Why can a decimal product be smaller?

start. Why is 0.5 × 0.8 less than 0.8?

Answer: Multiplying by 0.5 takes one half of 0.8. Half of eight tenths is four tenths, so the product is 0.4.

Teacher response: Listen for this idea: Multiplication always makes numbers larger. Follow up: That pattern holds for a positive number multiplied by a factor greater than one. A positive factor between zero and one takes only part of the amount.

example. Why is 0.5 × 0.8 less than 0.8?

1. Interpret 0.5 as one half. The question asks for half of the eight-tenths amount.
2. Show eight tenths of a whole, then split that shaded amount into two equal shares.
3. Each share contains four tenths. Record 0.5 × 0.8 = 0.4 and compare it with the original 0.8.
4. Keep the comparison rule precise: for positive amounts, multiplying by a factor below one shrinks, by one preserves, and by a factor above one enlarges. Zero needs its own case.

Worked example: Multiplying by 0.5 takes one half of 0.8. Half of eight tenths is four tenths, so the product is 0.4.

p1. Calculate 0.5 × 0.6.

Answer: 0.3

Teacher response: Half of six tenths is three tenths.

p2. Calculate 0.25 × 0.8.

Answer: 0.2

Teacher response: One quarter of eight tenths is two tenths.

p3. Calculate 1.5 × 0.4.

Answer: 0.6

Teacher response: One and a half copies of four tenths is six tenths.

e1. Compare 0.3 × 0.7 with 0.7. Enter the product.

Answer: 0.21

Teacher response: Three tenths of seven tenths is twenty-one hundredths, less than 0.7.

transfer. Without calculating, explain whether 0.9 × 0.65 is greater than, equal to, or less than 0.65.

Answer: It is less: 0.9 takes nine tenths of the positive amount 0.65. The missing tenth makes the product smaller.

next. If the learner only recalls whole-number multiplication, compare one copy, half a copy and two copies of the same amount.