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Weird Multiplication An Introduction to Various Multiplication Strategies Lynn West Bellevue, NE In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2011 Multiplication is one of the four basic operations of elementary arithmetic and is commonly defined as repeated addition. However, while this definition applies to whole number multiplication, some math researchers argue that it falls short for multiplication of fractions and other kinds of numbers. These mathematicians prefer to define multiplication as the scaling of one number by another, or as the process by which the product of two numbers is computed (Princeton University Wordnet, 2010). Despite the controversy, multiplication, by any definition, is an essential skill to students preparing for life in the mathematical world of the 21st century. It is an important tool in solving real-life problems and builds a firm foundation for proportional reasoning, algebraic thinking, and higher-level math. The standard algorithm for teaching the multiplication of larger numbers in this country is known as long multiplication and was originally brought to Europe by the Arabic-speaking people of Africa. In long multiplication, one multiplies the multiplicand by each digit of the multiplier and then adds up all the appropriately shifted results. This method requires memorization of the basic multiplication facts. However, a wide variety of efficient, alternative algorithms exist. Many students find these methods appealing and easier to navigate, even to the point of preferring them to the more traditional algorithm. Finger Multiplication Some of the oldest methods of multiplication involved finger calculations. One such method is believed to have come out of Italy and was widely used throughout medieval Europe (Rouse Ball, 1960 p. 189). The algorithm is fairly simple and can be used to calculate the product of two single digit numbers between five and nine. In order to use this method, one must understand that the closed fist represents five, and each raised finger adds one to that value. Thus, to determine the appropriate number of fingers to be raised on each hand, subtract five from each factor. For example, to find the product of 8 × 7, use the following steps: 1. Raise 3 fingers on the left hand; 8 - 5 = 3 2. Raise 2 fingers on the right hand; 7 - 5 = 2 3. Multiply each raised finger by 10; 5 × 10 = 50 4. Multiply the number of fingers in the down position on the left hand by the number of fingers in the down position on the right hand; 2 × 3 = 6 5. Add the two numbers; 50 + 6 = 56 × 8 7 Therefore, 8 × 7 = 56, but why does the algorithm work? Let’s rewrite the preceding process as an algebraic equation substituting x for eight and y for seven: 10[(x - 5) + (y - 5)] + [(10 - x)(10 - y)] Multiply the number of raised fingers (x-5 = 10x - 50 + 10y -50 + 100 -10x -10y + xy and y-5) by ten, and add the product of the = 10x - 50 + 10y -50 + 100 -10x -10y + xy fingers in the down position (10-x)(10-y). = xy Since all the terms cancel except for x and y, the equation gives the product of x and y. One advantage of this method is that it does not require memorization of multiplication facts beyond 5 × 5, so it is an effective tool for students who have not yet mastered the entire multiplication table. Introducing this algorithm to more advanced students offers them the opportunity to develop an understanding of the multiplication process and make connections to prior learning. Using mathematical reasoning to validate the algorithm fosters the development of conceptual understanding, an important component of proficiency (NCTM, 2000). Area Model of Multiplication All students need to be able to make connections between mathematical ideas and previously learned concepts in order to build new understandings. The area model of multiplication is an algorithm that uses multiple representations to explain the multiplication process, and can help students make connections to algebra and algebraic thinking. One can represent the multiplication of 14 × 12 as an area problem by drawing a rectangle with height 12 and width 14 on plain paper (or on graph paper to show the intermediate steps or in the case of multiplication by fractions). The area model is an application of the distributive property. 14 × 12 10 × 10 10 × 4 = [(10 + 2) × 10] + [(10 + 2) × 4] = 100 = 40 = (10 × 10) + (2 × 10) + (10 × 4) + (2 × 4) = 100 + 20 + 40 + 8 = 100 + 60 + 6 = 1= 2 × 10 = 20 2 × 4 = 8 The area model has theoretical limitations and cannot easily be used with irrational numbers; however, it is an excellent tool in helping students to establish a fundamental understanding of a variety of basic math concepts. For instance, it can be an effective tool in helping students to understand the concept of multiplication involving negative numbers. In the following example, we can represent the multiplication of 14 × 12 as (20 - 6) × (20 - 8): 400 -120 -160 48 This model clearly illustrates that the product of a positive number and a negative number is negative, while the product of two negative numbers must be positive. The area model also highlights the Distributive Property of Multiplication and using expanded notation. By allowing students to demonstrate graphically that 14 × 12 is the same as 12 × 14, the Commutative Property of Multiplication can be illustrated as well. In more advanced classes, this algorithm can be used to assist visual learners with the development of a relational understanding of polynomial multiplication and factoring. Lattice Multiplication Lattice multiplication, also known as sieve multiplication or the jalousia (gelosia) method, dates back to 10th century India and was introduced into Europe by Fibonacci in the 14th century (Carroll & Porter, 1998). It is algorithmically identical to the traditional long multiplication method, but breaks the process into smaller steps. For example, to multiply 453 × 25: 1. Draw a grid that has as many rows and columns as the multiplicand and the multiplier. 2. Draw a diagonal through each box from upper right corner to lower left corner. 3. Write the multipliers across the top and down the right side, lining up the digits with the boxes. 4. Record each partial product as a two-digit number with the tens digit in the upper left triangle and ones digit in the lower right triangle. (If the product does not have a tens digit, record a zero in the tens triangle.) 5. When all multiplications are complete, sum the numbers along the diagonals 6. Carry double digits to the next place, and record the answer. Therefore, 453 × 25 = 11,325. While some would argue that this algorithm ignores place value, it is easy to see that the diagonals actually represent the places that the digits occupy. Therefore, this multiplication essentially represents: (20 × 400) + (20 × 50) + (20 × 3) + (5 × 400) + (5 × 50) + (5 × 3) = 11,325 Multiplication of numbers beyond the single digits relies on three steps: multiplying, regrouping, and adding. The lattice method does each of these steps separately, so students are able to focus on the meaning of each part of the process. This method provides students with a structure for thinking about and recording their work. Lattice multiplication can also easily be extended to multiply decimal fractions and polynomials. Line Multiplication Another algorithm that is sometimes introduced to elementary school children is referred to as line multiplication. This algorithm presents students with a graphic representation of multiplication and can visually enhance their understanding of the multiplication process. Suppose you want to multiply 22 × 13: 1. First, draw two sets of vertical lines, two on the left and two on the right, to represent 22 (red lines). Next, draw two sets of horizontal lines, one on the top and three on the bottom, to represent 13 (blue lines). 2. Notice there are four sets of intersecting points (highlighted). To find the product, count up the intersection points in each of the highlighted sets and add diagonally. (Tanton, 2010) As with the traditional long multiplication algorithm, when a multiplication problem calls for regrouping, digits must be carried to the next place. Consider the multiplication problem 246 × 32. The answer, 6 thousands, 16 hundreds, 26 tens, and 12 ones is correct; however, numbers greater than or equal to 10 must be regrouped to write the answer in its standard form. Add the intersection points diagonally and then regroup numbers greater or equal to 10. 16,000 + 1,600 + 260 + 12 = 7,000 + 2600 + 260 + 12 = 7,000 + 800 + 160 + 12 = 7,000 + 800 + 70 + 2 = 7,872 (Tanton, 2010) Therefore, 246 × 32 = 7,872. This method works because diagonals of intersections of lines serve as placeholders (ones, 10’s, 100’s, etc.) and the number of points at each intersection represents the product of the number of lines. This is very similar to an area model of multiplication. When multiplying two two-digit numbers, as with 22 × 13, note that the problem can be rewritten as (20 + 2) × (10 + 3). 22 × 13 = (20 + 2) × (10 + 3) = (20 × 10) + (20 × 3) + (2 × 10) + (2 ×3) = 200 + 60 + 20 + 6 = 200 + 80 + 6 = 286 Note that the four sets of intersecting points are added diagonally. This serves to collect the points by place value (highlighted). This particular algorithm can be introduced to enhance the visual learner’s understanding of the multiplication process; however, all students should be exposed to a variety of strategies, models, and representations. Students should be able to explain the methods they use and understand there are multiple methods of efficiently solving any particular problem (NCTM, 2000).
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