What the nought is doing
Before you read anything
Write four hundred and seven. Then write four hundred and seventy.
One of those has a nought in the middle and one on the end. Say out loud what each nought is for.
They are not the same nought
Forty is four tens. So if the digits say what you have, why does forty need a second digit at all — and why that one?
wrote 407 as 47, because "there are no tens"
Because a column with nothing in it still has to be there. Take the nought out of 407 and the 7 slides into the tens place, and says seventy.
Nothing, or none of these?
:x nutshell-placeholder
A nought that says "none of this size", so the digits either side stay in their own columns.
A nought is not nothing. It is a :placeholder: four hundreds, no tens, seven ones. Reading it as nothing is what turns 407 into 47, and it is the commonest mistake there is with numbers.
The bundles say it plainly
Four hundred-bundles, no ten-bundles, seven loose sticks. You still have to leave the middle shelf empty — and an empty shelf is a thing you can point at.
Check yourself
Six to try
$7$ hundreds, no tens, and $6$ ones. Which number is that?
$2$ hundreds, no tens, and $7$ ones. Which number is that?
$4$ hundreds, no tens, and $4$ ones. Which number is that?
$3$ hundreds, no tens, and $1$ ones. Which number is that?
In $438$, what is the $3$ worth?
In $572$, what is the $5$ worth?
Show that you have it
Take it apart
$421 = 400 + \square + 1$. What goes in the box?
$753 = 700 + \square + 3$. What goes in the box?