Mathematics in several East Asian countries—including China, Japan, and Singapore—represent relationships among quantities in similar ways. These representations tend to be of two (closely related) kinds, illustrated below. (The relationship shown can be expressed symbolically in various ways, including a + b = 10 and 10 – a = b.)
See the 36 students problem for examples of bar model solutions. For examples in textbooks, see the Singapore textbooks (Textbook 3A, pp. 18–22, 37, 45–46; Textbook 5A, pp. 22–23, 56–59, 73, 75–78; Textbook 6A, pp. 21, 24–37, 78–80), Knowing Mathematics (chapters 1 and 2 for any grade level), and Japanese Grade 7 (pp. 56, 62, 65). For examples on video, see J3 (46:32), J4 (27:27–29:25), in which the teacher uses bar diagrams as pictorial models of algebra before introducing algebra formally. In Japan, teachers replace these diagrams with algebra, usually in grade 7 (which is why few examples appear in these lessons).
In S2 (062) blue and yellow strips represent lengths of five and three. Later in the lesson, students use directed line segments to represent lengths of x, y, and z.
Curriculum Planning & Development Division, Ministry of Education, Singapore (2003). Primary mathematics series. U.S. edition. Singapore: Times Media Private Limited.
Kodaira, K. (Ed.). (1992). Japanese grade 7 mathematics (H. Nagata, trans.). Chicago: University of Chicago School Mathematics Project (UCSMP). Available from UCSMP.
Ma, L., & Kessel, C. (2003). Knowing mathematics. [Intervention for students in grades 4, 5, and 6.] Boston, MA: Houghton Mifflin.
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