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# Pythagorean Theorem

The Pythagorean theorem—one of the great theorems in mathematics—is useful as well as surprising. Students should appreciate how to use it, but also learn about its origins.

Summaries of the TIMSS public release lessons dealing with the Pythagorean theorem follow. Some lessons feature this theorem as the main focus and include a derivation and/or applications, while others provide preliminary material or lesser applications. Additional resources are listed at the end.

HK1 Preparation: squares and square roots

In preparing to learn the Pythagorean theorem, students do whole-class and individual work focusing on squares and square roots, since knowledge of these is a prerequisite for using the theorem. This lesson provides an example of a teacher’s response to student error and misunderstanding.

N2 Introduction, proof, and applications of the theorem

In this introduction to the Pythagorean theorem, the teacher presents an incomplete proof of the theorem (the missing step is using the sum of the angles of a triangle to show that the inner figure is a square; see fig. 1 in Proofs). Whole-class and individual work also focuses on applications of the theorem.

C4 (38:00—49:00) Sample problem

At the end of a lesson on exponents, students work individually on a problem that applies the Pythagorean theorem. The teacher circulates.

The figure depicts a roof that is 5.5 meters high. The owner would like to lay new copper roofing to stop the leaks. Calculate how many square meters of copper paneling he will need in order to cover the entire roof, allowing for 15% extra for cutting and connecting the metal?

US2 (10:39—10:55) Sample problem

In a lesson on variable expressions, one warm-up problem is:

Students work on warm-up problems individually, and then one student claims the example above is a right triangle, because 25 + 144 = 169. (The teacher does not ask what led to this conclusion, but the students are likely to think it follows from the Pythagorean theorem. This reasoning is backwards, since the Pythagorean theorem starts with a right triangle, and the present question asks whether this is a right triangle. The theorem, which implies that it is a right triangle, is the converse of the Pythagorean theorem.)

S3 Sample problem

Asked to measure the distance between two points in three dimensions, students do this using the Pythagorean theorem twice. The two points are the end points of a diagonal of a rectangular parallelepiped. The students work in groups and then discuss their solutions.

C1 Sample problems

In a series of short tasks, students use the Pythagorean theorem in the plane and in three dimensions. They work at the board as well as in their seats.

Kodaira, K. (Ed.). (1992). Japanese Grade 9 Mathematics (H. Nagata, trans.). Chicago: University of Chicago School Mathematics Project (UCSMP). Available from UCSMP.

Chapter 6 gives a treatment of the Pythagorean theorem, its converse, and applications to plane figures and three-dimensional figures, including cones, spheres, and pyramids. This book can be used as a resource for other proofs and some applications; it contains two proofs and picture sketches of seven more proofs.

Polya, G. (1990). Mathematics and plausible reasoning, vol. 1. Princeton, NJ: Princeton University Press.

Section 5 of chapter 2 discusses a generalization of the Pythagorean theorem as an example of using analogy.

Shannon, A. (1999). Keeping score. Washington, DC: National Academy Press.

See pages 15—16 for two quite different right-triangle tasks. Students can do one by applying the Pythagorean theorem and the other using its converse. The latter was not straightforward for many students.

Shannon, A. (2003). Using classroom assessment tasks and student work as a vehicle for teacher professional development. In Next steps in mathematics teacher professional development, grades 9—12: Proceedings of a workshop. CD available from the National Academy Press.

This article discusses teachers’ use of and student reactions to an assessment task that involves multiple applications of the Pythagorean theorem. Like the problem involving the Pythagorean theorem in C4, this task was not straightforward for students. In both cases, the way the task is implemented influences how students respond.