Multiplying and dividing by 10 and 100
National Curriculum: this is in the Year 3 programme of study, part of key stage 2 (Years 3 to 6). Schools have to teach the key stage 2 programme by the end of Year 6, and can introduce content earlier or later within it.
Children in Year 3 are 7 or 8, and turn 8 during the school year.
What it is
Multiplying by 10 makes every digit worth ten times as much, so every digit moves one place to the left: 34 × 10 = 340, and the 3 that stood for 3 tens now stands for 3 hundreds. Multiplying by 100 moves every digit two places. Dividing does the same in the other direction: 340 ÷ 10 = 34, and 3400 ÷ 100 = 34.
Most adults were taught to add a zero, and on whole numbers that gives the right answer every time. It breaks in two places. On decimals: 2.5 × 10 is 25, but adding a zero produces 2.50, which is the number it started with. On division: taking a zero off works for 340 ÷ 10, and there is no zero to take off 34 ÷ 10, which is 3.4.
Moving the digits explains the zero rather than replacing it. When every digit in 34 moves one place left, the ones column is left empty, and the 0 in 340 is written to hold it. That one idea covers × 10, ÷ 10, × 100 and ÷ 100, and it still works in Year 5 on decimals and on × 1000.
When children learn it
This starts in Year 3, where children use the table facts they know to work out related ones: 3 × 4 = 12, so 3 × 40 = 120. Year 4 asks what dividing by 10 and 100 does to the digits, with answers in tenths and hundredths. Year 5 extends it to 1000 and to numbers with decimals, which is where a rule about zeros stops working.
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.
- Adding a zero instead of moving the digits. 2.5 × 10 answered as 2.50, or 4.7 × 100 answered as 4.700. The rule was applied exactly as it was taught and produced a number the same size as the one it started with. The same child is right on every whole-number question, so this stays hidden until decimals arrive.
- The digits move the wrong distance, or the wrong way. 34 × 100 answered as 340: the digits moved one place instead of two. Or 340 ÷ 10 answered as 3400: the right distance, the wrong direction. Both come from a remembered rule rather than from what multiplying by 100 does to each digit.
- The digits move on paper but not in the child's head. 34 × 10 is written as 340 correctly, but asked what the 3 is worth now, they say three, or thirty. The move was carried out as a writing rule, so the next question, 340 ÷ 10, gets guessed rather than worked as the same move backwards. What each digit is worth in a three-digit number is place value, and that is where this error belongs rather than in the times-ten rule.
How to help at home
- Say the move rather than the zero: every digit goes one place bigger for 10, and two places bigger for 100. Ask them where the 3 in 34 ends up.
- Find out which rule they are using by asking for 2.5 × 10. An answer of 2.50 is the zero rule showing, and Year 5 multiplies decimals by 10, 100 and 1000, so it is worth undoing now.
- Ask the division straight after the multiplication: 34 × 10 = 340, so 340 ÷ 10 = 34. Doing the pair together makes one move with two directions instead of two separate rules.
- Write the place value columns on paper, hundreds and tens and ones, put the number in, and move the digits across while the columns stay still. Coins do the same job: ten 1p coins make 10p, and ten 10p coins make £1.