algorithms for multiplication and division. in reality, no one can teach mathematics. effective...
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Algorithms for Multiplication and Division
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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
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How has this student misapplied the rules for multiplying?
Based upon the work above, what understandings and misunderstandings does this student have?
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Multiplication and DivisionWhat are the goals for students?
Develop conceptual understandingDevelop computational fluency
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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.
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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
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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
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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
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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)
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Multiplication Strategies12 cars with 4 wheels each. How many wheels?
Additive Strategies
Direct Modeling Repeated Addition Doubling
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Multi-digit Multiplication Strategies52 cards per deck. 18 decks of cards. How many cards?
Multiplicative Strategies
Single Number Partitioning Both Number Partitioning Compensating
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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
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Additional Multiplication Algorithms
Lattice MethodRussian Peasant MethodEgyptian Method
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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
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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
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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
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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
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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.
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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.
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Talisha solved this problem by subtracting groups of 32 from 1,780.
Talisha’s solution:
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Here is another division example.
Dana solved this problem by subtracting groups of 54 from 2,500.
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Walter solved this problem by multiplying groups of 54 to reach 2,500.
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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?
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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
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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
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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
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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
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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.
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INTEGERS AND MULTIPLICATION
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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
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MULTIPLICATION 2 x -3 means 2 groups of -3
2 x -3 = -6
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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
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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
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MULTIPLICATION To model -2 x -3 use 2 groups of the
opposite of -3
-2 x -3 = +6
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INTEGERS AND DIVISION
The University of Texas at Dallas
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DIVISION
Use tiles to model +12 ÷ +3 = ?
Divide 12 yellow tiles into 3 equal groups
+12 ÷ +3 = +4
4 yellow tiles in each group.
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DIVISION Use tiles to model -15 ÷ +5 =?
Divide -15 into 5 equal groups
-15 ÷ +5 = -3
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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.
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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?
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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
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Models for Reasoning About Multiplication
Fraction of a fraction Linear/measurement Area/measurement models Cross Shading
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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
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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
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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
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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
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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.
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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
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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
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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?
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Sample children’s strategies
1/4 of a cup
1 cup 2 cups 3 cups4 cups 5 cups
“…so 5 cups altogether.”
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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…
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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)
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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)
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Division With FractionsDivision With Fractions
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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
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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?
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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?
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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
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“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
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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
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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