Ten Times Bigger

Each place is ten times the one to its right, and that single rule generates the whole system all the way to a million and, later, all the way down past the decimal point.

Watch for large numbers read digit by digit. "Four hundred and seven thousand" is a structure; "four zero seven" is a phone number.

Have them say the number aloud before working with it. Most errors with big numbers are reading errors that the arithmetic then faithfully preserves.

The slip this lesson targets is giving the digit for its value: Sam's 5 in 2,590 is "worth 5". Ask "five of what?" and have them name the column before the value, which is the tip. Two others to watch: counting places from the left, so the ones place lands at the wrong end, and Maya's slip, that a bigger whole number must carry the bigger digit. Only the place sets a digit's value, and the 4 in 4,100 is worth ten times the 4 in 9,400.

One rule generates the whole system — Each place is worth ten times the place to its right: ones, tens, hundreds, thousands, ten thousands. The pattern never stops and never changes. Ten of anything bundles into one of the next thing up, which is why the columns march leftwards in a perfectly regular way.

The same digit, ten times the value — In 4,400 the two 4s look identical but the left one is worth 4,000 and the right one 400. The left is ten times the right, because it sits one place further left. Position is doing all the work; the digit only says how many.

The rule works in both directions — Going left multiplies by ten; going right divides by ten. A hundred is a tenth of a thousand. Reading the rule rightwards is what will let you continue past the ones place into tenths and hundredths later this year — the pattern simply keeps going.

Why ten, and not some other number — There is nothing mathematically special about ten; we use it because people have ten fingers. Computers bundle in twos and get along fine. Knowing our system is a choice rather than a law makes the bundling rule easier to see as a rule.

Each step left is ten times — Look at 4444. The right-hand 4 is 4 ones. Next left: 4 tens, ten times bigger. Next: 4 hundreds. Next: 4 thousands. Same digit, four different values. The place is doing all the work.