5 digits.
3. A number with 5 digits is greater than a number with 4 digits.

Answer: $$9{,}999 < 10{,}000$$

Tips for comparing whole numbers

• Always start with the digit farthest to the left.
• Compare one place at a time.
• If the digits are the same, move to the next place.
• If one number has more digits, it is greater.
• Use the correct symbol: \(>\), \(<\), or \(=\).

Lets look at a number line idea.

On a number line, numbers farther to the right are greater. Numbers farther to the left are less.

For example, 120 is to the right of 98 on a number line, so \(120 > 98\).

Common mistakes to avoid

• Do not compare only the last digit. You must start with the greatest place value.
• Do not forget that more digits usually means a greater number.
• Be careful with the symbols. The open side faces the greater number.

For example, when comparing \(2{,}105\) and \(2{,}099\), do not just look at the ones digits. Start at the thousands place, then the hundreds place.

• Thousands: 2 and 2, same
• Hundreds: 1 and 0
• So, \(2{,}105 > 2{,}099\)

Summary

To compare whole numbers, first check how many digits each number has. If they have the same number of digits, compare digits from left to right. Use \(>\) for greater than, \(<\) for less than, and \(=\) for equal to.

Ordering Multi-Digit Numbers means putting numbers in the correct order from least to greatest or from greatest to least.

To do this well, we use place value. Place value tells us that each digit has a different value depending on where it is in the number.

For example, in the number \(4,582\):

• The \(4\) is in the thousands place, so it means \(4,000\).
• The \(5\) is in the hundreds place, so it means \(500\).
• The \(8\) is in the tens place, so it means \(80\).
• The \(2\) is in the ones place, so it means \(2\).

When we order multi-digit numbers, we compare the digits from left to right. We start with the place that has the greatest value.

Ascending order means from smallest to largest.

Example: \(123, 245, 389\)

Descending order means from largest to smallest.

Example: \(389, 245, 123\)

Here is a step-by-step way to compare two or more numbers:

1. Look at the digit in the greatest place value.
2. If the digits are different, the number with the greater digit is greater.
3. If the digits are the same, move one place to the right.
4. Keep comparing until you find a difference.

Let’s see how this works.

Worked Example 1: Compare two 3-digit numbers

Which number is greater: \(426\) or \(389\)?

First, compare the hundreds digits.

• \(426\) has \(4\) hundreds.
• \(389\) has \(3\) hundreds.

Since \(4 > 3\), we know:

$$426 > 389$$

So in ascending order, they are:

$$389,\ 426$$

Worked Example 2: Compare numbers with the same first digit

Order these numbers from least to greatest: \(5,214\), \(5,189\), \(5,320\)

All three numbers have \(5\) in the thousands place, so we look at the hundreds place.

• \(5,214\) has \(2\) hundreds.
• \(5,189\) has \(1\) hundred.
• \(5,320\) has \(3\) hundreds.

Now we can order them by their hundreds digits: \(1\), \(2\), \(3\).

So the numbers from least to greatest are:

$$5,189,\ 5,214,\ 5,320$$

Worked Example 3: Keep comparing place values

Order these numbers from greatest to least: \(6,745\), \(6,754\), \(6,705\)

First, compare the thousands digits. They are all \(6\), so they are the same.

Next, compare the hundreds digits. They are all \(7\), so they are still the same.

Now compare the tens digits:

• \(6,745\) has \(4\) tens.
• \(6,754\) has \(5\) tens.
• \(6,705\) has \(0\) tens.

From greatest to least, the tens digits go \(5, 4, 0\).

So the order is:

$$6,754,\ 6,745,\ 6,705$$

Worked Example 4: One number has more digits

Order these numbers from least to greatest: \(999\), \(1,002\), \(875\)

Numbers with more digits are usually greater when we are comparing whole numbers.

• \(875\) has 3 digits.
• \(999\) has 3 digits.
• \(1,002\) has 4 digits.

So \(1,002\) is the greatest number.

Now compare \(875\) and \(999\):

• \(875\) has \(8\) hundreds.
• \(999\) has \(9\) hundreds.

Since \(8 < 9\), \(875\) is less than \(999\).

So the numbers from least to greatest are:

$$875,\ 999,\ 1,002$$

Helpful Tips

• Start comparing at the left.
• The first place where the digits are different tells you which number is greater.
• If one whole number has more digits than another, it is greater.
• Read the direction carefully: least to greatest is not the same as greatest to least.

Try to Think About It

If you compare \(4,321\) and \(4,318\), the thousands and hundreds digits are the same. The tens digits are different: \(2\) tens is greater than \(1\) ten, so \(4,321\) is greater.

That means:

$$4,318 < 4,321$$

Summary

To order multi-digit numbers, compare digits from left to right using place value. If the digits in one place are the same, move to the next place. This helps you put numbers in ascending order (smallest to largest) or descending order (largest to smallest).

Number Line Scaling and Intervals

A number line is a straight line that shows numbers in order. Numbers get larger as you move to the right and smaller as you move to the left.

Sometimes a number line shows every number, like 1, 2, 3, 4. But when we work with large numbers, number lines often skip by bigger amounts, such as 10s, 100s, or 1,000s. This is called scaling the number line.

To use a scaled number line correctly, we need to figure out the interval. An interval is the amount between two marks.

When you know the interval, you can place numbers in the right spot and read missing numbers between labeled points.

Why this matters

• It helps you understand how big numbers are.
• It helps you compare numbers.
• It helps you estimate where a number belongs.
• It builds strong place value skills.

Main Idea 1: Find the interval

To find the interval on a number line, look at two labeled marks. Then:

1. Find the difference between the two numbers.
2. Count how many equal spaces are between them.
3. Divide the difference by the number of spaces.

In math form:

$$\text{interval} = \frac{\text{difference between labeled numbers}}{\text{number of equal spaces}}$$

Main Idea 2: Count by the interval

Once you know the interval, move from mark to mark by adding that amount each time.

For example, if the interval is 100, the marks might be:

$$1{,}000,\ 1{,}100,\ 1{,}200,\ 1{,}300$$

If the interval is 1,000, the marks might be:

$$5{,}000,\ 6{,}000,\ 7{,}000,\ 8{,}000$$

Main Idea 3: Use place value to help

Large numbers can be easier to place when you think about place value.

• Counting by 10 changes the tens place.
• Counting by 100 changes the hundreds place.
• Counting by 1,000 changes the thousands place.

Example:

$$3{,}400,\ 3{,}500,\ 3{,}600,\ 3{,}700$$

Here, the hundreds place changes by 1 each step, so the interval is 100.

Main Idea 4: Open number lines

An open number line does not show every label. You may only see a few numbers and some marks. Your job is to use the given numbers to figure out the missing intervals.

Even on an open number line, the spaces between marks are equal unless the picture shows otherwise. Equal spaces mean equal intervals.

Steps for solving number line scaling problems

1. Look at the labeled numbers.
2. Find how much the numbers change.
3. Count the number of equal spaces between them.
4. Find the interval.
5. Count forward or backward to fill in missing numbers.
6. Place the target number where it belongs.

Worked Example 1: Find missing labels

A number line shows 2,000 at the first mark and 2,400 at the fifth mark. There are 4 equal spaces between them. What is the interval? What numbers go on the missing marks?

Step 1: Find the difference.

$$2{,}400 - 2{,}000 = 400$$

Step 2: Count the spaces.

There are 4 spaces.

Step 3: Find the interval.

$$400 \div 4 = 100$$

So the interval is 100.

Step 4: Count by 100.

$$2{,}000,\ 2{,}100,\ 2{,}200,\ 2{,}300,\ 2{,}400$$

Answer: The missing labels are 2,100, 2,200, and 2,300.

Worked Example 2: Place a number on a number line

A number line starts at 5,000 and ends at 6,000. It has 5 equal spaces. Where does 5,600 go?

Step 1: Find the total change.

$$6{,}000 - 5{,}000 = 1{,}000$$

Step 2: Divide by the number of spaces.

$$1{,}000 \div 5 = 200$$

The interval is 200.

Step 3: Count by 200.

$$5{,}000,\ 5{,}200,\ 5{,}400,\ 5{,}600,\ 5{,}800,\ 6{,}000$$

Answer: The number 5,600 goes on the fourth mark, after 5,000.

Worked Example 3: Work backward

On a number line, one mark is labeled 8,700. The next mark to the right is 8,800. What is the interval? What number is one mark to the left of 8,700?

Step 1: Find the change between the two marks.

$$8{,}800 - 8{,}700 = 100$$

So the interval is 100.

Step 2: Count backward by 100.

$$8{,}700 - 100 = 8{,}600$$

Answer: The interval is 100, and one mark to the left is 8,600.

Worked Example 4: Larger numbers with fewer labels

A number line has 10,000 at one mark and 14,000 at a mark 4 spaces to the right. What is the interval? What number is 2 spaces to the right of 10,000?

Step 1: Find the difference.

$$14{,}000 - 10{,}000 = 4{,}000$$

Step 2: Divide by 4 spaces.

$$4{,}000 \div 4 = 1{,}000$$

The interval is 1,000.

Step 3: Move 2 spaces to the right.

$$10{,}000 \rightarrow 11{,}000 \rightarrow 12{,}000$$

Answer: The interval is 1,000, and 2 spaces to the right of 10,000 is 12,000.

Tips to remember

• Look for equal spaces. Equal spaces mean equal intervals.
• Do not guess. Calculate the interval.
• Use subtraction to find the difference.
• Use division to find the value of each space.
• Count carefully forward or backward.
• Check that your last number matches the given label.

Common mistakes

• Mixing up marks and spaces: If there are 5 marks, there may be only 4 spaces between them.
• Skipping the division step: The whole difference is not the interval unless there is only 1 space.
• Counting by the wrong place value: Make sure you know if the line is counting by 10s, 100s, or 1,000s.

Quick check

• If 3,000 to 3,300 takes 3 spaces, the interval is 100.
• If 7,000 to 8,000 takes 5 spaces, the interval is 200.
• If 12,000 to 16,000 takes 4 spaces, the interval is 1,000.

Summary

Scaled number lines help us show large numbers without writing every number. To understand a scaled number line, find the difference between labeled numbers, count the equal spaces, and divide to find the interval.

Then count forward or backward by that interval to fill in missing numbers or place a number where it belongs. Using place value helps you see whether the line is counting by 10s, 100s, or 1,000s.

Rounding Concepts and Midpoints

Rounding means finding the nearest ten, hundred, or other place value for a number. We round when we want a number that is easier to read, compare, or use in quick math.

To round well, it helps to understand midpoints. A midpoint is the number exactly in the middle of two numbers. Midpoints help us decide whether a number is closer to the smaller value or the larger value.

For example, when rounding to the nearest ten, the number \(34\) is between \(30\) and \(40\). The midpoint between \(30\) and \(40\) is \(35\).

We can show that like this:

$$30 \qquad 35 \qquad 40$$

If a number is less than the midpoint, it rounds down. If a number is greater than the midpoint, it rounds up. If a number is exactly at the midpoint, it usually rounds up to the greater value.

Main Idea: Rounding is really about asking, Which friendly number is closer?

Using a Number Line

A number line is one of the best tools for rounding because it helps us see where a number belongs.

• Find the two tens or hundreds your number is between.
• Find the midpoint.
• Decide if the number is closer to the lower value or the higher value.

When rounding to the nearest ten:

• Numbers from \(0\) to \(4\) in the ones place round down.
• Numbers from \(5\) to \(9\) in the ones place round up.

Why does that work? Because \(5\) is the midpoint between two tens.

When rounding to the nearest hundred:

• Look at the tens digit.
• If the tens digit is \(0\) to \(4\), round down.
• If the tens digit is \(5\) to \(9\), round up.

That works because \(50\) is the midpoint between two hundreds.

Understanding Midpoints

A midpoint is halfway between two benchmark numbers.

• Between \(20\) and \(30\), the midpoint is \(25\).
• Between \(60\) and \(70\), the midpoint is \(65\).
• Between \(400\) and \(500\), the midpoint is \(450\).

If a number is exactly at the midpoint, it is equally far from both choices. In school rounding, we usually choose the greater place value.

So:

• \(25\) rounds to \(30\) when rounding to the nearest ten.
• \(450\) rounds to \(500\) when rounding to the nearest hundred.

Worked Example 1: Round \(42\) to the nearest ten

First, find the two tens around \(42\). They are \(40\) and \(50\).

The midpoint between \(40\) and \(50\) is \(45\).

Now compare \(42\) to \(45\). Since \(42\) is less than \(45\), it is closer to \(40\).

So, $$42 \approx 40$$

Worked Example 2: Round \(68\) to the nearest ten

\(68\) is between \(60\) and \(70\).

The midpoint is \(65\).

Since \(68\) is greater than \(65\), it is closer to \(70\).

So, $$68 \approx 70$$

Worked Example 3: Round \(35\) to the nearest ten

\(35\) is between \(30\) and \(40\).

The midpoint is \(35\).

This number is exactly halfway. When a number is exactly at the midpoint, we round up.

So, $$35 \approx 40$$

Worked Example 4: Round \(372\) to the nearest hundred

\(372\) is between \(300\) and \(400\).

The midpoint between them is \(350\).

Since \(372\) is greater than \(350\), it is closer to \(400\).

So, $$372 \approx 400$$

Another Way to Think About It

You can also use the digit to the right of the place you are rounding to.

• To round to the nearest ten, look at the ones digit.
• To round to the nearest hundred, look at the tens digit.

This shortcut works because that digit tells you whether the number is below the midpoint or at/above the midpoint.

For example:

• \(81\): ones digit is \(1\), so round down to \(80\).
• \(96\): ones digit is \(6\), so round up to \(100\).
• \(241\): tens digit is \(4\), so round down to \(200\).
• \(267\): tens digit is \(6\), so round up to \(300\).

Common Mistakes to Watch For

• Mixing up the place value: If you are rounding to the nearest hundred, do not look at the ones digit. Look at the tens digit.
• Forgetting the midpoint: The midpoint helps you know which side the number is on.
• Not finding the two benchmark numbers first: Always ask, “What two tens or hundreds is this number between?”

Try These Thinking Questions

1. What is the midpoint between \(70\) and \(80\)?
2. Would \(73\) round to \(70\) or \(80\)? Why?
3. What is the midpoint between \(500\) and \(600\)?
4. Would \(548\) round to \(500\) or \(600\)? Why?

Answers

1. The midpoint is \(75\).
2. \(73\) rounds to \(70\) because it is less than \(75\).
3. The midpoint is \(550\).
4. \(548\) rounds to \(500\) because it is less than \(550\).

Summary

Rounding means finding the nearest ten or hundred. To round, first find the two benchmark numbers the number is between. Then find the midpoint and decide which benchmark is closer.

If the number is below the midpoint, round down. If the number is above the midpoint, round up. If the number is exactly at the midpoint, round up to the greater value.

Rounding to Specific Place Values means finding the number that is closest to a certain place value, like the nearest ten, hundred, thousand, ten thousand, or hundred thousand.

Rounding helps us make numbers easier to understand and use. For example, if there are 4,982 people at an event, we might say there are about 5,000 people.

To round correctly, we need to understand place value. In a number, each digit has a value based on its place.

• In 482,731:
• The 4 is in the hundred thousands place.
• The 8 is in the ten thousands place.
• The 2 is in the thousands place.
• The 7 is in the hundreds place.
• The 3 is in the tens place.
• The 1 is in the ones place.

When you round to a specific place, you look at the digit in that place, and then you check the digit right next door to the right.

Here is the rounding rule:

• If the digit to the right is 0, 1, 2, 3, or 4, the rounding digit stays the same.
• If the digit to the right is 5, 6, 7, 8, or 9, the rounding digit goes up by 1.
• All digits to the right of the place you are rounding to become 0.

Think of it like this:

• 0 to 4: round down
• 5 to 9: round up

Let’s learn the steps.

1. Find the place value you are rounding to.
2. Look at the digit to the right of that place.
3. Decide whether to keep the digit the same or add 1.
4. Change all digits to the right into 0s.

Worked Example 1: Round 67 to the nearest ten.

The tens digit is 6. The digit to the right is 7.

Since 7 is 5 or more, we round up. The 6 becomes 7, and the ones digit becomes 0.

$$67 \approx 70$$

Worked Example 2: Round 342 to the nearest hundred.

The hundreds digit is 3. The digit to the right is 4 in the tens place.

Since 4 is less than 5, we keep the 3 the same. Then the tens and ones become 0.

$$342 \approx 300$$

Worked Example 3: Round 4,582 to the nearest thousand.

The thousands digit is 4. The digit to the right is 5 in the hundreds place.

Since 5 means round up, the 4 becomes 5. The hundreds, tens, and ones all become 0.

$$4,582 \approx 5,000$$

Worked Example 4: Round 482,731 to the nearest ten thousand.

The ten thousands digit is 8. The digit to the right is 2 in the thousands place.

Since 2 is less than 5, the 8 stays the same. All digits to the right become 0.

$$482,731 \approx 480,000$$

Now let’s try rounding to different place values in the same number.

Use the number 156,489.

• Nearest hundred: look at the tens digit.
• Nearest thousand: look at the hundreds digit.
• Nearest ten thousand: look at the thousands digit.

Round 156,489 to the nearest hundred.

The hundreds digit is 4. The tens digit is 8.

Since 8 is 5 or more, the 4 becomes 5.

$$156,489 \approx 156,500$$

Round 156,489 to the nearest thousand.

The thousands digit is 6. The hundreds digit is 4.

Since 4 is less than 5, the 6 stays the same.

$$156,489 \approx 156,000$$

Round 156,489 to the nearest ten thousand.

The ten thousands digit is 5. The thousands digit is 6.

Since 6 is 5 or more, the 5 becomes 6.

$$156,489 \approx 160,000$$

This shows that the same number can round to different answers depending on which place value you are asked to use.

A number line can help too.

Suppose we round 3,450 to the nearest thousand. The two thousands it is between are 3,000 and 4,000.

Since 3,450 is less than halfway to 4,000, it rounds to 3,000.

If the number were 3,500, it would be exactly halfway, so it would round up to 4,000.

Important idea: when the digit to the right is 5, we round up.

Here are some common mistakes to watch out for:

• Mistake 1: Looking at the wrong digit. Always look at the digit just to the right of the place you are rounding to.
• Mistake 2: Forgetting to change the digits to the right into 0s.
• Mistake 3: Rounding to the wrong place value. Read the directions carefully.

Quick Check

Try these on your own:

• Round 89 to the nearest ten.
• Round 761 to the nearest hundred.
• Round 12,349 to the nearest thousand.
• Round 698,201 to the nearest hundred thousand.

Answers

• 89 rounds to 90.
• 761 rounds to 800.
• 12,349 rounds to 12,000.
• 698,201 rounds to 700,000.

Summary

To round a number, first find the place you are rounding to. Next, look at the digit to the right. If that digit is 0 to 4, keep the rounding digit the same. If it is 5 to 9, add 1. Then change all digits to the right to 0.

With practice, rounding becomes quick and easy. It helps you estimate, check your work, and understand large numbers better.

Even and Odd Number Properties

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

When you understand even and odd numbers, you can sort numbers, spot patterns, and make smart guesses about answers when you add or subtract.

What is an even number?

An even number can be split into 2 equal groups with no leftovers.

Examples of even numbers are:

• (2, 4, 6, 8, 10)
• (12, 24, 36, 48)

If 8 apples are shared equally between 2 baskets, each basket gets 4 apples. There are no apples left over, so 8 is even.

What is an odd number?

An odd number cannot be split into 2 equal groups without 1 left over.

Examples of odd numbers are:

• (1, 3, 5, 7, 9)
• (11, 13, 25, 37)

If 7 apples are shared equally between 2 baskets, each basket gets 3 apples and 1 apple is left over. So 7 is odd.

How can you tell if a number is even or odd?

You can look at the ones digit. The ones digit tells whether the whole number is even or odd.

If a number ends in (0, 2, 4, 6, 8), it is even.

If a number ends in (1, 3, 5, 7, 9), it is odd.

Here are some examples:

• 42 ends in 2, so 42 is even.
• 135 ends in 5, so 135 is odd.
• 908 ends in 8, so 908 is even.
• 671 ends in 1, so 671 is odd.

Why does the ones digit matter?

In our base-ten number system, the ones digit tells how many ones are in the number. Tens, hundreds, and thousands are all made of groups of 10, and 10 is even.

That means the part that decides even or odd is the number of ones.

For example:

\(34 = 30 + 4\)
The 30 is even, and the 4 is even, so 34 is even.

\(57 = 50 + 7\)
The 50 is even, but the 7 is odd, so 57 is odd.

Patterns in even and odd numbers

Even and odd numbers make a pattern as you count:

\(1, 2, 3, 4, 5, 6, 7, 8, 9, 10\)

The pattern is:

• odd
• even
• odd
• even

This pattern keeps repeating. Every time you move 1 number forward, the type changes:

• even to odd
• odd to even

Adding even and odd numbers

We can predict whether a sum will be even or odd.

• Even + even = even
• Odd + odd = even
• Even + odd = odd

These rules help you check if an answer makes sense.

Why do these addition rules work?

If you put together 2 even groups, everything still pairs up, so the total is even.

If you put together 2 odd groups, each group has 1 extra. The 2 extras make a pair, so the total becomes even.

If you put together 1 even group and 1 odd group, the odd group still has 1 extra, so the total is odd.

Worked Example 1

Is 246 even or odd?

Step 1: Look at the ones digit.

The ones digit is 6.

Step 2: Decide if 6 is even or odd.

6 is even.

Answer: 246 is even.

Worked Example 2

Is 513 even or odd?

Step 1: Look at the ones digit.

The ones digit is 3.

Step 2: Decide if 3 is even or odd.

3 is odd.

Answer: 513 is odd.

Worked Example 3

Find whether the sum is even or odd: \(24 + 18\)

Step 1: Decide if each addend is even or odd.

• 24 is even because it ends in 4.
• 18 is even because it ends in 8.

Step 2: Use the rule.

Even + even = even

Step 3: Check by adding.

$$24 + 18 = 42$$

42 ends in 2, so it is even.

Answer: The sum is even.

Worked Example 4

Find whether the sum is even or odd: \(35 + 12\)

Step 1: Decide if each addend is even or odd.

• 35 is odd because it ends in 5.
• 12 is even because it ends in 2.

Step 2: Use the rule.

Odd + even = odd

Step 3: Check by adding.

$$35 + 12 = 47$$

47 ends in 7, so it is odd.

Answer: The sum is odd.

Subtracting even and odd numbers

You can also notice patterns when subtracting.

• Even - even = even
• Odd - odd = even
• Even - odd = odd
• Odd - even = odd

Here are quick examples:

• \(14 - 6 = 8\), so even - even = even
• \(15 - 7 = 8\), so odd - odd = even
• \(16 - 5 = 11\), so even - odd = odd
• \(13 - 4 = 9\), so odd - even = odd

Useful tips

• Only the last digit matters when deciding if a number is even or odd.
• If you count by 2s, you land on even numbers or odd numbers, depending on where you start.
• Even and odd numbers alternate as you count.
• You can use even and odd rules to check if an answer is reasonable.

Try thinking about these:

• Is 1,204 even or odd? It ends in 4, so it is even.
• Is 999 even or odd? It ends in 9, so it is odd.
• Will \(27 + 15\) be even or odd? Odd + odd = even.
• Will \(46 + 9\) be even or odd? Even + odd = odd.

Summary

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

To tell if a number is even or odd, look at the ones digit. Numbers ending in \(0, 2, 4, 6, 8\) are even. Numbers ending in \(1, 3, 5, 7, 9\) are odd.

You can also predict sums and differences:

• even + even = even
• odd + odd = even
• even + odd = odd

Knowing these patterns helps you understand numbers better and check your work.

Factor Identification means finding the numbers that divide evenly into another number.

If two whole numbers can be multiplied to make a product, then each of those numbers is a factor of the product.

For example, since \(3 \times 4 = 12\), both 3 and 4 are factors of 12.

Another way to think about factors is with rectangular arrays. A rectangular array uses rows and columns. If you can make a rectangle with a certain number of rows and columns, then those side lengths are factors of the total number of objects.

For 12 objects, you could make:

• 1 row of 12
• 2 rows of 6
• 3 rows of 4

These arrays show the factor pairs of 12: \((1,12)\), \((2,6)\), and \((3,4)\).

A factor pair is a pair of numbers that multiply to make the given number.

To find all factor pairs of a number, follow these steps:

1. Start with 1, because 1 is a factor of every whole number.
2. Ask: “What can I multiply by 1 to get the number?”
3. Then try 2, 3, 4, and so on.
4. Stop when the factor pairs start repeating in reverse order.

Let’s look at this with a multiplication list. For 18, we can test numbers:

• \(1 \times 18 = 18\)
• \(2 \times 9 = 18\)
• \(3 \times 6 = 18\)
• \(4\) does not work because 18 cannot be divided evenly into groups of 4.
• After that, the pairs repeat: \(6 \times 3\), \(9 \times 2\), and \(18 \times 1\).

So the factor pairs of 18 are \((1,18)\), \((2,9)\), and \((3,6)\).

The full list of factors of 18 is:

$$1, 2, 3, 6, 9, 18$$

Important idea: factors always divide the number evenly. That means there is no remainder left over.

For example, 5 is not a factor of 18 because \(18 \div 5\) does not make a whole number.

Arrays can help us see factors clearly. If 16 counters can be arranged into a rectangle, the side lengths of each rectangle are factor pairs of 16.

You can make these rectangles for 16:

• 1 by 16
• 2 by 8
• 4 by 4

So the factor pairs of 16 are \((1,16)\), \((2,8)\), and \((4,4)\).

Notice that \((4,4)\) is a special factor pair because both factors are the same. That makes a square array.

Here are some main things to remember about factors:

• Factors are numbers you multiply to make another number.
• Factors divide a number evenly.
• Factor pairs can be shown with rectangular arrays.
• Every whole number has at least 1 and itself as factors.

Worked Example 1

Find the factors of 10.

Test numbers that divide 10 evenly:

• \(1 \times 10 = 10\)
• \(2 \times 5 = 10\)
• \(3\) does not work
• \(4\) does not work

So the factor pairs are \((1,10)\) and \((2,5)\).

The factors of 10 are:

$$1, 2, 5, 10$$

Worked Example 2

Find the factor pairs of 15.

Try whole numbers in order:

• \(1 \times 15 = 15\)
• \(2\) does not work
• \(3 \times 5 = 15\)
• \(4\) does not work

So the factor pairs are:

$$ (1,15), (3,5) $$

The factors of 15 are:

$$1, 3, 5, 15$$

Worked Example 3

Find all factors of 24 using factor pairs.

Test numbers one at a time:

• \(1 \times 24 = 24\)
• \(2 \times 12 = 24\)
• \(3 \times 8 = 24\)
• \(4 \times 6 = 24\)
• After this, the pairs repeat in reverse.

So the factor pairs are:

$$ (1,24), (2,12), (3,8), (4,6) $$

The factors of 24 are:

$$1, 2, 3, 4, 6, 8, 12, 24$$

Worked Example 4

A student says 3 is a factor of 14 because 3 is less than 14. Is that correct?

No, that is not correct.

To be a factor, 3 must divide 14 evenly. But \(14 \div 3\) does not make a whole number.

So 3 is not a factor of 14.

The factor pairs of 14 are:

• \(1 \times 14 = 14\)
• \(2 \times 7 = 14\)

The factors of 14 are:

$$1, 2, 7, 14$$

Tips for finding factors

• Start with 1.
• Use multiplication facts you know.
• Think about arrays with rows and columns.
• Make sure the number divides evenly with no leftovers.
• List factor pairs so you do not miss any factors.

Let’s compare factors and non-factors for 20.

• 1 is a factor because \(1 \times 20 = 20\)
• 2 is a factor because \(2 \times 10 = 20\)
• 4 is a factor because \(4 \times 5 = 20\)
• 3 is not a factor because 20 cannot be divided evenly by 3
• 6 is not a factor because 20 cannot be divided evenly by 6

So the factors of 20 are:

$$1, 2, 4, 5, 10, 20$$

Summary

Factors are numbers that multiply together to make a given number.

You can find factors by testing which numbers divide evenly, or by drawing rectangular arrays with rows and columns.

Factor pairs help you organize your work and make sure you find all the factors.

When you know the factor pairs, you can list all the factors of the number.

Generating Multiples means making a list of numbers you get when you multiply a number by counting numbers like 1, 2, 3, 4, and so on.

For example, the multiples of 3 are found by multiplying 3 by 1, 2, 3, 4, and more:

So the first few multiples of 3 are 3, 6, 9, 12.

Learning multiples helps us with skip-counting, multiplication facts, division, and solving number patterns.

What is a multiple?

A multiple of a number is the result of multiplying that number by a whole number.

Here are some examples:

• 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, ...

Notice that multiples keep going. There is no last multiple.

How to generate multiples

There are two easy ways to generate multiples:

1. Multiply the number by 1, 2, 3, 4, and so on.
2. Skip-count by that number.

These two ways give the same answers.

For example, to generate multiples of 4:

• Multiply: \(4 \times 1, 4 \times 2, 4 \times 3, 4 \times 4\)
• Skip-count: 4, 8, 12, 16, ...

Factors and multiples are different

It is important to know the difference between a factor and a multiple.

• A factor is a number that divides another number evenly.
• A multiple is a number you get by multiplying.

Example with 12:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Multiples of 12: 12, 24, 36, 48, ...

Factors are usually a shorter list for one number. Multiples keep going forever.

Worked Example 1: Generate multiples of 2

Let’s find the first 6 multiples of 2.

Multiply 2 by 1 through 6:

So the first 6 multiples of 2 are 2, 4, 6, 8, 10, 12.

This is the same as skip-counting by 2.

Worked Example 2: Generate multiples of 5

Let’s find the first 5 multiples of 5.

So the first 5 multiples of 5 are 5, 10, 15, 20, 25.

A helpful pattern: multiples of 5 end in 0 or 5.

Worked Example 3: Generate multiples of 7

Let’s find the first 6 multiples of 7.

We can skip-count by 7:

7, 14, 21, 28, 35, 42

We can check with multiplication:

So the first 6 multiples of 7 are 7, 14, 21, 28, 35, 42.

Worked Example 4: Is it a factor or a multiple?

Look at the number 4 and the number 20.

Question: Is 20 a factor of 4, or is 20 a multiple of 4?

We ask: can we make 20 by multiplying 4 by a whole number?

Yes. So 20 is a multiple of 4.

Also, 4 is a factor of 20 because 20 can be divided evenly by 4.

Helpful patterns when generating multiples

• Multiples of 2 are even numbers: 2, 4, 6, 8, ...
• Multiples of 5 end in 0 or 5.
• Multiples of 10 end in 0.
• You can use repeated addition to help. For example, multiples of 6 can be found by adding 6 again and again: 6, 12, 18, 24, ...

Try this thinking

If you want the first 4 multiples of 8, multiply 8 by 1, 2, 3, and 4:

So the first 4 multiples of 8 are 8, 16, 24, 32.

Common mistake to avoid

Sometimes students mix up factors and multiples.

• If you are making a list by multiplying, you are finding multiples.
• If you are finding numbers that go into another number evenly, you are finding factors.

Example:

• Multiples of 3: 3, 6, 9, 12, 15, ...
• Factors of 15: 1, 3, 5, 15

Summary

Multiples are numbers you get when you multiply a number by 1, 2, 3, 4, and so on. You can generate multiples by multiplying or by skip-counting. Factors and multiples are not the same: factors divide evenly into a number, and multiples are made by multiplying.

Prime and Composite Classification

In math, numbers can be sorted into different groups. One important way to sort numbers is by looking at their factors.

A factor is a whole number that divides evenly into another number, with no remainder. For example, the factors of \(8\) are \(1, 2, 4, 8\).

Today we will learn how to tell if a number is prime or composite.

What is a prime number?

A prime number has exactly 2 factors: \(1\) and itself.

Examples of prime numbers are:

• \(2\), because its factors are \(1\) and \(2\)
• \(3\), because its factors are \(1\) and \(3\)
• \(5\), because its factors are \(1\) and \(5\)
• \(7\), because its factors are \(1\) and \(7\)

What is a composite number?

A composite number has more than 2 factors.

Examples of composite numbers are:

• \(4\), because its factors are \(1, 2, 4\)
• \(6\), because its factors are \(1, 2, 3, 6\)
• \(9\), because its factors are \(1, 3, 9\)
• \(12\), because its factors are \(1, 2, 3, 4, 6, 12\)

What about the number \(1\)?

The number \(1\) is neither prime nor composite.

Why? Because \(1\) has only 1 factor, and that factor is \(1\).

Remember:

• Prime = exactly 2 factors
• Composite = more than 2 factors
• \(1\) = neither

Using factor pairs

A helpful way to classify numbers is to look for factor pairs.

A factor pair is two whole numbers you multiply to get the number.

For example, the factor pairs of \(12\) are:

• \(1 \times 12 = 12\)
• \(2 \times 6 = 12\)
• \(3 \times 4 = 12\)

Since \(12\) has more than one factor pair, it has more than 2 factors. So \(12\) is composite.

Now look at \(11\):

• \(1 \times 11 = 11\)

That is the only factor pair. So \(11\) has exactly 2 factors, \(1\) and \(11\). That means \(11\) is prime.

How to tell if a number is prime or composite

1. Start with the number.
2. Find all the whole numbers that divide it evenly.
3. Count the factors.
4. If it has exactly 2 factors, it is prime.
5. If it has more than 2 factors, it is composite.
6. If the number is \(1\), it is neither.

Worked Example 1: Classify \(7\)

Let's find the factors of \(7\).

• \(1\) divides evenly into \(7\)
• \(7\) divides evenly into \(7\)

The factors are \(1\) and \(7\).

That is exactly 2 factors, so \(7\) is prime.

Worked Example 2: Classify \(10\)

Let's find the factors of \(10\).

• \(1 \times 10 = 10\)
• \(2 \times 5 = 10\)

So the factors are \(1, 2, 5, 10\).

Since \(10\) has more than 2 factors, \(10\) is composite.

Worked Example 3: Classify \(13\)

Check for factor pairs:

• \(1 \times 13 = 13\)

No other whole numbers multiply to make \(13\).

So the factors are \(1\) and \(13\).

That means \(13\) is prime.

Worked Example 4: Classify \(15\)

Check the factor pairs:

• \(1 \times 15 = 15\)
• \(3 \times 5 = 15\)

The factors are \(1, 3, 5, 15\).

Since there are more than 2 factors, \(15\) is composite.

A quick list of small prime numbers

Here are some prime numbers you may see often:

\(2, 3, 5, 7, 11, 13, 17, 19\)

Important note about \(2\)

The number \(2\) is a prime number.

It may look special because it is even, but it still has exactly 2 factors: \(1\) and \(2\).

Tips to help you

• If a number has only one factor pair, \(1\) and itself, it is probably prime.
• If you can find another factor pair, the number is composite.
• All even numbers greater than \(2\) are composite because they can be divided by \(2\).
• Always remember that \(1\) is neither prime nor composite.

Let's compare

• \(5\): factors are \(1, 5\) → prime
• \(8\): factors are \(1, 2, 4, 8\) → composite
• \(1\): factor is \(1\) → neither

Summary

To classify a number, look at its factors or factor pairs.

If a number has exactly 2 factors, it is prime. If it has more than 2 factors, it is composite. If the number is \(1\), it is neither prime nor composite.

Divisibility Rules for 2, 5, and 10

Sometimes we want to know if one number can be divided by another number without a remainder. A quick way to check is to use a divisibility rule.

In this lesson, you will learn the divisibility rules for 2, 5, and 10. These rules are easy because you only need to look at the last digit of the number.

What does divisible mean?

A number is divisible by another number if it can be split into equal groups with no leftovers.

For example, \(12 \div 2 = 6\). There is no remainder, so 12 is divisible by 2.

But \(13 \div 2 = 6\) remainder \(1\). Since there is a leftover, 13 is not divisible by 2.

Why do these rules work?

Our number system is based on tens. That means the digit at the end of a number tells us a lot. The last digit shows how many ones there are.

For divisibility by 2, 5, and 10, the ones digit is enough to tell us the answer.

Rule for 2

A number is divisible by 2 if its last digit is an even number.

The even digits are:

• 0
• 2
• 4
• 6
• 8

So if a number ends in 0, 2, 4, 6, or 8, it is divisible by 2.

Examples:

• 14 ends in 4, so 14 is divisible by 2.
• 36 ends in 6, so 36 is divisible by 2.
• 91 ends in 1, so 91 is not divisible by 2.

Rule for 5

A number is divisible by 5 if its last digit is 0 or 5.

Examples:

• 25 ends in 5, so 25 is divisible by 5.
• 70 ends in 0, so 70 is divisible by 5.
• 43 ends in 3, so 43 is not divisible by 5.

Rule for 10

A number is divisible by 10 if its last digit is 0.

Examples:

• 40 ends in 0, so 40 is divisible by 10.
• 130 ends in 0, so 130 is divisible by 10.
• 156 ends in 6, so 156 is not divisible by 10.

Important pattern to notice

If a number is divisible by 10, it is also divisible by 5 and by 2.

Why? Because numbers divisible by 10 always end in 0. A last digit of 0 matches the rule for 10, the rule for 5, and the rule for 2.

For example, 90 ends in 0.

• 90 is divisible by 10
• 90 is divisible by 5
• 90 is divisible by 2

Worked Example 1

Is 18 divisible by 2, 5, or 10?

Look at the last digit. The last digit of 18 is 8.

• 8 is even, so 18 is divisible by 2.
• 8 is not 0 or 5, so 18 is not divisible by 5.
• 8 is not 0, so 18 is not divisible by 10.

Answer: 18 is divisible only by 2.

Worked Example 2

Is 45 divisible by 2, 5, or 10?

Look at the last digit. The last digit of 45 is 5.

• 5 is not even, so 45 is not divisible by 2.
• 5 is 0 or 5, so 45 is divisible by 5.
• 5 is not 0, so 45 is not divisible by 10.

Answer: 45 is divisible only by 5.

Worked Example 3

Is 120 divisible by 2, 5, or 10?

Look at the last digit. The last digit of 120 is 0.

• 0 is even, so 120 is divisible by 2.
• 0 is 0 or 5, so 120 is divisible by 5.
• 0 means 120 is divisible by 10.

Answer: 120 is divisible by 2, 5, and 10.

Worked Example 4

Which numbers are divisible by 2, 5, and 10: \(32, 55, 87, 140\)?

Check each number by its last digit.

1. 32 ends in 2.
   It is divisible by 2.
   It is not divisible by 5 or 10.
2. 55 ends in 5.
   It is divisible by 5.
   It is not divisible by 2 or 10.
3. 87 ends in 7.
   It is not divisible by 2, 5, or 10.
4. 140 ends in 0.
   It is divisible by 2, 5, and 10.

A quick way to remember the rules

• Divisible by 2: last digit is 0, 2, 4, 6, or 8
• Divisible by 5: last digit is 0 or 5
• Divisible by 10: last digit is 0

Watch out for these mistakes

• Do not look at every digit. For these rules, only the last digit matters.
• A number ending in 5 is divisible by 5, but not by 10.
• A number ending in 0 is divisible by 10, and also by 5 and 2.
• A number ending in an odd digit like 1, 3, 5, 7, or 9 is not divisible by 2.

Let’s think with equations

If a number ends in 0, we can divide it by 10 with no remainder.

For example:

Since 120 ends in 0, it also works for 5 and 2:

Summary

Divisibility rules help you decide quickly if a number can be divided evenly.

• A number is divisible by 2 if the last digit is 0, 2, 4, 6, or 8.
• A number is divisible by 5 if the last digit is 0 or 5.
• A number is divisible by 10 if the last digit is 0.

When you check divisibility by 2, 5, or 10, remember to look at the ones digit. That one digit tells you the answer.

Common Place Value Misconceptions

Place value helps us understand what a digit is really worth in a number. In our base-ten system, the value of a digit depends on where it is placed.

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

• The \(4\) means 4 hundreds, or \(400\)
• The \(5\) means 5 tens, or \(50\)
• The \(2\) means 2 ones, or \(2\)

A common mistake is to look at digits as if they are just single numbers. But digits in a number are not all worth the same amount. A \(5\) in the tens place does not mean 5. It means \(50\).

In this lesson, we will learn some common place value mistakes and how to fix them.

1. Misconception: A digit always has the same value

This is not true. A digit can have different values depending on its place.

Look at the digit \(7\) in these numbers:

• \(7\) = 7 ones
• \(70\) = 7 tens = \(70\)
• \(700\) = 7 hundreds = \(700\)

Even though the digit is the same, its value changes with its position.

2. Misconception: The number with the bigger digit is always greater

Sometimes students compare only one digit and forget to look at the place value.

For example, compare \(398\) and \(421\).

Some students may think \(398\) is greater because \(9\) is bigger than \(2\). But we must compare from the greatest place first:

• \(398\) has 3 hundreds
• \(421\) has 4 hundreds

Since 4 hundreds is greater than 3 hundreds, \(421 > 398\).

3. Misconception: Zero does not matter

Zero is very important in place value. It shows that there are no groups in that place.

Look at these numbers:

• \(402\) means 4 hundreds, 0 tens, 2 ones
• \(420\) means 4 hundreds, 2 tens, 0 ones

These numbers use the same digits, but they are different because the digits are in different places.

4. Misconception: Expanded form is just writing the digits with plus signs

Expanded form shows the value of each digit, not just the digits themselves.

For example, the expanded form of \(634\) is:

$$634 = 600 + 30 + 4$$

It is not:

$$634 \neq 6 + 3 + 4$$

The digits must be changed into their place values.

5. Misconception: Reading a number is the same as saying each digit

When we read numbers, we read the value of the whole number, not each digit by itself.

For example:

• \(527\) is read as five hundred twenty-seven
• It is not read as five, two, seven

Reading numbers correctly helps you understand place value correctly.

How to avoid place value mistakes

Here are some helpful strategies:

• Look at the place of each digit: hundreds, tens, ones
• Ask, “What is this digit worth?”
• Use expanded form to show the value of each digit
• Compare numbers from left to right, starting with the greatest place
• Do not ignore zeros

Worked Example 1

What is the value of the digit \(6\) in \(364\)?

Step 1: Find the place of the digit \(6\).

In \(364\), the \(6\) is in the tens place.

Step 2: Find its value.

\(6\) tens = \(60\)

Answer: The value of the digit \(6\) is 60.

Worked Example 2

Write \(508\) in expanded form.

Step 1: Look at each digit and its place.

• \(5\) is in the hundreds place, so it means \(500\)
• \(0\) is in the tens place, so it means \(0\) tens
• \(8\) is in the ones place, so it means \(8\)

Step 2: Write the values as a sum.

$$508 = 500 + 0 + 8$$

Answer: The expanded form is \(500 + 0 + 8\).

Worked Example 3

Which number is greater: \(460\) or \(406\)?

Step 1: Compare the hundreds place.

Both numbers have 4 hundreds.

Step 2: Compare the tens place.

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

Since 6 tens is greater than 0 tens, \(460\) is greater.

$$460 > 406$$

Answer: \(460\) is greater.

Worked Example 4

A student says, “In the number \(2,145\), the digit \(1\) means 1.” Is the student correct?

Step 1: Find the place of the digit \(1\).

In \(2,145\), the digit \(1\) is in the hundreds place.

Step 2: Find its value.

\(1\) hundred = \(100\)

Answer: The student is not correct. The digit \(1\) means 100, not 1.

Important idea to remember

A digit by itself does not tell the whole story. Its place tells its value.

For example, in \(5,352\):

• \(5\) means 5 thousands = \(5,000\)
• \(3\) means 3 hundreds = \(300\)
• \(5\) means 5 tens = \(50\)
• \(2\) means 2 ones = \(2\)

The two digits \(5\) do not have the same value because they are in different places.

Brief Summary

Place value tells us how much each digit is worth. Common mistakes happen when students ignore the place, ignore zeros, compare the wrong digits, or write expanded form incorrectly. To avoid these mistakes, always ask: What place is the digit in, and what is its value?