Algorithms for Multiplication and Division

In reality, no one can teach mathematics 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 2

How has this student misapplied the rules for multiplying? Based upon the work above, what understandings and misunderstandings does this student have? Ask participants to discuss these two questions with a partner. Then have them form a group of 4 and discuss again. Finally, have several groups share with the whole group. Some ideas that are likely to come out: When applying the steps to the algorithm, the student added the 4 to the 1 before multiplying it by 6 – thus came up with 30 instead of 10. The student does seem to know multiples of 6 – at least through skip counting. The student is focused on the procedure of multiplying and not on the meaning. 308 isn’t a reasonable answer – if the problem were 20 x 6, the answer would be 120, so the answer to this problem must be less than 120. The student doesn’t have a clear understanding of the why behind the steps in the algorithm. 3

Multiplication and Division What are the goals for students? Develop conceptual understanding Develop computational fluency Our goals for students need to be more than just memorizing procedures. Students first need to develop strategies that make sense to them. From there, we can help them develop increasingly efficient strategies as they work toward computational fluency. 4

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.

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 Base 10 blocks helps visualize the repeated addition process 6

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 7

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 8

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)‏ To make sense of a multiplication problem, children have to understand what each number represents and be able to think of numbers in units. Unitizing means that students understand the concept of counting not only objects (in this case, wheels), but also groups of objects (in this case, cars). In addition, they must be able to keep track of both simultaneously. For example, when thinking about 12 x 4, students must be clear that one number stands for the number of groups and the other number stands for how many are in each group. It is important that this “big idea” is solid if students are to understand the process in multi-digit multiplication. 9

Multiplication Strategies 12 cars with 4 wheels each. How many wheels? Additive Strategies Direct Modeling Repeated Addition Doubling When given the opportunity to develop strategies for multiplying multi- digit number, most students will begin with Additive Strategies. Additive strategies for multi-digit multiplication are not the most efficient strategies, but they do help students develop an understanding of the concept of multi-digit multiplication. If you have students who struggle with multiplication clusters, then they probably are still at the point where they need to use additive strategies to build an understanding of the operation. Here are some examples of additive strategies related to the problem 12 cars, how many wheels? CLICK Direct modeling involves representing all of the parts of the problem and counting them out. In this example, the child drew all of the cars and then labeled the wheels with counting numbers. CLICK This example is a mixture of direct modeling and repeated addition. Although the cars aren’t drawn out, each 4 represents a car and the dots next to the 4s seem to indicate that the child counted them all. However, the child did use the notation of repeated addition. CLICK This example shows repeated addition. The child adds four more onto the sum each time. The dots next to the 4s seem to indicate that when he got to 32 the child counted to see how many 4s were on the paper. Then the child counted again at the end to be sure there were 12 cars represented. CLICK This is a slightly more efficient strategy than repeated addition. In this case the child represented a 4 for each car, and then used doubling to figure out how many wheels. On the last step, the child added 32 and 16 to get 48. CLICK Here is a more sophisticated doubling strategy. On the left side, the child is keeping track of the number of wheels and on the right side, she is keeping track of the number of cars that are represented. When she gets to 32, the child realized that she can’t just double 32 and instead looks for the value of 4 cars to bring the total to 12 cars and 48 wheels. Though this is a doubling strategy, it uses multiplicative thinking and shows an informal understanding of the distributive property – the child was thinking in terms of groups (cars) and was able to decompose 12 cars into 8 cars and 4 cars. 10

Multi-digit Multiplication Strategies 52 cards per deck Multi-digit Multiplication Strategies 52 cards per deck. 18 decks of cards. How many cards? Multiplicative Strategies Single Number Partitioning Both Number Partitioning Compensating Multiplicative strategies are built upon the “big idea” of partitioning. Partitioning means that students are able to break apart numbers into more useful chunks. For example, they might think of 52 as 50 and 2 or they might think of 18 as 10 and 8. An understanding of the distributive property is critical for partitioning in multiplication. Let’s look at some examples. CLICK In this example, the student left the number 52 intact and has partitioned or broken apart the number 18 into 10 + 4 + 4. This child started with 52 x 10. He used doubling to figure out that four 52s would equal 208. He then added 208 + 208 to get the total of eight 52s (416). He then added ten 52s plus eight 52s to get the total of 18 52s. This child clearly understand the distributive property. CLICK In this example, the child left 18 intact and partitioned or broke apart the number 52 into 25 + 25 + 2. The drawing at the right indicated that she was thinking about money when she did this and used the knowledge that 4 quarters (or 25s) is equal to a dollar (or 100). She used that information to figure out that 18 25s (or 25 18s) would be 450. She then doubled that to determine that 18 50s (or 50 18s). At this point the student first thought that they needed to add two more 52s (which is why the 104 is written and then crossed out). This is a common mistake – the child knows that they need 2 more of something, but they aren’t sure of what. This child does correct herself and realizes that she needs two more 18s not two more 52s. She then adds together the parts to get the total of 52 x 18. The way this child solved the problem does not logically follow the structure of the problem (18 deck with 52 cards in each deck), but it does indicate an understanding of the commutative property – the child understands that you can think of this problem either as 52 eighteens or as 18 fifty-twos. CLICK In this problem the child partitions both numbers – 52 into 50 + 2 and 18 into 10 + 8. The drawing is very similar to an open array, which is a very powerful model for visually showing multiplicative strategies. We’ll look at these some more later. It is very simple, yet very effective – and though it doesn’t follow the same steps, conceptually, it is exactly what is done in the standard algorithm for multiplication. CLICK This is an interesting example of both partitioning and compensating. The child keeps the number 52 intact and uses a table to figure out how many cards there are in one deck, ten decks, 20 decks, etc. In addition, he also uses compensating. Because 52 x 20 is easier than 52 x 18, he finds the answer to 52 x 20 and then subtracts two 52s. When he subtracts, rather than using a standard subtraction algorithm, he breaks the 104 apart – subtracting first 100 and then 4. 11

Multiplication Strategies As 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 12

Additional Multiplication Algorithms Lattice Method Russian Peasant Method Egyptian Method 13

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 14

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 Half Double 47 13 23 26 11 52 5 104 2 208 1 416 611 15

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 Count Double 1 47 √ 2 94 4 188 8 376 47 + 188 + 376 = 611 16

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 Because the focus of instruction has shifted away from procedures toward developing concepts, our role and the ways we know we’re doing a good job have to change. We still need to provide manipulatives and models like base-ten pieces, snap cubes and hundreds charts, but instead of modeling the process for students, we guide their thinking. Rather than showing the steps, we provide many opportunities for students to do their own thinking and share their strategies. As students are trying to make sense of multiplying or dividing larger numbers, we need to recognize the direction of their thinking and help them complete their approximations of a strategy. One of our most important roles is to help students learn mathematically accurate ways of recording their thinking. And finally, when we see that students are ready, it is our job to gently press them toward using more efficient strategies. If you are presenting this workshop in two sessions, stop here on day one. If you have time, have teachers go back to samples 3, 4 and 5 and try to model the student’s thinking using an open array. You can also go back to slide 9 and try modeling those examples using an open array. 17

Division Strategies The 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? The strategies students use for solving division problems will look much like those used with multiplication problems. When given the opportunity to develop strategies for dividing multi-digit numbers, most students will begin with Additive Strategies. As you look through the examples try to identify the kinds of strategies you see students using. 18

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.

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

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

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

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? 23

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 42 21 0 4 Thus, 84 ÷ 21 = 4 24

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. 84 ÷ 21 84 – 42 2 42 0 4 170 ÷ 14 170 – 140 10 30 – 28 2 2 12 Thus, 84 ÷ 21 = 4 Thus, 170 ÷ 14 = 12 Remainder = 2 25

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 to divide mentally.

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 27

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. We realizing that many teachers (and parents) feel compelled to have students do multiplication and/or division a standard way. If students are given time to develop a deep understanding of multiplication and division, most students will be able to make sense of a partial product method as a standard ways of multiplying and dividing by 5th grade. The real key is to delay “teaching” these methods until students are already using multiplicative strategies. It is also important not to force student to use these methods if they can’t make sense of them – these students still need time to develop understanding. 28

INTEGERS AND MULTIPLICATION

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

MULTIPLICATION 2 x -3 means 2 groups of -3 2 x -3 = -6

MULTIPLICATION -2 x +4 = -8 -2 x +4 = ? 4 groups of -2 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

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

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

INTEGERS AND DIVISION The University of Texas at Dallas

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

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

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.

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?

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?

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

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

The Linear Model with multiplication utilizes the number line and partitions the fractions How much is of ? Make another with 3/4 x 2/3

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

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

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.

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

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

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?

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

Another student strategy 1/4 of a cup So, 5, 6, 7, 8 -- that’s 2 cups. 9, 10, 11, 12 -- that’s 3 cups. 13, 14, 15, 16 -- that’s 4 cups. 17, 18, 19, 20 -- that’s 5 cups. 4 of these is 1 cup… …so 5 cups altogether.

Another student strategy 1/4 + 1/4 + 1/4 + 1/4 = 1 5 cups Q: What’s a number sentence for this problem? A: 20 x 1/4 = 5 (there are others)

Other Contexts for Multiplication of Fractions Finding part of a part (a reason why multiplication doesn’t always make things “bigger”) Pizza (pepperoni on ⅓ of ½ pizza) Recipes ( 1¾ cups of sugar is used but we want to make ½ a batch) Ribbon (you have ⅜ yd , ⅓ of the ribbon is used to make a bow)

Division With Fractions

Division with Fractions Sharing meaning for division: 1 One shared by one-third of a group? How many in the whole group? How does this work? Sh

Division With Fractions Repeated subtraction / measurement meaning 1 How many times can one-third be subtracted from one? How many one-thirds are contained in one? How does this work? How might you deal with anything that’s left? Sh - Which one do YOU connect with (RS or Sharing)

Division of Fractions examples How many quarters are in a dollar? Ground beef cost 2.80 for ½ pound. What is the price per pound? Maggie can walk the 2 ½ miles to school in 3/4 of an hour. How long would it take to walk 4 miles? Barb had ¾ of a pizza left over from her party. She wants to store it in plastic containers. Each container holds ⅓ of a pizza. How many containers will she use? How many will be completely full? How full will the last container be? Sh - Which one do YOU connect with (RS or Sharing)

Division of Fractions examples You have 1 cups of sugar. It takes cup to make 1 batch of cookies. How many batches of cookies can you make? How many cups of sugar are left? How many batches of cookies could be made with the sugar that’s left?

“How many one eighths are in three fourths?” Dividing Fractions “How many one eighths are in three fourths?” Our pizza is cut into 8 pieces. If three fourths of a pizza is left, how many slices remain? Recall: a slice represents one eighth of the pizza

Pizza How many one eighths are in three fourths? To find this we must first find 3/4 of the pizza. We then cut each fourth into halves to make eighths. We can see there are 6 eighths in three fourths.

Pizza Now only half of the pizza is left. How many slices remain? How many one eighths are in one half? Pizza Using a fraction manipulative, we show one half of a circle. To find how many one eighths are in one half, we cover the one half with eighths and count how many we use. We find there are 4. There are four one eighths in one half.