Grade 4 Mathematics Lesson Plan
Date:
June 27 (Wed.), 2007, Period 5
Teachers
A: Masako Adachi (Classroom 1)
B: Akihiko Suzuki (Classroom 2)
C: Masako Koizumi (Math Open)
Research Theme:
Nurturing students to become people who can be trusted in an international society: Developing instruction that will
foster students’ ability to communicate
1.
Name of the unit:
2.
About the unit
Let’s think about how to divide
Flow of the contents
Grade 3
Grade 4
(3) Division
• Meanings of
division and the
division sign
• Division that can be
solved by using the
basic multiplication
facts (no
remainder)
This unit: The division algorithm (1)
• The division algorithm for a 2- or
3-digit number divided by a 1-digit
number
• Methods of calculating math
sentences with
o both multiplication and
division operations
o consecutive division
Extending the meanings of division
and “times as much” (1st and 3rd uses
of times as much)
(7) Division with
remainders
• Division that can be
solved by using the
basic multiplication
facts (with
remainders)
• How to check
answers
• Meaning of
remainders
Division algorithm (2)
• Dividing by multiples of 10
• Division algorithm for 2- or 3-digit
numbers ÷ 1-digit numbers
• How to check answers for division
• Meaning of a tentative quotient and
how to adjust it
• Properties of division
Grade 5
(2) Multiplication and
division of decimals (I)
Division of whole numbers
with decimal quotients and
dividing decimals by whole
numbers
(4) Multiplication and
division of decimals (II)
• Meaning of dividing by a
decimal and how to
calculate
• The division algorithm
for dividing whole
numbers and s by
decimals
(6) Fractions and Decimals
• Quotients of whole
numbers can always be
written as fractions
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources for the
Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
Goals of the unit
 Students will understand the division algorithm for
dividing 2- and 3-digit numbers by 1-digit numbers and be
able to use it appropriately.
[Interest, Desire, Attitude]
• Students will try to use their previous study of division to
think about how to divide 2- and 3-digit numbers by
1-digit number.
[Mathematical Thinking]
• Students will be able to clearly explain that the division
algorithm proceeds from the tens digit and then the ones
when you are divide 2-digit numbers by 1-digit numbers.
• Students will be able to clearly explain that the division
algorithm for dividing 3-digit numbers by 1-digit
numbers works in the same way as was learned
previously (2-digit ÷ 1-digit).
[Representations, Procedures]
• Students will be able to accurately calculate 2- and
3-digit numbers ÷ 1-digit numbers.
• Students will be able to mentally calculate 2- and 3-digit
numbers ÷ 1-digit numbers with 2-digit quotients.
[Knowledge, Understanding]
• Students will understand how to divide 2- and 3-digit
numbers by 1-digit numbers using the division algorithm.
• Students will understand that they can use division to
determine “times as much”.
• Students will understand that one math sentence may be
used even when situations involve both multiplication
and division or two consecutive divisions.
Current state of the students
According to the results of the readiness
test, virtually all students can accurately
calculate division problems that use the basic
multiplication facts once, which was studied in
Grade 3. Although a few students missed
remainders on some problems, all problems
were correctly answered by at least 97% of the
students.
There were 2 problems that involve the
ideas to be studied in Grade 4: (1) 40 ÷ 2, and
(2) 600 ÷ 3. Although it seems like students
should be able to anticipate the answers, the
success rates were (1) 85% and (2) 70%.
Moreover, for both problems, 9% of the
students left them blank.
Given the success rates on the items
involving previously learned topics, we would
like to plan a lesson that will take advantage of
their ability to read and interpret math sentences
and further extend this ability. However, in the
prior unit on circles and spheres, some students
had difficulty using compasses appropriately,
indicating that some children still lack fine
motor skills.
Moreover, some students’ understanding
may still be rather superficial, and their
computational skills are more advanced than
their ability to think and reason. Therefore, we
want to provide opportunities for students to
enjoy solving problems by thinking carefully
and manipulating objects.
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
3.
Abilities we would like to foster in this unit and special instructional strategies
Target abilities for this unit
1. To make well-grounded
explanations
[Using prior understanding as the
foundation for reasoning]
• Ability to use properties of
division in explanations instead
of just learning the algorithm
procedurally.
• Ability to explain the reasons for
the steps of the division
algorithm instead of simply
reciting, “estimate, multiply,
subtract, and bring-down.”
2. To communicate effectively
[Creating concise representations,
diagrams and writing]
• Ability to describe situations
involving “times as much” and
division appropriately, utilizing
diagrams and math sentences
effectively.
3. To perceive and reason from
multiple perspectives
[Selecting more appropriate
reasoning]
• Ability to interpret math
sentences to know how others
thought about problems
involving both multiplication
and division.
• Ability to select more
appropriate reasoning using the
criteria: efficient, simple, and
accurate (ESA).
Instructional strategies
1. Level-raising strategies
The first 3 lessons of the unit will be used to discover the
properties of division.
To improve students’ ability to interpret math sentences, we
will frequently have them sort shared solutions and
strategies.
Repeatedly ask students to use diagrams and math sentences
as they explain.
To make number lines a more familiar tool, we will
frequently ask students to label the number lines.
Use missing number problems to promote algebraic
reasoning while exploring the algorithm and the properties
of division.
Employ small group instruction.
2. Effective learning tasks
• Use carefully selected numbers in the task to make it
more realistic (problem involving both x and ÷).
• Use a familiar setting to make the task more
interesting (times as much).
• Use a new type of problem to think deeply about the
division algorithm and the properties of division
(missing number).
3. Question posing
• In all lessons, include as many questions as possible
to encourage students to look for more appropriate
reasoning, foe example, “Which operation is more
‘ESA’ to calculate times as much/” or “Which
strategy is more ‘ESA’?”
• Carefully sequence students’ strategies to be shared
publicly, and ask students to share their observations
so that they may recognize similarities in reasoning.
4. Interpreting students’ reasoning.
• Throughout every lesson, ask students to record their
thinking in their notes to capture the change in their
thinking.
• Have students write in a learning journal.
Note: Arrows indicate how particular strategies may foster specific target abilitie
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
3. Group research – examining the research theme
(1) On “times as much”
(2) On the use of number line model
(3) Proposals on “times as much” and number line model
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
4. Plan of instruction
Plan for Class 1 (Ms. Adachi)
# Goals
Learning Task
(1) Properties of division
1
Readiness test
Administer the test.
To discover patterns in
• To discover patterns
division when the
from the math
dividend stays constant.
sentence, 24÷○=□.
2
To discover patterns in
• To discover patterns
division when the divisor
from the math
stays constant.
sentence, □÷ 3 = ○.
3
To discover patterns in
division when the
quotient stays constant.
• To discover patterns
from the math
sentence, □÷ ○ = 3.
(2) Dividing multiples of 10 and 100
4/
• To understand how the • To think about how to
5
division algorithm
calculate 80 ÷ 4.
works when dividing
• To practice similar
multiples of 10 and
problems.
100 by 1-digit
• To think about how to
numbers.
calculate 240 ÷ 6.
• To practice similar
problems.
(3) Division algorithm (I) [2-digit ÷ 1-digit]
6/
• To understand how the • To understand the the
7
division algorithm
problem situation and
works when dividing
write a math sentence.
2-digit numbers by
• To think about how to
1-digit numbers (no
calculate 52 ÷ 4.
remainder)
• To summarize how the
algorithm works with
52 ÷ 4.
• To check the answer
for 52 ÷ 4.
• To practice the
algorithm.
Evaluation
Preassessment
• Are Ss trying to discover
patterns in division?
(Interest)
• Can Ss discover that when
the dividend becomes △
times as much, the
quotient also becomes △
times as much (constant
divisor)? (Mathematical
Thinking)
• Can Ss discover patterns
and represent them clearly
(constant quotient)?
(Representation)
• Can Ss consider the
dividends using 10 as a
unit and apply the basic
multiplication facts to find
the quotient? (Math
Thinking)
• Can Ss divide multiples of
10 and 100 by a 1-digit
number by using the
relative sizes of the
dividends?
(Representations)
Class 2
(Suzuki)
(1)
(1) – 1
Open
(1)
(1) – 1
(1) – 2
(1) – 2
(1) – 3
(1) – 3
(2)
(2) –
4/5
(2)
(2)
(Koizumi)
(3)
(3)
• Are Ss trying to use their
(3) –
(3) –
prior learning to think
6/7
6/7
about how to calculate
2-digit ÷ 1-digit?
(Interest)
• Can Ss explain clearly
that the division algorithm
should proceed from the
tens digit and then the
ones when you are
dividing 2-digit numbers
by 1-digit numbers?
(Mathematical Thinking)
8
To understand how the
• To understand the
• Can Ss. calculate 2-digit ÷
(3) – 8
(3) – 8
division algorithm works
problem situation and
1-digit using the division
when dividing 2-digit
write a math sentence.
algorithm (with remainder
numbers by 1-digit
• To think about how to
and neither place is
numbers (with
use the division
divisible)?
remainder, neither place
algorithm for 76 ÷ 3.
(Representations)
is divisible).
• To know what the
• Do students know how to
“quotient” is.
divide 2-digit numbers by
• To check the answer
1-digit numbers (with
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
for 76 ÷ 3.
• To practice similar
problems.
• To read the “Math
Corner” in the text and
learn about the terms,
“sum,” “difference,”
and “product.”
• To think about how to
calculate 86 ÷ 4 and 62
÷ 3.
• To practice similar
problems.
• To practice dividing
2-digit numbers by
1-digit numbers (with
remainder and solvable
by using the basic
multiplication facts
once).
9
remainder and neither
place is divisible)?
(Knowledge)
• To understand how the
• Can Ss divide 2-digit
division algorithm
numbers by 1-digit
works when dividing
numbers (with remainder
2-digit numbers by
and solvable by using the
1-digit numbers (with
basic multiplication facts
remainder but the tens
once)? (Representations)
place is divisible).
• Do Ss understand how the
• To be able to divide
division algorithm works
2-digit numbers by
when dividing 2-digit
1-digit numbers by
numbers by 1-digit
using the algorithm
numbers (with remainder
(with remainder and
but the tens place is
solvable by using the
divisible)? (Knowledge)
basic multiplication
facts once).
(4) Patterns in the division algorithm and math sentences involving division
10 / • To understand that
• By using the
• Are Ss thinking logically
11
there are patterns in
relationships, divisor >
as they look for the digit
the division algorithm
remainder and quotient
in each □? (Mathematical
through examination
x divisor < dividend,
Thinking)
of missing-digit
find the numerals that
• Can Ss use the remainder
problems. (Lesson B
go in the □.
to figure out the quotient?
in Class 2)
• By thinking about the
(Representations)
digit in the ones place
• Can Ss discover the
of the dividend,
relationship between the
explore the
dividend and the divisor
relationship between
by looking at the three
the quotient and the
problems?
dividend.
(3) – 9
(3) – 9
(5) – 14
(4) – 12
(5) - 15
(4) - 13
12
(4) – 10
[lesson
A]
(5) – 14
[lesson
C]
• To understand that
situations that involve
both multiplication
and division or two
consecutive divisions
may be written as
single math
sentences. (Lesson A
in Class 1)
• Determine the number
of pencils each person
will receive when 4
dozen pencils are
shared among 6
people.
• To understand that
two-step problems
involving both
multiplication and
division or two
divisions can be
written as single math
sentences.
• Can Ss explain why
situations that involve
both multiplication and
division or two
consecutive divisions can
be written as single math
sentences? (Mathematical
Thinking)
• Do Ss understand how to
calculate math sentences
that contain both
multiplication and
division or two
consecutive divisions?
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
13
• To master what they
have been studying so
far.
• To practice problems
involving both
multiplication and
division and two
consecutive divisions.
• To complete “Let’s
master” in the
textbook.
(Knowledge)
• Are Ss appropriately using
what they have learned so
far to solve problems
correctly?
(Representations)
(4) – 11
(5) – 15
(5) Calculation involving “times as much”
14
To understand that
• To think about what
• Are Ss trying to utilize
(4) – 12 (4) – 10
division is used to
operation should be
diagrams such as number
determine how many
used to determine how
lines to grasp relationships
times as much a given
many times as long 15
among quantities?
quantity is as the base
m is as 3m by using
(Interest)
quantity. (Lesson C in
diagrams.
• Do Ss understand that
Open)
• To learn that division
division may be used to
can be used to
determine how many
determine how many
times as much?
times as much, and to
(Knowledge)
work on application
problems.
15
• To understand that
• To think about what
• Are Ss thinking about
(4) – 13 (4) – 11
division can be used
operation to use to
math sentences that will
to determine the base
determine the base
help them determine the
quantity given the
quantity given 72 kg is
base quantity?
compared quantity,
6 times as heavy as the
(Mathematical Thinking)
and how many times
base quantity.
• Do Ss understand that
it is as much as the
• To learn that division
division may be used to
base quantity.
can be used to
determine the base
determine the base
quantity? (Knowledge)
quantity, and to work
on application
problems.
(6) Division (3) (3-digit ÷ 1-digit)
16
• To understand how
• To understand the
• Are Ss thinking about
(6) – 16 (6) – 16
the division algorithm
problem situation and
3-digit ÷ 1-digit in the
works when dividing
write a math sentence.
same manner as 2-digit ÷
3-digit numbers by
• To think about how to
1-digit? (Mathematical
1-digit numbers (no
use the division
Thinking)
remainder, but the
algorithm for 734 ÷ 5.
• Can Ss accurately
tens and hundreds
• To summarize how to
calculate 3-digit
places are not
use the division
÷1-digit=3-digit?
divisible).
algorithm to calculate
(Representations)
734 ÷ 5.
• To practice similar
problems.
17
• To understand how
• To think about how to
• Can Ss calculate 3-digit ÷
(6) – 17 (6) – 17
the division algorithm
use the division
1-digit = 3-digit (with 0 in
works when dividing
algorithm to calculate
the quotient)?
3-digit numbers by
843 ÷ 4 and 619 ÷ 3.
(Representations)
1-digit numbers (with • To practice similar
0 in the quotient).
problems.
18
• To understand how
• To understand the
• Are Ss trying to relate to
(6) – 18 (6) – 18
the division algorithm
problem situation and
what they have learned
works when dividing
write a math sentence.
previously? (Interest)
3-digit numbers by
• To think about how to
• Can Ss accurately
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
1-digit numbers
(2-digit quotient, i.e.,
the leading digit <
divisor).
(7) Mental computation
19
• To understand how to
mentally calculate
2-digit ÷ 1-digit =
1-digit and divisions
of multiples of 10 and
100 by 1-digit
numbers, and to be
able to actually
perform such
calculations.
use the division
algorithm for 256 ÷ 4.
• To summarize how to
use the division
algorithm to calculate
256 ÷ 4.
• To check the answer
for 256 ÷ 4.
• To practice similar
problems.
calculate 3-digit ÷ 1-digit
= 2-digit?
(Representations)
• Do Ss understand how to
use long division to
calculate 3-digit ÷ 1-digit
= 2-digit? (Knowledge)
• To think about how to
calculate 74 ÷ 2
mentally.
• To practice similar
problems.
• To think about how to
calculate 740 ÷ 2
mentally.
• To practice similar
problems.
• Are students connecting
mental calculation of 2- or
3-digit ÷ 1-digit to their
prior learning by
decomposing the
dividends or considering
the relative sizes of the
dividends (i.e., using 10 or
100 as units)?
(Mathematical Thinking)
• Can Ss mentally calculate
2- or 3-digit ÷ 1-digit?
(Representations)
[ 1 ~ 2 lessons ]
• To complete the “Let’s • Do Ss understand the
check” section of the
contents? (Knowledge)
textbook.
(8) Summarizing
20
• To review and assess
students’
understanding of the
contents.
To deepen students’ understanding of the division algorithm by working on
“Challenge” on p. 96 of the textbook.
(7) – 19
(7) – 19
(8) – 20
(8) - 20
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
Lesson Plan B (Class #1, Ms. Adachi)
1. Goals of the lesson:
•
•
Students will be able to write math sentences for problems that require both multiplication and division
operations.
Students will understand that even when both multiplication and division operations are involved, the
situation can be represented in a single math sentence.
2. Flow of the lesson
Steps
Questions ()
Learning activities ()
Anticipated responses (C’s)
1. Understand the
task.
 Read the problem.
If you share  dozen pencils among  people, how many pencils
will each person receive?
 What operation do
we need? (Why?)
Understand
2. Solve the problem.
 Let’s think about it
in many different
ways.
Examine
T1 There are 8 dozen pencils. 6 people will share the
pencils.
C1 Maybe multiplication.
C2 Must be division because we are sharing equally.
C3 I think we need both because we don’t know the
number of pencils.
C4 I have a question. Do we need to think about the
number of pencils in the boxes?
T2 The question says “how many pencils,” so we do
have to think about the pencils in the boxes.
C5 How many pencils are in the boxes?
C6 It says 1 dozen, so there must be 12 pencils.
C7 Should we draw a diagram?
T3 It is up to you. If you think a diagram will be
helpful, please draw it.
Let’s work on this problem. Pay close attention to
what operation you will be using.
 Individual problem solving
C1 12 x 8 ÷ 6 = 16. Answer 16 pencils. (diagram)
C2 12 x 8 = 96 96 ÷ 6 = 16
C3 12 ÷ 6 x 8 = 16
C4 12 ÷ 6 = 2 2 x 8 = 16
C5 8 ÷ 6 = 1 rem. 2
12 x 2 = 24 24 ÷ 6 = 4
12 x 1 = 12 12 + 4 = 16
C6 8 ÷ 6 = 1 rem. 2 (confuses 2 dozen and 2 pencils)
12 x 1 + 2 = 14 Answer 14 pencils.
C7 When calculating the number of pencils, instead of
(# of pencils/dozen) x (# of dozens), writes (# of
boxes) x (# of pencils/dozen).
C8 Makes computational errors while using one of the
strategies for C1 ~ C4.
C9 Only draws a diagram.
C10 Cannot even get started.
Points of consideration (*)
Evaluation
Strategies for improving
communication ()
Materials: 8 dozens of pencils
 Problem that focuses
students’ attention on the
quantities in the situations.
 Problem statement is similar
to familiar division situations.
* C1 ~ C5 Encourage them to
think about other ways to solve
the problem.
* Write mainly math sentences
on the whiteboards.
* It is ok if not everyone solves
the problem correctly.
C7 Points out that the
multiplicand should be the
number of pencils in a box.
C8 Asks student to check
his/her calculation.
C9 Encourage student to think
about what math sentences can
be used.
C10 Provide concrete objects to
use.
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
3. Think about more
appropriate strategies.
 Let’s share our
solutions.
 Look at all the
solutions. Let’s
organize these
solutions.
 Let’s sort them
according to their
reasoning strategies.
 Listen to fellow classmates’ reasoning and compare
solution methods.
C1 Some of them start with 12 while others start with
8.
C2 Some start with multiplication while others start
with division.
C3 Some have only one math sentence while others
have 2.
* There may not be enough
space on the board to post all
students’ work.
 Start with the students seated
near the hallway, and have them
move the whiteboards around to
organize the solutions.
* We will not sort the strategies
based on the types of diagrams
used.
Deepen
Example 1: C1, C2, C3, C4, C5, and C6 are all
considered different.
Example 2: Those with one math sentence and
those with more than one.
Example 3: Those that start with 8 and those that
start with 12.
Example 4: Those that use multiplication first and
those that use division first.
Example 5: Focus on the objects involved in the
problem: those that shared pencils in each box first,
those that shared the boxes first then dealt with the
left over boxes, and those that figured out the total
number of pencils first.
C4 Some have different answers.
 Let’s explain.
The answer for 2-C6 is 14 pencils. There is no computation error, is there?
So, why is it different?
T1 Is there any other solution that used the same
reasoning as 2-C6?
C5 2-C5.
C6 They both found how many dozen pencils each
person will get.
T2 So is the problem the ways they looked at the
remainders?
C7 When we do 8 ÷ 6, we know that each person will
receive 1 dozen, and there will be 2 dozen left.
Since the remainder is not 2 pencils, I don’t think
we can add 2 to 12 pencils.
T3 So, the remainder tells us that there are 2 more
dozen, or 24 pencils, left to be shared. We
calculated how many dozen pencils each person
receives, so the remainder should also be in
dozens, as 2-C5 did.
* If no one does 2-C6, suggest
it as an example of an incorrect
answer. (To understand the
meaning of the remainder.)
 Use a callout and write,
“First ,find how many dozen
pencils there are for each
person.”
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
 Did you
notice
anything?
4. Summarize today’s
lesson and reflect.
 Which reasoning
or math sentence is
the best? Let’s use
“ESA.”
T4 Now that we have sorted these solutions, let’s hear
the explanation for each.
C8 I will explain 2-C2. First, we calculated the total
number of pencils. There are 96 pencils
altogether. Then, we shared them equally among
6 people.
C9 I will explain 2-C1. First, we calculated the total
number of pencils, then we shared them among 6
people.
C10 Their reasoning is the same.
T6 So can we call both of them “the method that first
calculates the total number of pencils”?
C11 Yes.
T7 2-C3, can you explain 2-C4’s solution?
C12 First, we shared the pencils in each box. Each
person will get 2 pencils. Since there are 8 dozen
pencils, each person will get 8 times as many as 2
pencils, or 16 pencils.
C13 I will explain 2-C3. The reasoning is exactly the
same as 2-C4, but it is written in one math
sentence.
T8 OK, so it is just like 2-C1 and 2-C2. One of them
uses one math sentence while the other uses 2
math sentences. So, what is the difference
between 2-C1/2 and 2-C3/4?
C14 The difference is whether you calculate the total
number of pencils first or the number of pencils
each person receives from each box first.
 Students will see the merit of writing one single
math sentence. Summarize today’s lesson.
Summarize
It is easy to make a mistake when you use
reasoning that involved remainders.
C2 When you write one math sentence, it looks
simpler.
T1 Is it easier to see the reasoning and solution in one
math sentence than 2 or more math sentences?
Please write down what you think is the best
approach.
T2 When both multiplication and division are in one
math sentence, can we start with the last
operation? (Example, 8 x 6 ÷ 12.)
C3 I don’t think we can because we will get different
answers.
T3 Let’s summarize.
 2-C1 explains.
 Callout, “Share the total
number of pencils.”
 2-C2 explains.
 2-C3 explains.
 Callout, “Calculate the
number of pencils for each box
first.”
 2-C4 explains.
Can Ss understand that the
solution with one math sentence
with both multiplication and
division and the one with 2 math
sentences are based on the same
reasoning? (Math Thinking)
 Think using the criteria,
“ESA.”
C1
Do Ss understand how to
calculate math sentences with
both multiplication and
division? (Knowledge)
We can write one math sentence even when both
multiplication and division are involved. Math
sentences with both multiplication and division
must be calculated in order from left to right.
 Let’s write a
journal entry.
C4
Writes his/her journal entry.
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
Lesson plan A (Class # 2, Mr. Suzuki) 2 lessons (today’s lesson / )
1. Goals of the lesson
•
By thinking about the missing digits, students will reaffirm their understanding of the properties of
division.
(First half: By thinking about the missing digits, students will recognize the relationship, b > e.)
(Second half: By thinking about the missing digits, students will understand that the properties of
division can be observed in the procedure of the division algorithm.)
2. Flow of the lesson
Steps
Questions ()
Learning activities ()
Anticipated responses (C’s)
1. Understand the
task.
 Read the problem.
T1 We have been studying division of 2-digit numbers with
remainders. Today’s lesson is based on that idea.
Write the numerals from 1 to 4 in the boxes
labeled A to D so that the division problem
Points of
consideration (*)
Evaluation
* Materials:
Numeral cards (1 ~ 4
and 1 ~ 9), several
sets and several
copies of the
problems (shown on
the left) on cards.
shown is correct.
Understand
 In the boxes A ~
D, you need to
write numerals 1 ~
4.
C1 Can we use them many times?
C2 What should go in the other boxes?
T2 You can use any numeral, but not everything will make
the division problem correct.
C3 I wonder if we can just pick numerals.
T3 For each numeral, there is reason why it has to be in a
specific box. For example, if you think about it very
carefully, you will know that 1 can only go into C. You
can use the numerals 1 4 once and only once. If you are
not sure, you can try putting a numeral in a box and see
what happens.
* Some students may
be hesitant getting
started. It is
important that
students actually start
trying.
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
2. Solve the
problem.
 Individual problem solving.
C1 (correct)
C2
* Explain how Ss
can use the hint card
(when you are not
sure, you can open it
up little by little).
C3
 Hint card
1.
Examine
C1
C2
C3
C4
C5
Correct answer.
Understands Divisor > Remainder.
Does not understand Divisor > Remainder.
Tries putting numerals in the boxes.
Cannot get started.
!
3. Think about
why.
 I thought like
this (incorrect
answer). What do
you think?
Deepen
 How about
these? They all
have B > D.
 Listen to other students’ ideas and think about how to get
the correct answer.
Show 2-C3 on the board.
C1 That’s not correct.
C2 Since the remainder is greater than the divisor, I don’t
think that is correct.
C3 It has to be B > D.
T1 Show answers like 2-C2 (show several of the same
pattern at once).
C1 If you do B x C, it will be greater than A in all of them.
C2 You can’t subtract from A.
T2 So what do we need to do?
C3 Explain by moving the cards around: Since we can use
each card once and only once, the only combinations
that will make it possible to subtract is B = 3 and D = 2.
So, the answer is, A = 4, B = 3, C = 1, and D = 2.
Summarize
T1 “We used the rule that the remainder must be less than
the divisor and solved the problem systematically.”
(Blackboard writing, too.)
C1 What numerals go in the other boxes? Will 5, 6, 7, 8 and
9 work?
What number
can go into the
remainder, D?
(Let’s think
about the
remainder first.)
2. Which is larger,
B or D? ( C3)
3. There is a
numeral that
cannot go into B.
( B " 1)
4. Try putting the
numerals in any
way.
5. Which is it, C <
A or C > A?
6. B = 3
* The answer to be
posted on the board
will be prepared by
the teacher.
* Have a student
who found the same
answer as 2-C3
explain. (Point to
focus.)
Blackboard: B > D
* All of them are
incorrect answers.
Are Ss reasoning
logically to determine
what numerals will
go in the boxes?
(Math Thinking)
* Acknowledge
students’ interest to
explore further.
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
1. Understand the
task.
Extension: Understand
 That is a very
good question.
What numerals will
go in E and F?
 Think about a new problem.
T1 We can use the numerals 1 to 9 in E and F, and we can
use the same numeral at most twice. What patterns can
you find when the division problem is correct?
We are not changing 1, 2, 3, 4 that are already shown.
E
F
2. Individual
problem solving
E
F
I
Extension: Examine
(4)
J
G H
K L
 Solve the problem individually.
C1 Cannot get started.
C2 I = 3 and G = 1.
C3 Could do the same as C2, but after that just uses random
trial and error.
C4 Keeps trying to put different numerals in E and F.
C5 I found one. By trial and error.
C6 I found two.
C7 I figured it out. It’s easy. There are 3 kinds.
C8 Understand that the number (4) , “1H,” must be two more
than a multiple of 3, and use that idea to find E and F.
Answer A
Answer B
Answer C
* We are not
changing the
numerals in A to D.
Are Ss trying to
examine whether
their previous
reasoning can be
extended to
determine the
numerals for boxes E
and F? (Interest)
 Hint card
1. What should be
I? J = 0, isn’t it?
2. What should be
G?
3. What can you
say about “KL”?
(K = 0 or 1).
4. How many more
is the number
“GH” than the
number “KL”?
What numeral
can go into E?
 Use whiteboards.
Can Ss figure out the
quotient by focusing
on the remainder?
(Representations)
3. Look for
patterns.
Extension: Deepen
Look at E (in
the quotient)
and the
divisor, “4F.”
 Look for patterns from the solutions to the missing digit
problem.
Answer A Answer B
Answer C (see above)
C1 The number, “1H,” must be 3 more than a multiple of 3.
So, 9 + 2 = 11, 12 + 2 = 14, 15 + 2 = 17 are the
possibilities. So, F must be 1, 4, or 7, and E must be 3,
4, or 5.
Display the three
correct solutions.
Can Ss discover the
pattern between the
dividend and the
divisor from the three
correct answers?
(Math Thinking)
C2 The quotients are increasing by 1.
C3 The dividends are increasing by 3.
C4 There are other patterns. 9, 12 and 15 are the 3’s facts in
the multiplication table.
C5 The numbers, (4), are also increasing by 3.
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
4. Summarize the
lesson.
 Summarize the patterns of division.
T1 41 ÷ 3 = 13 rem. 2
44 ÷ 3 = 14 rem. 2
47 ÷ 3 = 15 rem. 2
(dividend) ÷ (divisor) = quotient + remainder
Extension: Summarize
C1 We did this before.
C2 Even in the division algorithm, we have the same pattern
we saw in the 3’s facts—when one number increases by
1 the other increases by 3.
<Summary>
When the quotient increases by 1, the quotient will
increase by the divisor, 3.
 Let’s write
today’s journal
entry.
C3 Write his/her journal entry.
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
Lesson Plan C (Mathematics Open Room, Ms. Koizumi – small group for mathematics)
Goals of the lesson: Using a number line or tape diagram, students will think about methods to determine how
many times as much the base number is as the given number.
Flow of the lesson
Steps
Questions ()
Learning activities ()
Anticipated responses (C’s)
1. Understand the task for
the lesson.
I can swim 15 m without taking a breath. How many times as far as
______________ can I swim without taking a breath?
Understand
 This is today’s problem.
 That’s true. _______ can
swim 3m without taking a
breath.
C1 Something is wrong.
C2 We need to know how many meters _____
can swim without taking a breath.
C3 We can do this.
Points of consideration (*)
Evaluation
Strategies for improving
communication ()
 At first, display the problem
statement that does not include
the base amount.
 Pose a problem that is
appropriate for the season.
* Help students pay attention to
the base amount.
* Make sure students understand
that the base amount is 3 m.
I can swim 15 m without taking a breath. _______ can swim 3 m without taking a
breath. How many times as far can I swim without taking a breath as _______?
2. Solve the problem.
 Let’s use what we have
already studied to figure out
how many times as far as
_______I can swim without
taking a breath.
 Solve the problem using own strategies.
* Prepare the worksheet.
Cs Write their own strategies on the worksheet.
* For those students who cannot
even get started, provide tapes
that are 3 cm and 15 cm so that
they can draw a tape diagram.
Examine
Use multiplication to determine how many times.
3 x [5] = 15 5 times as far.
3x1=3 3m
3x2=6 6m
3x3=9 9m
3 x 4 = 12 12 m
3 x 5 = 15 15 m 5 times as far.
Think, “how many sets of 3 m will make 15 m?”
15 ÷ 3 = 5, 5 times as far.
Are students trying to solve the
problem? (Interest)
Can Ss solve the problem using
their own ideas and represent it
in a diagram, a math sentence, or
a chart? (Mathematical
Thinking)
* Have students write their math
sentences and diagrams on their
whiteboards.
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.
3. Read other students’
ideas.
 Let’s look at all the
solutions. I wonder if we can
sort them?
Deepen
 There is no space to show
“times as much” on a number
line, is there?
4. Summarize.
Summarize
 When you are figuring
out “how many times as
much,” what operation
should we use? Think
“ESA.”
 Using the idea from
today’s lesson, let’s try this
problem.
 Compare their own ideas with other students’
ideas. Explain their own reasoning.
Multiplication methods
Division methods
Number line methods
It shows only m, doesn’t it?
Where can we show “times as much”?
 Summarize today’s lesson on the worksheet.
To find how many times as much, we can
use division.
Draw a number line model.
 Students will attempt the problem using what
they learned in today’s lesson.
 Acknowledge and praise that
they were able to solve the
problem using many different
ideas.
 Have students explain other
students’ ideas, not their own.
* If there is any strategy that
cannot be grouped with others, or
other students cannot figure out
the strategy, have the student
who came up with the strategy
explain it.
 Based on students’ ideas,
orchestrate the discussion so that
students will recognize that
different ideas can be
summarized using number line
models.
* Make sure that students
understand the importance of
drawing a number line correctly
and using 1 as the base of
comparison.
* Review how to draw a number
line.
Can Ss draw a number line
correctly? (Mathematical
Thinking)
A mother whale is 24 m long.
A baby whale is 6 m long.
How many times as long is the mother whale as the baby whale?
5. Reflect on today’s lesson.
 Please write what you
noticed or realized in today’s
lesson.
 Write a journal entry.
* Encourage students to write
about the base amount in their
journal entries.
This lesson plan is originally written in Japanese and translated into English by the Global Education Resources
for the Lesson Study Immersion Program in Japan, June 21- July 5, 2007.
© 2007 Global Education Resources L.L.C. All rights reserved.