# Lesson 16

Methods for Multiplying Decimals

### Lesson Narrative

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.

### Learning Goals

Teacher Facing

• Coordinate area diagrams and vertical calculations that represent the same decimal multiplication problem.
• Interpret different methods for computing the product of decimals, and evaluate (orally) their usefulness.
• Justify (orally, in writing, and through other representations) where to place the decimal point in the product of two decimals with multiple non-zero digits.

### Student Facing

Let’s look at some ways we can represent multiplication of decimals.

### Student Facing

• I can use area diagrams to represent and reason about multiplication of decimals.
• I can use place value and fractions to reason about multiplication of decimals.

Building On