Whole Unit Lesson Study −6th Grade
Mathematics：
“Proportion”− How a study
(research) lesson fits in a unit ★Recording of all lessons taught in a unit★ 

Whole Unit Lesson Study Part II・Details −Contents
of Meetings (Record of the first lesson and its discussion)− 





8/9(Wed.) 
Meeting 1 



Agenda 
Item
1: Goal of mathematics education
at our school and its relationship to the whole unit lesson study Item 2: Discussion of the unit, “proportion
and inverse proportion” Item 3: Setting up agenda for the next
meeting 








Contents of
the discussion 



About item 1 
・
How far the 6^{th} grade students need to progress in
learning about the concept of “proportion and inverse proportion” ・
Helping students understand the concept of proportion using
charts, graphs, and mathematical expressions ・
Reading charts and understanding them are important skills that
are necessary not only in the field of mathematics but also other
fields. Therefore, we need to
help students foster such skills through manipulative activities. ・
Developing lessons and expanding them, focusing on the
understanding of two changing values. 



About item 2 
・
How far do the students need to progress in learning about making
graphs? The two periods reserved
in the unit instructional plan are too short to allow the students to acquire
the skills for reading and making graphs ・
The main goal of mathematics education at our school is fostering
students’ reasoning skills and mathematical thinking abilities, therefore,
these two items should be the focus and skills for making graphs should be
selective and only the important content should be touched. ・
It is better to provide a point of view of proportion and inverse
proportion to the students. ・
It is ideal to incorporate manipulative activities for understanding
of the characteristics of proportion and inverse proportion. 



☆ Confirmation of common understanding of item 1 and 2 




・
Teach proportion and inverse proportion at the same time. ・
Use charts as the center of learning activities. ・
Help the students understand the characteristics of proportion
and inverse proportion through manipulative activities. 



About item 3 
・
All the participants of the meeting will think about how to
develop the lessons that follow the first lesson. 







8/10(Thu.) 
Meeting
2 



Agenda 
Item 1: Confirmation of what they discussed
during the last meeting Item 2: About the instructional plan 




(1)
How to deal with
proportion and inverse proportion. (2)
About
experiments that students conduct during the second lesson. (3)
When to
summarize the characteristics of proportion Item 3: Confirmation of homework for the next
meeting. 








Contents of the discussion 



About item 2(1) 
・
Do not include the topic of “inverse proportion” in this unit and
focus on only the topic of “proportion” and develop the unit instructional
plan. ・
The main characteristic of proportion that the students need to
understand is regular (constant) changes of two values. 



About item 2(2) 
・
Is it necessary to have experiments during the second lesson? ・
Experiments should be done during science lessons and students
need to focus instead on investigating values. ・
It is okay to do experiments; creating charts is a necessary and
meaningful activity for students’ learning. 



About item 2(3) 
・
It is necessary to think about how far do we want the students to
understand the characteristics of proportion. 



About item 3 
・
Think about “hatsumon” (a question or action that promotes
students’ active thinking) to include during the introduction of the lesson. ・
Make a list of as many twovalue relationship that look like
“proportion” but actually are not 



☆ Confirmation of common understanding of item 2(1) 




・
Focus on only one kind of relationship between two values, i.e.
“one value increases and the other value also
increases,” (increase–increase twovalue relationship) and ask students to
find the rules of the relationship. ・
This unit will deal with only “proportion.” (“Inverse proportion” will be taught
some other time in a different unit. ・




☆ Confirmation of common understanding of item 2(2) 




Lesson 1: Ask students to find as many “increaseincrease twovalue
relationship” as possible. Lesson 2: Conduct experiments with weights and
springs. Summarize the results
in the charts and make a list of insights gained from the charts. Lesson 3: Investigate the characteristics of
“proportion.” Lesson 4: Solve application problems on “proportion” 



☆ Confirmation of common understanding of item 2(3) 




・
Help the students to understand the characteristics of
“proportion” in a certain order.
The mathematical expression that describes the relationship of
“proportion” will not be introduced until the end. 



※ We developed an instructional
plan for this unit based on the contents of discussions from the two
meetings. 








Instructional Plan（11 lessons） 


Small Units 

Goal 
Development 


Phase 1: Finding many different functions (relationship
between two values). （1 lesson） 
@ 
○ Be able to find
increase–increase twovalue relationship 
○ The students will be asked to look for phenomena that shows
increase–increase twovalue relationship from their surroundings. 

Phase 2: Finding out the relationship between the
stretch rate of a spring and the number of weights. （2 lessons） 
A B 
○ Be able to make
charts with dates collected from the experiment. ○ Students will
enthusiastically try to find rules of the relationship between the stretch
rate of a spring and the number of weights from the charts they create. 
○ Investigating the relationship between the stretch rate of a
spring and number of weights during the experiment and create charts using
the data collected. 

Phase 3: Investigating other kinds of twovalue
relationships by thinking about or comparing with the rules that the students
discovered through the experiment with springs and weights （4 lessons） 
C D E F 
○ Be able to
create charts based on many different relationships between values. ○ Be able to find
the relationship between values from charts. ○ Be able to
understand the relationship between two values: i.e.,when value x increases
by 2 times or 3 times the corresponding value of Y also increases by 2 times
and 3 times. ○ Be able to
write expressions to show a
“proportion.” 
○ Develop charts
with data and identify a chart that shows the same kind of relationship as
the relationship of two values that they studied during the previous lesson. ○ Think about the
reasons why we can say the relationship between two values showed on two
charts are similar. ○ Through
comparative activities, the teachers help the students organize the meaning
and characteristics of proportion.
The teacher also helps them think about or develop expressions that
show the proportional relationships. 

Graphs of
“proportion” (2 lessons) 
G H 
○ Be able to show
proportional relationships on graphs. ○ Be able to
understand a graph of “proportion”:
They are 
○ Show
“proportion” on a graph. ○ Think about the characteristics of graphs of proportion. 

Phase 4: Exercise problems dealing with “proportion” 
I J 
○ Make sure the
students learned the contents of this unit. 
○ Do the exercise
problems in the text book and aid materials. 






8/17(Thu.) 
Meeting 3 


Agenda 





Item 1: Confirmation of what they discussed. Item 2:
Discuss the introduction section of the first lesson. 


Contents of the discussion 



About Item 2 
・
We will ask students to put away their textbooks. Presenting some
kind of picture/diagram to the students to provoke images of changes of two
values. ・
Example of phenomena should be coming from the students’ school
life in order for them to associate the learning activity with their real
life, which is more meaningful for them. ・
We should not limited ourselves to only the phenomena that show
the proportional relationship of two values. We should include other kinds.. ・
We’d like to describe the experiment planned for the next lesson
in the introduction portion of the first lesson. ・
Whether or not it’s better to show some concrete materials to the
students. ・
If we show concrete materials, then we should provide some
opportunities for the students to experiment with them. In this case we would need two lesson
periods worth of time for phase 1. ・
If we present phenomena that do not show proportional
relationships, then the second lesson (experiment ) will be more meaningful
to the students. And the flow
between the two lessons will improve. 


☆ Confirmation of common
understanding of item 2 



・
Lesson #1 (Phase 1): Letting the students know that we are going to
study a relationship of two values where “if one value increases the other
value also increases,” (increaseincrease twovalue relationship). Also ask the students to think about
many examples of increaseincrease twovalue relationships. ・
Lesson #2 (the first lesson in phase 1): Select examples of increaseincrease
twovalue relationships from the pool of examples that the students found
during the first lesson. Then
have the students conduct experiments to investigate the nature of the
relationship between the two values. 







※ Lesson plan for lesson 1 was developed. 







Ideas for how to expand the
lesson 






Time 
Students’ main learning activities 
Important points relating to instruction 



1. Think about changes in two values. (1) The students
listen to the teacher’s examples and think about foresights of learning. 
・As an introduction to the unit, the teacher
will present examples of “two values that change at the same time” in order
to promote student foresight for learning 


・Use examples
from the students’ experience making potato chips in their home economics
class. 
・Use the cards
prepared for the blackboard when presenting the examples in order to
emphasize the that the concept “two values that change at the same time” will
be studied in today’s lesson 

8 min. 
・Baby’s growth
and height is a good example 



・Support students who are having trouble with
the exercise of finding examples 


Anticipated
students’ response: 
・Ask students to
find only two examples, choose one of them to write on their card to present
at the blackboard. Ask the
students to paste their card on the blackboard. 

18 min. 
・Speed of car and the
distance it traveled 


・number of goods and
their price 
・Ask students to
sit in front of the blackboard and categorize the cards according to how the
two values have changed (i.e., increase increase, etc.) 


・total time slept and
total time didn’t sleep in a day 

28 min. 
(3) Students will
list as many examples as possible, choose one, and then write it on a card
for presentation. 

・When
categorizing, have students think about the similarities and differences
between the various examples 


2. Each student will paste their cards on the board. Then they will categorize the cards. 

・After the
students complete the categorization, ask them to categorize their initial
two examples on their own at their desks 

(1) Categorizing the cards based on the students discussions. 

38 min. 
>Important
points for categorization: 

・If ○ increases then △ also increases. 
・At the end of
the lesson, the teacher will collect all the students’ notebooks ( with their
examples in them), in order to see each students’ learning and thinking
processes. This knowledge should
be used as a base for the next lesson. 

・If □ increases then × decreases. 

・If ■ increases then ▼ sometimes increases and sometimes decreases. 

45min. 
(2) Categorizing
the cards on the board 





8/23(Wed.) 
Implementation of the first lesson using the lesson
plan 





8/24(Thu.) 
Meeting 4 



Agenda 





Item１: First Meeting to “reflect” Item 2: Hearing
Thoughts of Observers about the lesson Item 3: Thinking
about how implement the subsequent lessons Item
4: Hearing Thoughts of the
Principal about the lesson 


Contents of the Discussion 




About
Item 1 
・
The introduction was
carefully planned and the instructional materials used for the introduction
were well prepared, therefore the lesson was thoughtprovoking for the
students and encouraged their participation. A variety of ideas were also forthcoming from the
students. However, students did
have some difficulty identifying the different relationships ・
Teacher’s encouragement
of students during their individual work was good. ・
Teacher’s use of
grouping was effective. ・
Necessary to pay better
attention to the allocation of time. 


About
Item 2 
・
The introduction was
very simple and engaging to the students. Also, the activity was closely related to the students’
life experiences, therefore it held the students’ interest well. Teacher’s encouragement of students
during their individual work was good.
It would have been better if the students had written their names on
the cards the presented at the blackboard. Student discussion during categorization was very active
and lively. 



・
The colored strips of
paper that the teacher used in the beginning of the lesson helped them
understand better the task they needed to perform for this lesson. Examples such as “if one increases
then the other decreases” came out of the students’ discussion, which
encouraged deeper learning. 



・
The introduction was
carefully planned, therefore the lesson was thoughtprovoking for the
students and encouraged their participation. Organization of the blackboard was good. The students not only noticed the
pattern evident in the concepts such as “if one increases, then the other
decreases”, but were able to notice how much the values changes as well. 



・
The teacher made sure
each student understood what they needed to do before they started to work on
the exercise. The use of
grouping was also effective. All
the students were interested in the lesson. Use of instructional materials such as instructional
materials was effective.
However, students did have some difficulty identifying the different
relationships. Although students
presented many examples, there were no good examples that represented
“proportion”. Better “hatsumon”
could have been used to encourage the students to pay attention to the change
in values, not just the pattern of the changes. The categories used in categorization came from the
teachers, but it would have been better if the students came up with the
categories themselves. 



・
The “hatsumon” used by
the teacher, “let’s find an example where if one value increases, then what
happens to the other value” was effective. Finishing the categorization within this lesson was too
much to expect. There were a few
students that had trouble finding examples. 



・
Value changes were not
emphasized, therefore the students came up with many interesting examples of
the relationship between two values.
Students will be able to notice examples that can be expressed with a
number, and those that can not be expressed with a number. It would have been better to spend
more time conducting student discussions. We should have considered more
carefully the following points:
time allocated to categorization; and how categorization should have
been conducted. Many students
wrote that they were able to understand the lesson. The teacher dealt very well with students who were having
difficulty understanding. 



・
Allocation of time was
difficult. Teacher providing the
categories was a good idea. 



・
Teacher focus on only
“Increaseincrease two value relationships” was a good idea. 







About
Item 3 
・
Convert words into
number expressions for the next experiment (dealing with springs and weights) ・
Students will make a
chart of the stretch rate of the spring and the number of weights, making
sure to pay attention to the change in values (not just the pattern changes). ・
Follow the basic idea of
the instructional plan developed previously ・
How to deal with
students who make mistakes in measuring during experiment. 


About
Item 4 
(Comments from the
Principal) ・
The introduction was
effective. Allocation of time
could have been better. It would
have been better to allocate more time to student discussions and summarizing
the lesson. ・
More attention should
have been paid to organization of the blackboard. Only two students were selected to help with
categorization, but three students should have been used because there were
three categories. A good
relationship between the students and the teacher existed and this was very
noticeable during the lesson. ・
The teacher spent too
much time talking, which prevented the students from taking initiative to
discover things on their own or summarizing what they learned. ・
With these comments in mind,
I would like the teacher to develop an even better lesson the next time. 












