Understanding Multiplication and Division with Whole Numbers

1
Understanding Multiplication and Divisionwith
Whole Numbers

• Adapted from Guiding Children's Learning of
  Mathematics by Leonard M. Kennedy (Author), Steve
  Tipps (Author), Art Johnson (Author)

2
Understanding Multiplication and Divisionwith
Whole Numbers

• Student Goals for Multiplication and Division
• Understands and explains the effects of addition,
  subtraction, and multiplication on whole numbers
  and the effects of division on whole numbers
  including the inverse relationship of
  multiplication and division.
• Selects the appropriate operations to solve
  specific problems
• Adds, subtracts, multiplies, and divides whole
  numbers to solve real world problems using
  appropriate methods of computing, such as mental
  mathematics, paper and pencil, and calculator.
  (5th grade, Sunshine State Standards)

3
Understanding Multiplication and Divisionwith
Whole Numbers

• Conceptual Understanding
• Real-life problem-solving stories provide a
  meaningful context for multiplication and
  division
• Children learn about multiplication and division
  facts and rules behind them
• by modeling and acting out with physical objects
• by representing the operations with pictures
• through naming and labeling the results of their
  actions

4
Understanding Multiplication and Divisionwith
Whole Numbers

• Multiplication Repeated Addition
• Joining or combining sets of equal size
• Tony has 4 packages of juice boxes. Each package
  has 3 juice boxes in it..
• Jaranda has 5 quarters in her pocket.
• Kayleigh earns 9 an hour babysitting and worked
  11 hours last week.

5
Understanding Multiplication and Divisionwith
Whole Numbers

• Multiplication Geometric Interpretation
• Objects are lined up in rows and columns
• Paolo arranged the chairs in 4 rows with 5 chairs
  on each row.
• Gina measured the room 9 feet long by 10 feet
  wide.
• The ceiling has 14 tiles across and 12 tiles
  front to back.

6
Understanding Multiplication and Divisionwith
Whole Numbers

• Multiplication Cartesian Product
• Combining each object from one set
• with each object from another set
• Molly wanted to know how many school outfits she
  had with 5 blouses and 3 pairs of pants..
• At her birthday party, Kara had 2 kinds of cake
  and 3 kinds of ice cream. How many combinations
  of cake and ice cream could she serve?.

7
Understanding Multiplication and Divisionwith
Whole Numbers

• Division Measurement
• Repeated subtraction of the same number
• Marisa served 5 cookies to each person at her
  party. She started with 25 cookies.
• Sandy had 93 saved. He spent 8 each week..

8
Understanding Multiplication and Divisionwith
Whole Numbers

• Division Partition or Sharing
• Distribute a set as evenly as possible
• into a known number of groups
• Kim had 19 strawberries to put into 4 boxes.
• The chef has 25 eggs. How many eggs can each of
  8 people have?

9
Understanding Multiplication and Divisionwith
Whole Numbers

• Properties of Multiplication and Division
• Commutative property order of factors does not
  matter
• Associative property factor may be paired in
  any order
• Identity element multiplication or division by
  1 does not change the original number value
• Distributive property over addition
  multiplication and division may be applied to
  factors represented as sums

10
Understanding Multiplication and Divisionwith
Whole Numbers

• The Meaning of Remainders in Division
• Introduce remainders as a natural result of
  division.
• Discuss the meaning of remainders and the various
  ways to handle remainders.
• Ignore the remainder
• Round up and adjust the solution
• Treat the remainder as a fraction
• Reread the problem to understand exactly what is
  wanted.

11
Understanding Multiplication and Divisionwith
Whole Numbers

• Strategic Learning of Basic Facts
• Children invent and learn strategies for the
  basic facts.
• skip counting
• commutative property (flip flop facts)
• multiplication by 0, 1 (identity), and 2
• division by 1
• square numbers, multiplication by 9
• associative and distributive property
• Children practice facts with the strategies to
  develop accuracy and reasonable speed.

12
Understanding Multiplication and Divisionwith
Whole Numbers

• Guidelines for Practice
• Accuracy is the primary goal. Speed is secondary.
• Practice strategies and facts students
  understand.
• Keep practice sessions brief 5 minutes.
• Use a variety of materials and procedures such as
  games, computer activities, and flash cards.
• Emphasize individual improvement rather than same
  goal for all children.
• Review and maintain basic facts to keep the skill
  active.

13
Understanding Multiplication and Divisionwith
Whole Numbers

• Transition to Larger Numbers
• The same situations introduced with numbers 1-100
  apply to multiplication and division situations
  with larger numbers.
• Multiplication and division with larger numbers
  requires
• understanding of the operations and basic facts
• understanding place value representation of
  larger numbers

14
Understanding Multiplication and Divisionwith
Whole Numbers

• Working with Numbers to 1000 and Beyond
• Problem-solving stories extend learning
  multiplication and division with larger numbers
• by modeling and acting out with base ten models
  up to 100, 500, and 1000
• by representing the operations with pictures of
  place value materials
• through recording the results of their actions
  and pictures with numerals and place value.

15
Understanding Multiplication and Divisionwith
Whole Numbers

• Developing Algorithms
• After actions with materials and recording
  results, students invent and/or learn alternative
  and standard algorithms as step by step
  procedures for multiplication and division with
  numbers to 1000 and larger than 1000.
• At the same time, students learn when and how to
  use other computational processes.
• Estimation
• Mental computation
• Technology- calculator and computer

16
Understanding Multiplication and Divisionwith
Whole Numbers

• The purpose of computation is to solve problems.
  Thus, although computation is important in
  mathematics and in daily life, our technological
  age requires us to rethink how computation is
  done today.
• (NCTM, 1989, p. 44)

17
Understanding Multiplication and Divisionwith
Whole Numbers

• Traditional and Alternative Multiplication
  Algorithms
• In multiplication, regrouping and naming is the
  traditional algorithm. Many children have trouble
  with order of operation and lining up numbers.
• Low stress algorithm start with the largest place
  value and works left to right.
• Transitional and historical algorithms provide
  students with other approaches to multiplication
  with larger numbers..

18
Understanding Multiplication and Divisionwith
Whole Numbers

• Traditional Transitional Low-stress
• 74 74 74
• x89 x89 x89
• 666 36 5600
• 5920 630 320
• 6586 320 630
• 5600 36
• 6586 6586

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Understanding Multiplication and Divisionwith
Whole Numbers

• Traditional and Alternative Division Algorithms
• Using trial divisors is the traditional
  subtraction algorithm.
• Ladder alternative algorithm is based on repeated
  subtraction procedures
• It allows children to start the division process
  with any convenient divisor..
• The value of the numbers is maintained throughout
  the ladder algorithm.

20
Understanding Multiplication and Divisionwith
Whole Numbers

• Traditional Ladder Algorithm
• 23 R 5 23 R 5
  
• 37 ) 856 37 ) 856
• 74 740 20
• 116 116
• 111 111 3
• 5 5

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Understanding Multiplication and Divisionwith
Whole Numbers

• Number Sense, Reasonableness, and Estimation
• Throughout development of multiplication and
  division concepts and procedures, number sense is
  developed.
• In problem situations, students ask themselves,
  Does this answer make sense?
• Estimation procedures are developed in situations
  that do not require exact answers.
• Front-end estimation
• Rounding estimation

22
Understanding Multiplication and Divisionwith
Whole Numbers

• Estimation for Multiplication
• Multiplication Front-End Rounding
• 74 70 70
• x 89 x 80 90
• 5600 6300
• Students should identify most reasonable answers
  using high and low estimates.

23
Understanding Multiplication and Divisionwith
Whole Numbers

• Estimation for Division
• Make an Easy Division Problem
• Using Rounding / Reasonableness
• ____ __20__ __ 22
• 37) 856 40) 800 40 ) 880
• Students should identify reasonable answers
  through several approximations.

24
Understanding Multiplication and Divisionwith
Whole Numbers

• Mental Computation
• Alternative algorithm and estimation procedures
  build confidence in mental computation.
• Understanding of multiplication and division of
  larger numbers is improved when students
  understand how to work with powers of ten.
• 20 x 400 2 x 4 x 10 x 100 8 x 1000 800
• 9000 divided by 30 is 9 divided by 3 x 1000
  divided by 10 300

25
Understanding Multiplication and Divisionwith
Whole Numbers

• Regardless of the particular computation method
  used, students should be able to explain their
  method, understand that many methods exist, and
  see the usefulness of methods that are efficient,
  accurate, and general. Students also need to be
  able to estimate and judge the reasonableness of
  results.
• (NCTM, 2000, p 32)

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Multiplication Possibilities

• Find as many ways as you can to fill in this
  problem so the arithmetic is correct
• __ __ __
• x __
• _______
• 9 6 6

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Division Problems

• Bring a loaf of sliced bread to class. Choose a
  loaf that is packaged in a clear wrap so children
  can count the slices. Ask them to figure how
  many sandwiches can be made from the loaf.

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Division Problems

• Tell the class that for a math activity, each
  student needs eight tiles. A box has two hundred
  tiles. Have them figure out if everyone in the
  class can participate in the activity at one time.

29
Division Problems

• Talk with the children about the way paper is
  packaged. Tell them that a ream contains five
  hundred sheets. Present this problem If each
  student needs 20 sheets to make a recording book,
  how many books can be made from a ream of paper.

30
Division Problems

• Tell the children about a person with an apple
  tree who had a giveaway celebration to get rid of
  fifty extra apples. She offered three apples to
  each person who asked. How many people could get
  free apples?