Lesson 16

In earlier grades, students have multiplied base-ten numbers up to hundredths (either by multiplying two decimals to tenths or by multiplying a whole number and a decimal to hundredths). Here, students use what they know about fractions and place value to calculate products of decimals beyond the hundredths. They express each decimal as a product of a whole number and a fraction, and then they use the commutative and associative properties to compute the product. For example, they see that \((0.6) \boldcdot (0.5)\) can be viewed as \(6 \boldcdot  (0.1) \boldcdot 5 \boldcdot (0.1)\) and thus as \(\left(6 \boldcdot \frac{1}{10}\right) \boldcdot \left(5 \boldsymbol \boldcdot \frac {1}{10}\right)\). Multiplying the whole numbers and the fractions gives them \(30 \boldsymbol \boldcdot \frac{1}{100}\) and then 0.3. 

Through repeated reasoning, students see how the number of decimal places in the factors can help them place the decimal point in the product (MP8).

Students continue to develop methods for computing products of decimals, including using area diagrams. Students have previously seen that, in a rectangular area diagram, the side lengths can be decomposed by place value. For instance, in an 18 by 23 rectangle, the 18-unit side can be decomposed into 10 and 8 units (tens and ones), and the 23-unit side can be expressed as 20 and 3 (also tens and ones), creating four sub-rectangles whose areas constitute four partial products. The sum of these partial products is the product of 18 and 23. Students extend the same reasoning to represent and find products such as \((1.8) \boldcdot (2.3)\). Then, students explore how these partial products correspond to the numbers in the multiplication algorithm.

Students connect multiplication of decimals to that of whole numbers (MP7), look for correspondences between geometric diagrams and arithmetic calculations, and use these connections to calculate products of various decimals.