Multiplication and Division of Decimals

1. Divide a decimal to the hundredths place by a decimal to tenths or hundredths that involve basic facts.
2. Reason about the placement of the decimal point in cases involving division of a number by some number of tenths or hundredths by thinking about the problem as equal-sharing, or quotative, division in which one partitions some number of units into an unknown number of groups of equal size, and then using any of the following lines of reasoning:
   1. E.g., $$7\div0.2$$ can be thought of as determining how many 2 tenths are in 7, which is the same as asking how many 2 tenths are in 70 tenths, i.e., $$7\div0.2$$ is the same as $$70\div 2$$ and thus, multiplying both the 7 and the 0.2 by 10 results in the same quotient,
   2. 0.2 is the same as $$2\times 0.1$$, so first divide 7 by 2, which is 3.5, and then divide that result by 0.1, which makes 3.5 ten times as large, namely 35, or
   3. Rewriting a division computation as a fraction to create an equivalent division expression, e.g., $$7 \div 0.2 = \frac{7}{0.2} = \frac{7 \times 10}{0.2 \times 10} = \frac{70}{2} = 35$$ (MP.3).
3. Check the answer to a division problem by using inverse operations, multiplying the quotient by the divisor, and seeing if it is equivalent to the dividend.
4. Estimate the quotient of a decimal divided by a whole number by finding nearby compatible numbers (e.g., $$ 1.32 \div 0.67 \rightarrow 1.4 \div 0.7$$).

After working with cases involving dividing by 0.1 and 0.01, “students can then proceed to more general cases. For example, to calculate 7 ÷ 0.2, students can reason that 0.2 is 2 tenths and 7 is 70 tenths, so asking how many 2 tenths are in 7 is the same as asking how many 2 tenths are in 70 tenths. In other words, 7 ÷ 0.2 is the same as 70 ÷ 2; multiplying both the 7 and the 0.2 by 10 results in the same quotient. Or students could calculate 7 ÷ 0.2 by viewing 0.2 as 2 × 0.1, so they can first divide 7 by 2, which is 3.5, and then divide that result by 0.1, which makes 3.5 ten times as large, namely 35. Dividing by a decimal less than 1 results in a quotient larger than the dividend and moves the digits of the dividend one place to the left. Students can summarize the results of their reasoning as specific numerical patterns, then as one general overall pattern such as ‘when the decimal point in the divisor is moved to make a whole number, the decimal point in the dividend should be moved the same number of places’” (NBT Progression, p. 20).

• Problem Set