This lesson is a review of linear equations and their graphs. The lesson was originally 46 minutes long; this version (42 minutes) does not include a video segment shown in the class. There are 24 students in the class.
The National Research Coordinator notes that in the Netherlands students have three main tracks: secondary vocational education, higher vocational education, and pre-university. The students in this lesson are in the higher vocational education track. In grade 8, the higher vocational education and pre-university tracks use the same textbook.
| 01:25–12:32 | (Whole class) The teacher asks students about the homework. She calls on students for answers to problem 23 and writes equations on an overhead transparency. Then she shows a transparency with four lines and four equations for problem 27 and asks students to match equations and graphs. |
| 12:32–14:36 | (Whole class) The class watches a video about mathematical relationships in real life (approximately 7.5 minutes; not shown in the TIMSS video). The video concerns relationships between: an automobile’s braking distance and speed, the number of alcoholic drinks a driver consumes and his or her automobile accidents, the number of days below freezing and the thickness of ice, the depth under water while scuba diving and the water pressure, and the number of cell phone calls and the phone bill. The teacher’s comments during the video concern whether or not relationships are linear or non-linear. |
| 14:36–16:03 | (Whole class) The class discusses the video. |
| 16:03–40:56 | The teacher circulates as students work individually on homework problems. |
| 40:56–42:41 | The teacher collects answer books as students leave. |
Like US1, this is a review lesson on graphing linear equations and mentions the slopes of perpendicular lines. Also like US1, the teacher engages students in what they notice with a question: “What do you notice about these last two equations?” (4:35–5:08). Students seem to be confused at times in understanding the y-intercept or “starting number” of the equation.
(4:35–4:56) The teacher calls attention to the fact that one can write the linear equation correctly in two ways: y = x + 1 and y = 1 + x.
The teacher uses the overhead projector at the beginning of the lesson and later in conversation with several students (25:16–27:40). Using video to make connections to real-life examples is an unusual aspect of this lesson, though it us unclear whether the video was effective. The teacher’s only comments about the video were pointing out whether the relationships were linear and those made immediately after showing the video (14:36–16:03).
Problem 27 has a “making connections” problem statement. The teacher expects students to analyze various relationships between equations and graphical representations. The implementation of the problem also involves making connections. (See the researcher’s comments at 6:11 and video 6:11–11:50.)
Like US2, the teacher uses “real-world examples” and tries to connect topics in class to students’ experiences (e.g., 14:36–15:16). The researcher notes that fifteen of the homework problems contain real-life connections, such as plotting the height over time of a burning candle and determining the number of colored tiles necessary for patios of varying size.
Technical detailsFile name: NB10311037.
The lesson graph appears to have a translation mistake: The problem about red and gray tiles should refer to “graphs” rather than “lines.”
The students need to be able to:
The topics in this lesson are preparation for learning about algebra, more work with linear equations and their graphs, and functions and relations and their graphs.
Questioning
In this lesson (unlike in US1), the teacher asks the students to go from graphs to equations as well as from equations to graphs and to explain specific connections between linear equations and their graphs. For example, at 3:30–12:15, the teacher asks questions about the significance of the number four, then about the fact that the line is descending: “Can you tell that from the equation?” Later in this segment, the teacher questions the students about four equations:
Teacher: Mark, which of these four is descending?
Mark: A and D.
Teacher: How can you see that? Ricardo?
Ricardo: It has a little minus mark.
Teacher: Yes, but where does the minus mark have to be?
Ricardo: In front of the slope.
Teacher: Yes, but I see three minuses here, and yet there are only two descending lines.
Both the form and the content of these questions differ from US1. In the segment above, the teacher seems to push the students to be more exact in their answers than does the US1 teacher. In the following example from US1 (2:05–2:11), the teacher seems to assume that the student did not mean to say “x-intercept”:
Teacher: Use the slope with the?
Student: y-intercept and x-intercept.
Teacher: y-intercept. Right? That's probably the way you'll probably want to do it, right Nick?
Student: Yeah.
Instructional strategies
Although students are working individually, the teacher talks with two students in one instance and gives the same assignment to several students (25:16–27:40).
Language
The numbers for slope and y-intercept in a linear equation are called “hellingsgetal” (slope-number) and “startgetal” (starting-number), so the words for the inclination of a line and the corresponding number in the slope-intercept form of its equation are different. The textbook refers to “formule” (formula) rather than “vergelyking” (equation).
The teacher discusses with students the order in which the terms of the equation are written (34:20–36:09), but the rationale for this order may have been unclear for students. There is a convention about coefficients with negative signs (e.g., –x + 3 is usually written 3 – x), but the teacher does not give rationales in her conversation with the students—such as that it is easier to compare two polynomials if they are in a standard form.
The teacher comments:
“There is some confusion about the notation. Sometimes I use another sequence in the equation. This is to teach students that equations can be written down in different ways. However, in retrospect, I think that the next time I teach this topic, I will stick to the method the book uses. Because students find this to be a difficult topic, the constant change in the sequence of the equations [order of the terms in the equation] causes too much confusion. That is why I advise the students seated a bit further down also to only select one method.”
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