Multiplying and dividing (age 10+)

Activities to try

1. 1

   Slide the digits across a place-value chart

   Why it works: Multiplying or dividing by a power of ten moves every digit a fixed number of place-value columns while the decimal point stays put; the number of columns equals the number of zeros in the power of ten, and empty columns must be filled with placeholder zeros.

   1. Set up a place-value chart with columns from thousands down to ten-thousandths, with the decimal point marked fixed at the ones boundary (ThHTO . tenths hundredths thousandths ten-thousandths); the extra low columns are needed because 45.2 ÷ 1000 lands a digit in the ten-thousandths place.
   2. Write 3.456 into the chart using digit cards, then multiply by 100: slide EVERY digit two columns to the LEFT in one move, keeping the point pinned in place, to land on 345.6.
   3. Now take 45.2 and divide by 1000: slide every digit three columns to the RIGHT (the tenths digit travels into the ten-thousandths column), and write a placeholder 0 into each emptied column that is written (ones and tenths) to get 0.0452.
   4. Say aloud for each step: 'the digits moved ___ places ___ because I multiplied/divided by ___', linking the number of places to the power of ten.
   5. Try a target challenge: starting from 3.456, work out which single operation reaches 345.6, and which reaches 0.3456.
2. 2

   Why ten-times moves each digit one column

   Why it works: Because each place-value column is worth ten times the column to its right, making a number ten times bigger promotes every digit into the next-higher column; this is WHY the shift happens and why ×100 shifts two columns and ÷1000 three.

   1. Pick a digit in a number, e.g. the 4 in 3.456 (worth four tenths), and state its value.
   2. Explain why, when the whole number is made ten times bigger, that digit must now be worth ten times as much (four ones) and therefore move one column to the left.
   3. Use a scaling chart or a 'ten-times' number line to show the whole quantity becoming ten times as large, not merely gaining an extra written digit.
   4. Chain the reasoning: ×100 is ×10 then ×10, so each digit moves one column left twice — two columns in total — and connect ÷10 as the exact inverse (one column right).
3. 3

   Fluency and error checks

   Why it works: Repeated practice with answers up to three decimal places (plus the four-decimal-place extension 45.2 ÷ 1000 = 0.0452) is intended to make the power-of-ten ↔ number-of-places mapping more automatic and to surface the common errors (missing placeholder zeros, wrong direction, miscounting by one).

   1. Complete fill-in-the-blank items such as 45.2 ÷ 1000 = ____ , writing all placeholder zeros, and 3.456 × 100 = ____ .
   2. Answer direction questions: when dividing by a power of ten, which way do the digits move, and how many places for ÷1000?
   3. Order a set of results by size, e.g. 2.5 × 10, 2.5 × 100, 2.5 × 1000, and explain the ordering by place-value shift.
   4. Self-check each answer by counting the columns crossed against the number of zeros in the multiplier or divisor.