204 CHAPTER 4 Systems of Numeration Lattice Multiplication Another method of multiplication is lattice multiplication . This method’s name comes from the use of a grid, or lattice, when multiplying two numbers. It is also known as the gelosia method. The lattice method was brought to Europe in the 1200s by Fibonacci, who likely learned it from the Egyptians, Arabs, or Hindus (see the Profile in Mathematics on page 283). Continue this process, dividing the number in the left column by 2, disregarding the remainder, and doubling the number in the right-hand column, as shown below. When a 1 appears in the left-hand column, stop. 39—23 19—46 9—92 4—184 2—368 1—736 Next, cross out all the even numbers in the left-hand column and the corresponding numbers in the right-hand column. 39—23 19—46 9—92 4—184 2—368 1—736 Now add the remaining numbers in the right-hand column to obtain 23 46 + + 92 736 897. + = The number obtained, 897, is the product of 39 23. × Thus, 39 23 897. × = 7 Now try Exercise 5 MATHEMATICS TODAY Pressmaster/Shutterstock The New Old Math Sometimes the new way to solve a problem is to use an old method. Such is the case in some elementary schools with regard to multiplying whole numbers. In addition to teaching traditional multiplication, some schools are also teaching the lattice method of multiplication. Lattice multiplication, along with the Hindu– Arabic numerals, was introduced to Europe in the thirteenth century by Fibonacci. (See the Profile in Mathematics on page 283.) Lattice multiplication is recommended by the National Council of Teachers of Mathematics as a way to help students increase their conceptual understanding as they are developing computational ability. Why This Is Important By learning lattice multiplication, we can better understand how the traditional multiplication algorithm works. Example 2 Using Lattice Multiplication Multiply 312 75 × using lattice multiplication. Solution To multiply 312 75 × using lattice multiplication, first construct a rectangle consisting of three columns (one for each digit of 312) and two rows (one for each digit of 75). Place the digits 3, 1, 2 above the boxes and the digits 7, 5 on the right of the boxes. Then place a diagonal in each box, as shown in Fig. 4.1 below. Complete each box by multiplying the number on top of the box by the number to the right of the box (Fig. 4.2). Place the units digit of the product below the diagonal and the tens digit of the product above the diagonal. 1 2 3 7 5 Figure 4.1 1 1 1 1 1 2 2 3 7 7 5 5 5 0 0 0 4 Figure 4.2
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