Place value

What it is

Place value is the rule that a digit is worth different amounts in different columns. In 462 the 4 is four hundreds, the 6 is six tens and the 2 is two ones, so that same 4 is worth 400 here and 40 in 45. Each column is worth ten of the column to its right.

Partitioning is that rule written out: 462 = 400 + 60 + 2. Schools also partition a number the other way round when a calculation needs it, so 462 becomes 300 + 160 + 2. A child who can move between those two is ready for the written methods. A child who cannot usually learns column addition and column subtraction as steps to copy, and the steps break as soon as a column needs exchanging.

A zero holds a column open. Three hundred and five is 305: there is nothing in the tens column, and the 0 is what keeps the 3 in the hundreds. Written as 35 the number has lost a column, and written as 3005 it has gained one.

When children learn it

Year 2 asks for two-digit numbers as tens and ones. Year 3 is where the National Curriculum asks for the place value of each digit in a three-digit number, reading and writing numbers up to 1000 in numerals and in words, and comparing and ordering them. Year 4 takes the same rule to four digits, Year 5 to a million and Year 6 to ten million. Each of those years adds columns rather than a new idea, so a gap here does not close on its own.

What it looks like when it goes wrong

These patterns come up again and again in children's answers on this skill. Each one points at a specific missing idea, not carelessness.

• The digit read at face value. Asked what the 4 is worth in 462, they answer 4. The number is being read as three separate digits rather than as hundreds, tens and ones. Every written method built on columns will be copied rather than understood while this is true.
• The value taken from the wrong column. Asked what the 6 is worth in 462, they answer 600, or 6. The idea that a column carries a value is there; which column this digit is standing in is not. Reading the number aloud from the left, naming each column, usually settles it.
• The zero dropped when writing a number. Asked to write three hundred and five, they write 35. The words name hundreds and ones and say nothing about tens, so the tens column is skipped rather than filled with a zero to hold it.
• Every part written out in full. Asked to write three hundred and five, they write 3005. Each part is written as it is heard, 300 and then 5, and the two are pushed together. This and the dropped zero are one gap seen from two sides: the number is being written from the sound of the words rather than column by column.

How to help at home

• Ask what a digit is worth, not what it is. In 462, what is the 6 worth? Sixty is the answer to listen for; six means the columns are not there yet.
• Say a number apart and back together: 462 is 400 and 60 and 2, and 400 and 60 and 2 is 462. Doing it in both directions is what partitioning is.
• Dictate a few numbers with a zero in the middle and ask them to write them: 305, 470, 1004. The zero is where place value shows.
• Compare two numbers made of the same digits, such as 462 and 426, and ask which is bigger and how they know. The answer you want names a column.

Other guides

• Column additionThe written method from Year 3: carrying, lining up digits, and the errors to watch for.
• Column subtractionThe written method from Year 3: exchanging, zeros in the middle, and the classic wrong answers.
• Multiplying and dividing by 10 and 100Why the add-a-zero rule works now and breaks later, and what to say instead.
• Four-digit numbersWhat the thousands column adds, why numbers with a zero inside them are the hard ones, and how Year 4 counts in 1,000s and compares beyond 1,000.