This lesson focuses on writing variable expressions. It is the introduction to a unit on equations. The lesson is 53 minutes long. There are 31 students in the class.
| 00:27–10:39 | The teacher circulates as students work individually on warm-up problems written on the board; she is checking that they have done the homework: three worksheets on ordering integers. |
| 10:39–13:58 | (Whole class) The teacher calls on students for answers to the warm-up problems. |
| 13:58–20:00 | (Whole class) The teacher calls on students for questions about the homework problems and displays some answers for problems on page 1 on the overhead projector. The teacher asks students for answers for problems on page 2, also displayed on the overhead. |
| 20:00–20:51 | A student reads “the quote of the week,” which is displayed on the easel. The teacher solicits student interpretations of the quote. |
| 20:54–24:05 | (Whole class) The teacher asks students to title their notes “Writing Variable Expressions.” She asks for examples of “when you hear words but in actually someone is talking about numbers.” Students give examples. The teacher gives an example of working different numbers of hours. |
| 24:05–26:47 | (Whole class) In discussing variable expressions, the teacher alters the first situation (working different hours) and asks for the corresponding variable expression. She then gives another example and asks for the corresponding expression. |
| 26:47–30:17 | (Whole class) In discussing evaluating expressions, the teacher uses the first situation as an example and then gives several other examples and answers questions. |
| 30:17–37:39 | (Whole class) The class discusses translating words into variable expressions and variable expressions into words. |
| 37:39–40:00 | (Whole class) The class discusses variable expressions with more than one operation. |
| 40:00–43:01 | (Whole class) The class solves practice problems from the textbook. The teacher calls on students to give solutions. |
| 43:01–49:10 | (Whole class) Students begin their homework assignment. |
| 49:10–53:00 | (Whole class) The class plays “24”: A student chooses four numbers, and another student comes to the board to use them in three equations, the last ending in “= 24.” |
Like several of the other TIMSS public release lessons, such as N1 and S4, this lesson includes a homework checking session, but unlike the other lessons, the teacher does not check the problems publicly. This lesson shows a classroom management technique involving students taking notes.
(20:54–26:43) The teacher devotes six minutes to introducing the concept of variable.
(20:54) The teacher begins discussion of the main topic of the lesson—variable expressions—about 21 minutes into the lesson.
There are several examples of a pattern known as IRE—Initiation (by the teacher), Response (by the student), Evaluation (by the teacher) (Schoenfeld, 1998): For example (12:26):
Teacher: Okay: number six, Patrick.
Patrick: Eight divided by four equals two, and two plus two equals four, and four times six equals 24.
Teacher: Good.
See Imprecise language below for examples.
This lesson has several examples of overhead projector use:
(13:58-–20:00) Display of answers to homework problems and answers,
(20:54–24:05) Introduction of new material
(24:05–26:47, 26:47–30:17, 37:39–40:00) Examples
(40:00–43:01) Display of practice problems
The Teaching Gap discusses differences in technology use with regard to different cultural systems of teaching (Stigler & Hiebert, 1999). See Using technology.
(20:54–33:22 and throughout) One student (Brandon) asks several questions that indicate lack of understanding, although he seems to be trying to make sense of what the teacher says and has written on the overhead. It is worthwhile to consider what this student does not seem to understand.
Technical detailsFile name: NB10311120.
There is no commentary from the national research coordinator. The Resources section of the CD includes the homework problems. The lesson graph features the warm-up problems.
Students should have learned:
Students will need to know:
Instructional decisions
(10:39–10:55) The teacher uses the Pythagorean theorem in a “warm-up” as one of a variety of problems:
One student says it is, since 25 + 144 = 169. The teacher asks where 25, 144, and 169 come from, and the student responds that 25 is 5 squared, etc. While this is true, it does not explain why showing that 5² + 12² = 13² proves that the triangle is a right triangle. This does not follow directly from the Pythagorean theorem, but from its converse.
Students have trouble with the use of a theorem and its converse, so when solving the given problem, the teacher should ask for an explanation of why 5² + 12² = 13² forces the triangle to be a right triangle.
(21:00–24:11) The teacher asks for examples of instances “when you hear words but in actuality someone is talking about numbers.” The examples students give do not involve variables (except of age). The teacher gives an example of a job: one can work 10 hours one week and 2 hours another week. The variable h represents the number of hours one works.
Students may have two different interpretations: h can change because numbers in an expression can replace it, or h can change because it represents a quantity that can change. Both of these are aspects of being a variable, but the second implies the first.
(37:38–39:57) This connection becomes a procedural review of PEMDAS. The students attempt to make connections between the order of operation rules they have learned and the procedures the teacher is introducing.
Giving results vs. an explanation
There are times when it is appropriate for a student to give an answer with no explanation for why it is correct, and other times when this is not appropriate. The teacher needs to be able to distinguish between the two and not ask for an explanation when it is not necessary.
Problem 6 from the warm-up is an example of a case in which explanations are not necessary: the teacher asks students to find three equations using 4, 8, 2, 6, with the last giving 24. The first solution was 8/4 = 2, 2 x 2 = 4, and, 4 x 6 = 24. This requires no explanation since the calculations are routine by eighth grade. The Pythagorean theorem problem described above is an example of one that does require explanation of the solution.
Curriculum considerations
The situations in part 1 of the homework concern “a number” that might be interpreted as an unknown quantity rather than a quantity that may vary as in the lesson situation of working h hours at $7 or $7.50 per hour or (as given on the overhead) “variable—a letter representing a value that can change.” One can certainly represent these by expressions with letters, but they are not closely related to the topic the lesson addresses: that a variable represents a quantity that may vary.
Another approach would be first to discuss “expressions” or “expressions with letters,” and later discuss variables, so that once students have experience using letters to represent unknown quantities the issue of representing an unknown versus representing a changing quantity is explicit. This approach is in at least one old U.S. textbook (Myers & Atwood, 1916) and one Japanese textbook (Kodaira, 1992).
“Letters and Expressions” is chapter 3 of the Japanese text, which begins with the example of how much one pays for various quantities of 50-yen stamps (the quantity ranges over positive integers in the example). Several questions in this U.S. lesson concern negative numbers, the use of parentheses, and order of operations. In Kodaira’s text, these topics occur in chapter 2 (“Positive and Negative Numbers”). Chapter 5 (“Functions and Proportions”) introduces variables with an example of the altitude of a rocket, which is called h (in the example, h ranges over non-negative real numbers): “Letters that take various values, like h, are called variables” (Kodaira, 1992, p. 97).
Describing a new term
The topic of this lesson is similar to the topic of S2. At the beginning of S2 the teacher tells the students that they will be studying terms and variables, but does not try to give definitions. The corresponding textbook page has a box with examples of expressions from the exercises above it and says they are called “terms.” The teacher has already introduced “term” in the exercise above.
The “Letters and Expressions” chapter of Japanese Grade 7 Mathematics asks students to write expressions for quantities (Kodaira, 1992). The chapter shows examples but does not discuss the meaning of “expression,” which suggests that students should know it already. This is likely, because grades 5 and 6 include expressions with letters, and students are introduced to writing mathematics expressions beginning in grade 1 (see Japanese Ministry of Education, Culture, and Sports, 2004).
Imprecise language
(37:30) The teacher’s response—that a quotient is the result of a division—is problematic.
The response seems to suggest that the teacher is focusing more on key words.
(37:38–39:57) In response to students’ questions (as they attempt to incorporate the procedures she demonstrates into those they have already learned), she incorrectly asserts that the term “negative” means the same as “minus.”
(42:32) The phrase “the quotient of three divided by w” should be “the quotient of three and w” or “three divided by w.”
Japanese Ministry of Education, Culture and Sports (2004). Elementary school teaching guide for the Japanese Course of Study (grades 1–6) (A. Takahashi, T. Watanabe, & M. Yoshida, trans.) Madison, NJ: Available from Global Education Resources. (Original edition published 1989)
Kodaira, K. (Ed.). (1992). Japanese Grade 7 Mathematics (H. Nagata, trans.). Chicago: University of Chicago School Mathematics Project (UCSMP). Available from UCSMP.
Myers, G. W., & Atwood, G. E. (1916). Elementary algebra. Chicago: Scott, Foresman.
Schoenfeld, A. H. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(1), 1–94.
Stigler, J. & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: Free Press.
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