An Introduction to Various Multiplication Strategies

Transcription

1 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

2 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 21 st 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.

3 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 = 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; = 56 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: [(x - 5) + (y - 5)] + [(10 - x)(10 - y)] = 10x y x -10y + xy = 10x y x -10y + xy = xy Multiply the number of raised fingers (x-5 and y-5) by ten, and add the product of the fingers in the down position (10-x)(10-y). 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

4 represent the multiplication of 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) = = 40 The area model is an application of the distributive property = [(10 + 2) 10] + [(10 + 2) 4] = (10 10) + (2 10) + (10 4) + (2 4) = = = 1= 2 10 = = 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 as (20-6) (20-8): 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

5 highlights the Distributive Property of Multiplication and using expanded notation. By allowing students to demonstrate graphically that 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 10 th 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 : 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, = 11,325.

6 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).

7 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 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. (Tanton, 2010) 1 6, , = 7, = 7, = 7, = 7,872 Therefore, = 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).

8 22 13 = (20 + 2) (10 + 3) = (20 10) + (20 3) + (2 10) + (2 3) = = = 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). Line multiplication could be introduced to students in the primary grades as one method of finding the product of two numbers. While line multiplication of extremely large numbers

9 would be rather cumbersome, for younger students and those who have not yet mastered the multiplication table, this method offers an opportunity for success with multiplying relatively large numbers. More advanced students can gain a better understanding of the multiplication process by investigating the algorithm and justifying the method mathematically. Circle/Radius Multiplication Another graphic multiplication algorithm, involves the drawing of concentric circles to represent the multiplier along with the drawing of radii to represent the multiplicand. For example, to multiply 3 4, first draw three concentric circles to model the multiplier three, and then add four radii to model the multiplicand four. Then count the number of separate pieces created within the circle. Since 12 pieces are created, 3 4 = 12. This algorithm is a little more complicated for larger numbers, as the next example shows. In order to calculate 21 34: 1. Draw two concentric circles to represent the two in the tens place of the multiplier (21), and duplicate the circles once for each digit in the multiplicand.

10 2. Draw a single circle to represent the one in the ones place of the multiplier; and also duplicate these circles once for each digit in the multiplicand. 3. To multiply by 34, draw three radii in the two circles on the left. 4. Then draw four radii in the two remaining circles. 5. Draw diagonal lines between the circles as shown, count the number of pieces in each section, and add diagonally. Therefore: = 714 This method works because the circles representing the digits in the multiplier are duplicated once for each digit in the multiplicand. This, in essence, allows for multiplication of each digit in the multiplier by each digit in the multiplicand. The diagonal lines serve to separate the digits into the appropriate place value.

11 Positive aspects of this method are that it can be used without knowledge of the multiplication table, it may appeal to visual learners, and it offers students another approach for multiplying, thus providing a deeper understanding of the multiplication process. However, drawing circles and radii to calculate products becomes more and more difficult as the size of the factors increases. Therefore, this algorithm could be introduced to students as one of several methods for multiplication, but they should not rely on this method alone. Paper Strip Multiplication Another way to perform the operation of multiplication involves writing the numbers to be multiplied on strips of paper, manipulating the order of the digits in one of the factors, and performing a series of one-digit multiplications. This alternative algorithm may appeal to handson learners. To multiply using this method: 1. First, write each factor on a strip of paper, reversing the order of the digits in one of the numbers ( ). 2. Place the strips of paper so that the inverted factor is above and to the right of the other factor. 3. Align the new first digit of the top number with the last digit of the bottom number, multiply, regroup the double digits, and write the product underneath. 4. Slide the bottom strip to right until the next set of digits line up, multiply the digits that are aligned, regroup the double digits, and write the sum underneath.

12 5. Continue sliding the bottom strip to the right, multiplying the aligned digits, regrouping the double digits, and writing the sum below the strips until all multiplications have been performed. 6. Therefore, = 271,296. But, how does reversing the order of the digits along with multiplying and adding pairs of numerals lead to the correct solution? Let's consider more closely what is actually happening as we carry out the steps of this algorithm. First, consider multiplying using the traditional long multiplication algorithm.

13 When the two methods are written side-by-side, it becomes apparent that they are actually the same algorithm with the multiplication steps carried out in a different order for each method. Because multiplication and addition are commutative, the order in which the steps are carried out makes no difference to the final product. Egyptian Multiplication Evidence of mathematical computation, including multiplication, dates back to about 2000 BC (O Connor & Robertson, 2011). Ancient Egyptian, Greek, Babylonian, Indian, and

14 Chinese civilizations are all credited with having methods for multiplying numbers. One of the earliest documented forms of multiplication dates back to Ancient Egypt and does not require memorization of the entire multiplication tables as it relies solely on the abilities to add and multiply by two. The following example of multiplying illustrates the Egyptian multiplication process. 1. First, make two columns of numbers, starting with 1 in the left column and one of the two factors (usually the greatest) in the right column. 2. Begin the process of decomposing the second factor into powers of 2 by doubling the numbers in both columns until the number in the left column represents the greatest power of 2 less than or equal to the second factor The numbers in the left column represent powers of 2. While the numbers on the right represent 46 these powers of 2. The greatest power of 2 28 is Decompose the second factor into powers of 2. The largest power of 2 less than or equal to 28 is = 12 The largest power of 2 less than or equal to 12 is = 4 The largest power of 2 less than or equal to 4 is 4. Thus, 28 is the sum of the powers of 2: Add the numbers in the right column that correspond to the powers of 2 indicated in Step = 1288 Therefore: = 1288

15 It is important to know why this method works. The algorithm is based on the distributive property of multiplication over addition and the ability to rewrite a product as a sum of powers of two. Since the numbers in the left hand column represent powers of two, the table can be modified as follows: 1 = = = = = Rewrite the problem as the sum of powers of 2: = ( ) 46 = (4 46) + (8 46) + (16 46) = (2 2 46) + (2 3 46) + (2 4 46) = = 1288 While the Egyptian multiplication method involves more steps than the long multiplication algorithm, its key advantage is users only need to know multiplication facts for two. It can also be used as a form of scaffolding for students who have not yet mastered all of their basic facts. This Algorithm could also be introduced in the classroom as a means of leading students into a discussion of the meaning of multiplication, the process, and why it works. For example, students could compare this multiplication method to the more traditional long multiplication method and then explain how both procedures lead to the correct answer. Introducing students to multiple algorithms also promotes conceptual understanding, and according to noted mathematics scholar Laping Ma (1999), when a problem is solved in multiple ways, it serves as a tie connecting several pieces of mathematical knowledge (p. 112). Russian Peasant Multiplication A variation of this Egyptian algorithm has also been linked to the peasants of early Russia and is still in use in some areas today (Bogomolny 2011). This method involves a process of halving and doubling, which reduces one factor to powers of two and uses the distributive

16 property of multiplication over addition to calculate a product. As with the Egyptian method, the process begins by arranging numbers in two columns. The following example of multiplying illustrates the Russian peasant multiplication process. 1. Create a column beneath each of the factors. 2. Repeatedly halve the number in the left hand column, dropping any remainder, until one is reached. 3. Repeatedly multiply numbers in the right hand column by two. 4. Cross off the rows that have an even number in the left hand column. 5. Add the remaining numbers in the right hand column to find the product = 1288 Therefore, = 1288 The justification for this algorithm is almost identical to the rationalization for the Egyptian method; however, let's look at it from a slightly different point of view. The fact that both of these procedures are dependent on multiplication/division by two, suggests that they are founded in the binary system. First, obtain the binary representation of 46 by following the subsequent procedure: 1. List the powers of two in a base two table from right to left starting at 2 0, and increase the exponent by one for each power. 2. Find the greatest power of two that fits into 46. Since 32 fits into 46, write a one for the leftmost binary digit. Subtract 32 from 46, which leaves Find the greatest power of two that fits into 14. Because 16 does not fit into 14, write a zero for the second binary digit.

17 4. Since eight fits into 14, write a one for the third binary digit. Subtract eight from 14, which leaves six. 5. Find the greatest power of two that fits into six. Since four fits into six, write a one for the fourth binary digit. Subtract four from six, which leaves two. 6. Find the greatest power of two that fits into two. Since two (a power of two) fits into two (the working decimal number), write one for the next binary digit and subtract two from two, which leaves zero. 7. Since no additional powers of two will fit into zero, write zero for the remaining binary digit. Power of Decimal Equivalent (32) (16) (8) (4) (2) (1) Binary Representation Next, add two columns to the original table; the first shows the exponent of two that gives the number and the second represents the binary digits of 46 written in reverse order Adding two additional columns clearly illustrates the relationship between this algorithm and the binary system: 46 = = (1 2 5 ) + (0 2 4 ) + (1 2 3 ) + (1 2 2 ) + (1 2 1 ) + (0 2 0 ) = Because the binary digit represents the remainder in division by powers of two, one corresponds to odd numbers in the column under 46 and zero corresponds to even numbers, thus only the rows with odd numbers will contribute to the multiplication. One corresponds to the odd numbers in the first column and zero corresponds to the even numbers in the first column. Thus only the columns with odd numbers in the first row will contribute to the multiplication.

18 Vedic Multiplication Vedic mathematics has its roots in the Vedas, which are ancient Indian texts first written in Sanskrit and thought to have originated around 2000 BC. Spiritual leader and mathematician, Sri Bharati Krsna Tirthaji reconstructed 16 sutras, or fundamental principles, derived from these ancient texts. The sutras, along with 13 sub-sutras, are the basis for the Vedic mathematics system (Nataraj & Thomas, 2006). Vedic multiplication is an efficient and simple form of mental calculation. For example to multiply using the vertically and crosswise sutra: 1. Multiply vertically on the right to find the units digit. 2. Multiply crosswise and add: (2 2) + (4 3) = 16. This gives the tens digit of the answer. (Notice that the one must be carried to the hundreds place.) 3. Multiply vertically on the left, and add the amount carried to find the hundreds digit. Therefore, = 768. This algorithm could be introduced in the classroom as a tool to support mental multiplication. When problems require multiple steps, solving them can become tedious, and students can easily make mistakes. Middle school and high school students would

19 especially appreciate this simple short cut to finding products, and they may even find the ability to calculate mentally, empowering. Let's take a deeper look at how this algorithm works. First of all, use the Distributive Property of Multiplication to rewrite the same problem: = (20 + 4)(30 + 2) = (20 30) + (20 2) + (4 30) + (4 2) = = = = 768 Notice how the highlighted portion of the problem corresponds to the vertical and crosswise algorithm. (4 2) (20 2) + (4 30) (20 30) This algorithm is based on the Distributive Property of Multiplication. Writing the factors in columns, along with adding the intermediary place holders between the first and last digit of the answer, assigns each digit the correct place value. To multiply larger numbers, one follows the same basic steps adding two more sets of crosswise multiplication using the following pattern:

20 For example, to multiply multiply and add as shown, regrouping values greater than 9 in the last step. Therefore, = Conclusion Multiplication is a basic mathematical skill, and understanding the process and its many applications is of fundamental importance to the future success of today s students. Education has grown beyond the point where all students were expected to learn in the same way and by the same instructional methods. Contemporary educators must be prepared to meet the widely varied and individual educational needs of each of the students that enter the classroom. Research has shown that when children are introduced to a variety of problem solving methods and strategies, they become more flexible and resourceful in their problem solving abilities (NCTM, 2000). As students gain knowledge of the history of the development of mathematical ideas, they are more likely to view mathematics as a discipline that continues to evolve as people look for faster and more efficient means of calculation in the quest to solve increasingly complicated problems.

21 References Bogomolny, A. (2011). Peasant multiplication. Retrieved June 10, 2011, from Carroll, W. M., & Porter, D. (1998). Alternative algorithms for whole-number operations. In The National Council of Teachers of Mathematics, The teaching and learning of algorithms in school mathematics (pp ). Reston, VA: The National Council of Teachers of Mathematics, Inc. Gray, E. D. (2001). Cajun multiplication: A history, description, and algebraic verification of a peasant algorithm. Louisiana Association of Teachers of Mathematics Journal, 1(1), article 6. Retrieved fromhttp:// Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates, Inc. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Nugent, P. M. (2007, September). Lattice multiplication in a preservice classroom. Mathematics Teaching in the Middle School, 13(2), O'Connor, J. J. & Robertson, E. F. (2011). An overview of the history of mathematics. In The MacTutor history of mathematics archive. Retrieved June 9, 2011, from School of Mathematics and Statistics, University of St Andrew Scotland website:

22 Princeton University "About WordNet" (2010). Retrieved June 9, 2011, from Rouse Ball, W. W. (1960). A short account of the history of mathematics (4th ed.). Mineola, NY: Dover Publications, Inc. (Original work published 1908) Rubenstein, R. N. (1998). Historical algorithm. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (pp ). Reston, VA: The National Council of Teachers of Mathematics, Inc. Saraswathy Nataraj, M., & Thomas, M. O. (2006). Expansion of binomials and factorisation of quadratic expressions: Exploring a Vedic method. Australian Senior Mathematics Journal, 20(2), Sgroi, L. (1998). An explanation of the Russian peasant method of multiplication. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (pp ). Reston, VA: The National Council of Teachers of Mathematics, Inc.