This is the first of five lessons on the Pythagorean theorem. 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 pre-university track.
| 1:05–27:09 | (Whole class) The teacher presents information and calls on students. Some students ask questions. |
| 1:05–1:48 | The teacher says, “We will begin with Pythagoras” and asks who has heard of Pythagoras and when would one use his theorem? |
| 1:48–4:51 | The class reviews triangles: right, isosceles, and equilateral. The teacher calls on students for characterizations of the triangles. |
| 4:51–5:50 | The class discusses names of the sides of a right triangle. The teacher presents information and calls on students. |
| 5:50–6:59 | The teacher says, “We will use Pythagoras to calculate the sides of a triangle” and mentions a calculation of square root of 37 from the previous lesson. |
| 7:03–10:28 | Problem 2 (see the textbook pages in the CD resources section and the translation) calculates the length of the side of a square inscribed in another square. The teacher draws the squares on a grid. (The area of the large square is nine. The area of the smaller square is the area of the larger square minus the areas of the four triangles. Each of the triangles has an area of one unit. Thus the area of the smaller square is five and its side length is  5.) The teacher says, “We have no problem figuring this out; we may not need Pythagoras at all.” |
| 10:29–16:01 | The teacher continues, “But it is kind of a hassle if you constantly have to draw on like this.” The teacher uses the overhead projector to do problem 5 (see textbook pages and translation), showing that one can rearrange a square inscribed in another square (as shown in the textbook). She illustrates the relationship between the hypotenuse of the triangle created by the smaller square and the legs of the triangle with a numerical example. |
| 17:42–27:09 | The teacher summarizes the preceding discussion and statement of the Pythagorean theorem. The class calculates the hypotenuse of a right triangle with legs of three and four. A student asks a question about the statement of the theorem. |
| 27:09 | (Whole class) The teacher assigns homework (see lesson graph for problems). |
| 29:20–50:20 | Students work individually on homework problems. |
(1:05–6:16) This segment shows a lesson setup that introduces new learning (about Pythagoras and the Pythagorean theorem) and elicits prior knowledge about properties of triangles and the Pythagorean theorem. (This is an example of formative assessment.) The teacher motivates the lesson (see Motivating a Topic) by suggesting the limitations of measurement tools.
Mathematical language(4:33–5:45) The teacher introduces the term “hypotenuse” and invites them to use the term that is more comfortable for them. This segment raises issues about the use of mathematical language, such as whether a teacher should use specific terms (e.g., hypotenuse) or allow students to develop language as they learn the material. A number of times the teacher refers to “area” as “surface area”, a term that in the U.S. that usually refers to area in three dimensions.
(6:26–6:45) The teacher makes a connection between the area of a square and the length of the side and the same for triangles. This lays the groundwork for the discussion in 10:29–16:01.
(8:54–9:01) This segment shows an example of questioning. A student responds to a question and is not fully able to explain himself. The teacher probes but then moves on, looking for different responses. Discussion of this clip provides an opportunity to consider the role of questions and how teachers can use them to support thinking, making connections, etc.
(5:50–6:59, 10:29–10:50) The teacher’s knowledge of students’ experiences in lower grades allows her to connect the topic of the lesson with experiences in previous grades.
(7:03–10:28). See the teacher’s comment about students’ prior experience with this method. The national research coordinator comments that in most schools in the Netherlands, teachers give lessons in all grades of lower or secondary school, or both, so teachers can make use in the lower grades of experience they gain teaching secondary grades.
The teacher touches on the idea of inverse in the “reverse calculation” (in U.S. terms this is “taking the square root”), mentioned in the textbook between problems 2 and 3.
(The entire lesson, particularly 21:46–27:01) The lesson and the problems students work on publicly and individually are consistent with the lesson introduction and the geometric proof of the Pythagorean theorem.
Technical detailsFile name: NB11121749.
Viewers may disagree with some statements in the lesson graph and the lesson index about the characterizations of segments 2, 3, and 4. There may be some mistakes in the transcript; at 8:02 something that might be “triangles” may have been translated as “angles.” It is hard to see what is on the board during much of the time, but the video captures most board writing at some point.
The textbook pages (in Dutch) for the lesson are in the Resources section of the CD. A translation is provided.
Students needed to know:
This lesson exposes students to ideas about proof and prepares them to learn:
Use of the overhead projector
(10:29–16:01) The teacher uses an overhead projector in discussing textbook problem 2. For more discussion of more general issues in the use of visual aids in classrooms, see Using Technology.
(11:14–13:47) It may be easier to identify the length of the sides of the two squares as a + b. There is no need to place one transparency on top of the other.
Proof
(10:29–16:01) Some viewers may think that the teacher’s actions in this segment are a proof of the Pythagorean theorem, but there are at least two reasons why it might not be. It seems that the small triangles fit together to create a large square, each of whose sides is a hypotenuse of a small triangle, but the teacher does not discuss whether this shape is in fact a square. The teacher does the “proof” for a specific right triangle, but we do not know whether it works for all right triangles.
(13:28–13:47) The teacher could have put more emphasis on “removing equal from equals leaves equals,” which is the crux of the proof. This should be considered an incomplete proof. The teacher makes the claim that the tilted figure inside the larger square is a square, based solely on the fact that the inner figure seems to be a square. The teacher says nothing about why this is the case, and no student asks, since the figure looks like a square. One way to show that it is a square is to use the fact that the sum of the angles of a triangle is 180 degrees:
Figure 1: Proof that the internal quadrilateral is a square
Pythagorean theorem
(5:50–18:06) This segment shows an extended and varied instructional session on the Pythagorean theorem using specific numbers or geometric reasoning that builds on students’ prior knowledge of geometry. The segment uses three different representations, which vary in complexity and concreteness. The segment shows connections between the numerical argument used with a specific right triangle given on a grid and a more geometric approach to the Pythagorean theorem, both built from the same picture. This clip could also be effective in helping teachers to build their content knowledge of the Pythagorean theorem. The clip gives an introduction to proving the theorem and provides an opportunity for students at the middle school level to consider the importance of proofs.
(18:43–21:14) This segment shows a summary of the meaning of the Pythagorean theorem.
A video of the Theorem of Pythagoras is available at Project MATHEMATICS!
Instructional decisions
(10:29–27:09) The teacher departs from what is in the textbook and explains why. She chooses not to label the legs and hypotenuse of the triangle, a, b, and c and instead writes the statement of the theorem in terms of sides of a triangle. Her reasons are suggested in the teacher commentary (19:26).
The teacher allocates time to different types of work. The extended work session (27:19–49:26) provides opportunities to discuss the use of individual work in this lesson and compare its use with that in other lessons, i.e. J3, C3, US1.
The Theorem of Pythagoras, Project MATHEMATICS!
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