Alternative Algorithms for Multiplication and Division

1
Alternative Algorithms for Multiplication and
Division

• If we dont teach them the standard way, how will
  they learn to compute?

2

• If we can convince students that mathematics is
  figure-out-able, that it is more than
  memorization, then we can increase student buy-in
  and confidence. If we can get students to think
  in class, instead of just trying to memorize
  series after series of steps, we can save time
  and decrease frustration because by building on
  understanding, we will have fewer misapplied and
  mixed-up rules.
• Why Numeracy for Secondary Students
• Harris Pope, 2005

3
How has this student misapplied the rules for
multiplying?

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

4
Multi-digit Multiplication and DivisionWhat are
the goals for students?

• Develop conceptual understanding
• Develop computational fluency

5
What is Computational Fluency?

• Fluency demands more of students than does
  memorizing a procedure. Fluency rests on a
  well-build mathematical foundation that involves
• Understanding implies that the student brings
  meaning to the operation being carried out. The
  student can explain the why of each step taken
  to solve the problem.
• Efficiency implies that the student does not get
  bogged down in the steps or lose track of the
  logic of the strategy. An efficient strategy is
  one that the student can carry out easily.
• Accuracy depends on careful recording, knowledge
  of basic number combinations and other important
  number relationships, as well as verifying the
  results.
• Flexibility requires the knowledge of more than
  one approach to solving a problem. Students need
  to be flexible to choose an appropriate strategy
  for a specific problem.

6
Conceptual Understanding
Computational Fluency
7
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)

8
Multi-digit Multiplication Strategies12 cars
with 4 wheels each. How many wheels?
? Multiplicative Strategies

• 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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Multi-digit 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

11
There are 18 ants with 6 legs each. How many
legs altogether?
Sample 1
Sample 2
12
Students collected cans to recycle. Each box
holds 12 cans. They filled 38 boxes with cans.
How many cans did they collect?
Sample 3
Sample 4
13
There are 62 fifth graders. It costs 38 per
student for outdoor school. How much do the
fifth graders need to earn so everyone can go?
Sample 5
Sample 6
14
There are 62 fifth graders. It costs 38 per
student for outdoor school. How much do the
fifth graders need to earn so everyone can go?
Sample 6
Sample 7
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Teachers Role

• Provide rich problems to build understanding
• Encourage the use of thinking tools
  (manipulatives like snap cubes or 300 charts)
  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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Two Contexts for Division

• Measurement Division (number of groups unknown)
• There are 54 children on a full bus. Each seat
  can hold 3 children. How many seats are there on
  the bus?
• Partition Division (size of groups unknown)
• There are 54 children on a full bus. There are
  18 seats. How many children are sitting on each
  seat?

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Multi-digit 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.

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

• Additive Strategies
• Direct Modeling
• Repeated Addition/Subtraction
• Doubling

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There are 54 children on a full bus. Each seat
can hold 3 children. How many seats are there on
the bus?
Sample 1
Sample 2
19
There are 54 children on a full bus. There are
18 seats. How many children are sitting on each
seat?
Sample 3
Sample 4
20
181 ? 15
Sample 5
Sample 6
21
2401 ? 27
Sample 7
Sample 8
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Teaching a standard way?

• Delay! Delay! Delay!
• Spend most of your time on invented strategies.
  The understanding students gain from working with
  invented strategies will make it much easier for
  them to understand a standard algorithm. For
  most students, this means delaying the teaching
  of a standard way of multiplying and dividing
  until 5th grade. Students who dont clearly
  understand the way should be allow to use a way
  that make sense to them.

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Which standard way?

• Partial Products for 52 x 18 (modeled
  by an open array)
• 52 52
• x 18 x 18
• 16 16
• 400 400
• 20 416
• 500 . 20
• 936 436
• 500
• 936

50
2
500
20
10
8
400
18
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Which standard way?

• Partial Products for 936 ? 18
• 18 936 18 936
  100 x 18 1800
• 180 10 x 18 900 50
• 756 36
• 360 20 x 18 36
  2
• 396 0
  52
• 360 20 x 18
• 36
• 36 2 x 18
• 0 52 x 18