Number & Operations for Teachers 

Copyright David & Cynthia Thomas, 2009

Arithmetic Operations with Decimals – Multiplication and Division

Multiplication

Multiplication of decimals is generally modeled using repeated addition or an area model. For instance, the product 4 x 1.50 may be computed as 1.50 + 1.50 + 1.50 + 1.50 = 6.00. To model the product 0.23 x 0.15, however, another approach is needed. Figure 6.4 shows an area model for decimal multiplication similar to that used in Chapter Three to model whole number multiplication. Note that the partial products indicated in the expanded algorithm are colored the same as their corresponding areas in the concept model. This approach works well when 1) both factors are two-digits in length, and 2) both factors have the same number of digits to the right of the decimal point. In other circumstances, this approach often leads to concept models of unacceptable complexity.

10 5

10 10 3

23 x 15 20 + 3 x 10 + 5 15 100 30 200 345

Note that § Large squares have area 100 § Rectangles have area 10 § Small squares have area 1. All areas and their corresponding partial products are multiples of 1.

1 0.5

1 1 0.3

2.3 x 1.5 2 + 0.3 x 1 + 0.5 0.15 1.00 0.30 2.00 3.45

Note that § Large squares have area 1 § Rectangles have area 0.10 § Small squares have area 0.01, or one-hundredth. All areas and their corresponding partial products are multiples of 0.01.

0.10 0.05

0 .10 0.10 0.03

0.23 x 0.15 0.2 + 0.03 x 0.1 + 0.05 0.0015 0.0100 0.0030 0.0200 0.0345

Note that § Large squares have area 0.01 § Rectangles have area 0.001 § Small squares have area 0.0001, or one ten- thousandth. All areas and their corresponding partial products are multiples of 0.0001.

Figure 6.4: Modeling Decimal Multiplication

In addition to modeling the products 23 x 15, 2.3 x 1.5, and 0.23 x 0.15, Figure 6.4 provides a context in which to explore the mathematical basis for a familiar rule: When multiplying two decimal numbers,

1. Ignore any decimal points associated with the factors and compute all partial products and their sum as if they were based on whole numbers. Call the sum s.

2. Count the number of digits to the right of the decimal point in each factor. Call those counts m and n.

3. Insert a decimal point m + n digits to the left of the right-most digit of s, using zeros as place holders as necessary.

In terms of the products modeled in Figure 6.4, it is clear that all three concept models have the same number of large squares, vertical rectangles, horizontal rectangles, and small squares. The models are different in that they are built on different scales. In the first product, 23 x 15, the area of each square and rectangle is a multiple of a square with area 1. In the second product, 2.3 x 1.5, the area of each square and rectangle is a multiple of a square with area 0.01. And in the third product, 0.23 x 0.15, the area of each square and rectangle is a multiple of a square with area 0.0001. Using the information in Table 6.6, explain the mathematical basis for the “rule” for multiplying decimals.

23 x 15 20 + 3 10 + 5 15 100 30 200 345 ones

2.3 x 1.5 2 + 0.3 1 + 0.5 0.15 = 15 x 0.01 1.00 = 100 x 0.01 0.30 = 30 x 0.01 2.00 = 200 x 0.01 3.45 = 345 hundredths

0.23 x 0.15 0.2 + 0.03 0.1 + 0.05 0.0015 = 15 x 0.0001 0.0100 = 100 x 0.0001 0.0030 = 30 x 0.0001 0.0200 = 200 x 0.0001 0.0345 = 345 ten-thousandths

Table 6.6: Applying the Rule for Multiplying Decimals