Multiplication and Division of Decimals
Multiplying and dividing decimals extends whole-number computation to the decimal system. 0.125 x 3 = 0.375: each hundredth and each thousandth is tripled. The place value logic is identical to whole-number multiplication; only the decimal point changes. 7.2 / 9 = 0.8: 72 tenths divided by 9 = 8 tenths = 0.8. Students who understand place value can derive any decimal computation rule rather than memorizing a procedure.
Decimal multiplication using place value
0.125 x 3: multiply as if 125 x 3 = 375. Count decimal places in the factors: 3 (in 0.125) + 0 (in 3) = 3. Place the decimal point 3 places from the right: 0.375. This works because 0.125 = 125/1000, and 125/1000 x 3 = 375/1000 = 0.375. The decimal place count rule comes directly from fraction multiplication.
Decimal division
7.2 / 9: think 72 / 9 = 8. But 7.2 has one decimal place, so the result has one: 0.8. Or: 7.2 tenths / 9 = 0.8 tenths. Wait: 7.2 is 7.2 ones. Rewrite as 72 tenths. 72 tenths / 9 = 8 tenths = 0.8. The key is naming the decimal in terms of the smallest place value, then dividing.
Estimation to verify
Before computing 4.7 x 2.3, estimate: approximately 5 x 2 = 10. Actual: 4.7 x 2.3 = 10.81. Close to 10: reasonable. If a student computes 108.1 or 1.081, the estimate of 10 immediately flags the error. Estimation is the most reliable checking tool for decimal operations where place-of-decimal-point errors are common.