homework & practice 1 2 understand whole number place value

Place Value Worksheets

Welcome to the Place Value worksheets page at Math-Drills.com! This page includes Place value worksheets for whole numbers and decimals including place value charts. Students who have a good understanding of place value will certainly excel at math. They will be better at estimating, calculating, and conceptualizing and may go on to great mathematical success in school and beyond.

Most Popular Place Value Worksheets this Week

$Decimal Place Value Chart (Hundred Millions to Millionths)$

Place Value Charts

$homework & practice 1 2 understand whole number place value$

Place value charts can be used to learn about place value. They might also be useful in correcting student thinking when they don't quite get the place holder concept (e.g. writing 132 as 100302 or 1004 as 14). Place value charts can also be used for addition, subtraction, multiplication and division. For example, to add two numbers, write each addend in its own row then add starting with the lowest place, regroup and keep moving to the left until the third row shows the sum. Each place value chart includes multiple lines for this purpose. Division on a place value chart parallels the long division algorithm, so it might be a good place to go if students are struggling with long division. There are plenty of place value charts in this section with different ranges both for whole and decimal numbers.

Place Value Charts Place Value Chart to Hundreds Place Value Chart to Thousands Place Value Chart to Ten Thousands Place Value Chart to Hundred Thousands Place Value Chart to Millions Place Value Chart to Ten Millions Place Value Chart to Hundred Millions
Decimal Place Value Charts Decimal Place Value Chart ( Ones to Hundredths ) Decimal Place Value Chart ( Ones to Thousandths ) Decimal Place Value Chart ( Hundreds to Hundredths ) Decimal Place Value Chart ( Thousands to Thousandths ) Decimal Place Value Chart ( Hundred Thousands to Thousandths ) Decimal Place Value Chart ( Hundred Millions to Millionths )

The main difference in the decimal place value charts in this section is that a comma is used as a decimal. Many countries outside of North America use this number format.

European Format Decimal Place Value Charts European Format Decimal Place Value Chart ( Ones to Hundredths ) European Format Decimal Place Value Chart ( Ones to Thousandths ) European Format Decimal Place Value Chart ( Hundreds to Hundredths ) European Format Decimal Place Value Chart ( Thousands to Thousandths ) European Format Decimal Place Value Chart ( Hundred Thousands to Thousandths ) European Format Decimal Place Value Chart ( Hundred Millions to Millionths )

$homework & practice 1 2 understand whole number place value$

Starting at the very beginning, students learn the names of numbers and probably how to count. They may start encountering individual numbers in writing, and learn how to identify them. Then things start getting complicated! Two- and three-digit numbers start to enter their lives and they must learn how to name those, and that is when a good knowledge of place comes in handy. In order to say the number 793, for example, children must know that the 7 is in the hundreds place, the 9 is in the tens place and the 3 is in the ones place to be able to say, "Seven hundred ninety-three."

"Determine Place and Value" worksheets are listed first in this section as they are the ones that are used most frequently. Students are asked to identify both the place and the value of an underlined digit. For example in 4 5,678, the underlined 4 is in the ten thousands place and has a value of 40,000. If students need a little more instruction to help them learn place value, the "Identify Place Only" worksheets might help. In those, students only have to determine which place is underlined. For example, in 4 5 ,678, they would say the thousands place is underlined.

There are also two worksheets for re-writing numbers with thousands separators. The first is not overly exciting, but the second also gets the student to follow a path to build the number before including the thousands separators.

Determine Place and Value Worksheets Determine Place and Value (to Hundreds ) ✎ Determine Place and Value (to Hundreds ) Large Print ✎ Determine Place and Value (to Thousands ) ✎ Determine Place and Value (to Thousands ) Large Print ✎ Determine Place and Value (to Ten Thousands ) ✎ Determine Place and Value (to Ten Thousands ) Large Print ✎ Determine Place and Value (to Hundred Thousands ) ✎ Determine Place and Value (to Hundred Thousands ) Large Print Determine Place and Value (to Millions ) ✎ Determine Place and Value (to Millions ) Large Print ✎ Determine Place and Value (to Ten Millions ) ✎ Determine Place and Value (to Ten Millions ) Large Print ✎ Determine Place and Value (to Hundred Millions ) ✎ Determine Place and Value (to Hundred Millions ) Large Print ✎
Identify Place Only Worksheets Identify Place Only (to Hundreds ) ✎ Identify Place Only (to Hundreds ) Large Print ✎ Identify Place Only (to Thousands ) ✎ Identify Place Only (to Thousands ) Large Print ✎ Identify Place Only (to Ten Thousands ) ✎ Identify Place Only (to Ten Thousands ) Large Print ✎ Identify Place Only (to Hundred Thousands ) ✎ Identify Place Only (to Hundred Thousands ) Large Print ✎ Identify Place Only (to Millions ) ✎ Identify Place Only (to Millions ) Large Print ✎ Identify Place Only (to Ten Millions ) ✎ Identify Place Only (to Ten Millions ) Large Print ✎ Identify Place Only (to Hundred Millions ) ✎ Identify Place Only (to Hundred Millions ) Large Print ✎
Re-Write Numbers with Thousands Separators Worksheets Re-Write Numbers with Thousands Separators Follow the Arrows to Build a Number and Insert Thousands Separators (Grid Numbers in Order) Follow the Arrows to Build a Number and Insert Thousands Separators (Grid Numbers Mixed)

Similar to the last section, but the numbers in this section are formatted with thin spaces for thousands separators as you might find in Canada or other English countries that have adopted the Metric (or S.I.) system. The final worksheet has students re-write numbers with spaces as thousands separators.

Determine Place and Value Worksheets (SI Format: Space-Separated Thousands) SI Format Determine Place and Value (to Thousands ) ✎ SI Format Determine Place and Value (to Thousands ) Large Print ✎ SI Format Determine Place and Value (to Ten Thousands ) ✎ SI Format Determine Place and Value (to Ten Thousands ) Large Print ✎ SI Format Determine Place and Value (to Hundred Thousands ) ✎ SI Format Determine Place and Value (to Hundred Thousands ) Large Print ✎ SI Format Determine Place and Value (to Millions ) ✎ SI Format Determine Place and Value (to Millions ) Large Print ✎ SI Format Determine Place and Value (to Ten Millions ) ✎ SI Format Determine Place and Value (to Ten Millions ) Large Print ✎ SI Format Determine Place and Value (to Hundred Millions ) ✎ SI Format Determine Place and Value (to Hundred Millions ) Large Print ✎
Identify Place Only Worksheets (SI Format: Space-Separated Thousands) SI Format Identify Place Only (to Thousands ) ✎ SI Format Identify Place Only (to Thousands ) Large Print ✎ SI Format Identify Place Only (to Ten Thousands ) ✎ SI Format Identify Place Only (to Ten Thousands ) Large Print ✎ SI Format Identify Place Only (to Hundred Thousands ) ✎ SI Format Identify Place Only (to Hundred Thousands ) Large Print ✎ SI Format Identify Place Only (to Millions ) ✎ SI Format Identify Place Only (to Millions ) Large Print ✎ SI Format Identify Place Only (to Ten Millions ) ✎ SI Format Identify Place Only (to Ten Millions ) Large Print ✎ SI Format Identify Place Only (to Hundred Millions ) ✎ SI Format Identify Place Only (to Hundred Millions ) Large Print ✎
Re-Write Numbers with Thousands Separators Worksheets (SI Format: Space-Separated Thousands) SI Format Re-Write Numbers with Thousands Separators

The worksheets in this section are similar to the previous two sections, but use a point-comma format for numbers where the point is used as a thousands separator and a comma is used as a decimal.

Determine Place and Value Worksheets (European Format: Period-Separated Thousands) European Determine Place and Value (to Thousands ) ✎ European Determine Place and Value (to Thousands ) Large Print ✎ European Determine Place and Value (to Ten Thousands ) ✎ European Determine Place and Value (to Ten Thousands ) Large Print ✎ European Determine Place and Value (to Hundred Thousands ) ✎ European Determine Place and Value (to Hundred Thousands ) Large Print ✎ European Determine Place and Value (to Millions ) ✎ European Determine Place and Value (to Millions ) Large Print ✎ European Determine Place and Value (to Ten Millions ) ✎ European Determine Place and Value (to Ten Millions ) Large Print ✎ European Determine Place and Value (to Hundred Millions ) ✎ European Determine Place and Value (to Hundred Millions ) Large Print ✎
Identify Place Only Worksheets (European Format: Period-Separated Thousands) European Identify Place Only (to Thousands ) ✎ European Identify Place Only (to Thousands ) Large Print ✎ European Identify Place Only (to Ten Thousands ) ✎ European Identify Place Only (to Ten Thousands ) Large Print ✎ European Identify Place Only (to Hundred Thousands ) ✎ European Identify Place Only (to Hundred Thousands ) Large Print ✎ European Identify Place Only (to Millions ) ✎ European Identify Place Only (to Millions ) Large Print ✎ European Identify Place Only (to Ten Millions ) ✎ European Identify Place Only (to Ten Millions ) Large Print ✎ European Identify Place Only (to Hundred Millions ) ✎ European Identify Place Only (to Hundred Millions ) Large Print ✎
Re-Write Numbers with Thousands Separators Worksheets (European Format: Period-Separated Thousands) European Re-Write Numbers with Thousands Separators

Decimal Place Value Worksheets

$homework & practice 1 2 understand whole number place value$

So, you've learned all the places before the decimal, how about the ones after the decimal? These worksheets might help. Unlike the numbers before the decimal, the ones after aren't usually in groups of three, so it is necessary to count very carefully and remember that the first place after the decimal is tenths. If you have trouble, try using a place value chart. As an added bonus, students can also practice their whole number place values as every question includes an underlined whole number digit and an underlined decimal digit.

Determine Place and Value of Decimal Numbers Worksheets Determine Place and Value of Decimal Numbers ( Hundredths to Tens ) ✎ Determine Place and Value of Decimal Numbers ( Hundredths to Tens ) Large Print ✎ Determine Place and Value of Decimal Numbers ( Thousandths to Hundreds ) ✎ Determine Place and Value of Decimal Numbers ( Thousandths to Hundreds ) Large Print ✎ Determine Place and Value of Decimal Numbers ( Ten Thousandths to Thousands ) ✎ Determine Place and Value of Decimal Numbers ( Ten Thousandths to Thousands ) Large Print ✎ Determine Place and Value of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) ✎ Determine Place and Value of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) Large Print ✎ Determine Place and Value of Decimal Numbers ( Millionths to Hundred Thousands ) ✎ Determine Place and Value of Decimal Numbers ( Millionths to Hundred Thousands ) Large Print ✎ Determine Place and Value of Decimal Numbers ( Ten Millionths to Millions ) ✎ Determine Place and Value of Decimal Numbers ( Ten Millionths to Millions ) Large Print ✎
Identify Place Only of Decimal Numbers Worksheets Identify Place Only of Decimal Numbers ( Hundredths to Tens ) ✎ Identify Place Only of Decimal Numbers ( Hundredths to Tens ) Large Print ✎ Identify Place Only of Decimal Numbers ( Thousandths to Hundreds ) ✎ Identify Place Only of Decimal Numbers ( Thousandths to Hundreds ) Large Print ✎ Identify Place Only of Decimal Numbers ( Ten Thousandths to Thousands ) ✎ Identify Place Only of Decimal Numbers ( Ten Thousandths to Thousands ) Large Print ✎ Identify Place Only of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) ✎ Identify Place Only of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) Large Print ✎ Identify Place Only of Decimal Numbers ( Millionths to Hundred Thousands ) ✎ Identify Place Only of Decimal Numbers ( Millionths to Hundred Thousands ) Large Print ✎ Identify Place Only of Decimal Numbers ( Ten Millionths to Millions ) ✎ Identify Place Only of Decimal Numbers ( Ten Millionths to Millions ) Large Print ✎

Very much the same as the previous section, but a thin space for a thousands separator is used. The lesser numbers are excluded from this section as there is no thousands separator involved, so the worksheets in the U.S. section can be used.

Determine Place and Value of Decimal Numbers Worksheets (SI Format: Space-Separated Thousands) SI Format Determine Place and Value of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) ✎ SI Format Determine Place and Value of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) Large Print ✎ SI Format Determine Place and Value of Decimal Numbers ( Millionths to Hundred Thousands ) ✎ SI Format Determine Place and Value of Decimal Numbers ( Millionths to Hundred Thousands ) Large Print ✎ SI Format Determine Place and Value of Decimal Numbers ( Ten Millionths to Millions ) ✎ SI Format Determine Place and Value of Decimal Numbers ( Ten Millionths to Millions ) Large Print ✎
Identify Place Only of Decimal Numbers Worksheets (SI Format: Space-Separated Thousands) SI Format Identify Place Only of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) ✎ SI Format Identify Place Only of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) Large Print ✎ SI Format Identify Place Only of Decimal Numbers ( Millionths to Hundred Thousands ) ✎ SI Format Identify Place Only of Decimal Numbers ( Millionths to Hundred Thousands ) Large Print ✎ SI Format Identify Place Only of Decimal Numbers ( Ten Millionths to Millions ) ✎ SI Format Identify Place Only of Decimal Numbers ( Ten Millionths to Millions ) Large Print ✎

The European Format uses a comma for a decimal and periods for thousands separators.

Determine Place and Value of Decimal Numbers Worksheets (European Format: Period-Separated Thousands and Comma Decimal) European Format Determine Place and Value of Decimal Numbers ( Hundredths to Tens ) European Format Determine Place and Value of Decimal Numbers ( Hundredths to Tens ) Large Print ✎ European Format Determine Place and Value of Decimal Numbers ( Thousandths to Hundreds ) ✎ European Format Determine Place and Value of Decimal Numbers ( Thousandths to Hundreds ) Large Print ✎ European Format Determine Place and Value of Decimal Numbers ( Ten Thousandths to Thousands ) ✎ European Format Determine Place and Value of Decimal Numbers ( Ten Thousandths to Thousands ) Large Print ✎ European Format Determine Place and Value of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) ✎ European Format Determine Place and Value of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) Large Print ✎ European Format Determine Place and Value of Decimal Numbers ( Millionths to Hundred Thousands ) ✎ European Format Determine Place and Value of Decimal Numbers ( Millionths to Hundred Thousands ) Large Print ✎ European Format Determine Place and Value of Decimal Numbers ( Ten Millionths to Millions ) ✎ European Format Determine Place and Value of Decimal Numbers ( Ten Millionths to Millions ) Large Print ✎
Identify Place Only of Decimal Numbers Worksheets (European Format: Period-Separated Thousands and Comma Decimal) European Format Identify Place Only of Decimal Numbers ( Hundredths to Tens ) ✎ European Format Identify Place Only of Decimal Numbers ( Hundredths to Tens ) Large Print ✎ European Format Identify Place Only of Decimal Numbers ( Thousandths to Hundreds ) ✎ European Format Identify Place Only of Decimal Numbers ( Thousandths to Hundreds ) Large Print ✎ European Format Identify Place Only of Decimal Numbers ( Ten Thousandths to Thousands ) ✎ European Format Identify Place Only of Decimal Numbers ( Ten Thousandths to Thousands ) Large Print ✎ European Format Identify Place Only of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) ✎ European Format Identify Place Only of Decimal Numbers ( Hundred Thousandths to Ten Thousands ) Large Print ✎ European Format Identify Place Only of Decimal Numbers ( Millionths to Hundred Thousands ) ✎ European Format Identify Place Only of Decimal Numbers ( Millionths to Hundred Thousands ) Large Print ✎ European Format Identify Place Only of Decimal Numbers ( Ten Millionths to Millions ) ✎ European Format Identify Place Only of Decimal Numbers ( Ten Millionths to Millions ) Large Print ✎

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Place Value Worksheets

And base 10 blocks.

Our place value worksheets focus on deepening a student's understanding of our base 10 system. In our " base 10 blocks " worksheets, students manipulate blocks (units of 1) and rods (groups of 10) to build, deconstruct or add numbers. Later worksheets focus on building or de-constructing multi-digit numbers.

Choose your grade / topic:

Grade 1: Base 10 blocks worksheets

Grade 1: Place value worksheets

Grade 2: Place value worksheets

Grade 3: Place value worksheets

Grade 4: Place value worksheets

Grade 5: Place value worksheets

Topics include:

Grade 1 base 10 blocks worksheets

Counting and making 10 with ten frames
Regrouping unit blocks into blocks of 10 ("rods")
Counting "tens" and "ones"
Breaking a number (11-99) into rods ("tens") and blocks ("ones")
Adding 2 digit numbers with base 10 blocks

Grade 1 place value worksheets

Identifying tens and ones from 2 digit numbers
Combining tens and ones into 2 digit numbers
Identifying a digit's place value (tens, ones)
Building a 2 digit number with missing addends
Write 2 digit numbers in expanded form
Write 2 digit numbers in normal form

Grade 2 place value worksheets

Building a 3-digit number from the parts
Missing place values in 3-digit numbers
Write 3-digit numbers in expanded form
Write 3-digit numbers in normal form
Hundreds, tens & ones - identify the underlined digit
Comparing and ordering numbers up to 100 and 1,000

Grade 3 place value worksheets

Building 3, 4 and 5-digit numbers from the parts
Missing place values in 3 and 4-digit numbers
Write 4-digit numbers in expanded form
Write 4-digit numbers in expanded notation
Write 4-digit numbers in standard form
Identify the place value of the underlined digit
Compare and order numbers up to 10,000 and 100,000

Grade 4 place value worksheets

Building 4, 5 and 6-digit numbers from the parts
Missing place values in 4, 5 and 6-digit numbers
Write 5-digit numbers in expanded form
Write 5 digit numbers in expanded notation
Write 5-digit numbers in standard form

Grade 5 place value worksheets

Building 5 and 6-digit numbers from the parts
Missing place values in 5 and 6-digit numbers
Build numbers from parts with decimals
Decimal numbers in expanded form
Decimal numbers in expanded notation

Place Value Worksheets Hub Page

Welcome to our Place Value Worksheets area.

Here you will find a wide range of place value activites and worksheets which will help your child gain a better understanding of how our number system and place value works.

You can also take a look at our printable place value charts or our place value sheets with decimals or BIG numbers over a million!

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This page contains links to other Math webpages where you will find a range of activities and resources.
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Place Value Worksheets and Resources

On this page you will find link to our range of place value resources, including online practice, charts and worksheets.

Quicklinks to Place Value ...

Online Place Value Practice
Place Value Charts and Number Grids
Place Value Tens and Ones

Place Value up to Hundreds

Place value up to thousands.

Big Number Place Value
Negative Number Place Value
Decimal Place Value
Randomly Generated Place Vaue Worksheets

Scientific Notation Support

Absolute Value Worksheets

Definition of Place Value

So what exactly is place value?

Place value refers to the value of the digits in any given number. In the number 482 for example, the value of the digit '8' is 80 and the value of the digit '4' is 400.

At a more advanced level, in the number 36.57, the value of the digit '5' is 0.5 and the value of the digit '7' is 0.07.

In our number system, each time you move a place to the right, the value of the digit gets ten times bigger. Each time you move a place to the left, the value of the digit gets ten times smaller.

Definition of Place Value Sheet
PDF version

Place Value Learning

Children start their learning journey in Math when they start to count. When they are confindent counting small groups of objects and getting beyond 10, they then begin to develop their understanding of place value up to 100 and beyond.

When they have understood how place value with whole numbers works, they can start learning about place value with decimals.

Our selection of free math place value worksheets has been split into different areas below so that you can more easily find the right sheet for your child.

Online Place Value Practice Zone

In our Math Place Value Practice area, you can practice your place value skills, adding thousands, hundreds, tens and ones. You can even use this area for adding on decimals: tenths and hundredths.

You can select the numbers you want to practice with, and print out your results when you have finished.

You can also use the practice zone for benchmarking your performance, or using it with a group of children to gauge progress.

Free Place Value Practice Zone 1 - Digit Values
Math Place Value Practice Zone 2 - Combining

Place Value Charts & Number Grids

Place value charts.

We have a selection of place value charts which are great for helping to develop understanding of place value.

The charts cover a range of numbers from billions to millionths.

They are a good supporting resource for children who are finding place value difficult.

We also have place value grids to help children convert to and from expanded form to support place value learning.

They are also a good way to get children to become familiar with how the number system works.

Place Value Charts (whole numbers)
Decimal Place Value Charts
Place Value Grids

Number Grids

These printable grids will help your child learn to read and write numbers and learn the number sequence.

Some of the charts are partially filled to help your child learn their place value.

Using these sheets will help your child to:

learn to count in tens and ones;
learn to read and write numbers.

1st Grade Place Value Charts

Free Number Chart 1-30
Number Chart to 50
Printable Number Charts 0-99
Hundred Number Charts (100 Squares)
Missing Number Chart 1-100
Number Grid up to 200
Number Grid up to 300

Place Value Games

Here you will find our selection of place value games.

We have a range of different games for helping to teach place value from 2-digit games up to decimal place value games.

There are a range of games are suitable for all ages, from kindergarten and upwards.

Games to teach Place Value

Place Value and Number Sense Tens and Ones

Here you will find our selection of 2 digit Place Value worksheets.

Using these Math Worksheets Place Value will help your child to:

learn their place value to 100;
understand the value of each digit in a 2 digit number;
Round numbers up to 100 to the nearest 10
learn to read and write numbers to 100.
Place Value Worksheets for Kindergarten
Place Value to 20 Worksheets
Place Value Ones and Tens Models
Place Value Tens and Ones (standard and expanded form)
Basic Math Worksheets - Ordering numbers 2 Digits
Rounding to the nearest 10 Worksheets
Add and Subtract 10 Worksheets

Here you will find our selection of 3 digit Place Value worksheets.

learn their place value with 3 digit numbers;
understand the value of each digit in a 3 digit number;
learn to read and write numbers up to 999.
2nd Grade Place Value Models up to hundreds
Second Grade Place Value Worksheets 3 Digits
Ordering numbers worksheets up to 999

Here you will find our selection of Place Value up to 4 Digits worksheets.

Using these sheets will help your child learn to:

learn their place value with 4 digit numbers;
use place value models to understand how to combine thousands, hundreds, tens and ones;
understand the value of each digit in a 4 digit number;
learn to use standard and expanded form with 4 digit numbers.
Place Value Models 4 Digits
Place Value 4 Digit Numbers Worksheets (conversion)
Ordering Numbers up to Thousands Sheets

Place Value BIG Numbers

Welcome to our BIG Number Place Value area.

Here you will find sheets to help your child learn their place value to 10 million.

Know how to read and write numbers to 10 million;
Understand place value to 10 million.
Solve place value problems.

All the 4th grade math worksheets in this section support elementary math benchmarks.

4th Grade Place Value up to 6 digits
5th Grade Place Value Worksheets to 10 million
5th Grade Ordering Large Numbers up to 100 million

Place Value Negative Numbers

learn to order negative numbers;
learn to position numbers from -10 to 10 on a number line.
ordering and comparing rational numbers
Ordering Negative Numbers -10 to 10
Ordering and Comparing Rational Numbers

Place Value Decimals

Here you will find our selection of Place Value involving Decimals with up to 2 decimal places (2dp).

learn their place value with decimals up to 2dp;
understand the value of each digit in a decimal number;
learn to read and write numbers with up to 2dp.
Decimal Place Value Worksheets to 2dp
Place Value to 3dp
Ordering Decimals Worksheets

Rounding Numbers Worsheets

As well as our random number worksheet generator, we have a wide range of graded rounding worsheets.

These sheets are great for supporting less able students, or for giving an extra challenge to more able students.

Using the link below will take you to our Rounding numbers hub page where you will find links to all our rounding numbers worksheets.

Rounding Numbers Hub page

Place Value Random Worksheet Generator

Here is our generator for generating your own place value worksheets.

Our generator will create the following worksheets:

digit values
comparing numbers
converting between standard and expanded form

These sheets involve saying the value of the underlined digit.

Digit Place Value Worksheets
Standard Form to Expanded Form Worksheets
Expanded Form to Standard Form Worksheets
Rounding Off Numbers Worksheets
Greater Than Less Than Worksheets

We have a selection of worksheets designed to help students learn about asbolute value.

Topics covered include:

absolute value and opposite numbers
comparing absolute values
absolute value arithmetic
solving absolute value equations

Our Convert to scientific notation calculator will take a number and convert it to scientific notation and e-notation.

It shows you all the working out along the way too.

$Convert to Scientific Notation Calculator image$

Convert to Scientific Notation Calculator
Standard Notation to Scientific Notation Support page
Scientific Notation to Standard Notation support page

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Course: 4th grade > Unit 1

Place value: FAQ
Place value blocks
Place value tables

Finding place value

Identify value of a digit
Creating the largest number
Creating largest or smallest number

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homework & practice 1 2 understand whole number place value

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Place Value: Whole Numbers

Concepts Whole Numbers Decimal Places

Our fingers are called "digits", and so also are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 . ("Numerals" means "number characters".) We use these ten digits, along with the concept of place value, in exactly the same sense that we were using our fingers and our piles of marbles on the previous page: a certain "place" tells us what unit we're working with, and the digit tells us how many we need of that unit.

So, for instance, the expression " 264 " means "two 100 s, plus six 10 s, plus four 1 s", because hundreds, tens, and ones are what are stored in those particular places. In what is called "expanded notation", the number " 264 " can be written as:

200 + 60 + 4 = 2×100 + 6×10 + 4×1

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Place Value

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The commas marking off sets of three digits, like the comma between the 1 to the left and the 0 to the right in " 1,000 ", are used to make it easier for people to read the numbers. Properly, if you spell out a number in words, you should use commas at those same spots. So " 1,234 " would be spelled out as "one thousand, two hundred thirty-four".

Note: There should not be an "and" between the "hundred" and the "thirty-four" in " 234 "; you should not pronounce " 234 " as "two hundred 'and' thirty-four". Yes, I know that's how most people say it; it's still wrong.

Write the number 32,067 in expanded notation.

To "expand" this number, I need to split it up into its different places. If I'm not sure of my places, I'll count them out, starting from the right-hand digit.

This number has five digits. From the table above (if I haven't memorized this information yet), I know that this means that I'm dealing with tens of thousands. The one comma tells me that I'm dealing with thousands, too; one comma means I'm into the thousands, two commas would have meant I'd have been into the millions, and so forth.

So I've got three 10,000 s, two 1,000 s, zero 100 s, six 10 s, and seven 1 s. Usually, we ignore the zeroes in expanded notation, so this gives me:

30,000 + 2,000 + 60 + 7

Write the standard form for the number which, in expanded notation, is written as follows: 9,000 + 300 + 2

I've got nine thousands, three hundreds, and two ones. I don't have any tens in the expanded form, so I'll need to use a zero in the tens place to keep that slot open. Then my standard form is:

9,000 + 300 + 0 + 2 = 9,302

For the number 52,973 , (a) state the place held by the 2 , and (b) state the digit in the tens place.

a) The 2 is immediately to the left of the comma. I only have the one comma, so I know this number only goes into the thousands. In the thousands, I've got " 52 ", so the 2 is in the thousands place. (The 5 is in the ten-thousands place.)

b) The tens place is the second place, just to the left of the 3 in the ones place. There is a 7 in this second place, so the digit in the tens place is 7 .

State the number 622,937,285 in words.

The commas break this number into digestible pieces. The " 285 " is in terms of ones, tens, and hundreds of ones (or of just "regular numbers", in extremely informal language). The " 937 " is in terms of ones, tens, and hundreds of thousands. And the " 622 " is in terms of ones, tens, and hundreds of millions.

So the number they gave me, when I spell it out in words, is:

six hundred twenty-two million, nine hundred thirty-seven thousand, two hundred eighty-five.

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Place Value Worksheets

The concept of place value is often tough for a number of young learners. We have plenty of practice to help you. We start out very basic and give you a great deal of helpful reference tables and charts. This section of our website features a wide range of skills based worksheets that explore just about every aspect of using place values to better understand the value of all different types of values.

Working With Values (Tens and Ones)

Place Value Tens and Ones - A great stepping stone for students. A great starter page for students. It focuses and guides them through tens and ones places.
Ones and Tens Place Value Cubes - The visual of the 3-D cubes makes it easier for students to understand the concepts. We break out the visual three dimensional cubes to represent ones and tens places. This is a fun one for students that have problems envisioning the concept.
Place Value to 99 Review Sheet - A deep review of all the basics of the core concept. A very deep and complete review of the ones and tens place value.
Know Your Place (Tens and Ones) - Identify if it's a ones or tens place. We identify the place value of an integer within a preexisting number. There are only two different possible answers in this one.
Write the names of the numbers (Tens and Ones) - Name the numbers. We ask students to produce the names of the given numbers now. Again all numbers are under one hundred here too.
Compare Physical Numbers (Ones and Tens Place Values) - You compare just the place value, not the integers themselves. This is a bit of unique greater than, less than activity. You compare the value of the places and not the total numbers.
Matching Names to Numbers (Ones and Tens Place) - This is a commonly tested skill. Match the names and the numbers. The numbers are ninety-nine or less.
Cut and Paste Order Cards (Tens and One) - A fun cooperative review activity for kids. This can be an individual or group activity. There are 3 version present in this printable. You could also mix all three versions together for even more fun.

Working With Larger Values (Ones to Thousands)

Ones, Tens, and Hundreds Ordered Place Value Cubes - The visual is a great way to introduce them to it. This one covers almost all the skills we need with place value. Including writing numbers in words and expanded format.
Know Your Place (Hundreds, Tens, and Ones) - Identify the places. What is the value of the underlined place. Is it a ones, tens, or hundreds place?
Place Value Hundreds, Tens, and Ones - This is a pretty easy one. This is a nice way to introduce the hundreds place. We have students spilt up the numbers in a methodic pattern.
Missing Place Value Worksheet to 100 - Insert the values to make both sides equal. This should be one of the first place sheets students work with. It's very basic.
Missing Place Value Worksheet to 1000 - Same as the one above, we just go a place higher. Fill in the missing integer to get equivalent numbers.
Write Place Values Worksheet to 1000 - Ditto from the one sheet above this. Separate numbers that are in the hundreds.
Compare Physical Numbers (1s, 10s, 100s) - This is outstanding for focusing students on the importance of place value. You can always use the old analogy, "Would you rather have $100, $10, or $1?" That always seems to instantly click with kids.
Matching Names to Numbers - Not as easy as you would think for most kids. Matching names and numbers sounds easy, but 32% of national tested fourth graders did not master this skill last year.
Naming Numbers (Hundreds, Tens, and Ones) - We ask students to generate names. Students have to actually generate and write the names. This gives many children trouble, so tell them to not get frustrated.
Write Place Values Worksheet to 100 - A core skill that only 65% mastered last year. Place the correct place in the value slots. This makes a good transition for students that don't fully understand the skill.
Cut and Paste Order Cards (Ones, Tens, Hundreds) - A fun game you can play with a partner or yourself. 3 versions in 1. This can be a great cooperative or individual hands-on activity for you and your students. There are 3 full versions for you.

Expanded Form of Numbers

Standard to Expanded Form (Ones and Tens Places) - This is your basic introductory sheet. This is the first time that we work on expanded form. We just need to remind students that it is just reverse engineering the addition of the numbers. Wait, that sounds complicated; doesn't it!
Expanded to Standard Form (Ones and Tens) - Students have an easier time with this format. Kids have a very easy time going to standard form. They usually look at it as just finding the sum of the expanded form.
Expanded to Standard Form with Decimals - This can be tricky for some. We practice writing decimal numbers in expanded form. This is where we will lose a few students. Don't let the decimal point scare them.
Standard to Expanded Form (Ones, Tens, Hundreds) - We jump a place higher. We work with 100s and start writing expanded forms. The expanded form just needs to add up to equal the standard form.
Expanded to Standard Form (Ones, Tens, Hundreds) - When in doubt, just find the sum. Students convert expanded numbers into standard form. To them, it feels like basic addition.
Standard to Expanded Form with Decimals (Thousands) - This starts to get a bit cramped. You will find that most students will have difficulty with the decimal portion of this sheet. If you walk them through the first one, the second one will come with ease.
Writing Larger Numbers in Expanded Form Standard - This is a skill that carries into middle school. Students that work off of the first problem will have an easy time with this. It is more of a guided worksheet.

Working With Larger Places

Place Value Puzzle - We ask you to guess the numbers that we describe. We suggest that you only tackle this one after you have a solid understanding of the unit. If not, you will be lost. If you ace this one, you are good to go!
Missing Place Value Worksheet to 10,000 - Fill in the missing places. You need to create matching values on both sides of the equal sign.
Missing Place Value Worksheet to 100,000 - Same as above just a jump up in value. This one steps up the skill a bit more. We randomly mix the level of the values.
Place Value Worksheet to 9,999 - We ask you to name the underlined place. We move up to the thousands place and see how we do. Hopefully we do great!
Place Value Worksheet to 99,999 - Same as above just up in the ten-thousands. We ask you to write the actual vale in words.
Place Value Worksheet to 999,999 - Just under a million, but same as above. We arrive at the highest value recognized by the standards for this skill.
Write Place Values Worksheet to 10,000 - This is an easy, but lengthy worksheet. Place the correct values for each place on the lines. This can really get you into a rhythm.
Write Place Values Worksheet to 100,000 - We jump up the value. We go up to the ten-thousands place again with this sheet. We keep a standard format for students to grow into.

Quick Reference Helpers

Place Value Chart (4 places) - From thousands to thousandths, this lays it out for you. This one will help you learn your hundreds to hundreths or vice versa.
Place Value Chart (6 places) - A very large set of places and it includes decimals. This one covers 6 places on both sides of the decimal. A big help for middle level students.
Billions Whole Number Place Value Chart - We go over every major whole place value. We only work with whole numbers here. We cover every place up to a billion.
Where Do the Commas Go? - This is a skill we often forget about. We give you random numders of digits and ask you to place the commas in the proper place. Remember commas every three integers from the end.

Grade Leveled Work

Tens and Ones Place Chart - Grade 1
Number Form From Place Value - Grade 2 (easy)
Place Value to Hundreds Chart - Grade 2 guided
Rounding to the 10s and 100s Place - Grade 3
Round to the Nearest Place - Grade 4
Compare the Value of Places - Grade 4 Mastery Skill

Science Related Notation

Scientific Notation - This is the level of difficulty the basic curriculum covers under math standards. Science standards take it a bit further.
Irregular Scientific Notation - You usually don't see that in a lab, unless you need to re-verify data.

Place Value Related Teacher Resources

Estimate The Sum Worksheet | Answers
How to Teach the Concept of Place Value
Teacher Resources For Math
Word Problems

How to Teach Students the Concept of Place Value

Math is an important subject as it's ubiquitous. No single aspect of life is complete without math. However, the basics of math are the most crucial part of learning mathematics in a way to remember and apply for a lifetime. And since math is significantly reliant on numbers, this is one of the most basic concept of math to be taught in primary classes to this day.

Keep reading to find effective strategies for teaching this concept that can help your students stay engaged in class and provide a solid base for future mathematical concepts.

What Is It?

Place value is the value of a digit relative to its position in the number. For example, the number "123" has a "1" in the hundreds place, a "2" in the tens place, and a "3" in the ones place. The value of each digit increases by a power of ten as you move to the left in the number. So, the value of the "1" in "123" is 1 x 100 = 100, the value of the "2" is 2 x 10 = 20, and the value of the "3" is 3 x 1 = 3. The place value of a number can be used to represent the number in different ways. For example, the number "123" can be represented as 1 hundred + 2 tens + 3 ones, or as 1 x 100 + 2 x 10 + 3 x 1.

Strategies for Teaching It

Teachers can do a few key things to teach this concept to their students effectively.

1. It is crucial to provide a clear and concise definition of place value. Teachers can introduce the topic with a game or colorful and interactive presentations to keep students engaged during the class.

2. Once the concept has been introduced and defined, teachers should provide ample opportunities for students to practice identifying the value of each digit within numbers. This practice can be done through various activities and games specifically designed to help students understand this concept. Teachers must ensure that students get more hands-on experience. You can also provide a worksheet for practice worksheets, like you will see above.

3. Teachers need to assess their students' understanding of place value on a regular basis. It will ensure that students are mastering this important mathematical concept. Simple classroom questions can also show which students are struggling and might need more assistance.

Activities for Teaching It

Several activities and games can be practical for teaching this concept to students. Let's take a look at those now:

Rearranging Students

Divide students divided into groups. Line up half of the students in front of the class, holding a particular number in their hand. The other half of the group has to place these students in order from least to greatest or greatest to least.

Number Lines

Another activity is to have students put numbers in order on a number line. You can play this game collectively in a classroom or provide a chance for individual practice by giving out a worksheet for place value.

Roll the Dice

Another activity is to have students roll dice and create numbers, then compare them based on their place value. This activity can help them to understand how this concept works more concretely. Such activity is perfect for pair work and can assist in practical peer evaluation.

Place Value War

This game can effectively teach this concept effectively, as they provide more fun and interactive ways for students to learn.

- To play the game, you will need a deck of cards, a whiteboard, or a piece of paper. The object of the game is to get the highest score by correctly identifying the place value of the numbers on the cards.

- To start the game, each player is dealt seven cards. The player with the highest score starts the game and plays passes to the left. On their turn, the player draws a card from the deck and places it face up in front of them. The other players then have to guess the place value of the number on the card.

- If the player guesses correctly, they score a point whereas an incorrect guess can lead to point deduction. The game continues until all the cards have been drawn. The player who scores the highest number of points wins the game.

The Importance of Teaching the Concept

Place value is the foundation of our number system; without it, we would not be able to understand or use numbers effectively. Teaching this principle to students is essential because:

Basic Number Sense

It helps students adopt the basic number sense. Students can understand the value of numbers by identifying a digit's place in the number. Students can also learn how to use these numbers correctly.

Reading and Writing Numbers

Students can learn to read and write numbers once they've grasped the grassroots concept of place value. Students can write the numbers in figures and words, learning what numbers are called out in everyday life. For instance, $1,000 is a thousand dollars, whereas $1,200 is commonly called twelve hundred dollars.

Count and Compare Numbers

Students can learn to effectively count the numbers and compare them with other numbers to identify which is greater and which is smaller. Students can do so by comparing the digits in each place, determining the digit's value, and ultimately showing which number is greater. For instance, 1,220 and 1,231 may have the same first two digits, but the last two digits show that the latter is greater due to 3 at tens place and 1 at ones.

Perform Basic Operations

As the levels progress, students learn basic math skills and operations such as addition and subtraction. To correctly add or subtract a number, students must understand the alignment of numbers which can be done with the knowledge of place value.

To Sum It Up

The above-discussed strategies for teaching place value can come in handy when you're struggling to keep your class engaged and attentive. Since place value in math is such a fundamental concept, students should grasp it without errors or confusion to ensure a solid base for future mathematical concepts and knowledge.

Using worksheets for place value practice can also be practical as it gives students hands-on experience. You can also arrange games in your classroom to reinforce place value concepts while evaluating students' progress.

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The Complete Guide to Place Value Lessons

By Mary Montero

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This FREE place value guide includes a ton of ideas, place value lessons, essential skills, book list, free activities, and more!

I always start the year with teaching place value, whether I’m teaching 2nd, 3rd, 4th, or 5th grade! I compiled a ton of ideas, an introductory lesson, essential skills, book list, freebies and more into one gigantic (and FREE!) PDF Guide to help you plan your place value lessons. This massive freebie is 27 pages long and includes Place Value and Rounding Anchor Charts, Place Value Mystery Number Task Cards, A Fun Introductory Activity, and a Place Value Choice Board. I have also linked half a dozen extra freebies inside it!

Download Everything You Need– for free!

This free PDF guide will allow you to have all of the place value resources right at your fingertips.

Place Value Standards

An understanding of the place value system is the backbone of success in math. The skills required of students progress as they move through the grade levels, but many students require a review of prior grade level skills well beyond the age at which they appear in the standards.

Standards and Skills – Numbers and Operations in Base Ten – Place Value Understanding

Note: I have only included place value-specific skills. Operation strategies based on place value skills are not included.

Teacher Tip: Use these standards for vertical alignment and differentiation. For example, if I’m teaching gifted 3rd graders, not only do I dive more deeply into grade level standards, but I look at the next grade level standard to accelerate their learning as well. For intervention, I typically look back at the previous grade level standards to be sure there are no gaps in knowledge preventing students from moving forward.

2nd Grade Expectations:

Understand the value of the digits in a three-digit number
Count within 1000; skip count by 5s, 10s, and 100s
Read and write numbers to 1000 using base-ten numbers , names , and expanded form .
Compare two three-digit numbers using >, <, and =

3rd Grade Expectations:

Use place value understanding to round to nearest 10 or 100

4th Grade Expectations:

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
Read and write multi-digit whole numbers using bas—ten numerals, number names, and expanded form
Compare two multi-digit numbers using >,<, and =
Use place value understanding to round multi-digit whole numbers to any place

5th Grade Expectations:

Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Use whole number exponents to denote powers of 10.
Read, write, and compare decimals to thousandths.
Use place value understanding to round decimals to any place.

Lesson Introduction

Place value suggested introduction lesson: 3rd-5th grade.

Note: This lesson plan assumes a basic understanding of the place value system from 1st and 2nd grade standards. This lesson is relevant to grades 3rd-5th. For 5th grade, begin the lesson using a whole number with a decimal.

1. Introduction and Hook

Place a two or three-digit number on the board using magnetic numbers (you can also just write the number). Make sure all the digits in the number are different. Ask students to tell you everything they can about the number they see on the board. Write down EVERYTHING they say, even if it is not accurate. Their statements typically start out basic, but as you push them a little bit more, they will be able to give more and more information. Be sure to ask them about all the different ways the number can be written, if they don’t produce this themselves. If there are any misconceptions that you notice right away, place a star next to them and tell students you’ll come back to them.

2. Base Ten Exploration and Place Value Discs

At this point, students should have already shown the multiple ways to write the number, including using base ten blocks. Have them use base ten blocks ( which they should be familiar with from previous grades ) to represent the number. 3rd Graders may find a place value mat helpful.
For older students (4-5th grade), ask them to show ANOTHER way to represent the number using base ten blocks. These are called equivalent groupings . For example, if the number is 125, they can swap out their hundreds block for ten extra tens and represent the number with 12 tens and 5 ones.
Then, give students place value discs and have them place the appropriate discs under the base ten blocks. Don’t skip this step. Place value discs are an amazing tool, but since the discs are all the same size, it’s critical that students are able to see the conceptual size difference between values using the base ten blocks.
Repeat this process with several numbers, using place value specific language even if you haven’t specifically introduced it yet. (Example: The digit seven is in the tens place, and its value is seventy.)
_____ cube = _____ flats = ______ rods = ______ units

3. Anchor Chart, Direct Instruction, and Interactive Notebooks

At this point, I tell students that we are going to be reviewing and learning important place value skills. I use the anchor chart (see page 6) and interactive notebooks to complete this portion of the lesson. This is when our conceptual math vocabulary comes into play, including the meaning of PLACE, VALUE, DIGIT, and NUMBER. This is also when I review all the ways to write a number. This takes multiple days!

Suggested Lesson Sequence

Place value lessons suggested skill sequence: 3rd-5th grade.

Note: This is a suggested sequence in which I teach my students these skills. This is a 12-day sequence, but I typically spend about 3 full weeks on these skills in grades 3 and 5 and about two weeks in fourth grade.

Day 1-2: Place Value Introduction Lesson

Between the introduction, base ten block and place value disk activity, and anchor chart direct instruction, this will likely take 2-3 days and serve as a solid foundation of place value skills.

Day 3-4: Place Value Skill Reinforcement

By this point, you will have likely introduced the major skills that are addressed during the direct instruction portion. I spend at least 1-2 days reinforcing these skills, and I typically use my place value flipbook, task cards, and other assorted activities. I also spend a GREAT deal of time working on whiteboards practicing writing numbers, converting to different forms, reading numbers, modeling numbers, etc. CLICK HERE TO VIEW MY PLACE VALUE FLIPBOOK

$place value flipbook lessons$

Day 5-6: Rounding & Estimating

Lessons are includes in the free PDF! CLICK HERE TO VIEW MY ROUNDING ON A NUMBER LINE CARDS

Day 7: Comparing & Ordering Numbers

Lessons are includes in the free PDF!

Day 8-9: Small Group Review, Enrichment, and Remediation

During this time, I do small, mini lessons to reinforce a variety of skills. Then, I have students work in centers for practice while I work with small groups on enrichment or remediation.

Day 10 & 11: Place Value Detectives Cumulative Project.

CLICK HERE TO VIEW PLACE VALUE DETECTIVES

Day 12: Assessment

CLICK HERE FOR THE FREE ASSESSMENT OPTIONS

Place Value Anchor Charts

I highly recommend creating a template for this anchor chart and FILLING IT IN with you students during the lesson. The blue digits that create the large number are sticky notes so that we can create multiple numbers using our place value chart. Click here to grab the template for free .

free teaching place value anchor chart templates

Important Considerations During Direct Instruction

Be sure to teach the specifics of writing numbers in word form, including commas, place names, hyphens, and the absence of the word “and” if a decimal is not present.
Give students plenty of practice with zeros appearing in a variety of places and formats.
Verbally saying a large number for students to write in standard form.
Showing a model using base ten blocks or disks for students to write in word form, standard form, or expanded form.
Writing a number in expanded form for students to represent using base ten blocks, standard form, or word form.
Giving clues such as: A number with a 3 in the tens place, a 2 in the hundreds place, and a 1 in the ones place OR A number with 3 hundreds, 7 tens, and 6 ones.

Exposure Checklist

It is vital that students can work with place value skills in a variety of ways. I have compiled a list of visuals, skills, and vocabulary to incorporate, as well as questions to pose to students as you plan your place value unit. You can find this as part of the free PDF download.

Skill-Based Mini Lessons

The free PDF also includes many place value supplemental activity and lesson ideas.

Place Value Book List

Teacher Tip: Use The Guinness Book of World Records (any edition) to help students practice place value skills. Give them criteria to hunt for specific numbers. For example, have students find a record with a six in the tens place.

I also created an Amazon Affiliate List of my favorite place value books . The free PDF includes activity suggestions to use with each book to make planning your place value lessons easier!

teaching place value book recommendations

Related Resources

Place Value Task Card Bundle
Place Value Flipbook
Place Value Detectives
Place Value Topple Blocks
Place Value Puzzlers
Place Value Mystery Numbers
Place Value Error Analysis
The Place Value Candy Rush
Place Value Millions and Billions Task Cards

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

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7 Whole Number Place Value

Janet Stramel

The focus of chapter 6 was on number sense and a relational understanding of numbers. Number sense is linked to place value and an understanding of the base-ten number system. The progression of place value across grades K-5 is critical for understanding:

Decomposing numbers in base ten
Reading and writing numbers
Rounding numbers
Comparing numbers and quantities in base ten.

By the end of kindergarten, students are expected to count to 100 and count sets of 20 (KSDE, 2017a). In kindergarten, counting is based on a ones approach – the number 15 means 15 ones. According to Wright, et. al. (2006), there is a progression to understanding ten:

children understand ten as ten ones,
children see ten as a unit, and
children easily work with units of ten.

Consider the phrase, “ten ones make one ten.” And now think like a child. How does this make sense? Students in phase one count a set of items and think of that set as ones. When students move to phase two, they begin to see a group of ten as a unit, such as a group of ten ones.

We use the base-ten system; using ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Every number can be represented using these digits. In the base-ten system, the value of each place is always 10 times the value of the place to the immediate right. When moving one place left, the value of the place is multiplied by 10.

In the base-ten system, the “places” are ones, tens, hundreds, thousands, etc. And the digit in each place represents 0-9 of those units. Students learn that ten ones makes a unit, called a ten. They then learn that two-digit numbers are composed of ones and tens.

Place value is a fundamental concept in the elementary grades, and understanding place value is essential in learning mathematics. Place value is the value of the digit in its position. For example, the number 358 has three columns or “places,” each with a specific value. In 358, the 3 is in the “hundreds” place, the 5 is in the “tens” place, and the 8 is in the “ones” place. But more importantly, the value of each digit is 3 hundreds (or 300), 5 tens (or 50), and 8 ones.

Base-Ten Blocks

Base-ten blocks provide a hands-on model of our base-ten number system. The smallest cubes are called units. The long, narrow blocks are called rods. The flat, square blocks are called flats. The large cubes are called cubes. The size relationships of the base-ten blocks are perfectly designed to help children discover that it takes 10 units to make one rod, or 10 ones to make one ten; and 10 tens to make one hundred.

Base-ten blocks are ideal for students to physically manipulate “the numbers” so they can conceptually understand the concepts of place value. Nothing can replace the physical manipulatives used to help students make the connections from concrete to abstract understanding.

An example showing the relationship among the manipulatives, the place or position, and the place value of the digits in a number is below.

The number 235 is written in standard notation . This shows the “value” of each digit. There are two hundreds, three tens, and five ones. Additionally, in the number 235: 2 is in hundreds place and its place value is 200, 3 is in tens place and its place value is 30, 5 is in ones place and its place value is 5.

Understanding the place value of digits in numbers helps in writing numbers in their expanded form . For instance, the expanded notation of the number 235 is 200 + 30 + 5.

Read more about the Base-Ten Number System by clicking here.

Developing Whole Number Place Value Concepts across the Grades

When thinking about place value and base-ten understanding, first consider the 5 Strands of Mathematical Proficiency from the National Research Council (2001).

Mathematical proficiency is based upon five interwoven components:

Conceptual understanding – comprehension of mathematical concepts, operations, and relations
Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
Strategic competence – ability to formulate, represent, and solve mathematical problems
Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification
Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy (NRC, 2001, p. 116).

Read more about the 5 Strands of Mathematical Proficiency by downloading the book, “Helping Children Learn Mathematics.”

According to James Brickwedde (2012), “place value and the base ten system is an early and easy entry point for students to begin to explore this agility. Without this level of flexibility and fluency, students are limited to inefficient strategies or are overly dependent upon tactical procedures they know only through rote application.”

Your language matters when teaching students about place value. Speak in value, not in digits. For example, the value of the 2 in 26 is two tens or 20.

Kindergarten

Kindergarten is the first time many students work with numbers greater than 10 using manipulatives and/or drawings. Kindergarten students separate (decompose) a set of 11-19 objects into a group of 10 and some other ones. Experiences with double ten frames will help students to understand this concept.

Notice the grouping of the ten. It is obvious that there is one group of ten with some others left over. Students will immediately see that 14 is made up of one ten and four ones.

The “teen” numbers are one group of ten and some more ones. Notice the “teen” numbers that do not follow a particular pattern in the counting sequence. First, think of the number names 11-19: eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen.

Eleven and twelve are special number words that do not have “teen” as a suffix.
The verbal counting of “teen” numbers is backwards: the ones digit is said before the tens digit. For example, 37 is read as thirty-seven; tens to ones. But the number 14 is read as fourteen; read ones to tens.

When teaching the “teen” numbers, ask students to read the number as well as describe the quantity. For the number 18, students should read “eighteen” and then say, “18 is one group of ten and eight ones.” Additionally, students should record the number sentence 18 = 10 + 8.

By the end of kindergarten, students should be able to compose and decompose numbers between 11 and 19 into tens and some ones. When teaching to compose and decompose numbers, students must use manipulatives and drawings.

Source: KSDE Flip Book Kindergarten (2017b)

Check out the task “ What Makes a Teen Number? ” at the Illustrative Mathematics website. This activity helps students decompose teen numbers using ten-frames and number sentences.

First Grade

In the first grade Domain Number and Operations in Base Ten, the second Cluster is understand place value. Thinking about “10 ones makes 1 ten” is confusing for students. In first grade, students should see 10 ones as 1 ten and that 1 ten has the same value as 10 ones.

Students need many opportunities to practice grouping 10 ones into a bundle of 10. In first grade, students begin to unitize . They see that a group of ten objects is also one ten.

Watch this video “ One Is One, Or Is It? ” narrated by Christopher Danielson for more information on unitizing.

Students must be given numerous experiences using ten-frames, snap cubes, or other groupable models to help develop the concept of place value.

Suggested Activity using Ten-Frames:

Give students 13 counters.

Teacher: “Do you have enough to make a ten?” “Would you have any left over?” “If so, how many left overs do you have?”

Student: “I have filled up one ten-frame and have 3 counters left over. The number 13 has 1 ten and 3 ones.

In first grade, students explore the decade numbers (10, 20, 30, 40, 50, 60, 70, 80, 90) as groups of ten with no ones left over. Students use manipulatives to group, or bundle, groups of ten. As students count any number up to 99, they should group/bundle tens and some more. Furthermore, they should focus on the mathematical language associated with the quantity. For example, the number 62 can be expressed as 62 ones; or 6 groups of ten with 2 left over.

In addition, students should read numbers in standard form AND using place value concepts. Read the number 62 as “sixty-two” as well as 6 tens and 2 ones.

First grade students “compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the relational symbols >, <, =, and ≠”. According to Dougherty, Flores, Louis, & Sophian (2010), students should determine whether quantities are equal or not equal before using “greater than” and “less than.” Once students know the two quantities are not equal, then teachers can have a discussion about “how” they are not equal by asking the following questions. “Which one is greater?” “Which one is less?” These discussions must occur before using the greater than and less than symbols.

Give students extensive experiences to explain their thinking using words and models before using the symbols. Students can use manipulatives to model the two numbers, as well as pictures or number lines. Again, it is critical that students develop the concept of composing and decomposing tens and ones. This is foundational to understanding place value and involves number relationships and promotes a flexibility with mental computation. Furthermore, students must learn through a progression of representations. They begin with concrete models, then move to pictorial models, and then abstract models.

Second Grade

In second grade, students work on decomposing numbers by using place value. Give students extensive experiences with manipulatives as well as pictorial representations of numbers. Students also should be able to decompose numbers into hundred, tens, and ones in several different combinations. For example, 268 could be shown as 2 hundreds, 6 tens, and 8 ones. But is also correct to show 268 as 26 tens and 8 ones, OR 1 hundred, 16 tens, and 8 ones. There are several ways to represent the number 268:

Building on their work in first grade, second grade students need multiple opportunities to count and bundle groups of hundreds. It is critical that students understand that 100 is 10 tens as well as 100 ones.

Students in second grade also learn to read, write, and represent a number of objects in various forms

Base-ten numerals – 268
Number names – two hundred sixty-eight
Expanded notation – 200 + 60 + 8
Unit form – 2 hundreds, 6 tens, 8 ones

Notice, in the number names, the word “and” is not used between any of the whole number words. Save the word “and” for decimals.

Second grade students build on their work in first grade to compare two numbers and use the relational symbols of <, >, =, and ≠. Give students extensive experiences to explain their thinking using words and models before using the symbols. Students can use manipulatives to model the two numbers, as well as pictures or number lines.

Another way to ensure that students develop a conceptual understanding of the relational symbols (<, >, =, and ≠), use a number line. Be sure to put the greater than and less than symbols above the number line.

Then share the two numbers you want to compare; for example 14 and 7. Explain to students that you want to compare where 14 is in relation to the number 7. So you can move the two relational symbols above the 7.

Ask students where 14 is in relation to 7. Is it to the left, which would be lesser number? Or is it to the right, which would be greater number?

Using relationships truly focuses on the numbers and on tricks.

Additionally, second grade students fluently add using strategies based on place value. For example, using the place value strategy for 56 + 27, a student might say:

I decomposed 56 and 27 into tens and ones.
5 tens plus 2 tens is 7 tens.
Then I added the ones; 6 ones and 7 ones is 13 ones.
Then I combined the tens and ones. 7 tens plus 13 ones is 83.

Source: 2017 Kansas Mathematics Standards Flip Book 2nd Grade

Rounding Whole Numbers

The first time students encounter rounding whole numbers is in third grade. Students “use place value understanding to round whole numbers to the nearest 10 or 100.” Additionally, students should have a deep understanding of place value and number sense in order to explain and reason about their rounded answers.

When teaching students about rounding , you are deciding which number is closest. For example, 127 is closer to 100 than it is 200. If students don’t know this, then there is a gap in their place value understanding.

A strong understanding of place value is critical for students to understand rounding of whole numbers. Don’t begin by telling students the rules; use a number line so that students can develop a conceptual understanding of rounding. Students need to learn when and why to round numbers by first identifying possible answers and the halfway point.

Using an open number line to teach students about rounding will help both their rounding skills and strengthen place value understanding. For example, round 167 to the nearest ten.

Source: KSDE Flip Book 3rd Grade (2017c)

Teaching Greater Than and Less Than

Don’t teach greater than and less than using methods such as “Pac Man” or “Alligator” or other aids. When you use these, students don’t grasp the full meaning of the relational symbol. If you must use one of these “learning aids,” be sure to include the mathematical name and symbol with it. But first, explicitly teach that the symbols have names. Second, ask students to read the entire inequality, reading the numbers and symbols left to right, in the same way they would read a sentence.

For example, when students write 5 < 8, they would read this sentence as “five is less than eight.” This is a critical step in learning these relational symbols. Additionally, when students read the inequality aloud, they can recognize errors. For example, 5 < 8, if students say five is greater than eight, that does not make sense and they can correct their mistake.

Calculators in the Elementary Mathematics Classroom

The NCTM (2015) Position Statement, “Calculator Use in Elementary Grades,” states:

Calculators in the elementary grades serve as aids in advancing student understanding without replacing the need for other calculation methods. Calculator use can promote the higher-order thinking and reasoning needed for problem solving in our information- and technology-based society. Their use can also assist teachers and students in increasing student understanding of and fluency with arithmetic operations, algorithms, and numerical relationships and enhancing student motivation. Strategic calculator use can aid students in recognizing and extending numeric, algebraic, and geometric patterns and relationships.

Calculators in the elementary classroom are essential in helping students make sense of mathematics and reason mathematically. Teachers need to plan for the strategic use of calculators that will support student thinking and assist them in making connections to real-world situations.

Read NCTM’s Position Statement “ Calculator Use in Elementary Grades .”

the value of a digit in its position

physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics

writing a number with one digit in each place value

writing a number and showing the place value of each digit

a concept that a group of 10 objects is also one ten

making a number simpler but keeping its value close to what it was

A number line that has no numbers. Students fill in the number line based on the problem they are solving.

Share This Book

1.1 Introduction to Whole Numbers

Learning objectives.

By the end of this section, you will be able to:

Use place value with whole numbers
Identify multiples and apply divisibility tests
Find prime factorizations and least common multiples

Be Prepared 1.1

A more thorough introduction to the topics covered in this section can be found in Prealgebra in the chapters Whole Numbers and The Language of Algebra .

As we begin our study of elementary algebra, we need to refresh some of our skills and vocabulary. This chapter will focus on whole numbers, integers, fractions, decimals, and real numbers. We will also begin our use of algebraic notation and vocabulary.

Use Place Value with Whole Numbers

The most basic numbers used in algebra are the numbers we use to count objects in our world: 1, 2, 3, 4, and so on. These are called the counting number s . Counting numbers are also called natural numbers . If we add zero to the counting numbers, we get the set of whole number s .

Counting Numbers: 1, 2, 3, …

Whole Numbers: 0, 1, 2, 3, …

The notation “…” is called ellipsis and means “and so on,” or that the pattern continues endlessly.

We can visualize counting numbers and whole numbers on a number line (see Figure 1.2 ).

Manipulative Mathematics

Our number system is called a place value system, because the value of a digit depends on its position in a number. Figure 1.3 shows the place values . The place values are separated into groups of three, which are called periods. The periods are ones, thousands, millions, billions, trillions , and so on. In a written number, commas separate the periods.

Example 1.1

In the number 63,407,218, find the place value of each digit:

Place the number in the place value chart:

ⓐ The 7 is in the thousands place. ⓑ The 0 is in the ten thousands place. ⓒ The 1 is in the tens place. ⓓ The 6 is in the ten-millions place. ⓔ The 3 is in the millions place.

For the number 27,493,615, find the place value of each digit:

ⓐ 2 ⓑ 1 ⓒ 4 ⓓ 7 ⓔ 5

For the number 519,711,641,328, find the place value of each digit:

ⓐ 9 ⓑ 4 ⓒ 2 ⓓ 6 ⓔ 7

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period, followed by the name of the period, without the s at the end. Start at the left, where the periods have the largest value. The ones period is not named. The commas separate the periods, so wherever there is a comma in the number, put a comma between the words (see Figure 1.4 ). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

Name a Whole Number in Words.

Step 1. Start at the left and name the number in each period, followed by the period name.
Step 2. Put commas in the number to separate the periods.
Step 3. Do not name the ones period.

Example 1.2

Name the number 8,165,432,098,710 using words.

Name the number in each period, followed by the period name.

Put the commas in to separate the periods.

So, 8 , 165 , 432 , 098 , 710 8 , 165 , 432 , 098 , 710 is named as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Name the number 9 , 258 , 137 , 904 , 061 9 , 258 , 137 , 904 , 061 using words.

Name the number 17 , 864 , 325 , 619 , 004 17 , 864 , 325 , 619 , 004 using words.

We are now going to reverse the process by writing the digits from the name of the number. To write the number in digits, we first look for the clue words that indicate the periods. It is helpful to draw three blanks for the needed periods and then fill in the blanks with the numbers, separating the periods with commas.

Write a Whole Number Using Digits.

Step 1. Identify the words that indicate periods. (Remember, the ones period is never named.)
Step 2. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
Step 3. Name the number in each period and place the digits in the correct place value position.

Example 1.3

Write nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine as a whole number using digits.

Identify the words that indicate periods. Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas. Then write the digits in each period.

The number is 9,246,073,189.

Write the number two billion, four hundred sixty-six million, seven hundred fourteen thousand, fifty-one as a whole number using digits.

Write the number eleven billion, nine hundred twenty-one million, eight hundred thirty thousand, one hundred six as a whole number using digits.

In 2013, the U.S. Census Bureau estimated the population of the state of New York as 19,651,127. We could say the population of New York was approximately 20 million. In many cases, you don’t need the exact value; an approximate number is good enough.

The process of approximating a number is called rounding . Numbers are rounded to a specific place value, depending on how much accuracy is needed. Saying that the population of New York is approximately 20 million means that we rounded to the millions place.

Example 1.4

How to round whole numbers.

Round 23,658 to the nearest hundred.

Round to the nearest hundred: 17,852 . 17,852 .

Round to the nearest hundred: 468,751 . 468,751 .

Round Whole Numbers.

Step 1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
Step 2. Underline the digit to the right of the given place value.
Yes–add 1 1 to the digit in the given place value.
No–do not change the digit in the given place value.
Step 4. Replace all digits to the right of the given place value with zeros.

Example 1.5

Round 103,978 103,978 to the nearest:

ⓒ ten thousand

Round 206,981 to the nearest: ⓐ hundred ⓑ thousand ⓒ ten thousand.

Try It 1.10

Round 784,951 to the nearest: ⓐ hundred ⓑ thousand ⓒ ten thousand.

In algebra, we use a letter of the alphabet to represent a number whose value may change or is unknown. Commonly used symbols are a , b , c , m , n , x , and y . Further discussion of constants and variables appears later in this section.

Identify Multiples and Apply Divisibility Tests

The numbers 2, 4, 6, 8, 10, and 12 are called multiples of 2. A multiple of 2 can be written as the product of 2 and a counting number.

Similarly, a multiple of 3 would be the product of a counting number and 3.

We could find the multiples of any number by continuing this process.

Table 1.1 shows the multiples of 2 through 9 for the first 12 counting numbers.

Multiple of a Number

A number is a multiple of n if it is the product of a counting number and n .

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 15 by 3, we get a counting number. In fact, 15 ÷ 3 15 ÷ 3 is 5, so 15 is 5 · 3 . 5 · 3 .

Divisible by a Number

If a number m is a multiple of n , then m is divisible by n .

Look at the multiples of 5 in Table 1.1 . They all end in 5 or 0. Numbers with last digit of 5 or 0 are divisible by 5. Looking for other patterns in Table 1.1 that shows multiples of the numbers 2 through 9, we can discover the following divisibility tests:

Divisibility Tests

A number is divisible by:

2 if the last digit is 0, 2, 4, 6, or 8.
3 if the sum of the digits is divisible by 3.
5 if the last digit is 5 or 0.
6 if it is divisible by both 2 and 3.
10 if it ends with 0.

Example 1.6

Is 5,625 divisible by 2? By 3? By 5? By 6? By 10?

Try It 1.11

Determine whether 4,962 is divisible by 2, by 3, by 5, by 6, and by 10.

Try It 1.12

Determine whether 3,765 is divisible by 2, by 3, by 5, by 6, and by 10.

Find Prime Factorizations and Least Common Multiples

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n , we can say that m is divisible by n . For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Since 8 · 9 = 72 , 8 · 9 = 72 , we say that 8 and 9 are factors of 72. When we write 72 = 8 · 9 , 72 = 8 · 9 , we say we have factored 72.

Other ways to factor 72 are 1 · 72 , 2 · 36 , 3 · 24 , 4 · 18 , and 6 · 12 . 1 · 72 , 2 · 36 , 3 · 24 , 4 · 18 , and 6 · 12 . Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72.

In the expression a · b a · b , both a and b are called factors . If a · b = m a · b = m and both a and b are integers, then a and b are factors of m .

Some numbers, like 72, have many factors. Other numbers have only two factors.

Prime Number and Composite Number

A prime number is a counting number greater than 1, whose only factors are 1 and itself.

A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.

The counting numbers from 2 to 19 are listed in Figure 1.5 , with their factors. Make sure to agree with the “prime” or “composite” label for each!

The prime number s less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.

A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful later in this course.

Prime Factorization

The prime factorization of a number is the product of prime numbers that equals the number. These prime numbers are called the prime factors.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime!

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

Example 1.7

How to find the prime factorization of a composite number.

We say 2 · 2 · 2 · 2 · 3 2 · 2 · 2 · 2 · 3 is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer!

If we first factored 48 in a different way, for example as 6 · 8 , 6 · 8 , the result would still be the same. Finish the prime factorization and verify this for yourself.

Try It 1.13

Find the prime factorization of 80.

Try It 1.14

Find the prime factorization of 60.

Find the Prime Factorization of a Composite Number.

Step 1. Find two factors whose product is the given number, and use these numbers to create two branches.
Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
Step 3. If a factor is not prime, write it as the product of two factors and continue the process.
Step 4. Write the composite number as the product of all the circled primes.

Example 1.8

Find the prime factorization of 252.

Try It 1.15

Find the prime factorization of 126.

Try It 1.16

Find the prime factorization of 294.

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominator s. Two methods are used most often to find the least common multiple and we will look at both of them.

The first method is the Listing Multiples Method. To find the least common multiple of 12 and 18, we list the first few multiples of 12 and 18:

Notice that some numbers appear in both lists. They are the common multiples of 12 and 18.

We see that the first few common multiples of 12 and 18 are 36, 72, and 108. Since 36 is the smallest of the common multiples, we call it the least common multiple. We often use the abbreviation LCM.

Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

The procedure box lists the steps to take to find the LCM using the prime factors method we used above for 12 and 18.

Find the Least Common Multiple by Listing Multiples.

Step 1. List several multiples of each number.
Step 2. Look for the smallest number that appears on both lists.
Step 3. This number is the LCM.

Example 1.9

Find the least common multiple of 15 and 20 by listing multiples.

Notice that 120 is in both lists, too. It is a common multiple, but it is not the least common multiple.

Try It 1.17

Find the least common multiple by listing multiples: 9 and 12.

Try It 1.18

Find the least common multiple by listing multiples: 18 and 24.

Our second method to find the least common multiple of two numbers is to use The Prime Factors Method. Let’s find the LCM of 12 and 18 again, this time using their prime factors.

Example 1.10

How to find the least common multiple using the prime factors method.

Find the Least Common Multiple (LCM) of 12 and 18 using the prime factors method.

Notice that the prime factors of 12 ( 2 · 2 · 3 ) ( 2 · 2 · 3 ) and the prime factors of 18 ( 2 · 3 · 3 ) ( 2 · 3 · 3 ) are included in the LCM ( 2 · 2 · 3 · 3 ) . ( 2 · 2 · 3 · 3 ) . So 36 is the least common multiple of 12 and 18.

By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

Try It 1.19

Find the LCM using the prime factors method: 9 and 12.

Try It 1.20

Find the LCM using the prime factors method: 18 and 24.

Find the Least Common Multiple Using the Prime Factors Method.

Step 1. Write each number as a product of primes.
Step 2. List the primes of each number. Match primes vertically when possible.
Step 3. Bring down the columns.
Step 4. Multiply the factors.

Example 1.11

Find the Least Common Multiple (LCM) of 24 and 36 using the prime factors method.

Try It 1.21

Find the LCM using the prime factors method: 21 and 28.

Try It 1.22

Find the LCM using the prime factors method: 24 and 32.

Access this online resource for additional instruction and practice with using whole numbers. You will need to enable Java in your web browser to use the application.

Sieve of Eratosthenes

Section 1.1 Exercises

Practice makes perfect.

In the following exercises, find the place value of each digit in the given numbers.

51,493 ⓐ 1, ⓑ 4, ⓒ 9, ⓓ 5, ⓔ 3

87,210 ⓐ 2 ⓑ 8 ⓒ 0 ⓓ 7 ⓔ 1

164,285 ⓐ 5, ⓑ 6, ⓒ 1, ⓓ 8, ⓔ 2

395,076 ⓐ 5 ⓑ 3 ⓒ 7 ⓓ 0 ⓔ 9

93,285,170 ⓐ 9 ⓑ 8 ⓒ 7 ⓓ 5 ⓔ 3

36,084,215 ⓐ 8 ⓑ 6 ⓒ 5 ⓓ 4 ⓔ 3

7,284,915,860,132 ⓐ 7 ⓑ 4 ⓒ 5 ⓓ 3 ⓔ 0

2,850,361,159,433 ⓐ 9 ⓑ 8 ⓒ 6 ⓓ 4 ⓔ 2

In the following exercises, name each number using words.

In the following exercises, write each number as a whole number using digits.

four hundred twelve

two hundred fifty-three

thirty-five thousand, nine hundred seventy-five

sixty-one thousand, four hundred fifteen

eleven million, forty-four thousand, one hundred sixty-seven

eighteen million, one hundred two thousand, seven hundred eighty-three

three billion, two hundred twenty-six million, five hundred twelve thousand, seventeen

eleven billion, four hundred seventy-one million, thirty-six thousand, one hundred six

In the following, round to the indicated place value.

Round to the nearest ten.

ⓐ 386 ⓑ 2,931

ⓐ 792 ⓑ 5,647

Round to the nearest hundred.

ⓐ 13,748 ⓑ 391,794

ⓐ 28,166 ⓑ 481,628

ⓐ 1,492 ⓑ 1,497

ⓐ 2,791 ⓑ 2,795

ⓐ 63,994 ⓑ 63,940

ⓐ 49,584 ⓑ 49,548

In the following exercises, round each number to the nearest ⓐ hundred, ⓑ thousand, ⓒ ten thousand.

Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, 3, 5, 6, and 10.

In the following exercises, find the prime factorization.

In the following exercises, find the least common multiple of the each pair of numbers using the multiples method.

In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.

Everyday Math

Writing a Check Jorge bought a car for $24,493. He paid for the car with a check. Write the purchase price in words.

Writing a Check Marissa’s kitchen remodeling cost $18,549. She wrote a check to the contractor. Write the amount paid in words.

Buying a Car Jorge bought a car for $24,493. Round the price to the nearest ⓐ ten ⓑ hundred ⓒ thousand; and ⓓ ten-thousand.

Remodeling a Kitchen Marissa’s kitchen remodeling cost $18,549, Round the cost to the nearest ⓐ ten ⓑ hundred ⓒ thousand and ⓓ ten-thousand.

Population The population of China was 1,339,724,852 on November 1, 2010. Round the population to the nearest ⓐ billion ⓑ hundred-million; and ⓒ million.

Astronomy The average distance between Earth and the sun is 149,597,888 kilometers. Round the distance to the nearest ⓐ hundred-million ⓑ ten-million; and ⓒ million.

Grocery Shopping Hot dogs are sold in packages of 10, but hot dog buns come in packs of eight. What is the smallest number that makes the hot dogs and buns come out even?

Grocery Shopping Paper plates are sold in packages of 12 and party cups come in packs of eight. What is the smallest number that makes the plates and cups come out even?

Writing Exercises

Give an everyday example where it helps to round numbers.

If a number is divisible by 2 and by 3 why is it also divisible by 6?

What is the difference between prime numbers and composite numbers?

Explain in your own words how to find the prime factorization of a composite number, using any method you prefer.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
Publisher/website: OpenStax
Book title: Elementary Algebra 2e
Publication date: Apr 22, 2020
Location: Houston, Texas
Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
Section URL: https://openstax.org/books/elementary-algebra-2e/pages/1-1-introduction-to-whole-numbers

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1.1: Place Value and Names for Whole Numbers

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Learning Objectives

Identify counting numbers and whole numbers
Model whole numbers
Identify the place value of a digit
Use place value to name whole numbers
Use place value to write whole numbers
Round whole numbers

Identify Counting Numbers and Whole Numbers

Learning algebra is similar to learning a language. You start with a basic vocabulary and then add to it as you go along. You need to practice often until the vocabulary becomes easy to you. The more you use the vocabulary, the more familiar it becomes.

Algebra uses numbers and symbols to represent words and ideas. Let’s look at the numbers first. The most basic numbers used in algebra are those we use to count objects: $1, 2, 3, 4, 5, …$ and so on. These are called the counting numbers . The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly. Counting numbers are also called natural numbers.

Definition: Counting Numbers

The counting numbers start with $1$ and continue.

$1, 2, 3, 4, 5 \ldots $

Counting numbers and whole numbers can be visualized on a number line as shown in Figure $\PageIndex{1}$.

$An image of a number line from 0 to 6 in increments of one. An arrow above the number line pointing to the right with the label “larger”. An arrow pointing to the left with the label “smaller”.$

Figure $\PageIndex{1}$: The numbers on the number line increase from left to right, and decrease from right to left.

The point labeled $0$ is called the origin . The points are equally spaced to the right of 0 and labeled with the counting numbers. When a number is paired with a point, it is called the coordinate of the point.

The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the whole numbers .

Definition: Whole Numbers

The whole numbers are the counting numbers and zero.

$0, 1, 2, 3, 4, 5 \ldots$

We stopped at $5$ when listing the first few counting numbers and whole numbers. We could have written more numbers if they were needed to make the patterns clear.

Example $\PageIndex{1}$: Number Identification

Which of the following are

counting numbers
whole numbers

\[0, \dfrac{1}{4}, 3, 5.2, 15, 105 \nonumber\]

The counting numbers start at $1$, so $0$ is not a counting number. The numbers $3$, $15$, and $105$ are all counting numbers.
Whole numbers are counting numbers and $0$. The numbers $0, 3, 15,$ and $105$ are whole numbers. The numbers $\dfrac{1}{4}$ and $5.2$ are neither counting numbers nor whole numbers. We will discuss these numbers later.

Exercise $\PageIndex{1}$

\[0, \dfrac{2}{3}, 2, 9, 11.8, 241, 376 \nonumber \]

$2, 9, 241, 376$

$0, 2, 9, 241, 376$

Exercise $\PageIndex{2}$

\[0, \dfrac{5}{3}, 7, 8.8, 13, 201 \nonumber \]

$7, 13, 201$

$0, 7, 13, 201$

Model Whole Numbers

Our number system is called a place value system because the value of a digit depends on its position, or place, in a number. The number $537$ has a different value than the number $735$. Even though they use the same digits, their value is different because of the different placement of the $3$ and the $7$ and the $5$.

Money gives us a familiar model of place value. Suppose a wallet contains three $$100$ bills, seven $$10$ bills, and four $$1$ bills. The amounts are summarized in Figure $\PageIndex{2}$. How much money is in the wallet?

$An image of three stacks of American currency. First stack from left to right is a stack of 3 $100 bills, with label “Three $100 bills, 3 times $100 equals $300”. Second stack from left to right is a stack of 7 $10 bills, with label “Seven $10 bills, 7 times $10 equals $70”. Third stack from left to right is a stack of 4 $1 bills, with label “Four $1 bills, 4 times $1 equals $4”.$

Figure $\PageIndex{2}$

Find the total value of each kind of bill, and then add to find the total. The wallet contains $$374$.

$An image of “$300 + $70 +$4” where the “3” in “$300”, the “7” in “$70”, and the “4” in “$4” are all in red instead of black like the rest of the expression. Below this expression there is the value “$374”. An arrow points from the red “3” in the expression to the “3” in “$374”, an arrow points to the red “7” in the expression to the “7” in “$374”, and an arrow points from the red “4” in the expression to the “4” in “$374”.$

Base-$10$ blocks provide another way to model place value, as shown in Figure $\PageIndex{3}$. The blocks can be used to represent hundreds, tens, and ones. Notice that the tens rod is made up of $10$ ones, and the hundreds square is made of $10$ tens, or $100$ ones.

$An image with three items. The first item is a single block with the label “A single block represents 1”. The second item is a horizontal rod consisting of 10 blocks, with the label “A rod represents 10”. The third item is a square consisting of 100 blocks, with the label “A square represents 100”. The square is 10 blocks tall and 10 blocks wide.$

Figure $\PageIndex{3}$

Figure $\PageIndex{4}$ shows the number $138$ modeled with base-$10$ blocks.

$An image consisting of three items. The first item is a square of 100 blocks, 10 blocks wide and 10 blocks tall, with the label “1 hundred”. The second item is 3 horizontal rods containing 10 blocks each, with the label “3 tens”. The third item is 8 individual blocks with the label “8 ones”.$

Figure $\PageIndex{4}$: We use place value notation to show the value of the number 138.

$An image of “100 + 30 +8” where the “1” in “100”, the “3” in “30”, and the “8” are all in red instead of black like the rest of the expression. Below this expression there is the value “138”. An arrow points from the red “1” in the expression to the “1” in “138”, an arrow points to the red “3” in the expression to the “3” in “138”, and an arrow points from the red “8” in the expression to the “8” in 138.$

Example $\PageIndex{2}$: place value notation

Use place value notation to find the value of the number modeled by the base-$10$ blocks shown.

$An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is one horizontal rod containing 10 blocks. The third item is 5 individual blocks.$

Figure $\PageIndex{5}$

There are $2$ hundreds squares, which is $200$.

There is $1$ tens rod, which is $10$.

There are $5$ ones blocks, which is $5$.

$An image of “200 + 10 + 5” where the “2” in “200”, the “1” in “10”, and the “5” are all in red instead of black like the rest of the expression. Below this expression there is the value “215”. An arrow points from the red “2” in the expression to the “2” in “215”, an arrow points to the red “1” in the expression to the “1” in “215”, and an arrow points from the red “5” in the expression to the “5” in 215.$

The base-$10$ blocks model the number $215$.

Identify the Place Value of a Digit

By looking at money and base-10 blocks, we saw that each place in a number has a different value. A place value chart is a useful way to summarize this information. The place values are separated into groups of three, called periods. The periods are ones, thousands, millions, billions, trillions , and so on. In a written number, commas separate the periods.

Just as with the base-$10$ blocks, where the value of the tens rod is ten times the value of the ones block and the value of the hundreds square is ten times the tens rod, the value of each place in the place-value chart is ten times the value of the place to the right of it.

Figure $\PageIndex{8}$ shows how the number $5,278,194$ is written in a place value chart.

$A chart titled 'Place Value' with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 8 values in the next row are blank. Starting with the ninth column, the values are '5', '2', '7', '8', '1', '9', and '4'.$

Figure $\PageIndex{8}$

The digit $5$ is in the millions place. Its value is $5,000,000$.
The digit $2$ is in the hundred thousands place. Its value is $200,000$.
The digit $7$ is in the ten thousands place. Its value is $70,000$.
The digit $8$ is in the thousands place. Its value is $8,000$.
The digit $1$ is in the hundreds place. Its value is $100$.
The digit $9$ is in the tens place. Its value is $90$.
The digit $4$ is in the ones place. Its value is $4$.

Example $\PageIndex{3}$: place value

In the number $63,407,218$; find the place value of each of the following digits:

Write the number in a place value chart, starting at the right.

$A figure titled “Place Values” with fifteen columns and 2 rows, with the colums broken down into five groups of three. The first row has the values “Hundred trillions”, “Ten trillions”, “trillions”, “hundred billions”, “ten billions”, “billions”, “hundred millions”, “ten millions”, “millions”, “hundred thuosands”, “ten thousands”, “thousands”, “hundreds”, “tens”, and “ones”. The first 7 values in the second row are blank. Starting with eighth column, the values are “6”, “3”, “4”, “0”, “7”, “2”, “1” and “8”. The first group is labeled “trillions” and contains the first row values of “Hundred trillions”, “ten trillions”, and “trillions”. The second group is labeled “billions” and contains the first row values of “Hundred billions”, “ten billions”, and “billions”. The third group is labeled “millions” and contains the first row values of “Hundred millions”, “ten millions”, and “millions”. The fourth group is labeled “thousands” and contains the first row values of “Hundred thousands”, “ten thousands”, and “thousands”. The fifth group is labeled “ones” and contains the first row values of “Hundreds”, “tens”, and “ones”.$

Figure $\PageIndex{9}$

The $7$ is in the thousands place.
The $0$ is in the ten thousands place.
The $1$ is in the tens place.
The $6$ is in the ten millions place.
The $3$ is in the millions place.

Exercise $\PageIndex{6}$

For each number, find the place value of digits listed: $519,711,641,328$

ten thousands

hundred thousands

hundred millions

Use Place Value to Name Whole Numbers

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period followed by the name of the period without the ‘s’ at the end. Start with the digit at the left, which has the largest place value. The commas separate the periods, so wherever there is a comma in the number, write a comma between the words. The ones period, which has the smallest place value, is not named.

$An image with three values separated by commas. The first value is “37” and has the label “millions”. The second value is “519” and has the label thousands. The third value is “248” and has the label ones. Underneath, the value “37” has an arrow pointing to “Thirty-seven million”, the value “519” has an arrow pointing to “Five hundred nineteen thousand”, and the value “248” has an arrow pointing to “Two hundred forty-eight”.$

So the number $37,519,248$ is written thirty-seven million, five hundred nineteen thousand, two hundred forty-eight. Notice that the word and is not used when naming a whole number.

How to: Name a Whole Number in Words.

Step 1. Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.

Step 2. Use commas in the number to separate the periods.

Example $\PageIndex{4}$: name whole numbers

Name the number $8,165,432,098,710$ in words.

$An image with five values separated by commas. The first value is “8” and has the label “trillions”. The second value is “165” and has the label “bilions”. The third value is “432” and has the label “millions”. The fourth value is “098” and has the label “thousands”. The fifth value is “710” and has the label “ones”. Underneath, the value “8” has an arrow pointing to “Eight trillion”, the value “165” has an arrow pointing to “One hundred sixty-five billion”, the value “432” has an arrow pointing to “Four hundred thirty-two million”, the value “098” has an arrow pointing to “Ninety-eight thousand”, and the value “710” has an arrow pointing to “seven hundred ten”.$

Putting all of the words together, we write $8,165,432,098,710$ as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Exercise $\PageIndex{7}$

Name each number in words: $9,258,137,904,061$

nine trillion, two hundred fifty-eight billion, one hundred thirty-seven million, nine hundred four thousand, sixty-one

Example $\PageIndex{5}$: name whole numbers

A student conducted research and found that the number of mobile phone users in the United States during one month in 2014 was $327,577,529$. Name that number in words.

Identify the periods associated with the number.

$An image with three values separated by commas. The first value is “327” and has the label “millions”. The second value is “577” and has the label “thousands”. The third value is “529” and has the label “ones”.$

Name the number in each period, followed by the period name. Put the commas in to separate the periods.

Millions period: three hundred twenty-seven million

Thousands period: five hundred seventy-seven thousand

Ones period: five hundred twenty-nine

So the number of mobile phone users in the Unites States during the month of April was three hundred twenty-seven million, five hundred seventy-seven thousand, five hundred twenty-nine.

Exercise $\PageIndex{9}$

The population in a country is $316,128,839$. Name that number

three hundred sixteen million, one hundred twenty-eight thousand, eight hundred thirty nine

Use Place Value to Write Whole Numbers

We will now reverse the process and write a number given in words as digits.

How to: Use Place Value to Write Whole Numbers

Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)

Step 2. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Step 3. Name the number in each period and place the digits in the correct place value position.

Example $\PageIndex{6}$: write whole numbers

Write the following numbers using digits.

fifty-three million, four hundred one thousand, seven hundred forty-two
nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine
Identify the words that indicate periods.

Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Then write the digits in each period.

$An image with three blocks of text pointing to numerical values. The first block of text is “fifty-three million”, has the label “millions”, and points to value 53. The second block of text is “four hundred one thousand”, has the label “thousands”, and points to value 401. The third block of text is “seven hundred forty-two”, has the label “ones”, and points to value 742.$

Put the numbers together, including the commas. The number is $53,401,742$.

$An image with four blocks of text pointing to numerical values. The first block of text is “nine billion”, has the label “billions”, and points to value 9. The second block of text is “two hundred forty-six million”, has the label “millions”, and points to value 246. The third block of text is “seventy-three thousand”, has the label “thousands”, and points to value 742. The fourth block of text is “one hundred eighty-nine”, has the label “ones”, and points to the value 189.$

The number is $9,246,073,189.$

Notice that in part (b), a zero was needed as a place-holder in the hundred thousands place. Be sure to write zeros as needed to make sure that each period, except possibly the first, has three places.

Exercise $\PageIndex{11}$

Write each number in standard form:

fifty-three million, eight hundred nine thousand, fifty-one

$53,809,051$

Exercise $\PageIndex{12}$

two billion, twenty-two million, seven hundred fourteen thousand, four hundred sixty-six

$2,022,714,466$

Example $\PageIndex{7}$: write standard form

A state budget was about $$77$ billion. Write the budget in standard form.

Identify the periods. In this case, only two digits are given and they are in the billions period. To write the entire number, write zeros for all of the other periods.

$An image with four blocks of text pointing to numerical values. The first block of text is “77 billion”, has the label “billions”, and points to value “77”. The second block of text is null, has the label “millions”, and points to value “000”. The third block of text is null, has the label “thousands”, and points to value “000”. The fourth block of text is null, has the label “ones”, and points to the value “000”.$

So the budget was about $$77,000,000,000$.

Exercise $\PageIndex{14}$

The total weight of an aircraft carrier is $204$ million pounds.

$204,000,000\: pounds$

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.

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1.7: Whole Number Place Value

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Janet Stramel
Fort Hays State University

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Decomposing numbers in base ten
Reading and writing numbers
Rounding numbers
Comparing numbers and quantities in base ten.

children understand ten as ten ones,
children see ten as a unit, and
children easily work with units of ten.

Base-Ten Blocks

An example showing the relationship among the manipulatives, the place or position, and the place value of the digits in a number is below.

Understanding the place value of digits in numbers helps in writing numbers in their expanded form . For instance, the expanded notation of the number 235 is 200 + 30 + 5.

Read more about the Base-Ten Number System by clicking here.

Developing Whole Number Place Value Concepts across the Grades

When thinking about place value and base-ten understanding, first consider the 5 Strands of Mathematical Proficiency from the National Research Council (2001).

Mathematical proficiency is based upon five interwoven components:

Conceptual understanding – comprehension of mathematical concepts, operations, and relations
Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
Strategic competence – ability to formulate, represent, and solve mathematical problems
Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification
Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy (NRC, 2001, p. 116).

Read more about the 5 Strands of Mathematical Proficiency by downloading the book, “Helping Children Learn Mathematics.”

Your language matters when teaching students about place value. Speak in value, not in digits. For example, the value of the 2 in 26 is two tens or 20.

Kindergarten

Notice the grouping of the ten. It is obvious that there is one group of ten with some others left over. Students will immediately see that 14 is made up of one ten and four ones.

Eleven and twelve are special number words that do not have “teen” as a suffix.
The verbal counting of “teen” numbers is backwards: the ones digit is said before the tens digit. For example, 37 is read as thirty-seven; tens to ones. But the number 14 is read as fourteen; read ones to tens.

Source: KSDE Flip Book Kindergarten (2017b)

Check out the task “ What Makes a Teen Number? ” at the Illustrative Mathematics website. This activity helps students decompose teen numbers using ten-frames and number sentences.

First Grade

Students need many opportunities to practice grouping 10 ones into a bundle of 10. In first grade, students begin to unitize . They see that a group of ten objects is also one ten.

Watch this video “ One Is One, Or Is It? ” narrated by Christopher Danielson for more information on unitizing.

Students must be given numerous experiences using ten-frames, snap cubes, or other groupable models to help develop the concept of place value.

Suggested Activity using Ten-Frames:

Give students 13 counters.

Teacher: “Do you have enough to make a ten?” “Would you have any left over?” “If so, how many left overs do you have?”

Student: “I have filled up one ten-frame and have 3 counters left over. The number 13 has 1 ten and 3 ones.

In addition, students should read numbers in standard form AND using place value concepts. Read the number 62 as “sixty-two” as well as 6 tens and 2 ones.

Second Grade

Students in second grade also learn to read, write, and represent a number of objects in various forms

Base-ten numerals – 268
Number names – two hundred sixty-eight
Expanded notation – 200 + 60 + 8
Unit form – 2 hundreds, 6 tens, 8 ones

Notice, in the number names, the word “and” is not used between any of the whole number words. Save the word “and” for decimals.

Ask students where 14 is in relation to 7. Is it to the left, which would be lesser number? Or is it to the right, which would be greater number?

Using relationships truly focuses on the numbers and on tricks.

Additionally, second grade students fluently add using strategies based on place value. For example, using the place value strategy for 56 + 27, a student might say:

I decomposed 56 and 27 into tens and ones.
5 tens plus 2 tens is 7 tens.
Then I added the ones; 6 ones and 7 ones is 13 ones.
Then I combined the tens and ones. 7 tens plus 13 ones is 83.

Source: 2017 Kansas Mathematics Standards Flip Book 2nd Grade

Rounding Whole Numbers

Using an open number line to teach students about rounding will help both their rounding skills and strengthen place value understanding. For example, round 167 to the nearest ten.

Source: KSDE Flip Book 3rd Grade (2017c)

Teaching Greater Than and Less Than

Calculators in the Elementary Mathematics Classroom

The NCTM (2015) Position Statement, “Calculator Use in Elementary Grades,” states:

Read NCTM’s Position Statement “ Calculator Use in Elementary Grades .”

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Homework & Practice: 1-2; Understand Whole-Number Place Value
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Place value charts can be used to learn about place value. They might also be useful in correcting student thinking when they don't quite get the place holder concept (e.g. writing 132 as 100302 or 1004 as 14). Place value charts can also be used for addition, subtraction, multiplication and division. For example, to add two numbers, write each ...
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Grade 3 place value worksheets. Building 3, 4 and 5-digit numbers from the parts. Missing place values in 3 and 4-digit numbers. Write 4-digit numbers in expanded form. Write 4-digit numbers in expanded notation. Write 4-digit numbers in standard form. Identify the place value of the underlined digit. Compare and order numbers up to 10,000 and ...
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Here you will find our selection of 2 digit Place Value worksheets. Using these Math Worksheets Place Value will help your child to: learn their place value to 100; understand the value of each digit in a 2 digit number; Round numbers up to 100 to the nearest 10. learn to read and write numbers to 100.
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In a whole number, the digit farthest to the right is always in the ones place. The next farthest to the right is in the tens place. The remaining digits continue to fill in the place values until there are no digits left. Example: 459. The number 4 5 9 is made up of 4 hundreds, 5 tens, and 9 ones. We can also write this as: 4 5 9 = 400 + 50 + 9 .
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These place value worksheets are great for testing children on names of place values for decimals. You may select 4, 5, or 6 digit numbers with 1, 2, or 3 numbers to the right of the decimal. These place value worksheets are appropriate for Kindergarten, 1st Grade, and 2nd Grade. Place and Value for Money Worksheets
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Start quiz. Level up on all the skills in this unit and collect up to 1,400 Mastery points! Start Unit test. It's time to master place value! In this unit, you'll use blocks and tables to learn all about expanding, standardizing, and writing numbers. You'll also sharpen your skills in comparing and regrouping whole numbers.
PDF Name Lesson 1.2 Place Value of Whole Numbers
at the top of the page. Write five numbers that are at least 8 digits long under Standard Form. Write the expanded form and the word form for each number under the appropriate heading. Practice and Homework COMMON CORE STANDARD—5.NBT.A.1 Understand the place value system. Lesson 1.2
1.1 Introduction to Whole Numbers
5. Yes—add 1. 1. 1 to the digit in the given place value. If that digit is 9, replace it with 0 and add 1 to the digit immediately to its left. If that digit is also a 9, repeat. No—do not change the digit in the given place value. Step 4. Replace all digits to the right of the given place value with zeros.
Finding place value (video)
4 years ago. Negative numbers use the same place values as positive numbers. The numbers are just on the other side of 0 on a number line. The values are just negative. If you are familiar with expanded form: 27 = 20+7. -27 = -20+ (-7) In both cases we have tens place and ones place. Hope this helps.
Place Value: Whole Numbers
I only have the one comma, so I know this number only goes into the thousands. In the thousands, I've got "52", so the 2 is in the thousands place. (The 5 is in the ten-thousands place.) b) The tens place is the second place, just to the left of the 3 in the ones place. There is a 7 in this second place, so the digit in the tens place is 7.
1.1: Introduction to Whole Numbers (Part 1)
Figure 1.1.1 1.1. 1: The numbers on the number line increase from left to right, and decrease from right to left. The point labeled 0 0 is called the origin. The points are equally spaced to the right of 0 and labeled with the counting numbers. When a number is paired with a point, it is called the coordinate of the point.
understand whole number place value
5.NBT.1 Practice Sheets: Understanding Place Value. by . Kathleen and Mande' 4.9 (483) $3.00. PDF; Activity. This CCSS aligned place value packet includes 12 practice sheets to teach/reinforce identifying place value of whole numbers and decimals as well as comparing place values (10 times the value, 100 times the value, 1/10 the value, and 1/ ...
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The value of each digit increases by a power of ten as you move to the left in the number. So, the value of the "1" in "123" is 1 x 100 = 100, the value of the "2" is 2 x 10 = 20, and the value of the "3" is 3 x 1 = 3. The place value of a number can be used to represent the number in different ways.
1.1.1: Place Value and Names for Whole Numbers
A man owes $2, 562 on a car. Write the word name for this. Solution. The word name is two thousand, five hundred sixty-two. For word names of greater numbers, begin at the left with the greatest period. For each period, write the one- to three-digit number in the period, and then the period name. See the example below.
The Complete Guide to Place Value Lessons
Use place value understanding to round multi-digit whole numbers to any place; 5th Grade Expectations: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Use whole number exponents to denote powers of 10.
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Place value is a fundamental concept in the elementary grades, and understanding place value is essential in learning mathematics. Place value is the value of the digit in its position. For example, the number 358 has three columns or "places," each with a specific value. In 358, the 3 is in the "hundreds" place, the 5 is in the "tens ...
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Doing the Manipulative Mathematics activity "Number Line-Part 1" will help you develop a better understanding of the counting numbers and the whole numbers. Our number system is called a place value system, because the value of a digit depends on its position in a number.
PDF Whole Number Place Value: Chapter Opener
You can use your understanding of place-value patterns and a place-value chart to write numbers that are 10 times as much as or __ 1 10. of any given number. Use the steps below to complete the table. STEP 1. Write the given number in a place-value chart. STEP 2. Use the place-value chart to write a number that is 10 times as much
5.NBT.1 Practice Sheets: Understanding Place Value
This CCSS aligned place value packet includes 12 practice sheets to teach/reinforce identifying place value of whole numbers and decimals as well as comparing place values (10 times the value, 100 times the value, 1/10 the value, and 1/100 the value). These sheets can be used to introduce a concept,...
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Browse place value whole number and decimal resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources.
1.1: Place Value and Names for Whole Numbers
Example 1.1.4 1.1. 4: name whole numbers. Name the number 8, 165, 432, 098, 710 8, 165, 432, 098, 710 in words. Solution. Begin with the leftmost digit, which is 8 8. It is in the trillions place. eight trillion. The next period to the right is billions. one hundred sixty-five billion.
1.7: Whole Number Place Value
Additionally, in the number 235: 2 is in hundreds place and its place value is 200, 3 is in tens place and its place value is 30, 5 is in ones place and its place value is 5. Understanding the place value of digits in numbers helps in writing numbers in their expanded form. For instance, the expanded notation of the number 235 is 200 + 30 + 5.

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Course: 4th grade > Unit 1

Finding place value

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Place Value: Whole Numbers

MathHelp.com

Write the number 32,067 in expanded notation.

Write the standard form for the number which, in expanded notation, is written as follows: 9,000 + 300 + 2

For the number 52,973 , (a) state the place held by the 2 , and (b) state the digit in the tens place.

State the number 622,937,285 in words.

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Place Value Worksheets

Working With Values (Tens and Ones)

Working With Larger Values (Ones to Thousands)

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Expanded Form of Numbers

Working With Larger Places

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Place Value Standards

Standards and Skills – Numbers and Operations in Base Ten – Place Value Understanding

2nd Grade Expectations:

3rd Grade Expectations:

4th Grade Expectations:

5th Grade Expectations:

Lesson Introduction

1. Introduction and Hook

2. Base Ten Exploration and Place Value Discs

3. Anchor Chart, Direct Instruction, and Interactive Notebooks

Suggested Lesson Sequence

Day 1-2: Place Value Introduction Lesson

Day 3-4: Place Value Skill Reinforcement

Day 5-6: Rounding & Estimating

Day 7: Comparing & Ordering Numbers

Day 8-9: Small Group Review, Enrichment, and Remediation

Day 10 & 11: Place Value Detectives Cumulative Project.

Day 12: Assessment