This lesson focuses on linear inequalities in one variable. It is the first in a sequence of seven lessons on this topic. The lesson plan, which was developed during the teacher’s participation in a lesson study cycle, discusses the subject matter, the connection between the lesson and the “research subject” at the teacher’s school, goals for the lesson, anticipated student responses at various stages of the lesson, and what the teacher would do in response (see Further discussion section below).
| 1:00–2:02 | (Whole class) The teacher reads the problem aloud: “It has been one month since Ichiro’s mother has entered the hospital. He has decided to say a prayer with his smaller brother at a local temple every morning so that she will be well soon. There are 18 10-yen coins in Ichiro’s wallet and 22 five-yen coins in his smaller brother’s wallet. They have decided every time to take one coin from each of them, and put them in the offertory box, and continue their prayers until either wallet becomes empty. One day after they were done with their prayers, when they looked into each other's wallets, the smaller brother's amount of money was greater than Ichiro's. How many days has it been since they started praying?” | ||||||||||
| 2:02–4:39 | (Whole class) The teacher represents the problem on the board. | ||||||||||
| 4:39–4:55 | The teacher asks the students to think about how they will solve the problem. | ||||||||||
| 4:55–18:32 | The teacher circulates while students work individually on the problem. | ||||||||||
| 18:32–31:39 | (Whole class) The teacher selects students to come to the board to present solutions in the order of increasing sophistication of the solution method. After each solution is presented, the teacher posts a label for the solution method on the board and asks students who used that solution method to raise their hands. The solutions are as follows:
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| 31:39–40:41 | (Whole class) The teacher assigns another task: Using the inequality from the 5th solution 180 – 10x < 110 – 5x, students are to compare values of the left and right sides of the inequality at x = 13, 14, . . . , 19. | ||||||||||
| 40:41–46:02 | (Whole class) The class discusses the results. | ||||||||||
| 46:02–46:32 | (Whole class) The teacher reads the next problem aloud: “The prayer was answered and their mother was able to leave the hospital safely. That night they gave a toast with juice. Currently there are 50 milliliters of juice in Ichiro’s cup, and 80 milliliters in the smaller brother’s cup. When the mother poured juice into Ichiro’s cup it made Ichiro’s amount become greater . . . how many milliliters had she poured?” | ||||||||||
| 46:32–50:57 | (Whole class) The teacher provides further elaboration of the problem on the board and works with the students to solve it using an inequality. | ||||||||||
| 51:00–53:28 | (Whole class) The teacher asks what kinds of numbers the solution represents. |
(31:39–46:02) A similar table occurs in the textbook Japanese Grade 8 (Kodaira, 1992, pp. 27–28). The strategy of comparing the left and right sides for particular values of x is used also in HK4 (6:35–10:12, 11:18–13:08) for an equation that turns out to be an identity.
(1:00–2:02 and 46:32) There are graphical representations of word problems, one concerning a discrete situation, the other a continuous situation.
Students are able to solve the problem at whatever level they understand it. The solutions illustrate multiple entry points to the problem. This lesson demonstrates a way in which to differentiate instruction by using one problem. Students are not expected to solve the problem in just one way, and in the process, they all learn different strategies from each other. To ensure access for all students, the teacher then selects a second problem for practice that is easier, allowing all students the opportunity to set up an inequality algebraically.
Having students write responses, then choosing students to present their work and choosing the order in which the work is presented affords different kinds of decisions on the part of the teacher. (See C2 for an example of a whole-class discussion in which students do not first respond in writing.)
The teacher asks the last four students to write their solutions for the first problem on particular parts of the blackboard (20:44–23:34, 24:35–24:44, 26:15–26:54, 29:20–29:25) so that once the students have presented their work all of the solutions are visible (31:39). The teacher still has room to write the table for values of the inequality (31:39–40:41) and to post the representation for the juice problem (46:02–46:32). At the end of the lesson (54:11), the entire board is a record of problems and solutions. For additional discussion of more general issues in the use of visual aids, see Using Technology. (For discussion of Japanese teachers’ use of the blackboard, see Yoshida, 2005).
Technical detailsFile name: NB11131323.
There may be some translation difficulties with the lesson plan. For example, “domain function” in paragraph 2 should perhaps be “domain of a function.”
In the lesson, “futoh-shiki” has been translated as “inequality equation,” but might be more accurately translated as “expression of inequality,” “inequality expression,” or “inequality.” The word consists of three characters when written in Chinese characters. The first character (“fu”) signifies “not.” The second (“toh”) signifies equality. The third (“shiki”) signifies expression in a general sense; it includes, for example, 3 + 5, 2x + 3, 2x + 3 =, 2x + 3 = 4, and x < 2.
The lesson plan is detailed and extensive. According to the national coordinator’s comments (18:52), most Japanese teachers are able to create detailed lesson plans such as the one given for this lesson.
Subject matter
The lesson plan has a separate section on subject matter, discussing what students have learned at various grades that are related to learning the topic of the lesson. This view of subject matter is similar to the “knowledge packages” discussed by Ma (1999), particularly the remark about leading students to see a common element of equations and inequalities—that they are both mathematical statements that constrain the values of an unknown x. For an example of a “Systematic Chart” used in Japan that seems to support this view of students’ learning, see Lee (2001). For a detailed discussion of when and how topics such as mathematical expressions (with numbers and letters) are taught to students, see, for example, Japanese Ministry of Education, Culture and Sports, 2004, pp. 38–39. In upper elementary grades, students begin to learn to interpret and understand expressions with letters.
Research subject
In the lesson plan there is mention of “research subject at this school,” which is an overarching theme for lesson study at the teacher’s school. For more discussion of lesson study, see Fernandez and Yoshida (2004), Lewis (2002), and Wang-Iverson and Yoshida (2005).
Four-column format for student responses
This lesson plan is created in a four-column format, but lesson plans can vary from three- to five-column formats. The teaching of Japanese teachers is due in large measure to these detailed lesson plans, which serve as scripts teachers use during a lesson to anticipate student responses and misunderstanding. For further discussion, see Fernandez and Yoshida (2004), Stigler, Fernandez, and Yoshida (1996), and Wang-Iverson and Yoshida (2005).
Fernandez C. & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Mahwah, NJ: Lawrence Erlbaum Associates.
Japanese Ministry of Education, Culture and Sports. (2004). Elementary school teaching guide for the Japanese Course of Study: Arithmetic (Grades 1–6). (A. Takahashi, T. Watanabe, & M. Yoshida, Trans.). Madison, NJ: Global Education Resources. (Original work published 1989).
Kodaira, K. (Ed.). (1992). Japanese Grade 8 Mathematics (H. Nagata, trans.). Chicago: University of Chicago School Mathematics Project. Available from UCSMP.
Lee, S-Y. (2001). Student curriculum materials: Japanese teachers’ manuals. In Knowing and learning mathematics for teaching: Proceedings of a workshop (pp. 78–85). Washington, DC: National Academy Press.
Lewis, C. (2002). Lesson study: A handbook of teacher-led instructional change. Philadelphia: Research for Better Schools.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
Stigler, J., Fernandez, C., & Yoshida, M. (1996). Cultures of mathematics instruction in Japanese and American elementary classrooms. In T. Rohlen & G. LeTendre (Eds.), Teaching and learning in Japan (pp. 213–247). New York: Cambridge University Press.
Wang-Iverson, P. & Yoshida, M., eds. (2005) Building our understanding of lesson study. Philadelphia, PA: Research for Better Schools.
Yoshida, M. (2005). Using lesson study to develop effective blackboard practices. In P. Wang-Iverson & M. Yoshida (Eds.), Building our understanding of lesson study (pp. 93–100). Philadelphia, PA: Research for Better Schools.
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