Grade 4 · 4.NBT.A.2

Numbers to one million

A park welcomed three hundred four thousand, six hundred twenty-seven visitors. How do six digits hold a number that big, and how do you read it, write it and tell it apart from another big number?

You need

Reading and writing numbers to 1,000Grade 2

Unlocks

Rounding to any place, including decimalsGrade 4 The standard addition algorithmGrade 4

Where you are 15 in Place value

Number9

Place value15

▴ 5 above Comparing and ordering whole numbers Decimal place value4

Numbers to one million

The place value chart Expanded form ▾ Show all 15 (5 more)

When you reach for it

You already know how to read and write numbers up to 1{,}000: hundreds, tens and ones (Reading and writing numbers to 1,000). But many real counts are much bigger. A national park can have hundreds of thousands of visitors in one summer. A state fair can sell tickets to more than half a million people over its run. To read a sign that says 304{,}627 visitors, to write a number someone says aloud, or to tell which of two big counts is larger, you need the places past the thousands.

The good news is that nothing new has to be invented. The same three places, hundreds, tens and ones, simply repeat. Once you see the pattern, a six-digit number is two three-digit numbers sitting side by side, with a name in between.

This lesson takes three days. Part 1 meets the thousands period and the rule that every place is worth ten times the place to its right. Part 2 names the periods, uses commas and zeros, and reaches one million. Part 3 compares, orders and builds big numbers.

The procedure

Part 1: the thousands period, and ten times as much

Ten makes the next place. Ten ones make one ten. Ten tens make one hundred. Ten hundreds make one thousand. The pattern does not stop there. Ten thousands make one ten thousand, which is 10{,}000. Ten ten thousands make one hundred thousand, which is 100{,}000. And ten hundred thousands make one million, 1{,}000{,}000.

Figure 1Every step up the ladder is worth ten times the step below. Ten hundred thousands make one million.

So each place is worth ten times the place just to its right. Two places to the left is ten times ten, which is 100 times as much. This one rule is what lets ten digits, 0 to 9, write any whole number at all: when a place fills up with ten, those ten are traded for one in the next place to the left.

A digit’s value. The value of a digit is the digit times what its place is worth. In 304{,}627, the 4 sits in the thousands place, so it is worth 4 thousands, which is 4{,}000. The 3 sits in the hundred thousands place, so it is worth 300{,}000. A The place value chart lays the places out in columns so you can read each digit’s place at a glance.

Figure 2304{,}627 in a place value chart. Each period has its own hundreds, tens and ones; the comma marks where the thousands period ends.

Three ways to write one number. The number in the chart can be written in three forms.

Standard form is the digits themselves: 304{,}627.
Expanded form writes the number as a sum of what each digit is worth (Expanded form): 300{,}000 + 4{,}000 + 600 + 20 + 7. The 0 in the ten thousands place is worth nothing, so it adds no term.
Word form says the number the way you would read it aloud: three hundred four thousand, six hundred twenty-seven.

All three name exactly the same amount. Expanded form shows the place values; standard form is the short way to write them; word form is what you hear.

Part 2: periods, commas and one million

Periods. Look again at the chart. The six places come in two groups of three. The right group is hundreds, tens, ones. The left group is hundred thousands, ten thousands, thousands: it is the same hundreds, tens and ones again, but each one counts thousands. Each group of three places is a period. The right one is the ones period and the left one is the thousands period.

This is why a big number is easy to read. You read the thousands period as an ordinary three-digit number, say the word “thousand”, then read the ones period as an ordinary three-digit number. For 304{,}627: the thousands period holds 304, read “three hundred four”, then “thousand”; the ones period holds 627, read “six hundred twenty-seven”. The name of the ones period is never said.

Commas. A comma is written between periods, counting three places from the right. It is a reading aid: it shows where the word “thousand” goes. To place it, count three digits in from the right end, never from the left. In word form, a comma goes in the same spot, after the word “thousand”. Whole number names never use the word “and”.

Zeros hold places. A zero means “none of this place”, and it must still be written whenever there are digits to its left. In 304{,}627 the zero says there are no ten thousands. Without it the digits 3 and 4 would slide one place to the right, and 34{,}627 is a very different number. When you write a number from words, every period after the first must have exactly three digits, using zeros to fill any place the words skip.

One million. The biggest six-digit number is 999{,}999. Add one more and every place fills up with ten in turn: ten ones become a ten, which makes ten tens, which become a hundred, and so on, all the way to ten hundred thousands. Those become one million, written 1{,}000{,}000. A million is 1{,}000 thousands, the way a thousand is 1{,}000 ones. The digit 1 sits in a new place, the ones place of the next period, the millions period. This lesson stops at that one number; the millions period itself comes later.

Part 3: comparing, ordering and building numbers

Comparing. To decide which of two whole numbers is greater, first count digits. A whole number with more digits (and no zeros at the front) is greater: any six-digit number is at least 100{,}000, and any five-digit number is at most 99{,}999. If they have the same number of digits, line them up on the ones place and read from the left. The first place where the digits differ decides: the number with the larger digit there is greater, because that one place is worth more than all the places to its right put together. Write a > b for “a is greater than b” and a < b for “a is less than b”. The symbol opens toward the greater number. Comparing and ordering whole numbers covers this for numbers of every size.

Ordering. To put a list in order, compare in the same way: sort by number of digits first, then by the leftmost digit, then by the next digit for numbers that tie, and so on.

Building. When you are given digits to arrange, the leftmost place matters most. To make the greatest number, put the largest digit in the leftmost place, the next largest in the next place, and so on down. To make the least, put the smallest digits on the left, except that zero cannot go first, or the number would have fewer digits.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3, 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.

Example 1

A park sign in three forms

Lerato sees a sign at a national park: “Visitors this summer: 304{,}627.” Give the place and value of each digit, then write the number in expanded form and in word form.

Solution

Put the digits into the chart from the right: 7 in the ones place, 2 in the tens, 6 in the hundreds, 4 in the thousands, 0 in the ten thousands, 3 in the hundred thousands.

digit	place	value
3	hundred thousands	300{,}000
0	ten thousands	0
4	thousands	4{,}000
6	hundreds	600
2	tens	20
7	ones	7

Each value is the digit times its place. Adding the values, and leaving out the zero because it adds nothing, gives the expanded form:

\begin{aligned} 304{,}627 &= 300{,}000 + 4{,}000 \\ &\quad + 600 + 20 + 7 \end{aligned}

For word form, read the thousands period, 304, as “three hundred four” and add “thousand”. Then read the ones period, 627, as “six hundred twenty-seven”. So the sign reads three hundred four thousand, six hundred twenty-seven.

Check: 300{,}000 + 4{,}000 = 304{,}000, and 304{,}000 + 627 = 304{,}627.

Example 2

The same digit, two values

At another park, Alani’s guidebook lists 382{,}815 visitors in a year. The digit 8 appears twice. What is each 8 worth, and how many times as much is the first 8 as the second?

Solution

From the right, the places hold 5 ones, 1 ten, 8 hundreds, 2 thousands, 8 ten thousands and 3 hundred thousands. So the left 8 is in the ten thousands place, worth 80{,}000, and the right 8 is in the hundreds place, worth 800.

The ten thousands place is two places to the left of the hundreds place. One step left makes it ten times as much: 800 becomes 8{,}000. A second step makes it ten times as much again: 8{,}000 becomes 80{,}000. So the left 8 is worth 10 \times 10 = 100 times the right one. Check: 100 \times 800 = 80{,}000.

The same digit can be worth very different amounts. Its place, not the digit alone, decides.

Example 3

Writing a number from words

The state fair’s newsletter says that six hundred eight thousand, forty people came this year. Arash wants to write the number in standard form.

Solution

The word “thousand” says the number has a thousands period and a ones period. The part before “thousand” is six hundred eight, which is 608, so the thousands period holds the digits 6, 0, 8.

The part after the comma is forty. The ones period must have exactly three digits, because it sits after another period: forty is 0 hundreds, 4 tens and 0 ones, so the ones period is 040.

Putting the periods together with a comma between them gives 608{,}040. Both zeros in the ones period are needed: without them the number would be 60{,}840 or 6{,}084, and each digit would slide into the wrong place.

Example 4

One more makes a million

Arash’s job at the fair is counting tickets, which come in rolls of 1{,}000. What number comes just after 999{,}999? How many rolls of 1{,}000 tickets make that many?

Solution

Add 1 to 999{,}999. The ones place now holds ten ones, so trade them for one ten. That makes ten tens, traded for one hundred, which makes ten hundreds, traded for one thousand. The trading runs on through every place: ten thousands become a ten thousand, ten ten thousands become a hundred thousand, and ten hundred thousands become one million. Every place is left at 0:

999{,}999 + 1 = 1{,}000{,}000

For the rolls, look at the thousands period. In 1{,}000{,}000 the digits to the left of the last comma are 1{,}000, and they count thousands. So one million is 1{,}000 thousands, and it takes 1{,}000 rolls of 1{,}000 tickets. In the same way it is 10 hundred thousands, since the ladder says each hundred thousand is one tenth of a million.

Example 5

A number given in odd pieces

A park ranger tallies visitors by bundles and writes: 2 hundred thousands + 15 thousands + 9 hundreds + 3 ones. Write the number in standard form.

Solution

Fifteen thousands is too many for one place, since a place can hold only 0 to 9. Trade ten of the thousands for one ten thousand: 15 thousands is 1 ten thousand and 5 thousands. Now each place has one digit:

place	digit
hundred thousands	2
ten thousands	1
thousands	5
hundreds	9
tens	0
ones	3

There are no tens, so the tens place holds 0. The number is 215{,}903. As a check, add the values: 200{,}000 + 15{,}000 + 900 + 3 = 215{,}903.

Example 6

Ordering four parks

Lerato compares summer visitor counts at four parks: 304{,}627, 340{,}267, 98{,}950 and 304{,}672. Put them in order from least to greatest.

Solution

Line the numbers up on the ones place.

Figure 3Four visitor counts lined up on the ones place. Read down each column from the left to find the first place where two numbers differ.

First count digits. 98{,}950 has five digits and the others have six, so 98{,}950 is the least: it is less than 100{,}000, and each of the others is at least that.

The three six-digit numbers all have 3 in the hundred thousands place, a tie. Move one place right to the ten thousands: 0, 4, 0. The 4 is largest, so 340{,}267 is the greatest.

That leaves 304{,}627 and 304{,}672. They agree in the hundred thousands (3), ten thousands (0), thousands (4) and hundreds (6). In the tens place one has 2 and the other 7, and 2 < 7, so 304{,}627 < 304{,}672. The ones place never gets a say, because the tens place already decided.

In order: 98{,}950 < 304{,}627 < 304{,}672 < 340{,}267.

Example 7

Building numbers from six digits

Alani has six cards: 0, 2, 5, 6, 8, 9. Using every card once, she makes six-digit numbers. What is the greatest she can make? The least? Which one is closest to 500{,}000?

Solution

For the greatest, the leftmost place is worth the most, so it gets the largest digit, 9. The next place gets the next largest, and so on: 986{,}520. Any other choice would put a smaller digit in some place where this number has a larger one, with all places to the left equal, so it would be less.

For the least, the smallest digit goes first, except that 0 cannot lead, or the number would have only five digits. So 2 goes first, then the rest from smallest to largest, with 0 next: 205{,}689.

For the closest to 500{,}000, think about the first digit. A number that starts with 5 is at least 500{,}000. The smallest such number puts the remaining cards in increasing order: 502{,}689, which is 2{,}689 above 500{,}000. A number that starts with a digit less than 5 must start with 2 (the 0 cannot lead), and the largest of those is 298{,}650, which is 201{,}350 below. Numbers that start with 6, 8 or 9 are at least 600{,}000, more than 100{,}000 away. So the closest is 502{,}689.

Boundaries

This lesson covers whole numbers, 0, 1, 2, 3 and so on, up to 1{,}000{,}000. The same pattern goes on past a million: a millions period, then billions, each a new group of three places. Digits to the right of the ones place, after a decimal point, also follow the ten-times rule, but they are not whole numbers.
A place can hold only one digit, 0 to 9. A count of ten or more in one place, as in Example 5, must be traded into the next place before the number can be written in standard form.
The leftmost period may have one, two or three digits (4{,}000, 40{,}000, 400{,}000). Every period after it must have exactly three.
The rule “more digits means greater” needs numbers written without zeros at the front. Nobody writes 004{,}627, and if they did, the extra zeros would add digits without adding value.
Commas group digits by threes in the United States. Many other countries write the same number as 304\,627 or 304.627; the places and their values do not change, only the mark between periods.
The ten-times rule belongs to our base-ten system. Tally marks and Roman numerals have no places at all, so an \text{X} is worth ten wherever it stands.

Where this goes next

Big numbers are the ground for most of the arithmetic ahead. With every place known, Rounding to any place, including decimals replaces a number like 304{,}627 with a nearby, simpler one such as 300{,}000. The The standard addition algorithm adds big numbers column by column, and its carrying is the same trade of ten in one place for one in the next that made 999{,}999 + 1 into a million. Multiplying and dividing by ten, one hundred and one thousand turns the ten-times rule into multiplication: times 10 moves every digit one place to the left. Later, the same rule runs the other way, past the ones place, into tenths, hundredths and Thousandths: reading, writing and expanded form, and Powers of ten gives a short way to write the place values themselves.

Exercises

Exercises 1 to 8 go with Part 1, 9 to 16 with Part 2, 17 to 22 with Part 3, and 23 to 25 ask you to explain.

Part 1: places, values and three forms

A park had 271{,}958 visitors in a year. Name the place of the digit 7 and give its value.
In 516{,}394, give the value of the digit 5, the digit 1 and the digit 6.
Write 463{,}215 in expanded form.
Write 729{,}604 in word form.
Write 800{,}000 + 50{,}000 + 7{,}000 + 300 + 90 + 1 in standard form.
In 636{,}363, the value of the 3 in the ten thousands place is how many times the value of the 3 in the hundreds place?
Ten thousands make one ten thousand. How many thousands make one hundred thousand?
Alani’s number has 4 hundred thousands, 0 ten thousands, 9 thousands, 2 hundreds, 0 tens and 6 ones. Write it in standard form and in word form.

Part 2: periods, commas, zeros and one million

Put the comma in each number: 250078 and 70306.
On its busiest day the state fair counted seventy thousand, six hundred five people. Write the number in standard form.
Write five hundred thousand, eighty in standard form.
Write 603{,}050 in word form.
Write 900{,}000 + 7{,}000 + 40 in standard form.
Write 3 hundred thousands + 12 ten thousands + 6 hundreds in standard form.
One million is how many hundred thousands? How many thousands? How many hundreds?
What number is 1 more than 459{,}999? What number is 10{,}000 more than 990{,}000?

Part 3: comparing, ordering and building

Write <, > or = between 487{,}352 and 478{,}532.
Write <, > or = between 69{,}999 and 100{,}000.
State fair attendance on four days was 215{,}480, 251{,}084, 215{,}408 and 52{,}148. Put the numbers in order from least to greatest.
Using the digits 1, 3, 4, 7, 0, 8 once each, write the greatest and the least six-digit numbers you can.
Arash thinks of a six-digit number. Its thousands digit is 5. Its hundred thousands digit is 2 more than its thousands digit. Its ones digit is 4 less than its thousands digit. Every other digit is 0. What is the number?
One digit is missing: 34\square{,}512. Which digits can go in the box so that the number is greater than 348{,}900?

Reasoning

Explain why every six-digit whole number is greater than every five-digit whole number.
Lerato writes five hundred six thousand, thirty-one as 56{,}031. Explain what went wrong and why the zero matters.
Explain why a place is worth 1{,}000 times the place three to its right. Use this to explain why 352 thousands is written 352{,}000.

Answers

Ten thousands; its value is 70{,}000. From the right: 8 ones, 5 tens, 9 hundreds, 1 thousand, 7 ten thousands.
The 5 is worth 500{,}000, the 1 is worth 10{,}000 and the 6 is worth 6{,}000. They sit in the hundred thousands, ten thousands and thousands places.
400{,}000 + 60{,}000 + 3{,}000 + 200 + 10 + 5. Each term is a digit times its place.
Seven hundred twenty-nine thousand, six hundred four. The thousands period is 729 and the ones period is 604.
857{,}391. The terms give one digit per place: 8, 5, 7, 3, 9, 1.
100 times. The 3 in the ten thousands place is worth 30{,}000 and the 3 in the hundreds place is worth 300; two places left is 10 \times 10 = 100, and 100 \times 300 = 30{,}000.
100 thousands. Ten thousands make a ten thousand, and ten ten thousands make a hundred thousand, so it takes 10 \times 10 = 100 thousands.
409{,}206, four hundred nine thousand, two hundred six. The zeros hold the ten thousands and tens places.
250{,}078 and 70{,}306. Count three digits in from the right and put the comma there.
70{,}605. The thousands period is seventy, 70; the ones period is six hundred five, 605.
500{,}080. The thousands period is 500; eighty in the ones period needs three digits, 080.
Six hundred three thousand, fifty. The thousands period is 603 and the ones period, 050, is fifty.
907{,}040. Zeros fill the ten thousands, hundreds and ones places.
420{,}600. Twelve ten thousands is 1 hundred thousand and 2 ten thousands, so there are 4 hundred thousands and 2 ten thousands: 300{,}000 + 120{,}000 + 600 = 420{,}600.
10 hundred thousands, 1{,}000 thousands, 10{,}000 hundreds. Each step down the ladder multiplies the count by 10: 10, then 10 \times 10 = 100 ten thousands, 1{,}000 thousands, 10{,}000 hundreds.
460{,}000 and 1{,}000{,}000. Adding 1 to 459{,}999 trades ten ones, ten tens, ten hundreds and ten thousands in turn, making 46 ten thousands. And 99 ten thousands plus 1 ten thousand is 100 ten thousands, which is one million.
487{,}352 > 478{,}532. Both have 4 hundred thousands; in the ten thousands place 8 > 7.
69{,}999 < 100{,}000. The first has five digits and the second has six.
52{,}148 < 215{,}408 < 215{,}480 < 251{,}084. The five-digit number is least. Of the others, 251{,}084 has 5 ten thousands against 1. The two that start 215{,}4 differ first in the tens place, 0 < 8.
Greatest 874{,}310, least 103{,}478. For the greatest the digits go from largest to smallest; for the least 1 goes first because 0 cannot lead, then the rest from smallest to largest.
705{,}001. The thousands digit is 5, the hundred thousands digit is 5 + 2 = 7, the ones digit is 5 - 4 = 1, and the ten thousands, hundreds and tens digits are 0.
Only 9. The numbers agree in the hundred thousands and ten thousands (3 and 4). With 9 in the box, 349{,}512 > 348{,}900 because 9 > 8 in the thousands place. With 8, the number 348{,}512 ties in the thousands and then loses in the hundreds, 5 < 9. Any digit below 8 loses in the thousands place.
A six-digit number has a digit from 1 to 9 in the hundred thousands place, so it is at least 100{,}000. A five-digit number is at most 99{,}999, and 99{,}999 + 1 = 100{,}000. So the smallest six-digit number is already greater than the largest five-digit one.
Lerato left out the zero that holds the ten thousands place. The number is 506{,}031: the thousands period must be 506, three digits. Without the 0, the 5 slides from the hundred thousands place to the ten thousands place and is worth 50{,}000 instead of 500{,}000. Her number is only about one ninth of the real one.
Each step to the left is worth ten times as much, so three steps are worth 10 \times 10 \times 10 = 1{,}000 times as much. The thousands period is three places to the left of the ones period, so a digit there counts thousands instead of ones. Writing 352 in the thousands period, with zeros to hold the ones period, gives 352{,}000, which is 352 thousands.