Algorithms for Multiplication and Division

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

How has this student misapplied the rules for multiplying?

Based upon the work above, what understandings and misunderstandings does this student have?

Multiplication and DivisionWhat are the goals for students?

Develop conceptual understandingDevelop computational fluency

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

Partial Products Algorithm

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)

4 7

x 1 32 1(7 x 3)

1 2 0 (40 x 3)

7 0(7 x 10)

+4 0 0 (40 x 10)

6 1 1Use the Partial Products Algorithm to show

124 × 135 = 16,740

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

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)

Multiplication Strategies12 cars with 4 wheels each. How many wheels?

Additive Strategies

Direct Modeling Repeated Addition Doubling

Multi-digit Multiplication Strategies52 cards per deck. 18 decks of cards. How many cards?

Multiplicative Strategies

Single Number Partitioning Both Number Partitioning Compensating

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.

Additive Strategies

Direct Modeling Repeated Addition Doubling

Multiplicative Strategies

Single Number Partitioning Both Number Partitioning Compensating

Additional Multiplication Algorithms

Lattice MethodRussian Peasant MethodEgyptian Method

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

0

7 0

7

4

4

2 1

1 3 2

1 0

6 1 1 Use the Lattice Multiplication

Algorithm to show 124 × 135 = 16,740

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.

Half Double

47 13

23 26

11 52

5 104

2 208

1 416

47 × 13

611

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

Count Double

1 47 √

2 94

4 188 √

8 376 √

47 × 13

47 + 188 + 376 = 611

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

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.

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?

Here is an example of a division problem.

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?

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 4Thus, 84 ÷ 21

= 4

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– 42 2

0 4

Thus, 84 ÷ 21

= 4

170 ÷ 14

170– 140 10

30– 28 2

2 12

Thus, 170 ÷ 14 =

12Remainder

= 2

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.

Scaffold Algorithm

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

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.

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 = ?

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

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

Use tiles to model +12 ÷ +3 = ?

Divide 12 yellow tiles into 3 equal groups

+12 ÷ +3 = +4

4 yellow tiles in each group.

DIVISION 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 FractionsConsider:

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?

2

3

3

4

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.

34

We use a fraction square to represent the fraction .3

4Then, we shade of

We can see that it is the same as .

23

34

612

2

3

3

4

1

2

How much is of ?

2

3

3

4

The Linear Model with multiplication utilizes the number line and partitions the fractions

4

3

3

2of

4

3

3

1of 4

3

3

3of

How much is of ?

2

3

3

4

4

4

1

4

2

4

3

4

1

2

0

2

3

3

4

1

2

3

4

We can also use the linear model with shapes and partition accordingly

How much is of ?

2

3

3

4

Identify ¾ of the circleBreak into 3 pieces

Take 2 pieces

2

3

3

4

1

2

Answer is ½

In the third method, we will think of multiplying fractions as multiplying a length times a length to get an area.

23

34

This area is X = 612

Width is

Length is

Area

Number of square units

Is 6 out of 12

3 4

2 3

2

3

3

4

1

2

How much is of ?

2

3

3

4

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.

34is

23is

How much is of ?

2

3

3

423

34

2

3

3

4

1

2

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

1 cup 2 cups 3 cups4 cups 5 cups

“…so 5 cups altogether.”

Another student strategy

1/4 of a cupSo, 5, 6, 7, 8 -- that’s 2 cups.

…so 5 cups altogether.

9, 10, 11, 12 -- that’s 3 cups.

17, 18, 19, 20 -- that’s 5 cups.

13, 14, 15, 16 -- that’s 4 cups.

4 of these is 1 cup…

Another student strategy

1/4 + 1/4 + 1/4 + 1/4 = 1

1/4 + 1/4 + 1/4 + 1/4 = 1

1/4 + 1/4 + 1/4 + 1/4 = 1

1/4 + 1/4 + 1/4 + 1/4 = 1

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 FractionsDivision 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?

13

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?

13

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?

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?

1

2

1

3

“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?

?8

1

4

3

Recall: a slice represents one eighth of the 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.

?8

1

4

3

Pizza

68

1

4

3

Now only half of the pizza is left. How many slices remain?

How many one eighths are in one half?

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.

Pizza

We find there are 4. There are four one eighths in one half.

48

1

2

1

?8

1

2

1