Place Value to 1,000,000
A digit does not have a value of its own. It has the value of the column it is standing in, and that one idea carries the whole of arithmetic.
A digit and its column
A digit is worth what its column says it is worth.
Write a 4 on its own and it means four. Write the same 4 one column to the left and it means forty, and two columns left it means four hundred, and the digit itself never changed at all. This is the idea that separates our number system from counting on fingers, and a child who has it can read a number they have never seen before. A child who does not have it is memorising, and memorising runs out at about four digits.
The columns up to a million
Each column to the left is worth ten times the one before it. Ones, then tens, then hundreds, and then the pattern simply repeats itself with the word thousand attached: thousands, ten thousands, hundred thousands. After that comes millions.
The pattern worth spotting
Ones, tens, hundreds. Then thousands, ten thousands, hundred thousands. Then millions.
It is the same three words each time. Once a child notices that, seven columns stop being seven separate things to learn.
What is the 6 worth in 462,180?
- Start at the right and name the columns going left: ones, tens, hundreds, thousands, ten thousands, hundred thousands.
- The digits are 4, 6, 2, 1, 8, 0 reading left to right.
- The 6 is the second digit from the left, so it sits in the ten thousands column.
- Six lots of ten thousand.
The 6 is worth 60,000.
Why zero has to be there
Zero is the column that has nothing in it, and it is the most useful digit in the whole number. Take 407. There are four hundreds, no tens and seven ones, and if the zero were left out you would write 47, which is a completely different number. The zero is not saying "nothing here to see", it is holding the tens column open so the 4 stays where it belongs.
That is why it gets called a placeholder. It holds a place.
Where this bites
Ask a child to write "three hundred thousand and forty" and a great many write 300,40 or 30,040. Working through the columns one at a time, out loud, gets 300,040 every time. Guessing gets it right about half the time, which in a test is the same as not knowing it.
Reading a big number aloud
Split the number into groups of three, starting from the right. That is what the commas are for, and they are not decoration. 3,407,000 splits into 3 | 407 | 000, and now you read each group and say which group it was.
Three digits at a time
Read the group, then say its name. "Three million, four hundred and seven thousand." The last group needs no name, because ones is where we already are.
Read 250,608 out loud.
- Group in threes from the right: 250 | 608.
- The left group is the thousands group, so read it and say thousand: two hundred and fifty thousand.
- Read the right group as it stands: six hundred and eight.
- Notice the zero in the tens column, which is why it is eight and not eighty.
Two hundred and fifty thousand, six hundred and eight.
Common mistakes
1. Counting columns from the left
Reading 462,180 and starting at the 4, calling it the ones. Columns are named from the right, always, because that is where ones live and everything else is built outwards from there. Start at the right hand digit and work left, even when the number is long and it feels slow.
2. Dropping a zero when writing from words
Writing "sixty thousand and five" as 60,5 or 6005. The words skip the empty columns because saying "and no hundreds and no tens" would be absurd in speech, but the digits cannot skip them. Sixty thousand and five is 60,005, with three zeros doing real work.
3. Judging size by how many digits look impressive
Deciding 89,999 is bigger than 100,001 because it is full of nines. Count the digits first: five against six. More digits wins, every time, for whole numbers. Only when the counts match do you compare from the left.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
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What to learn next
Place value is the floor everything else stands on. With it secure, rounding stops being a rule to memorise and becomes obvious, and multiplying by ten stops being about adding zeros.
For parents
Read house numbers, prices and odometer readings out loud in the car, and ask which digit is worth the most. It takes ten seconds and it costs nothing, and it is the closest thing there is to practice that does not feel like practice. If your child hesitates on a six-digit number, that hesitation is the thing worth working on, not the answer.
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