MULTIDIGIT MULTIPLICATION AND DIVISION - All Children

APPENDIX B

MULTIDIGIT MULTIPLICATION AND DIVISION

Participants examine multidigit multiplication and division strategies that
build on students’ understanding of single-digit multiplication and division.
Participants learn about area models for multiplication and division. They
examine the meaning of the traditional multiplication and division
algorithms and variations of these algorithms.

Lesson Goals

Identify advanced strategies that children use to solve multidigit
multiplication and division problems in sense-making ways
Connect concepts of multiplication and division to standard procedures
Discuss teaching strategies that enhance a child’s understanding of
multidigit multiplication and division

Word Bank

skip counting
area model
expanded notation
commutative property of multiplication
associative property of multiplication
distributive property of multiplication

Focus Questions

How does an understanding of the meaning of multiplication and
division affect the appropriate and accurate use of these operations?
What are some interim strategies students can use to develop
multiplication and division procedures?
How do interim strategies connect to the standard algorithms?

UCLA Math Content Program for Teachers COMP5 – PG1
Multidigit Multiplication and Division Appendix B

MULTIDIGIT MULTIPLICATION AND DIVISION

(Estimated Time: 4 hours)

Lesson Summary Lesson Goals Word Bank

Participants examine multidigit • Identify advanced strategies that skip counting
multiplication and division strategies children use to solve multidigit area model
that build on students’ understanding multiplication and division problems expanded notation
of single-digit multiplication and in sense-making ways commutative property of multiplication
division. Participants learn about area associative property of multiplication
models for multiplication and division. • Connect concepts of multiplication distributive property of multiplication
They examine the meaning of the and division to standard procedures
traditional multiplication and division Strategies for Special Needs
algorithms and variations of these • Discuss teaching strategies that
algorithms. enhance a child’s understanding of • Use physical models (ALL)
multidigit multiplication and division • Ask students to explain strategies to

Materials Reproducibles each other (ELL, L-R, L-E)
• Encourage students to use the fourfold

way (pictures, numbers, symbols,
words) to explain strategies in writing
(ALL)

Prepare Ahead

Blank paper R1-3*: Teaching and Learning Notes
3-4 LUCIMATH Video *distribute as desired
TV/VCR
Base-10 blocks Overhead Transparencies Focus Questions
OH base-10 blocks
Overhead transparencies and pens
Chart paper and markers

Participant Pages

PP1: Summary Page OH1: Focus Questions • How does an understanding of the
PP2: Selected CA Math Standards OH2: CA Math Standards – In Brief meaning of multiplication and
PP3: A Warmup OH3: A Warmup division affect the appropriate and
PP4: 27 X 4 OH4: Multiplication and Division accurate use of these operations?
PP5-6: Multiplication and Division Strategies
Strategies OH5: Division Questions • What are some interim strategies
PP7: Analyze and Assess students can use to develop
PP8-9: Student Work Journal Idea multiplication and division
PP10: Strategies that Promote Classroom procedures?
Discourse
PP11: The Art of Questioning in • How do interim strategies connect to
Mathematics the standard algorithms?
PP12: The Importance of Recording
PP13: Using Area Models • How might we increase parental
PP14-17: Algorithms from Around the involvement for children?
World
PP18: Journal 1: Algorithms Around the Assessment Idea
World
PP19: Brown and Green
PP20-22: Multidigit Division Student Work
PP23: Journal 2: Base-10 Block Division
PP24: Problem Starters
PP25-30: Multicultural Mathematics Article

Problem of the Week

Problem Starters* Select one of the algorithms from Write a story problem to go along with
*Use as preview or review around the world. Explain how it
works and why it works. 380÷20 = 19. Solve problem two different
ways. Identify problem type and
strategies used to solve it.

UCLA Math Content Program for Teachers COMP5 – PG2
Multidigit Multiplication and Division Appendix B

MULTIDIGIT MULTIPLICATION AND DIVISION

This module builds on the concepts developed in the single-digit multiplication and
division module (COMP4). Students extend strategies involving direct modeling,
counting, and derived facts to multidigit operations. An important new strategy that
students use is a grouping strategy. The grouping strategy is based on a stronger
content knowledge of place value, which is further developed in third grade. Grouping
strategies lead naturally to the traditional algorithm for multiplication. The area model
provides a concrete picture for visualizing grouping strategies and for interpreting the
laws of arithmetic.

Formal strategies for solving multidigit multiplication and division problems depend on
base-10 number concepts. It has usually been assumed that it is necessary for children
to develop base-10 number concepts before they add, subtract, multiply, and divide
two- and three-digit numbers. According to research in the field of Cognitively Guided
Instruction, this assumption is not valid. As long as children can count, they can solve
problems involving two-digit numbers even when they have limited notions of grouping
by ten. By encouraging the use of sense-making strategies for computation, teachers
can help children develop computational proficiency as they develop meaning for the
number system.

R1-3* (Teaching and Learning Notes) are reproductions of many FYI boxes in
this lesson. They provide an additional reference for participants. Instructors
may distribute them as desired.

Preview (5 minutes)

Use OH1 (Focus Questions) and PP1 (Summary Page) to introduce the goals of the
lesson.

Use OH2, PP2 (California Math Standards) to introduce standards addressed in the
lesson. Participants may want to make a note of them on appropriate participant
pages.

Some participants may need a fast review of some of the properties of real

numbers:

• Commutative Property of multiplication:

ab = ba 3(4) = 4(3)

• Associative Property of multiplication:

(ab)c = a(bc) (2 x 3) x 4 = 2 x (3 x 4)

• Distributive Property (connects multiplication and addition):

a (b + c) = ab + ac 3(5 + 2) = 3(5) + 3(2)

UCLA Math Content Program for Teachers COMP5 – PG3
Multidigit Multiplication and Division Appendix B

Part One: Multiplication

Warmup (15 minutes)

Use OH3, PP3 (A Warm-up) to remind participants of useful estimation and mental
math strategies for multiplication. (Useful procedural strategies include determining
the number of digits in a solution, rounding, and understanding how to multiply by
powers of 10.)

Introduce (20 minutes)

(Pairs) Use PP4 (27 X 4). Invite participants to compute 27 X 4 two different ways
using pictures, words, or numbers; explain one of their solutions strategies to their
partner; and be prepared to explain their partner’s solution to the whole group.

Think-Pair-Share: A Strategy for Students with Special Needs

Activities where students first work on a problem individually, then exchange
ideas with a partner, and finally discuss with a larger group are called “think-
pair-share” activities. Think-pair-share gives students opportunities to talk in a
safe environment. It is especially recommended for English language learners
and students with receptive or expressive language disorders.

(Whole group) Participants share the varied strategies of their partners with the
whole group. Demonstrate appropriate recording strategies on chart paper. Name
invented strategies (i.e. Lindsay’s way).

Use OH4, PP5-6 (Multiplication and Division Strategies). Identify participant
strategies used in the 27 X 4 example and arrange the various solutions in a
developmental sequence. Use the “Possible Developmental Sequence” chart below
as a backup to discuss any strategies not demonstrated.

POSSIBLE DEVELOPMENTAL SEQUENCE FOR MULTIPLICATION
Direct Modeling
(record by 1s)

Direct Modeling Counting
(count in chunks) (repeated addition,

Modeling skip counting)
(using 10s and 1s)

Written Records Other invented strategies
(using 10s and 1s) (Double half, benchmarks, distributive property)

Algorithm COMP5 – PG4
(compact recording) Appendix B

UCLA Math Content Program for Teachers
Multidigit Multiplication and Division

Explore: Student Work (20 minutes)

Use PP7 (Analyze and Assess) and PP8-9 (Student Work). Participants analyze
student work. Encourage participants to discuss the mathematical knowledge that
each student knows and an appropriate question to ask each student that would
help to clarify or extend their thinking.

ABOUT THE STUDENT WORK

This student work came from a third grade classroom in Phoenix, Arizona.
The class contained 29 students: 11 girls and 18 boys. It was predominately
Caucasian, but had an equal share of African American, Native American,
Asian, and Hispanic students. The socio-economic level of this class ranged
from affluent to free lunch status with the majority of the students falling in the
middle range. The math class included 3 resource students, 6 ELL (minimal or
no English) students, 3 severe behavior students, and 4 gifted students. All
participated in the daily math lessons.

• What does the child know? [A: Advanced Grouping; B: Advanced Counting; C: Direct

Modeling; D: Advanced Counting; E: Advanced Grouping; F: Standard Algorithm; G: Advanced
Direct modeling; H: Advanced Grouping.]

• What might be a good next step? [See “Next Step” Talking Points.]

NEXT STEP TALKING POINTS

Although base-10 number concepts are not prerequisites for solving multidigit
problems, this knowledge increases efficiency of finding solutions. Encourage
children to solve problems using tens.

Multiplication problems normally place the larger number above the smaller
number when using a vertical alignment. For some student strategies, the
work is not lined up in traditional columns. Most students will eventually move
to the more conventional-looking alignment. It is important for the teacher to
focus attention on the meaning of the operation, the student’s understanding of
the problem, and the student’s ability to solve the problem, rather than simply
the procedure to get the answer.

Students will develop mathematical thinking and reasoning by solving
problems in two ways and sharing strategies with the class. Identify properties
of arithmetic (such as commutative property) that are illustrated by the various
student strategies.

Encourage children to show multiplication with rectangular arrays. This is a
good way to show partial products and the distributive property.

Classroom Connection (20 minutes)

Use PP10 (Strategies that Promote Classroom Discourse). Show Video Clip 1
(24 x 9). Ask participants to pay close attention to the teacher’s role in facilitating
classroom discussions about mathematics.

UCLA Math Content Program for Teachers COMP5 – PG5
Multidigit Multiplication and Division Appendix B

# Who Problem Strategy Talking Points

Sofia (girl with pigtails) Notice how teacher records Sofia’s

benchmark numbers, repeated strategy.

addition, compensation.

Ken (boy with red shirt) Ken: Comment on teacher recording

Susan benchmark numbers, doubles, strategy (note error in writing

1 Aldridge’s 24 X 9 incremental addition. equalities; teacher might use arrows
4th Grade
instead).

Class

Alison (gray sweatshirt) Explain how Alison uses distributive

grouping, distributive property. property to multiply.

Rachel (girl with glasses) Explain how Rachel uses

grouping, distributive property. distributive property to multiply.

Permission for limited use of this video clip was granted by Creative Publications, 2001.

• What is mathematical discourse? [Discourse is a process where students present

mathematical explanations and evaluate strategies by verifying, challenging, and comparing

them.]

• The teacher does not correct wrong answers or tell how to carry out the
computation. What message do you think this communicates to students?

[Rather than give students the message that “teaching is telling,” the teacher models a disposition
towards mathematics and a way of thinking that she wants her students to develop.]

• What are some of the benefits of having the teacher record the steps of
students’ solutions? What did you like or dislike about her recordings?

[Teacher recordings help discussion move more quickly. Skillful recordings help students follow
each other’s explanations and connect symbolic notation to informal language and language of
mathematics. Teachers may find it beneficial to record class strategies on chart paper for future
reference.]

• At one point in the lesson, the teacher wrote “225-10=215+1=216”. Do you
think this notation is problematic? Why or why not? [The statements are not all

equal to each other. Arrows indicating sequential thinking would be more appropriate here.]

Use PP11 (The Art of Questioning in Mathematics) and PP12 (The Importance of
Recording) to discuss appropriate classroom questioning and recording techniques.

UCLA Math Content Program for Teachers COMP5 – PG6
Multidigit Multiplication and Division Appendix B

Part Two: Area Models

Introduce (15 minutes)

This part of the lesson focuses on area models, another sense-making strategy
that draws upon students understanding of base-10 number concepts and
expanded notation.

Use base-10 blocks. Put 14 X 3 on the overhead as 3 rows of 14.

• How can we use this model to help us solve this problem? [3 groups of 14 ones,

14 groups of 3, area of rectangle that is 3 by 14, 3 groups of 10 and 3 groups of 4.]

• Does the mathematical meaning change if the array is rotated 90 degrees?

[No, by Commutative Property.]

Using base-10 blocks, guide participants through the use of an area model to find
12 X 13, emphasizing how the distributive property is used. See “12 X 13 – Area
Models” below.

12 X 13 – AREA MODELS

12 rows of 13 13 rows of 12

12 X 10 = 120 13 X 10 = 130
12 X 3 = 36 13 X 2 = 26

156 156

12 rows of 13 Rectangle 12 X 3

10 X 13 = 130 100 + 20 + 30 + 6
2 X 13 = 26 = 156
156

Many people learned the FOIL method (First, Outer, Inner, Last), which applies
the distributive property of multiplication twice. Note that the order of
multiplication does not matter as long as all of the products are found.
(10 + 2)(10 + 3)

UCLA Math Content Program for Teachers COMP5 – PG7
Multidigit Multiplication and Division Appendix B

Explore (10 minutes)

(Pairs/Tables) Use PP13 (Using Area Models) and base-10 blocks. Participants
use area models (blocks or drawings) to find products. They record the solution
both pictorially and numerically.

Summarize (10 minutes)

Invite participants to make overheads to share their approaches to the problem. Be
sure a variety of approaches are included.

• How did the blocks help you to solve the problems?

• How might the base-10 blocks help a student connect concepts of
multiplication to written recordings of multiplication? [Many common student

errors for multiplication center around place value, and confusion between the name of the digit
and the value of the digit. In the numeral 24, the first digit has a name of 2 (two), but it has a
value of 20. Base-10 blocks make this distinction clear.]

Connect the area models to the traditional algorithm. See 12 X 13 – Connections to
the Traditional Algorithm.

• Identify some specific mathematical ideas that are used in the traditional
algorithm? [Place value, derived facts, addition, use of distribute property.]

• Do you think that using base-10 blocks helps to give meaning to the
multiplication algorithm? How? [One common concern when using models is that

students will not make connections between the concrete models, their representations, and the
mathematical concept. Base-10 blocks as an area model emphasize distributive property and
provide a visual representation to the partial products of the multiplication algorithm.]

12 X 13 – Connections to Traditional Algorithm

12 rows of 13 Traditional Algorithm Rectangle

12 x10 = 12 12 12 =2x3
12 x 3 = x 13 x 13 x 13 = 10 x 3
120 = 10 x 2
36 6 = 10 x 10
36 12 30
156 156 20
100

156

UCLA Math Content Program for Teachers COMP5 – PG8
Multidigit Multiplication and Division Appendix B

Extend (5 minutes)

Demonstrate the area multiplication recording model for a larger problem. Students
eventually outgrow the usefulness of manipulatives. However, drawing on
their experiences with area models, students can now represent problems
conceptually with expanded notation.

Multiply 23 X 143 100 + 40 + 3

143 20 2000 800 60
x 23 + 120 9
2000 3 300
800
300

60
120
+9

3289

Show how the area model extends for multiplying polynomials.

x +2

Multiply (x+2) (x+3) x x2 2x
= x2 + 2x + 3x + 6 + 6
= x2 + 5x + 6 3 3x

Use PP14-17 (Algorithms from Around the World), PP18 (Journal 1), and PP25-30
(Multicultural Mathematics Article). Invite participants to read about algorithms from
around the world and explain one or more of them.

Part Three: Division

Introduce (15 minutes)

(Pairs) Use PP19 (Brown and Green). Solve each problem two different ways. Try
to use “child” strategies such as doubles, halves, and easy multiplication facts to
solve the problems.

• What did you do to solve these problems?

• What operations were involved?

• Do you consider the situation multiplication or division? Why?

UCLA Math Content Program for Teachers COMP5 – PG9
Multidigit Multiplication and Division Appendix B

• Compare your strategies others. How are they alike? How are they
different? Did the problem type (partitive or measurement division) change
your strategy?

Record participant’s strategies exactly as given on chart paper (i.e. drawing pictures,
repeated addition, repeated subtraction, manipulatives, multiplication facts,
traditional algorithm). If necessary, ask clarifying questions to help these recordings
make sense to others. Reinforce the importance of recording.

Even when children explain their thinking orally, they may have difficulty
putting their ideas into written form. Teachers who record as students explain
strategies help them learn to represent their ideas in writing. In the classroom,
recordings make student explanations public, serve as models, allow for
comparisons and discussions, and help other children develop alternative
strategies.

Learning to record children’s thinking takes practice. Teachers can develop
recording systems by watching how children record their thinking, and helping
them to refine their methods.

Explore 1/Summarize (30 minutes)

(Whole group) Use PP20-22 (Multidigit Division Student Work). Participants analyze
student work. Encourage participants to discuss the mathematical knowledge that
each student knows, evaluate the efficiency of the student strategies for 3rd graders
or 4th graders, and the extent the strategies can be generalized.

Discuss strategies.

• What number facts and strategies were most important in the student
solutions? [Doubling, multiplying by 10, addition.]

• What mathematical ideas did students use to solve the problems? [Kept

numbers intact, kept the problem in mind, and performed actions of fair share division or
measurement division.]

• What mathematics do the students need to understand to “do” these
interim algorithms? [Multiplication strategies, addition, subtraction, and derived facts.]

• To what extent do these strategies generalize? offer good transitions
toward a more traditional division algorithm? [Most interim strategies are based on

writing the total amount for each step of the procedure; the traditional algorithm shortcuts this
step by using carrying and addition for the next step thus reducing four steps in a double digit
problem to two steps.]

UCLA Math Content Program for Teachers COMP5 – PG10
Multidigit Multiplication and Division Appendix B

Explore 2 (10-20 minutes)

(Pairs/Tables) Use PP23 (Journal 2: Base-10 Block Division). Invite participants to
think about how they might connect the manipulative to the standard division
algorithm.

Show Video Clip 2 (Marilyn Burns Division). Pay special attention to the language
Marilyn uses to explain the algorithm.

# Who Problem Strategy Talking Points

Pay attention to the questions

Marilyn Marilyn asks:

2 Burns with 435 divided by 3 Connects base-10 blocks • How many did you put in each
three 4th to division algorithm group?

graders • How much did you use
altogether?

• How much is left on your board?

Permission for limited use of this video clip was granted by Marilyn Burns and Associates. 2001

Summarize (10 minutes)

(Pairs) Use OH5 (Division Questions). Allow time after the video clip for participants
to role-play (Marilyn – Student) to practice modeling the algorithm as Marilyn did with
the boys.

Use PP25 (Problem Starters) as a homework problem of the week if desired.

Closure (5 minutes)

Use OH2, PP2 (California Math Standards) to revisit standards. Connect module
activities to student outcome goals.

Activity Grade 2 Grade 3 Grade 4 Grade 5
NS 1.3, 1.5, 2.4 NS 1.3
Estimation NS 3.1, 3.2, 3.3
Warm up AF 1.1 NS 2.1, 2.3, 2.4 NS 3.2, 3.3, 3.4
NS 3.1 AF 1.1, 1.5
Multidigit strategies for MG 1.0 AF 1.3
multiplication & division AF 1.5
NS 1.3, 1.5 NS 3.2, 3.3, 3.4
Area model-
multiplication NS 1.3, 1.5, 2.3, 2.4
AF 1.5
Multiplication & Division
algorithms

Use OH1 (Focus Questions) and PP1 (Summary Page) to revisit the goals for the
lesson. Tie up loose ends.

UCLA Math Content Program for Teachers COMP5 – PG11
Multidigit Multiplication and Division Appendix B

TEACHING AND LEARNING NOTES

This module builds on the concepts developed in the single-digit multiplication and division
module (COMP4). Students extend strategies involving direct modeling, counting, and derived
facts to multidigit operations. An important new strategy that students use is a grouping
strategy. The grouping strategy is based on a stronger content knowledge of place value, which
is further developed in third grade. Grouping strategies lead naturally to the traditional algorithm
for multiplication. The area model provides a concrete picture for visualizing grouping strategies
and for interpreting the laws of arithmetic.

Formal strategies for solving multidigit multiplication and division problems depend on base-10
number concepts. It has usually been assumed that it is necessary for children to develop
base-10 number concepts before they add, subtract, multiply, and divide two- and three-digit
numbers. According to research in the field of Cognitively Guided Instruction, this assumption is
not valid. As long as children can count, they can solve problems involving two-digit numbers
even when they have limited notions of grouping by ten. By encouraging the use of sense-
making strategies for computation, teachers can help children develop computational
proficiency as they develop meaning for the number system.

Some participants may need a fast review of some of the properties of real

numbers:

• Commutative Property of multiplication:

ab = ba 3(4) = 4(3)

• Associative Property of multiplication:

(ab)c = a(bc) (2 x 3) x 4 = 2 x (3 x 4)

• Distributive Property (connects multiplication and addition):

a (b + c) = ab + ac 3(5 + 2) = 3(5) + 3(2)

POSSIBLE DEVELOPMENTAL SEQUENCE FOR MULTIPLICATION

Direct Modeling
(record by 1s)

Direct Modeling Counting
(count in chunks) (repeated addition,

Modeling skip counting)
(using 10s and 1s)

Written Records Other invented strategies
(using 10s and 1s) (Double half, benchmarks, distributive property)

Algorithm
(compact recording)

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – R1
Multidigit Multiplication and Division Appendix B

ABOUT THE STUDENT WORK

This student work came from a third grade classroom in Phoenix, Arizona.
The class contained 29 students: 11 girls and 18 boys. It was predominately
Caucasian, but had an equal share of African American, Native American,
Asian, and Hispanic students. The socio-economic level of this class ranged
from affluent to free lunch status with the majority of the students falling in the
middle range. The math class included 3 resource students, 6 ELL (minimal or
no English) students, 3 severe behavior students, and 4 gifted students. All
participated in the daily math lessons.

12 X 13 – PART 1 – AREA MODELS

12 rows of 13 13 rows of 12

12 X 10 = 120 13 X 10 = 130
12 X 3 = 36 13 X 2 = 26

156 156

12 rows of 13 Rectangle 12 X 3

10 X 13 = 130 100 + 20 + 30 + 6
2 X 13 = 26 = 156
156

Many people learned the FOIL method (First, Outer, Inner, Last) which applies
the distributive property of multiplication twice. Note that the order of
multiplication does not matter as long as all of the products are found.
(10 + 2)(10 + 3)

12 X 13 – Part 2: Connections to Traditional Algorithm

12 rows of 13 Traditional Algorithm Rectangle

12 x10 = 12 12 12 =2x3
12 x 3 = x 13 x 13 x 13 = 10 x 3
120 = 10 x 2
36 6 = 10 x 10
36 12 30
156 156 20
100
156

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – R2
Multidigit Multiplication and Division Appendix B

Multiply 23 X 143 100 + 40 + 3

143 20 2000 800 60
x 23 + 120 9
2000 3 300
800
300

60
120
+9
3289

x +2

Multiply (x+2) (x+3) x x2 2x
= x2 + 2x + 3x + 6 + 6
= x2 + 5x + 6 3 3x

Even when children explain their thinking orally, they may have difficulty
putting their ideas into written form. Teachers who record as students explain
strategies help them learn to represent their ideas in writing. In the classroom,
recordings make student explanations public, serve as models, allow for
comparisons and discussions, and help other children develop alternative
strategies.

Learning to record children’s thinking takes practice. Teachers can develop
recording systems by watching how children record their thinking, and helping
them to refine their methods.

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – R3
Multidigit Multiplication and Division Appendix B

FOCUS QUESTIONS

• How does an understanding of the
meaning of multiplication and division
affect the appropriate and accurate use of
these operations?

• What are some interim strategies students
can use to develop multiplication and
division procedures?

• How do interim strategies connect to the
standard algorithms?

• How does parental involvement affect the
teaching and learning of mathematics?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – OH1
Multidigit Multiplication and Division Appendix B

CA MATH STANDARDS – IN BRIEF

Grade 2

• Strategies for multiplication and division
• Concept of division

Grade 3

• Multiplication and division as inverses
• Procedures for multiplication and division
• Commutative and associative properties
• Represent quantities with expressions, sentences,

inequalities

Grade 4

• Rounding
• Multiplication and division algorithms
• Long division (1 digit divisors)

Grade 5

• Distributive property

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – OH2
Multidigit Multiplication and Division Appendix B

A WARMUP

Find each answer:
1) 3 X 4 =
2) 30 X 4 =
3) 300 X 40
4) 3 X 0.4
What strategies did you use to find the answers?

Estimate each answer:
5) 34 x 56 =
6) 383 X 420
7) 15.09 X 3.4
8) 6390 divided by 32
What strategies did you use to estimate?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – OH3
Multidigit Multiplication and Division Appendix B

MULTIPLICATION AND DIVISION
STRATEGIES

Direct Modeling Strategies

Counting Strategies

Derived Fact Strategies

Grouping Strategies

Other Invented Strategies

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – OH4
Multidigit Multiplication and Division Appendix B

DIVISION QUESTIONS
To connect division using blocks to the
division algorithm, ask:

1. How many did you put in each group?

2. How much did you use altogether?

3. How much is left on your board?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – OH5
Multidigit Multiplication and Division Appendix B

MULTIDIGIT MULTIPLICATION AND DIVISION

Participants examine multidigit multiplication and division strategies that
build on students’ understanding of single-digit multiplication and division.
Participants learn about area models for multiplication and division. They
examine the meaning of the traditional multiplication and division
algorithms and variations of these algorithms.

Lesson Goals

Identify advanced strategies that children use to solve multidigit
multiplication and division problems in sense-making ways
Connect concepts of multiplication and division to standard procedures
Discuss teaching strategies that enhance a child’s understanding of
multidigit multiplication and division

Word Bank

skip counting
area model
expanded notation
commutative property of multiplication
associative property of multiplication
distributive property of multiplication

Focus Questions

How does an understanding of the meaning of multiplication and
division affect the appropriate and accurate use of these operations?
What are some interim strategies students can use to develop
multiplication and division procedures?
How do interim strategies connect to the standard algorithms?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP1
Multidigit Multiplication and Division Appendix B

SELECTED CA MATH STANDARDS

Grade 2

NS3.1 Use repeated addition, arrays, and counting by multiples to do multiplication.
NS3.2 Use repeated subtraction, equal sharing, and forming equal groups with remainders to do
division.
NS3.3 Know the multiplication tables for 2s, 5s, and 10s (“two times ten”) and commit them to
memory.
AF1.1 Use the commutative and associative rules to simplify mental calculations and to check
results.

Grade 3

NS1.3 Identify the place value for each digit in numbers to 10,000.
NS1.5 Use expanded notation to represent numbers (e.g. 3,206 = 3,000 + 200 +6).
NS2.1 Find the sum or difference of two whole numbers between 0 and 10,000.
NS2.3 Use the inverse relationship of multiplication and division to compute and check results.
NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers
(3671 x 3 = ).
AF1.1 Represent relationships of quantities in the form of mathematical expressions, equations,
and inequalities.
AF1.5 Recognize and use the commutative and associative properties of multiplication.

Grade 4

NS1.3 Round whole numbers through the millions to the nearest ten, hundred, thousand, ten
thousand, or hundred thousand.
NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for
multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a
one-digit number; use relationships between them to simplify computations and to check results.
NS3.3 Solve problems involving multiplication of multidigit numbers by two-digit numbers.
NS3.4 Solve problems involving division of multidigit numbers by one-digit numbers.
MG1.0 Students understand perimeter and area.

Grade 5

AF1.3 Know and use the distributive property in equations and expressions with variables.

Activity Grade 2 Grade 3 Grade 4 Grade 5
NS 1.3, 1.5, 2.4 NS 1.3
Estimation

Warm up

Multidigit strategies for NS 3.1, 3.2, 3.3 NS 2.1, 2.3, 2.4 NS 3.2, 3.3, 3.4
AF 1.1, 1.5
multiplication & division AF 1.1
NS 1.3, 1.5
Area model- NS 3.1 AF 1.5 MG 1.0 AF 1.3

multiplication NS 1.3, 1.5, 2.3, 2.4
AF 1.5
Multiplication & Division NS 3.2, 3.3, 3.4

algorithms

NS: Number Sense AF: Algebra and Functions

MG: Measurement and Geometry

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP2
Multidigit Multiplication and Division Appendix B

A WARMUP

Find each product:
9) 3 x 4 =
10) 30 x 4 =
11) 300 x 40
12) 3 x 0.4
What strategies did you use to find the products?

Estimate each answer:
13) 34 x 56
14) 383 x 420
15) 15.09 x 3.4
16) 6390 divided by 32

What strategies did you use to estimate?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP3
Multidigit Multiplication and Division Appendix B

27 X 4

Compute 27 x 4 two different ways using pictures, words, or numbers.

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP4
Multidigit Multiplication and Division Appendix B

MULTIPLICATION AND DIVISION STRATEGIES

Direct Modeling Strategies

These typically involve the use of manipulatives (fingers, tally marks, counters, base-10
blocks) to represent the problem. For multiplication, children model using “groups of,” “rows
of,” or “arrays.” For division, modeling takes the form of “dealing” or “measuring.”

Counting strategies

Skip Counting
3, 6, 9, 12, 15…so 3 X 5 = 15

Repeated addition
6 + 6 + 6 =18 so 6 X 3 = 18

Repeated subtraction
18 – 6 = 12 and 12 – 6 = 6 and 6 – 6 = 0 so 18 ÷ 6 = 3

Doubling
8 + 8 = 16 and 8 + 8 = 16 and 16 + 16 = 32 so 8 X 4 = 32

Counting on, counting back
3 X 3 = 9…10,11,12…so 3 X 4 =12
4 X 4 = 16…15,14,13,12…so 4 X 3 = 12

Derived Fact Strategies

Doubling
4 X 6 = 24… so 8 X 6 = 48

Squaring
6 X 6 = 36…+ 7 = 42…so 6 X 7 = 42

Add-on
4 X 6 = 24, 24 + 4 = 28, SO 4 X 7 = 28

Take-away
9 X 10 = 90…- 9 = 81…so 9 X 9 = 81

Grouping strategies

Place value understanding is developed as children group by 1s, 10s, 100s, etc.
20 X 4 = 80 and 3 X 4 = 12 and 80 + 12 = 92 so 23 X 4 = 92

Breaking one number into smaller, more manageable groups
4 X 8 = 32 and 3 X 8 = 24 and 32 + 24 = 56 so 7 X 8 = 56

Other invented strategies

Double/half, estimation, compensation, student inventions – children often invent strategies
which defy classification. Naming strategies in honor of the inventor reinforces respect for
good thinking.

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP5
Multidigit Multiplication and Division Appendix B

UCLA Math Programs for Teachers/LUCIMATH Project MULTIDIGIT MULTIPLICATION AND DIVISION STRATEGIES
Multidigit Multiplication and Division
Problem Advanced Modeling Advanced Counting Advanced Derived Advanced Grouping Other Invented
Strategies Strategies
Strategies Strategies Strategies

24 X 6 = Build 6 rows of 24 using 24 + 24 = 48 Round 24 to 25. 24 is two tens and “I know
base-10 blocks. (That's 2) “25 X 6 = 150. four ones or 20 + 4. 1/2 of 24 is 12.
_____ _____ ....
_____ _____ .... 48 + 48 = 96 But that's too big. 20 X 6 = 120 and
_____ _____ .... (That's 4) (6 groups of 20) 2 X 6 = 12
_____ _____ .... So take away one
_____ _____ .... 48 + 96 = 144 6. 4 X 6 = 24 So
_____ _____ .... (That's 6) (6 groups of 4) 12 X 12 = 144"
150 - 6 = 144”
Then add rectangular 120 + 24 = 144
pieces together

6 X 20 = 120
(6 rows of 20)
6 X 4 = 24
(6 rows of 4)
120 + 24 = 144

Take 120 blocks. Build 5 I want to know how
rows of blocks by dealing
(partitive division) until all many 5s in 120
blocks are used up. Trade
100 block for 10s and 10s Skip count while “I know 5 X 25 = 125. (measurement “I know
for 1s as needed. keeping track of groups. But that's too big. division) 120 ÷ 10=12
_____ _____ ....
120 ÷ 5 = _____ _____ .... “5, 10, 15, 20...120. 20 X 5 =100 and
_____ _____ .... So take one 5 away 4 X 5 = 20 12 X 2=24
_____ _____ ....
_____ _____ .... That's 24 groups.” to get 120. 120 So
120 ÷ 5 = 24”
So it's 24.”

COMP5 – PP6 So 24 5's make 120
Appendix B
Count the number in each
row (24).

Extended from Research in Cognitively Guided Instruction
Created by Shelley Kriegler (4/96)

ANALYZE AND ASSESS

Student Strategies Next step?
A

B

C

D

E

F

G

H

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP7
Multidigit Multiplication and Division Appendix B

STUDENT WORK

What mathematics does this student demonstrate in his/her strategy?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP8
Multidigit Multiplication and Division Appendix B

STUDENT WORK

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP9
Multidigit Multiplication and Division Appendix B

STRATEGIES THAT PROMOTE
CLASSROOM DISCOURSE

STRATEGIES COMMENTS/QUESTIONS

• Provide wait time • Who needs more time?
• Promote full inclusion • Turn to your neighbor.
• Teach listening skills • Listen to your classmates.

• Foster communication skills • Teacher asks presenter to prove it.

• Connect listening with participation • Who thought about it a different way?

• Use questioning to highlight • How did you think about that part in
strategies your head?

• Focus on reasoning • Were you able to follow his thinking
from beginning to end?

• Encourage students to monitor their • Give students a chance to catch their

own explanation own errors. Are you done?

• Facilitate the exchange of ideas • What questions do you have for
Allison?

• Request clarification • Request clarification in order to help
the class follow the presenters

thinking. Where did that come from?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP10
Multidigit Multiplication and Division Appendix B

THE ART OF QUESTIONING
IN MATHEMATICS

(From The NCTM Professional Teaching Standards)

HELP STUDENTS WORK TOGETHER TO MAKE SENSE OF MATHEMATICS:

“What do others think about what ____________ said?”
“Do you agree? Disagree? Why or why not?”
“Does anyone have the same answer but a different way to explain it?”
“Would you ask the rest of the class that question?”
“Do you understand what they are saying?”
“Can you convince the rest of us that that makes sense?”

HELP STUDENTS TO RELY MORE ON THEMSELVES TO DETERMINE WHETHER
SOMETHING IS MATHEMATICALLY CORRECT

“Why do you think that?”
“Why is that true?”
“How did you reach that conclusion?”
“Does that make sense?”
“Can you make a model and show that?”

HELP STUDENTS TO LEARN TO REASON MATHEMATICALLY

“Does that always work? Why or why not?”
“Is that true for all cases? Explain?”
“Can you think of a counter example?”
“How could you prove that?”
“What assumptions are you making?”

HELP STUDENTS LEARN TO CONJECTURE, INVENT, AND SOLVE PROBLEMS

“What would happen if ____________? What if not?”
“Do you see a pattern? Explain?”
“What are some possibilities here?”
“Can you predict the next one? What about the last one?”
“How did you think about the problem?”
“What decision do you think he/she should make?”
“What is alike and what is different about your method of solution and his/hers?”

HELP STUDENT TO CONNECT MATHEMATICS, ITS IDEAS, AND ITS APPLICATIONS

“How does this relate to __________?”
“What ideas that we have learned before were useful in solving this problem?”
“Have we ever solved a problem like this one before?”
“What uses of mathematics did you find in the newspaper last night?”
“Can you give me an example of ___________?”

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP11
Multidigit Multiplication and Division Appendix B

THE IMPORTANCE OF RECORDING

Even when children explain their thinking orally, they may have difficulty putting their
ideas into written form. Teachers who record as students explain their strategies help
them learn to represent their ideas in writing. In the classroom, recordings make
student explanations public, serve as models, allow for comparisons and discussions,
and help other children develop alternative strategies.

Learning to record children’s thinking takes practice. Teachers can develop recording
systems by watching how children record their thinking, and helping them to refine their
methods.

Consider the product: 14 X 3

STUDENT WORDS / IDEAS TEACHER RECORDINGS

USE A MODEL: Use base-10 blocks on overhead or write
_____….
“Three rows of fourteen” _____….
_____….
ADDITION: HORIZONTAL FORM
14 + 14 + 14
“10 + 10 + 10 = 30
4 + 4 + 4 = 12 10 10 10
30 + 12 = 42“
30 12
ADDITION: VERTICAL FORM
42
“10 plus 10 plus 10 is 30. 4 plus 4 plus 4 is 12.
30 and 12 is 42.” 10
10
MULTIPLICATION: 10
HORIZONTAL FORM 30
12
“3 tens is 30. 3 fours is 12. 30 and 12 is 42.” 42
MULTPLICATION: VERTICAL FORM
3 10s = 30 3 x 10 = 30
“3 times 10 is 30. 3 times 4 is 12. 30 + 12 3 4s = 12 3 x 4 = 12
equals 42.” 30 + 12 = 42
GROUPING 42
(BREAK APART NUMBERS)
14
“Think of 14 as 10 + 4. 10 times 3 is 30. x3
4 times 3 is 12. 30 and 12 is 42.” 30
12
42

14 = 10 + 4

10 x 3 = 30 or 10 x 3 = 30

4 x 3 = 12 4 x 3 = 12

30 + 12 = 42 42

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP12
Multidigit Multiplication and Division Appendix B

USING AREA MODELS

Find each product using an area model. Record both pictures and numbers
below.

6 x 12

14 x 15

23 x 13

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP13
Multidigit Multiplication and Division Appendix B

ALGORITHMS FROM AORUND THE WORLD

From a school in Southeast Asia

Right to Left

14 5 x 4 = 20
x25 5 x 10 = 50
20 x 4 = 80
20 20 x 10 = 200
50
80
200
350

From a teacher in France

Left to Right

14 20 x 10 = 200
x25 20 x 4 = 80
200 5 x 10 = 50
5 x4 = 20
80
50
20
350

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP14
Multidigit Multiplication and Division Appendix B

VISUAL MULTIPLICATION ALGORITHMS

From a 6th grade classroom in California

Area Model

20 + 5 20 x 10 = 200
10 200 50 20 x 4 = 80
+ 10 x 5 = 50
4 80 20 5 x 4 = 20

From a popular adopted textbook in the United States

Lattice Multiplication

25 2 hundreds = 200
001 150
15 tens =
25 0
024 0 tens = 350

80
2

15 0

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP15
Multidigit Multiplication and Division Appendix B

BRITISH ALGORITHM

from an Italian student attending a school in Iran
taught by British nuns

648 648 x 200 = 129600
x 279 648 x 70 = 45360
648 x 9= + 5832
180,792

INTERESTING ALGORITHM
From a teacher in Germany

“Nike Math”

(20 + 5)(10 + 4)

First: 20 x 10 = 200

Outside: 20 x 4 = 80

Inside: 5 x 10 = 50

Last: 5 x4 = 20

350

This application of the distributive property is sometimes referred to
as “FOIL” – a procedure for multiplying binomials in algebra

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP16
Multidigit Multiplication and Division Appendix B

RUSSIAN PEASANT METHOD OF
MULTIPLICATION

In the 1800’s, peasants in a remote area of Russia were discovered multiplying
numbers using a remarkably unusual process. This process, known as the
“Russian peasant method” of multiplication, is said to be still in use in some
parts of Russia.

Assume you want to multiply 18 x 25.

Halve this column: Double this
Discard remainders column

18 x 25

9 50 Cross out all the rows which have

an even number on the left, then

4 100 add up all the remaining numbers

on the right.

2 200

1 400
450

Use the Russian peasant method of multiplication to compute these
products.

1) 20 x 25 ________ 3) 12 x 25 _________

2) 16 x 30 __________ 4) 22 x 75 _____________

Why and how does the Russian Peasant Method work?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP17
Multidigit Multiplication and Division Appendix B

JOURNAL 1: ALGORITHMS AROUND THE WORLD

The algorithms on the previous pages represent different ways that students from
around the world were taught to multiply. Select one of the algorithms from
around the world. Explain how it works and why it works.

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP18
Multidigit Multiplication and Division Appendix B

BROWN AND GREEN

Solve each problem in two different “kid” ways. Be prepared to share your
strategies.
1. Mrs. Brown has a bag of 258 candies. She wants to share them equally among

her 22 students. How many candies will each student get?
[Partitive Division]

2. Mrs. Green has 35 yards of fabric, which she is using to make jerseys for the
soccer team. Each jersey requires 2 yards of fabric. How many jerseys can
Mrs. Green make?
[Measurement Division]

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP19
Multidigit Multiplication and Division Appendix B

BROWN

Mrs. Brown has a bag of 258 candies. She wants to share them equally among her 22
students. How many candies will each student get?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP20
Multidigit Multiplication and Division Appendix B

GREEN

Mrs. Green has 35 yards of fabric, which she is using to make jerseys for the
soccer team. Each jersey requires 2 yards of fabric. How many jerseys can Mrs.
Green make?

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP21
Multidigit Multiplication and Division Appendix B

MORE STUDENT WORK

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP22
Multidigit Multiplication and Division Appendix B

JOURNAL 2: BASE-10 BLOCK DIVISION

Describe how you might connect base-10 block model to the division
algorithm.

UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP23
Multidigit Multiplication and Division Appendix B

PROBLEM STARTERS

Find the solution to each problem in a non-traditional way using the suggested starters.
Can you think of another non-traditional way to computer the answer?

Problem: Compute 14 X 9

Start with 10 X 9
Start with 7 X 9
Start with 14 X 10

Problem: Compute 18 X 43
Start with 10 X 43
Start with 2 X 43 and 20 X 43
Start by listing the first four multiples of 18

Problem: Compute 703 ÷ 17

Start with 17 X 10
Start by listing a few multiples of 17 and finding the largest one less than 703
Start by drawing a rectangle with one side of 17

Problem: Compute 504 ÷ 70

Start with 70 + 70
Start with 7 X 7
Start with 70 X 10

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Multidigit Multiplication and Division Appendix B

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Multidigit Multiplication and Division Appendix B

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UCLA Math Programs for Teachers/LUCIMATH Project COMP5 – PP30
Multidigit Multiplication and Division Appendix B