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Standard place value and column mechanics

Numbers up to 7 digits: identifying values, and multiplying and dividing by 10, 100 and 1000.

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Numbers are made from ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. With just those ten, you can write any number you like, from 7 to 7 million.

The secret is where each digit sits. In this lesson you will learn:

  • what each digit in a number is worth
  • how to read and write numbers up to 7 digits long (millions)
  • how to compare big numbers and put them in order
  • what happens to the digits when you multiply or divide by 10, 100 and 1000
Red: the millions Blue: the thousands Dark blue: the ones

1. Every digit has a job

Key idea A digit is worth a different amount depending on its place in the number. This is called place value.

Look at the digit 2 in each of these numbers:

  • In 52, the 2 is worth 2.
  • In 325, the 2 is worth 20.
  • In 4,207, the 2 is worth 200.

Same digit, different place, different value.

Each place is called a column. For numbers up to the millions you need seven columns. Start from the right:

ColumnShort nameA 1 here is worth
OnesO1
TensT10
HundredsH100
ThousandsTh1,000
Ten thousandsTTh10,000
Hundred thousandsHTh100,000
MillionsM1,000,000

We put the columns side by side in a place value chart. Here is 3,584,061. The 8 is in the ten thousands column, so it is worth 80,000.

Millions Thousands Ones M HTh TTh Th H T O

3,584,061

three million, five hundred and eighty-four thousand and sixty-one

2. Each column is worth 10 times more

Key idea Move one column to the left and the value gets 10 times bigger.
  • 10 ones make 1 ten.
  • 10 tens make 1 hundred.
  • 10 hundreds make 1 thousand.
  • And so on, all the way up: 10 hundred thousands make 1 million.
Fun fact If you counted one number every second, without stopping to sleep or eat, it would take you about 11 and a half days to count to one million.
Millions Thousands Ones M HTh TTh Th H T O 1 0 0 0

1,000

1 thousand is 1,000. That is the same as 10 hundreds.

3. Reading big numbers

Key idea Split a big number into groups of three digits, starting from the right. The commas show you where the groups are.

Each group has a name. From the right: the ones, the thousands, then the millions.

How to read a big number

  1. Find the commas
    No commas? Count three digits from the right and put one in. Count three more and put another.
  2. Read the first group, then say its name
    “three million”
  3. Read the next group, then say its name
    “five hundred and eighty-four thousand”
  4. Read the last group on its own
    “and sixty-one”

Example: read 3,584,061

3,584,061

  • 3 → three million
  • 584 → five hundred and eighty-four thousand
  • 061 → sixty-one
Answerthree million, five hundred and eighty-four thousand and sixty-one
Watch out You don’t say the 0 in 061 out loud, but you must write it. Without it the number would be 358,461, which is a completely different number.

4. Writing numbers from words

Key idea Every column needs a digit. If a column is empty, write 0 in it. This zero is called a placeholder.

This is where lots of children slip up. Try it: write four thousand and six.

It is tempting to write 406. But 406 is four hundred and six. Four thousand needs four columns, so draw them first:

Thousands Ones Th H T O 4 0 0 6

The answer is 4,006. The two zeros are doing an important job: they keep the 4 in the thousands column and the 6 in the ones column.

How to write a number from words

  1. Find the biggest part of the number
    “four thousand” means you need the thousands column.
  2. Draw the columns from there down to the ones
  3. Put each digit in its column
  4. Fill every empty column with a 0

Example: two million, forty thousand and nine

Millions Thousands Ones M HTh TTh Th H T O 2 0 4 0 0 0 9
Answer2,040,009

5. Comparing and ordering numbers

Key idea Compare the digits column by column, starting from the left. The first column that is different tells you which number is bigger.

How to compare two numbers

  1. Count the digits
    A whole number with more digits is always bigger. 100,001 is bigger than 99,999.
  2. Same number of digits? Line them up and start on the left
  3. Stop at the first column that is different
    The number with the bigger digit there is the bigger number.

Example: which is bigger, 45,812 or 45,821?

  • Ten thousands: 4 and 4. The same.
  • Thousands: 5 and 5. The same.
  • Hundreds: 8 and 8. The same.
  • Tens: 1 and 2. Different! 2 is bigger.
Answer45,821 is bigger. You never needed to look at the ones.

Two symbols help you write it down:

  • < means is less than: 45,812 < 45,821
  • > means is greater than: 45,821 > 45,812

The open side always faces the bigger number, like a hungry crocodile. The point aims at the smaller one.

To put a list in order, find the smallest number first, then the next smallest, and so on. Ascending order means smallest first. Descending order means biggest first.

6. Multiplying and dividing by 10, 100 and 1000

Key idea When you multiply, the digits move to the left. When you divide, they move to the right. The number of zeros tells you how many places.
CalculationZerosThe digits move
× 1011 place left
× 10022 places left
× 100033 places left
÷ 1011 place right
÷ 10022 places right
÷ 100033 places right

Example 1: 352 × 100

  1. 100 has two zeros, so every digit moves 2 places left
    The 3 moves from hundreds to ten thousands. The 5 moves from tens to thousands. The 2 moves from ones to hundreds.
  2. Fill the empty tens and ones columns with 0
Thousands Ones TTh Th H T O 3 5 2 0 0
Answer352 × 100 = 35,200

Example 2: 46,000 ÷ 100

  1. 100 has two zeros, so every digit moves 2 places right
    The 4 moves from ten thousands to hundreds. The 6 moves from thousands to tens.
  2. The last two zeros move past the ones column and drop away
Thousands Ones TTh Th H T O 4 6 0
Answer46,000 ÷ 100 = 460
Watch out You may hear “to times by 10, just add a zero”. It gives the right answer for whole numbers, but it is not what really happens, and it goes wrong with decimals (Lesson 1.2). Think: the digits move, and zeros fill the empty columns.

In this lesson every answer is a whole number. When a digit moves past the ones column, you need decimals. That is Lesson 1.2.

352

Millions Thousands Ones M HTh TTh Th H T O 3 5 2

7. Working backwards

Key idea Multiplying and dividing undo each other. They are inverses.

So if a question hides the start number, do the opposite to find it.

Example 1: ? × 100 = 34,000

  1. The opposite of × 100 is ÷ 100
    34,000 ÷ 100 = 340

Check: 340 × 100 = 34,000.

Answer340

Example 2: how many hundreds are there in 45,600?

  1. “How many hundreds?” means “divide by 100”
    45,600 ÷ 100 = 456
Answer456 hundreds
Watch out The answer is not 6. The 6 is just the digit in the hundreds column. All the thousands and ten thousands are made of hundreds too.

The mistakes that cost marks

  1. Leaving out a placeholder zero. Four thousand and six is not 406. draw the columns and put a 0 in every empty one: 4,006.
  2. Giving the digit instead of its value. In 3,584,061 the 8 is not worth 8. say which column it is in. The 8 is in the ten thousands, so it is worth 80,000.
  3. Counting the columns from the left. always start at the ones column, on the right.
  4. Comparing the first digits of numbers that are different lengths. 99,999 is not bigger than 100,001. count the digits first. More digits means bigger.
  5. Mixing up < and >. remember the open side faces the bigger number.
  6. Moving the digits the wrong way. remember that multiplying makes a number bigger, so the digits move left. Dividing makes it smaller, so they move right.
  7. Moving the wrong number of places. count the zeros: 10 is 1 place, 100 is 2 places, 1000 is 3 places.

Quick checklist

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