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Algorithms for Multiplication and Division

Algorithms for Multiplication and Division.

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Algorithms for Multiplication and Division

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  1. Algorithms for Multiplication and Division

  2. In reality, no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics. Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding Everybody Counts National Research Council, 1989

  3. How has this student misapplied the rules for multiplying? Based upon the work above, what understandings and misunderstandings does this student have?

  4. Multiplication and DivisionWhat are the goals for students? • Develop conceptual understanding • Develop computational fluency

  5. Multiplication • Teaching multiplication to kids can be less challenging when you relate it to a skill they already have, such as addition. • Students who learn a variety of algorithms and possibly who are even given a chance to invent their own will develop into powerful users of numbers.

  6. Multiply 4 x 23 • 4 groups of 23 will be • Now there are 8 longs and 12 ones • Regroup: 9 longs and 2 ones for 92

  7. Partial Products Algorithm 4 7 x 1 3 2 1 (7 x 3)‏ 1 2 0 (40 x 3)‏ 7 0 (7 x 10)‏ +4 0 0(40 x 10)‏ 6 1 1 • Similar to the partial sums algorithm for addition. The procedure is to multiply one pair of digits at a time. • Note that with this algorithm it does not matter the order in which digits are multiplied. (commutative property)‏ Use the Partial Products Algorithm to show 124 × 135 = 16,740

  8. Standard Multiplication Algorithm • This is basically an abbreviation of the partial products algorithm 4 7 x 1 3 1 4 1 (47 x 3)‏ +4 7 0(47 x 10)‏ 6 1 1 Use the Standard Multiplication Algorithm to show 124 × 135 = 16,740

  9. Multiplicative Thinking Multiplication is more complex than addition because the two numbers (factors) in the problem take different roles. 12 cars with 4 wheels each. How many wheels? 12 x 4 = 48 cars wheels/car wheels (groups) (items per group) (total number of items)‏ (multiplier) (multiplicand) (product)‏

  10. Multiplication Strategies12 cars with 4 wheels each. How many wheels? Additive Strategies • Direct Modeling • Repeated Addition • Doubling

  11. Multi-digit Multiplication Strategies52 cards per deck. 18 decks of cards. How many cards? Multiplicative Strategies • Single Number Partitioning • Both Number Partitioning • Compensating

  12. Multiplication StrategiesAs you look at student work, try to identify the kinds of strategies you see students using. While this list is not comprehensive, it will give you a place to begin. Often you will see evidence of more than one strategy being used. Multiplicative Strategies • Single Number Partitioning • Both Number Partitioning • Compensating Additive Strategies • Direct Modeling • Repeated Addition • Doubling

  13. Additional Multiplication Algorithms • Lattice Method • Russian Peasant Method • Egyptian Method

  14. Lattice Multiplication Algorithm • This is basically the partial products algorithm recorded in a different format. • Multiply row by column • Sum the diagonals 47 × 13 = 611 Use the Lattice Multiplication Algorithm to show 124 × 135 = 16,740

  15. Half Double 47 13 23 26 11 52 5 104 2 208 1 416 Russian Peasant Multiplication • The procedure is to create two lists by taking half the first factor and double the second factor (dropping the remainder each time) until the value of the column for the first factor is one. • Then, cross out the terms in the second column that correspond to the values in the first column that are even. • Finally, add the remaining values in the second column. 47 × 13 611

  16. Count Double 1 47 √ 2 94 4 188 √ 8 376 √ Egyptian Multiplication • Start with 1 and a number of the multiplication (47)‏ • Then we double each number and write the results under the originals. Proceed till the counting column exceeds the other multiplication number (13)‏ • At this point we start down the left side looking for a total of the other number (13). • Each time we can add the number without exceeding our goal of 13, we put a check mark by the number opposite • Sum the values in the double column 47 × 13 47 + 188 + 376 = 611

  17. Teacher’s Role • Provide rich problems to build understanding • Encourage the use of “thinking tools” (manipulatives) when needed • Guide student thinking • Provide multiple opportunities for students to share strategies • Help students complete their approximations • Model ways of recording strategies • Press students toward more efficient strategies

  18. Division StrategiesThe strategies students use for division will be very similar to those they used for multiplication. As you look at student work, try to identify the kinds of strategies you see students using. This is not a comprehensive list, and often you will see evidence of more than one strategy being used. Here is an example of a division problem. • Janet has 1,780 marbles. She wants to put them into bags, each of which holds 32 marbles. How many full bags of marbles will she have?

  19. Samantha solved this problem by multiplying groups of 32 to reach 1,780. Samantha’s solution: 1,760 is as close as she can get to 1,780 using groups of 32. 1,780 ÷ 32 = 55 R20 Janet can fill 55 bags, and she will have 20 extra marbles.

  20. Talisha solved this problem by subtracting groups of 32 from 1,780. Talisha’s solution:

  21. Here is another division example. Dana solved this problem by subtracting groups of 54 from 2,500.

  22. Walter solved this problem by multiplying groups of 54 to reach 2,500.

  23. Direct Model • You can use objects to help you think about division. • You have 12 cookies • Think of division as sharing. Suppose you are sharing 12 cookies with 3 friends. How many cookies would each person receive?

  24. Repeated Subtraction Algorithm • The procedure is to subtract the divisor repeatedly from the dividend, then the quotient is the number of times the divisor was subtracted. • The algorithm is easy to apply, but the process may take a lot of steps 84 ÷ 21 84 – 21 1 63 – 21 1 42 – 21 1 21 – 21 1 0 4 Thus, 84 ÷ 21 = 4

  25. Scaffold Algorithm • This is a more efficient version of repeated subtraction. The procedure is to subtract multiples of the divisor. • Note that the multiple chosen maybe any number that is less than the dividend. 170 ÷ 14 170 – 140 10 30 – 28 2 2 12 84 ÷ 21 84 – 42 2 42 – 42 2 0 4 Thus, 84 ÷ 21 = 4 Thus, 170 ÷ 14 = 12 Remainder = 2

  26. Scaffold Algorithm There are many advantages of using scaffolding: • It's fun and it makes sense. • It develops estimation skills. • Students are engaged in mental arithmetic – they are thinking throughout the process, not just following an algorithm. • Students develop number sense. • The more number sense that students possess, the more efficient the process. • There are many correct ways to arrive at a solution. • There are fewer opportunities for error than with long division. • Students who practice scaffolding are better able todivide mentally.

  27. Long Division • Long division, which is used to divide numbers of more than one digit, is really just a series of simple division, multiplication, and subtraction problems. The number that you divide is called the dividend. The number you divide the dividend by is the divisor. The answer to a division problem is called a quotient. take a lot of steps Divide 564 by 12 The quotient is 47

  28. Teaching Division • Although division can be a confusing concept for many students, the more simply it is taught, the easier it will be. • Make sure that your students understand the concept of basic division before moving on to long division. • Almost all math becomes easier to master for any student when they can see a relationship between the math and their own life.

  29. INTEGERS AND MULTIPLICATION

  30. MULTIPLICATION • Red and yellow tiles can be used to model multiplication. • Remember that multiplication can be described as repeated addition. • So 2 x 3 = ? 2 groups of 3 tiles = 6 tiles

  31. MULTIPLICATION • 2 x -3 means 2 groups of -3 2 x -3 = -6

  32. MULTIPLICATION • -2 x +4 = ? 4 groups of -2 -2 x +4 = -8 Use the fact family for -2 x +4 = ?  We can’t show -2 groups of +4 +4 x -2 = ? we can show 4 groups of -2

  33. MULTIPLICATION • +1, -1 are opposites the products are opposite • Since +2 and -2 are opposites of each other, • +2 x -3 and -2 x -3 have opposite products. +1 x +3 = +3 -1 x +3 = -3

  34. MULTIPLICATION • To model -2 x -3 use 2 groups of the opposite of -3 -2 x -3 = +6

  35. INTEGERS AND DIVISION The University of Texas at Dallas

  36. DIVISION • Use tiles to model +12 ÷ +3 = ? 4 yellow tiles in each group. Divide 12 yellow tiles into 3 equal groups +12 ÷+3 = +4

  37. DIVISION • Use tiles to model -15 ÷ +5 =? Divide -15 into 5 equal groups -15 ÷+5 = -3

  38. Operating With Fractions • Meaning of the denominator (number of equal-sized pieces into which the whole has been cut); • Meaning of the numerator (how many pieces are being considered); • The more pieces a whole is divided into, the smaller the size of the pieces; • Fractions aren’t just between zero and one, they live between all the numbers on the number line; • Understand the meanings for operations for whole numbers.

  39. A Context for Fraction Multiplication • Nadine is baking brownies. In her family, some people like their brownies frosted without walnuts, others like them frosted with walnuts, and some just like them plain. So Nadine frosts 3/4 of her batch of brownies and puts walnuts on 2/3 of the frosted part. How much of her batch of brownies has both frosting and walnuts?

  40. Multiplication of Fractions Consider: • How do you think a child might solve each of these? • Do both representations mean exactly the same thing to children? • What kinds of reasoning and/or models might they use to make sense of each of these problems? • Which one best represents Nadine’s brownie problem?

  41. Models for Reasoning About Multiplication • Fraction of a fraction • Linear/measurement • Area/measurement models • Cross Shading

  42. 2 3 6 12 How much is of ? 3 4 3 3 4 4 We will think of multiplying fractions as finding a fraction of another fraction. We use a fraction square to represent the fraction . Then, we shade of We can see that it is the same as .

  43. How much is of ? The Linear Model with multiplication utilizes the number line and partitions the fractions

  44. How much is of ? We can also use the linear model with shapes and partition accordingly Break into 3 pieces Identify ¾ of the circle Answer is ½ Take 2 pieces

  45. 3 4 How much is of ? 2 3 2 6 3 4 12 3 In the third method, we will think of multiplying fractions as multiplying a length times a length to get an area. Length is Area Width is Number of square units Is 6 out of 12 This area is X =

  46. Modeling multiplication of fractions using the length times length equals area approach requires that the children understand how to find the area of a rectangle. A great advantage to this approach is that the area model is consistently used for multiplication of whole numbers and decimals. Its use for fractions, then is merely an extension of previous experience.

  47. How much is of ? 2 2 3 3 3 3 is 4 4 In the fourth method, we will represent both fractions on the same square. is

  48. Modeling multiplication of fractions using the cross shading approach does produce correct answers. However, many elementary students may not grasp the “because it is shaded in both directions” overlapping concept. This may require some additional explanations

  49. Classroom Problem • Eric and his mom are making cupcakes. Each cupcake gets 1/4 of a cup of frosting. They are making 20 cupcakes. How much frosting do they need?

  50. 1/4 of a cup Sample children’s strategies 5 cups 4 cups 1 cup 2 cups 3 cups “…so 5 cups altogether.”

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