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Lessons · Teaching · place value, and why the position is the whole idea

Place value: the position is doing the work

In base ten, a digit's value is the digit multiplied by the place it sits in, and each place to the left is worth ten times the one to its right. 4 in the tens place is not a 4, it is 40. That is the whole of the number system, and every algorithm taught after it is a consequence of it.

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In beta. This lesson was written for Hone and has not yet been checked by a state certification officer. Practice material, not professional advice. What that means.

What it is for

A child who writes 305 as 3005 has not made a careless slip. They have heard 'three hundred and five' and written down what they heard: three hundred, then five. The zero is a placeholder saying 'no tens here', and nobody said it out loud. You cannot fix that error by marking it wrong, only by teaching the position.

How to think about it

Read any number by naming the place of the leading digit, then say the number, then check the digit count against the place. Three hundred and five: leading digit is in the hundreds, so the number has three digits before any decimal point. Write the places as a row of empty boxes first and drop digits in, and a missing ten becomes a visible empty box rather than a silent one.

Worked example

In 3,072 the 3 is worth 3,000 and the 7 is worth 70
Digit times its place, every time. The 0 in the hundreds says there are none.
Each place left is ten times the place right: 1, 10, 100, 1,000
Which is why moving a digit one place left multiplies it by ten.
To the right of the decimal point the same rule runs on: 0.1, 0.01, 0.001
Tenths, hundredths, thousandths. Ten times smaller each step, not ten times larger.
0.3 is three tenths and 0.03 is three hundredths, so 0.3 is the larger
The longer decimal is not the bigger number, which is the commonest adult error here.

Your turn

Name what the 0 is doing in 305.

The 0 in 305 is a  saying there are no tens

The trap

Reading a decimal as if it were a whole number. 0.45 is not 'bigger than 0.5 because 45 is bigger than 5'. Compare place by place from the left: four tenths against five tenths settles it at the first digit.

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