Download Reasoning Algebraically

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 Division
What are the goals for students?
Develop conceptual understanding
 Develop 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


47
Similar to the partial sums
algorithm for addition. The
x 13
procedure is to multiply one
21
pair of digits at a time.
Note that with this algorithm it does not
matter the order in which digits are
multiplied. (commutative property)
120
70
+4 0 0
611
Use the Partial Products Algorithm to show
124 × 135 = 16,740
(7 x 3)
(40 x 3)
(7 x 10)
(40 x 10)
Standard Multiplication Algorithm

This is basically an
abbreviation of the partial
products algorithm
47
x 13
141
+4 7 0
611
(47 x 3)
(47 x 10)
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
cars
x
4
=
wheels/car
(groups) (items per group)
(multiplier)
(multiplicand)
48
wheels
(total number of items)
(product)
Multiplication Strategies
12 cars with 4 wheels each. How many wheels?
Additive Strategies



Direct Modeling
Repeated Addition
Doubling
Multi-digit Multiplication Strategies
52 cards per deck. 18 decks of cards. How many cards?
Multiplicative Strategies



Single Number Partitioning
Both Number Partitioning
Compensating
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.
Additive Strategies



Direct Modeling
Repeated Addition
Doubling
Multiplicative Strategies



Single Number Partitioning
Both Number Partitioning
Compensating
Additional Multiplication Algorithms
Lattice Method
 Russian Peasant Method
 Egyptian Method

Lattice Multiplication Algorithm



This is basically the partial
products algorithm recorded in a
different format.
Multiply row by column
4
Sum the diagonals
47 × 13 = 611
Use the Lattice Multiplication
Algorithm to show
124 × 135 = 16,740
7
0
0 4 07 1
1 2 3
6 2 1
1 1
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
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).


47 × 13
Count
Double
1
47
2
94
4
188
√
8
376
√
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 + 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 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?
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
84 ÷ 21


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
63
21
42
21
21
21
0
1
1
1
1
4
Thus,
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
170 ÷ 14
84
– 42
42
– 42
0
170
– 140
30
– 28
2
2
2
4
Thus,
84 ÷ 21
=4
10
2
12
Thus,
170 ÷ 14
= 12
Remainder
=2
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
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
+1
are opposites
x +3 = +3
-1 x +3 = -3

the products are opposite
Since +2 and -2 are opposites of each other,
 +2
x -3 and -2 x -3 have opposite products.
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 = ?
4 yellow tiles in each group.
Divide 12 yellow
tiles into 3 equal
groups
+12
÷ +3 = +4
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 Fractions
Consider:
2 3

3 4

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.
2
3
How much is of ?
3
4
We use a fraction
square to represent
the fraction 34 .
Then, we shade
2
3
3
4
of
We can see that it is
6 .
the same as 12
3
4
2 3 1
 
3 4 2
The Linear Model with multiplication utilizes
the number line and partitions the fractions
3
4
2
3
How much is of ?
3
4
1
4
0
1
3
of
3
4
3
4
2
4
2
3
of
3
4
1
2
3
3
of
3
4
2 3 1
 
3 4 2
4
4
We can also use the linear model with
shapes and partition accordingly
Identify ¾ of the circle
2
3
How much is of ?
3
4
Take 2 pieces
Break into
3 pieces
Answer is ½
2 3 1
 
3 4 2
In the third method, we will think of
multiplying fractions as multiplying a
length times a length to get an area.
Length is
3
4
2
3
How much is of ?
3
4
2
3
Area
Width is
Number of square units
Is 6 out of 12
This area is 2
3
2 3 1
 
3 4 2
X
3 = 6
4
12
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.
2
3
2
3
How much is of ?
3
4
2
is 3
3
4
3
is 4
2 3 1
 
3 4 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
“…so 5 cups altogether.”
3 cups
4 cups
5 cups
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.
4 of these is 1 cup…
17, 18, 19, 20 -- that’s 5 cups.
…so 5 cups altogether.
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
5 cups
1/4 + 1/4 + 1/4 + 1/4 = 1
1/4 + 1/4 + 1/4 + 1/4 = 1
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

1 3
One shared by one-third of a group?
• How many in the whole group?
•
•
How does this work?
Division With Fractions

Repeated subtraction / measurement meaning
1
•
•
•
•
1

3
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?
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

1
12
You have
cups of sugar. It takes
cup to make 1 batch of cookies.
1
3

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?”
3 1
 ?
4 8
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?
3 1
 ?
4 8
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.
3 1
 6
4 8
Pizza
1 1
 ?
2 8
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.
We find there are 4. There are four one eighths in one half.
1 1
 4
2 8