Understanding
Multiplication and Division
with Whole Numbers
Adapted from: Guiding Children's Learning of Mathematics
by Leonard M. Kennedy (Author), Steve Tipps (Author), Art Johnson
(Author)
Understanding Multiplication and Division
with Whole Numbers
Student Goals for Multiplication and Division
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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. (5 grade, Sunshine State Standards)
th
Understanding Multiplication and Division
with Whole Numbers
Conceptual Understanding
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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
Understanding Multiplication and Division
with Whole Numbers
Multiplication – Repeated Addition
Joining or combining sets of equal size
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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.
Understanding Multiplication and Division
with Whole Numbers
Multiplication – Geometric Interpretation
Objects are lined up in rows and columns
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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.
Understanding Multiplication and Division
with Whole Numbers
Multiplication – Cartesian Product
Combining each object from one set
with each object from another set
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
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?.
Understanding Multiplication and Division
with Whole Numbers
Division – Measurement
Repeated subtraction of the same number
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Marisa served 5 cookies to each person at her party.
She started with 25 cookies.
Sandy had $93 saved. He spent $8 each week..
Understanding Multiplication and Division
with Whole Numbers
Division – Partition or Sharing
Distribute a set as evenly as possible
into a known number of groups
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Kim had 19 strawberries to put into 4 boxes.
The chef has 25 eggs. How many eggs can each of 8
people have?
Understanding Multiplication and Division
with Whole Numbers
Properties of Multiplication and Division
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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
Understanding Multiplication and Division
with Whole Numbers
The Meaning of Remainders in Division
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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.
Understanding Multiplication and Division
with Whole Numbers
Strategic Learning of Basic Facts
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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.
Understanding Multiplication and Division
with Whole Numbers
Guidelines for Practice
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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.
Understanding Multiplication and Division
with Whole Numbers
Transition to Larger Numbers
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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
Understanding Multiplication and Division
with Whole Numbers
Working with Numbers to 1000 and Beyond
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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.
Understanding Multiplication and Division
with 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
Understanding Multiplication and Division
with 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)
Understanding Multiplication and Division
with Whole Numbers
Traditional and Alternative Multiplication
Algorithms
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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..
Understanding Multiplication and Division
with Whole Numbers
Traditional
Transitional
Low-stress
74
74
74
x89
x89
x89
666
5920
6586
36
630
320
5600
6586
5600
320
630
36
6586
Understanding Multiplication and Division
with Whole Numbers
Traditional and Alternative Division
Algorithms
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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.
Understanding Multiplication and Division
with Whole Numbers
Traditional
23 R 5
37 ) 856
74
116
111
5
Ladder Algorithm
23 R 5
37 ) 856
740
116
111
5
20
3
Understanding Multiplication and Division
with Whole Numbers
Number Sense, Reasonableness, and Estimation
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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.
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Front-end estimation
Rounding estimation
Understanding Multiplication and Division
with Whole Numbers
Estimation for Multiplication
Multiplication
74
x 89
Front-End
70
x 80
5600
Rounding
70
+90
6300
Students should identify most reasonable answers
using high and low estimates.
Understanding Multiplication and Division
with 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.
Understanding Multiplication and Division
with Whole Numbers
Mental Computation
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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.
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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
Understanding Multiplication and Division
with 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)
Multiplication Possibilities
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Find as many ways as you can to fill in
this problem so the arithmetic is
correct:
__ __ __
x __
_______
9 6 6
Division Problems
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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.
Division Problems
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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.
Division Problems
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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.
Division Problems
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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?