This lesson focuses on solving linear equations in one unknown applying “theorems of equivalence” (in the teacher’s words). The lesson is taught in Swiss French and is 45 minutes long. There are 22 students in the class.
| 00:17–08:18 | (Whole class) The teacher goes over the homework: eight problems reviewing how to solve linear equations using an idea introduced earlier, namely how to change an equation to an equivalent equation by adding or subtracting terms from both sides of the equal sign. The teacher asks students to give solution methods and solutions orally, which students do for the first three problems, all of which have one linear term with a coefficient of 1. |
| 1:54 | One student does not give a solution method for problem d, 3x = 2x + 5. With input from the class the teacher writes a solution for problem d on the board, mentions the three theorems of equivalence in connection with each problem solution and writes solutions for:
|
| 08:18–17:03 | The teacher circulates as students work individually. |
| 17:03–23:56 | (Whole class) The class discusses:
|
| 23:56–30:38 | The teacher circulates as students continue working individually. |
| 30:38–34:46 | (Whole class) The teacher discusses one student's work: the student has attempted to solve –4x + 15 = 11 by adding 5x – 15. The teacher writes this on the board and points out the difficulty: adding 5x to both sides puts a term with x on the second side, where (in this case) there was no such term before. |
| 34:46–37:53 | The teacher circulates as students continue working individually. |
| 37:53–42:02 | (Whole class) The teacher discusses the next two problems:
|
| 42:02–45:00 | The teacher circulates as students continue working individually. |
Periods of whole-class discussion alternate with periods of individual work, during which the teacher circulates among the students. This structure allows the teacher to monitor student understanding as expressed in writing and in questions. The teacher uses individual student difficulties as a beginning for whole-class discussion (see 16:54–23:49).
The teacher sometimes gives students wait time that is insufficient for thm to think and to respond with an answer. (1:27) The teacher asks if there are questions and then immediately looks down to the problem set and calls on a student. The teacher also generally confirms or invalidates students' answers immediately and often develops the reasoning behind them himself.
The teacher draws a vertical bar to the right of an equation followed by a constant or a term to denote what is added, subtracted, multiplied, or divided on both sides of the equation. (3:38) The use of the notation “S = {5}” to denote the solution, rather than just “x = 5” could lay the groundwork for the idea that an equation can have more than one solution.
File name: NB10311052.
There are some minor translation errors. “Fait” is fairly frequently translated as “makes.” Sometimes this is correct, but “fait” has other translations (such as “ça fait” = “that is” or “on fait” = “one does”) that are more accurate in certain contexts. Note that “parentheses” was translated as “brackets” (in this video and in some others). Although “brackets” is not incorrect, it would be more usual (and help get rid of some of the “foreign feeling” of the video) to write “parentheses” for the translation.
00:33: “ou” is translated as “or” but makes more sense as “where” (there are French words meaning “where” and “or” that are both pronounced “ou”), so the sentence would be “where one can practice the theorems of equivalence.”
Other mistranslations include:
(13:47–13:49)
- Teacher: Then it's not divided by four, [the] fifteen?
- Student: Yes, but it does have decimal points.
(21:45) “not comfortable”—a better translation would be “not convenient.”
(25:56) S should be translated as X in the following: “S equals minus five. Anyway, one writes S equals, so whether one has X equals minus 16 or minus 16 equals X.”
(39:02) “The G part” should be “part G.”
(40:24) “G of R of R” should be “G from R to R.”
(42:01) “Function of the right side” should be “function on the right.”
Students need to know:
Much of algebra requires the ability to solve linear equations. The topics in this lesson also contribute to later learning of solving equations, solving linear inequalities, and algebraic manipulations.
Equivalence theorems
Solving equations in one variable is sometimes associated with metaphors involving balance or scales (see for example Kodaira, 1992, pp. 80–81), which is sometimes paraphrased as “doing the same thing to both sides of the equation.” In this lesson, “doing the same thing” is formulated in terms of the equivalence theorems. In solving other equations, “doing the same thing” may include squaring both sides, which—unlike using the equivalence theorems—may not lead to an equivalent equation.
Instructional decisions
The teacher gives some justifications by appeal to theorems of equivalence, others by appeal to geometrical interpretation of equations. Two examples of the latter include:
The different segments of the lesson build up by introducing new complexities. In the first segment: (00:17–8:18), the problems the teacher presents (see lesson graph) are represented by equations of the form.
x + a = b
cx + a = dx, and one instance of
cx + a = dx + b,
where in all cases c = d + 1, so it will not be necessary to divide by the coefficient of x to solve the problem, as is necessary in the next segment.
In the second segment: (8:18–34:46) equations of the form cx + a = b where c is not 1.
In the third segment: (34:46–45:00) equations of the form cx + a = dx + b where c is not d + 1.
Meaning of “procedure,” student understanding
(1:40–3:16) The statements of these problems are classified as “using procedures” in the TIMSS study, but for students such as Katia, the lesson may not be only a matter of “using procedures,” but rather of solving a problem—which begs the question of the meaning of procedural knowledge and student understanding. (See the first meaning of “problem” in Problems and Problem Solving.)
Specific teacher moves of interest
Students are sometimes encouraged to think of getting rid of a positive term by adding its negative and getting rid of an integer by multiplying by its reciprocal, but not always (see, for example, 12:51).
(16:30 and 16:37) The teacher mentions why it may be desirable to put the terms with x’s on the right (to avoid negative coefficients), and has a later conversation (24:55–25:11) about the position of x that mentions the alternative position of always wanting to get the final solution as x = something, with x on the left side:
In several instances the teacher uses a concrete model to clarify a mathematical point:
(14:47–15:14)
(23:17–23:24)
(35:02–35:07)
(27:09–27: 21) The teacher gives a reason for why Ø is not equal to {Ø}.
(20:50–21:13) The teacher provides clarification for a rule:
“You can’t do times half x . . . You can't do times letters . . . It's forbidden, we’ve seen. Do you want me to show you again on the transparency what happens? Do you remember? Suddenly it has curves. We have raised one solution. Therefore we only have three possibilities to do something . . . All numbers, add letters, or multiply by a number which is not equal to zero . . . We can’t multiply by a literal part . . . We're not allowed to do that.”
(17:01–18:32)
The teacher interrupts student work to discuss problems he discovered they were having. In the context of an example (second set of exercises, B), he tries to clarify that “first thing, isolate everything unknown on the same side.” (It would have been clearer to say: “First, combine all the terms that contain the unknown on the same side.”) Then he could have illustrated dividing by the coefficient to finish the solution.
In the discussion of 4x = 4x + 1, the teacher uses the empty set sign and draws the picture of the two lines. He does not explicitly point out that there is no number x that would satisfy the equation, although he does recall why the parallelism of the graphs is related to the nonexistence of a solution (41: 33–41:45). This explanation may have been too fast for students, and he does not explain the connection between the solution and the geometry (students should have recalled it, but probably many did not).
Kodaira, K. (Ed.). (1992). Japanese Grade 7 Mathematics (H. Nagata, trans.). Chicago: University of Chicago School Mathematics Project (UCSMP). Available from UCMSP.
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