Reasoning Algebraically

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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
3
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 understanding
• Develop 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
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
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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)? (multipl
ier) (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.

• Multiplicative Strategies
• Single Number Partitioning
• Both Number Partitioning
• Compensating

• Additive Strategies
• Direct Modeling
• Repeated Addition
• Doubling

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Additional Multiplication Algorithms

• Lattice Method
• Russian Peasant Method
• Egyptian 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

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.

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

47 13
47 188 376 611
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Teachers 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.
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?

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Samantha solved this problem by multiplying
groups of 32 to reach 1,780. Samanthas
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. Talishas 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 4
Thus, 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.

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
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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.

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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 cant
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 ?

4 yellow tiles in each group.
Divide 12 yellow tiles into 3 equal groups

• 12 3 4

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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 arent 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 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 Nadines brownie
  problem?

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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.
We use a fraction square to represent the
fraction .
Then, we shade of We can see that it is
the same as .
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The Linear Model with multiplication utilizes the
number line and partitions the fractions
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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
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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.
Length is
Area
Width is
Number of square units Is 6 out of 12
This area is X
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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.
is
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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 childrens strategies
5 cups
4 cups
1 cup
2 cups
3 cups
so 5 cups altogether.
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Another student strategy
So, 5, 6, 7, 8 -- thats 2 cups.
9, 10, 11, 12 -- thats 3 cups.
13, 14, 15, 16 -- thats 4 cups.
17, 18, 19, 20 -- thats 5 cups.
4 of these is 1 cup
so 5 cups altogether.
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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 Whats 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 doesnt always make things
  bigger)
• Pizza (pepperoni on ? of ½ pizza)
• Recipes ( 1¾ cups of sugar is used but we want to
  make ½ a batch)
• Ribbon (you have ? yd , ? of the ribbon is used
  to make a bow)

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Division 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?

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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 thats 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 ? 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 thats left?

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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?
Recall a slice represents one eighth of the
pizza
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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.
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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.