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 two-value 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 two-value 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 “increase-increase two-value 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 two-value relationship		○ The students will be asked to look for phenomena that shows increase–increase two-value 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 two-value 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,” (increase-increase two-value relationship). Also ask the students to think about many examples of increase-increase two-value relationships. ・ Lesson #2 (the first lesson in phase 1): Select examples of increase-increase two-value 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
	(2) Let’s find two values that change at the same time					・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 thought-provoking 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 thought-provoking 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 “Increase-increase 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.