Download 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 Content Program for Teachers
Multidigit Multiplication and Division
COMP5 – PG1
Appendix B
MULTIDIGIT MULTIPLICATION AND DIVISION
(Estimated Time: 4 hours)
Lesson Summary
Lesson Goals
Word Bank
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.
• 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
skip counting
area model
expanded notation
commutative property of multiplication
associative property of multiplication
distributive property of multiplication
Materials
Reproducibles
Blank paper
3-4 LUCIMATH Video
TV/VCR
Base-10 blocks
OH base-10 blocks
Overhead transparencies and pens
Chart paper and markers
R1-3*: Teaching and Learning Notes
Participant Pages
Overhead Transparencies
Focus Questions
PP1: Summary Page
PP2: Selected CA Math Standards
PP3: A Warmup
PP4: 27 X 4
PP5-6: Multiplication and Division
Strategies
PP7: Analyze and Assess
PP8-9: Student Work
PP10: Strategies that Promote Classroom
Discourse
PP11: The Art of Questioning in
Mathematics
PP12: The Importance of Recording
PP13: Using Area Models
PP14-17: Algorithms from Around the
World
PP18: Journal 1: Algorithms Around the
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
OH1: Focus Questions
OH2: CA Math Standards – In Brief
OH3: A Warmup
OH4: Multiplication and Division
Strategies
OH5: Division 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 might we increase parental
involvement for children?
Problem of the Week
Journal Idea
Assessment Idea
Problem Starters*
Select one of the algorithms from
around the world. Explain how it
works and why it works.
Write a story problem to go along with
380÷20 = 19. Solve problem two different
ways. Identify problem type and
strategies used to solve it.
*Use as preview or review
Strategies for Special Needs
• Use physical models (ALL)
• Ask students to explain strategies to
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
*distribute as desired
UCLA Math Content Program for Teachers
Multidigit Multiplication and Division
COMP5 – PG2
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
Multidigit Multiplication and Division
COMP5 – PG3
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 “thinkpair-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
(count in chunks)
Counting
(repeated addition,
skip counting)
Modeling
(using 10s and 1s)
Written Records
(using 10s and 1s)
Other invented strategies
(Double half, benchmarks, distributive property)
Algorithm
(compact recording)
UCLA Math Content Program for Teachers
Multidigit Multiplication and Division
COMP5 – PG4
Appendix B
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
Multidigit Multiplication and Division
COMP5 – PG5
Appendix B
#
Who
1
Susan
Aldridge’s
4th Grade
Class
Problem
24 X 9
Strategy
Talking Points
Notice how teacher records Sofia’s
Sofia (girl with pigtails)
benchmark numbers, repeated strategy.
addition, compensation.
Ken: Comment on teacher recording
Ken (boy with red shirt)
benchmark numbers, doubles, strategy (note error in writing
equalities; teacher might use arrows
incremental addition.
instead).
Explain how Alison uses distributive
Alison (gray sweatshirt)
grouping, distributive property. property to multiply.
Explain how Rachel uses
Rachel (girl with glasses)
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
Multidigit Multiplication and Division
COMP5 – PG6
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
12 X 10 = 120
12 X 3 = 36
156
12 rows of 13
10 X 13 = 130
2 X 13 = 26
156
13 rows of 12
13 X 10 = 130
13 X 2 = 26
156
Rectangle 12 X 3
100 + 20 + 30 + 6
= 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
Multidigit Multiplication and Division
COMP5 – PG7
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
12 x10 =
12 x 3 =
Traditional Algorithm
12
x 13
120
36
156
UCLA Math Content Program for Teachers
Multidigit Multiplication and Division
12
x 13
36
12
156
Rectangle
12
x 13
6
30
20
100
156
=2x3
= 10 x 3
= 10 x 2
= 10 x 10
COMP5 – PG8
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
143
x 23
2000
800
300
60
120
+ 9
3289
100
20
+
40
+
3
2000
800
60
300
120
9
+
3
Show how the area model extends for multiplying polynomials.
x
Multiply (x+2) (x+3)
x
2
= x + 2x + 3x + 6
+
2
x2
2x
3x
6
+
= x2 + 5x + 6
3
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
Multidigit Multiplication and Division
COMP5 – PG9
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
Multidigit Multiplication and Division
COMP5 – PG10
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 asks:
Marilyn
• How many did you put in each
Burns with
Connects base-10 blocks
group?
2
435 divided by 3
to division algorithm
three 4th
• How much did you use
graders
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
Estimation
Warm up
Multidigit strategies for
multiplication & division
Area modelmultiplication
Multiplication & Division
algorithms
Grade 2
NS 3.1, 3.2, 3.3
AF 1.1
NS 3.1
Grade 3
NS 1.3, 1.5, 2.4
Grade 4
NS 1.3
NS 2.1, 2.3, 2.4
AF 1.1, 1.5
AF 1.5
NS 1.3, 1.5
NS 1.3, 1.5, 2.3, 2.4
AF 1.5
NS 3.2, 3.3, 3.4
MG 1.0
Grade 5
AF 1.3
NS 3.2, 3.3, 3.4
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
Multidigit Multiplication and Division
COMP5 – PG11
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 sensemaking 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
(count in chunks)
Counting
(repeated addition,
skip counting)
Modeling
(using 10s and 1s)
Written Records
(using 10s and 1s)
Other invented strategies
(Double half, benchmarks, distributive property)
Algorithm
(compact recording)
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – R1
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
12 X 3 = 36
156
13 X 10 = 130
13 X 2 = 26
156
Rectangle 12 X 3
12 rows of 13
100 + 20 + 30 + 6
= 156
10 X 13 = 130
2 X 13 = 26
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
12 x10 =
12 x 3 =
Traditional Algorithm
12
x 13
120
36
156
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
12
x 13
36
12
156
Rectangle
12
x 13
6
30
20
100
156
=2x3
= 10 x 3
= 10 x 2
= 10 x 10
COMP5 – R2
Appendix B
Multiply 23 X 143
143
x 23
2000
800
300
60
120
+ 9
3289
100
20
+
40
800
60
300
120
9
+
3
x
= x + 2x + 3x + 6
2
= x + 5x + 6
3
2000
Multiply (x+2) (x+3)
2
+
x
+
2
x2
2x
3x
6
+
3
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
Multidigit Multiplication and Division
COMP5 – R3
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
Multidigit Multiplication and Division
COMP5 – OH1
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
Multidigit Multiplication and Division
COMP5 – OH2
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
Multidigit Multiplication and Division
COMP5 – OH3
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
Multidigit Multiplication and Division
COMP5 – OH4
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
Multidigit Multiplication and Division
COMP5 – OH5
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
Multidigit Multiplication and Division
COMP5 – PP1
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
Estimation
Warm up
Multidigit strategies for
NS 3.1, 3.2, 3.3
multiplication & division
AF 1.1
Area modelNS 3.1
multiplication
Multiplication & Division
algorithms
NS: Number Sense
MG: Measurement and Geometry
Grade 3
NS 1.3, 1.5, 2.4
Grade 4
NS 1.3
NS 2.1, 2.3, 2.4
AF 1.1, 1.5
NS 1.3, 1.5
AF 1.5
NS 1.3, 1.5, 2.3, 2.4
AF 1.5
NS 3.2, 3.3, 3.4
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
MG 1.0
Grade 5
AF 1.3
NS 3.2, 3.3, 3.4
AF: Algebra and Functions
COMP5 – PP2
Appendix B
A WARMUP
Find each product:
9)
3x4=
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
Multidigit Multiplication and Division
COMP5 – PP3
Appendix B
27 X 4
Compute 27 x 4 two different ways using pictures, words, or numbers.
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP4
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
Multidigit Multiplication and Division
COMP5 – PP5
Appendix B
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP6
Appendix B
Count the number in each
row (24).
Take 120 blocks. Build 5
rows of blocks by dealing
(partitive division) until all
blocks are used up. Trade
100 block for 10s and 10s
for 1s as needed.
_____ _____ ....
_____ _____ ....
_____ _____ ....
_____ _____ ....
_____ _____ ....
6 X 20 = 120
(6 rows of 20)
6 X 4 = 24
(6 rows of 4)
120 + 24 = 144
Then add rectangular
pieces together
Build 6 rows of 24 using
base-10 blocks.
_____ _____ ....
_____ _____ ....
_____ _____ ....
_____ _____ ....
_____ _____ ....
_____ _____ ....
Advanced Modeling
Strategies
150 - 6 = 144”
So take away one
6.
But that's too big.
Round 24 to 25.
“25 X 6 = 150.
Advanced Derived
Strategies
120 + 24 = 144
4 X 6 = 24
(6 groups of 4)
20 X 6 = 120
(6 groups of 20)
24 is two tens and
four ones or 20 + 4.
Advanced Grouping
Strategies
So it's 24.”
So 24 5's make 120
I want to know how
many 5s in 120
“I know 5 X 25 = 125. (measurement
Skip count while
division)
keeping track of groups.
But that's too big.
20 X 5 =100
“5, 10, 15, 20...120.
4 X 5 = 20
So take one 5 away
to get 120.
120
That's 24 groups.”
48 + 96 = 144
(That's 6)
48 + 48 = 96
(That's 4)
24 + 24 = 48
(That's 2)
Advanced Counting
Strategies
Extended from Research in Cognitively Guided Instruction
Created by Shelley Kriegler (4/96)
120 ÷ 5 =
24 X 6 =
Problem
12 X 12 = 144"
2 X 6 = 12
So
and
120 ÷ 5 = 24”
12 X 2=24
“I know
120 ÷ 10=12
So
and
“I know
1/2 of 24 is 12.
Other Invented
Strategies
MULTIDIGIT MULTIPLICATION AND DIVISION STRATEGIES
ANALYZE AND ASSESS
Student
Strategies
Next step?
A
B
C
D
E
F
G
H
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP7
Appendix B
STUDENT WORK
What mathematics does this student demonstrate in his/her strategy?
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP8
Appendix B
STUDENT WORK
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP9
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
strategies
• How did you think about that part in
your head?
• Focus on reasoning
• Were you able to follow his thinking
from beginning to end?
• Encourage students to monitor their
own explanation
• Give students a chance to catch their
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
Multidigit Multiplication and Division
COMP5 – PP10
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
Multidigit Multiplication and Division
COMP5 – PP11
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
TEACHER RECORDINGS
STUDENT WORDS / IDEAS
USE A MODEL:
“Three rows of fourteen”
Use base-10 blocks on overhead or write
_____….
_____….
_____….
14 + 14 + 14
ADDITION: HORIZONTAL FORM
10
“10 + 10 + 10 = 30
4 + 4 + 4 = 12
30 + 12 = 42“
30
3 10s
3 4s
30 + 12
=
=
=
“3 times 10 is 30. 3 times 4 is 12. 30 + 12
equals 42.”
“Think of 14 as 10 + 4. 10 times 3 is 30.
4 times 3 is 12. 30 and 12 is 42.”
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
30
12
42
3 x 10 = 30
3 x 4 = 12
42
14
x3
30
12
42
MULTPLICATION: VERTICAL FORM
GROUPING
(BREAK APART NUMBERS)
12
10
10
10
30
12
42
“10 plus 10 plus 10 is 30. 4 plus 4 plus 4 is 12.
30 and 12 is 42.”
“3 tens is 30. 3 fours is 12. 30 and 12 is 42.”
10
42
ADDITION: VERTICAL FORM
MULTIPLICATION:
HORIZONTAL FORM
10
14 = 10 + 4
10 x 3 = 30 or
4 x 3 = 12
30 + 12 = 42
10 x 3 = 30
4 x 3 = 12
42
COMP5 – PP12
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
Multidigit Multiplication and Division
COMP5 – PP13
Appendix B
ALGORITHMS FROM AORUND THE WORLD
From a school in Southeast Asia
Right to Left
14
x25
20
50
80
200
350
5
5
20
20
x
x
x
x
4
10
4
10
=
=
=
=
20
50
80
200
From a teacher in France
Left to Right
14
x25
200
80
50
20
350
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
20
20
5
5
x
x
x
x
10
4
10
4
=
=
=
=
200
80
50
20
COMP5 – PP14
Appendix B
VISUAL MULTIPLICATION ALGORITHMS
From a 6th grade classroom in California
Area Model
10
+
4
20 + 5
200 50
80
20
20
10
5
20
x
x
x
x
10 =
4 =
5 =
4 =
200
80
50
20
From a popular adopted textbook in the United States
Lattice Multiplication
2
0
2
0
8
5
0
5
2
0
15
0
1
4
2 hundreds =
15 tens
=
0 tens
=
200
150
0
350
2
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP15
Appendix B
BRITISH ALGORITHM
from an Italian student attending a school in Iran
taught by British nuns
648
x 279
648
648
648
x
x
x
200
70
9
=
=
=
129600
45360
+ 5832
180,792
INTERESTING ALGORITHM
From a teacher in Germany
“Nike Math”
(20 + 5)(10 + 4)
First:
Outside:
Inside:
Last:
20
20
x
x
10 =
4 =
200
80
5
x
10
=
50
5
x
4
=
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
Multidigit Multiplication and Division
COMP5 – PP16
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:
Discard remainders
18
Double this
column
x
25
9
50
4
100
Cross out all the rows which have
an even number on the left, then
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
Multidigit Multiplication and Division
COMP5 – PP17
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
Multidigit Multiplication and Division
COMP5 – PP18
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
Multidigit Multiplication and Division
COMP5 – PP19
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
Multidigit Multiplication and Division
COMP5 – PP20
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
Multidigit Multiplication and Division
COMP5 – PP21
Appendix B
MORE STUDENT WORK
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP22
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
Multidigit Multiplication and Division
COMP5 – PP23
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
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP24
Appendix B
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP25
Appendix B
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP26
Appendix B
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP27
Appendix B
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP28
Appendix B
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP29
Appendix B
UCLA Math Programs for Teachers/LUCIMATH Project
Multidigit Multiplication and Division
COMP5 – PP30
Appendix B