Multiplying by 10, 100 and 1000
Nothing gets a zero added to it. The digits move, and a zero turns up afterwards to hold the empty column. Get that the right way round and decimals never catch you out.
The digits move
Multiplying by ten makes every digit worth ten times more, so every digit moves one column to the left.
Take 35. The 3 is worth thirty and the 5 is worth five. Make the whole thing ten times bigger and the 3 has to be worth three hundred, while the 5 has to be worth fifty. Each of them has moved one column left. Nothing was added to the number. The digits are the same digits and there are still only two of them; they are simply standing somewhere more valuable.
The zero at the end is a consequence of the move, not the cause of it. That distinction sounds pedantic right up until the first decimal question, at which point it becomes the whole thing.
How far they move
Count the zeros, move that many columns
x 10 moves every digit one column left.
x 100 moves every digit two columns left.
x 1000 moves every digit three columns left.
What is 47 x 1000?
- 1000 has three zeros, so every digit moves three columns left.
- The 4 goes from tens to ten thousands.
- The 7 goes from ones to thousands.
- Three columns are now empty, and each needs a zero to hold it.
47,000
Dividing is the same move
Dividing by ten makes everything worth a tenth as much, so the digits move one column to the right instead. 350 divided by 10 sends the 3 from hundreds to tens and the 5 from tens to ones. The zero that was holding the ones column now has nothing to hold, so it goes.
It is genuinely the same idea running backwards, and teaching them together saves a child from learning two unrelated rules and then mixing them up under pressure.
The trouble with adding a zero
Almost every child is told at some point to multiply by ten by adding a zero. It is quick, it works, and for two or three years nothing goes wrong. Then decimals arrive and the child writes 3.5 x 10 = 3.50, which is not ten times bigger, it is the same number with a pointless zero on the end.
What actually happens to 3.5
The 3 moves from ones to tens and the 5 moves from tenths to ones, giving 35. There is no empty column left over, so no zero appears anywhere. The rule that produced 3.50 was never really a rule about multiplying; it was a description of what happens to whole numbers.
Common mistakes
1. Adding the wrong number of zeros
Multiplying by 100 and adding one zero, or by 1000 and adding two. Counting the zeros in the number you are multiplying by fixes it. Moving the digits fixes it permanently, because you cannot move a digit two columns and think you moved it one.
2. Moving the decimal point instead of the digits
Both descriptions give the same answer, and only one survives contact with place value. The columns do not move: ones is always ones. Teaching the point as the thing that slides leaves a child with a picture in which the whole number system shuffles about, which makes the next topic harder than it needs to be.
3. Removing a zero from the middle when dividing
Working out 305 divided by 10 and writing 35, because a zero was taken out. Only the zero at the end can go, because only that one was holding an empty column. The right answer is 30.5, and the 0 in the middle is doing a job.
Practice questions
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What to learn next
This is the move that long multiplication is built out of. Multiplying by 47 is multiplying by 40 and by 7 and adding, and the "by 40" part is exactly what this page is about.
For parents
Ask what a price would be if everything cost ten times more, then a hundred. Then ask about something that costs $4.50, and if your child says $4.500, you have found the exact thing to work on. It is not carelessness, it is a rule that has quietly stopped working, and it is much easier to fix now than in grade seven.
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