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..