Why multiplying by ten changes each place value

start. Why is 0.24 × 10 equal to 2.4, not 0.240?

Answer: Each digit represents ten times as much: two tenths become two ones, and four hundredths become four tenths.

Teacher response: Listen for this idea: Multiplying by ten always means adding a zero at the end. Follow up: A final zero in 0.240 does not change its value. Multiplication by ten changes the amount. Track the value represented by each digit.

example. Why is 0.24 × 10 equal to 2.4, not 0.240?

1. Read 0.24 as two tenths plus four hundredths.
2. Multiply each part by ten. Ten copies of two tenths make two ones; ten copies of four hundredths make four tenths.
3. Combine two ones and four tenths to make 2.4. The decimal point continues to mark the boundary between ones and tenths.
4. Check with an estimate: ten copies of about one quarter should be about two and a half, so 2.4 is sensible.

Worked example: Each digit represents ten times as much: two tenths become two ones, and four hundredths become four tenths.

p1. Calculate 0.36 × 10.

Answer: 3.6

Teacher response: Three tenths become three ones; six hundredths become six tenths.

p2. Calculate 2.07 × 10.

Answer: 20.7

Teacher response: The zero keeps the empty ones place in the result.

p3. Calculate 0.05 × 100.

Answer: 5

Teacher response: One hundred copies of five hundredths make five wholes.

e1. Calculate 0.004 × 10.

Answer: 0.04

Teacher response: Four thousandths become four hundredths.

transfer. Explain why 3.2 × 100 can be found by multiplying by ten twice.

Answer: 100 = 10 × 10. Each multiplication makes the value ten times as large: 3.2 becomes 32, then 320.

next. If the learner appends zeros, compare 0.24 with 0.240 in the chart. Ask them to build ten copies before using a shortcut.