This lesson focuses on geometrical solids. It is the first lesson in which students use the Pythagorean theorem to explore solids, although they know and can apply it to two-dimensional polygons. The lesson is taught in Swiss Italian and is 50 minutes long. There are 17 students in the class.
| 00:09–03:1 | (Whole class) The teacher poses the day’s problem: “We want to send a present to our friends, that has a long and narrow shape. At the post office there are boxes with six different shapes. And these are the six shapes you can see over there [on a table]. Take one of these straws; try to find out the position in which the straw can be placed to obtain the longest straw.” |
| 03:11–36:19 | Students work in groups of two or three to solve the problem and complete a worksheet that the teacher has prepared (see National Coordinator commentary). Each group has a different box to use, as well as straw, measurement tools (carpenter’s hinged measuring stick), and a calculator. The teacher circulates, talking with different groups of students. After the measurements and calculations, and completion of the worksheet, each group prepares a transparency for presenting the results. |
| 36:19–37:42 | (Whole class) The class begins a discussion of the worksheet. |
| 37:42–38:20 | (Whole class) The students insert “Pythagorean Theorem in solid geometry” as the title of the worksheet. |
| 38:20–40:37 | (Whole class) The class begins a discussion of methods for finding a solution. The teacher describes method 1: “Calculate the diagonal of the base of the box, then calculate the diagonal of the box using its height and the diagonal of the base.” (All the groups use this method.) |
| 40:37–41:59 | (Whole class) A student comes to the board to present method 2, which uses one less step than method 1: “Calculate the diagonal of the box by using the square of the diagonal of the base plus the square of the height of the box, then taking the square root.” |
| 41:59–43:18 | (Whole class) The teacher discusses method 2. |
| 43:18–48:47 | Students complete the worksheet. Those who finish begin work on a sheet of exercises with a partner. The teacher circulates, working with individual students. |
The National Research Coordinator comments: “The methodology used by the teacher is based on situated teaching…, with important moments of development … in which pupils take responsibility for their own learning and are free to act in group work. This methodology is explicitly taught to math teachers and supported by the math experts in the canton [district] of Ticino [the Italian part of Switzerland].” (Balacheff et al., 1997; see also review by Kieran, 1998).
The entire class focuses on one rather elaborate problem that involves geometric thinking about three-dimensional uses of the Pythagorean theorem, significant measurements and numerical calculations, and algebraic thinking about how to simplify the calculations strategically. The teacher challenges students to observe how to streamline the calculation by taking only one square root instead of two.
The fact that the teacher does not state the objective up front allows students to discover the goals of the lesson through the problem. The teacher comments, “I haven’t written the title on the worksheet because it would have been too easy for the students to understand the topic of the lesson. If it had been written, they couldn’t have made their discovery” [of applying the Pythagorean theorem].
(9:45–11:02) The teacher reasons with the students in one of the groups: “Try to think about if you utilize the fact that this one is high . . . Now reason within the group; explain it to them.”
Technical detailsFile name: NB11121751.
All worksheets except the second are included in the Resources section of the CD.
Students should know the following:
Students will build upon this knowledge to learn how to:
Manipulatives and the Pythagorean theorem
This lesson presents a common 8th grade topic in a context that allows students to manipulate materials and make connections to the Pythagorean theorem.
Real-world connection
The lesson is based in the real world without extraneous information to distract students’ attention from the mathematics.
Level of difficulty
(36:30–49:00) The teacher recapitulates student explanation of calculation simplification and uses language of “inverse operation.” The numbers are sufficiently complicated (two digits plus one decimal place) so that the numerical work is not trivial. Calculators are used to do the numerical work.
Change from the numerical solution to the algebraic form
The way the pupils have written their calculations is already highly formal—a numerical expression with use of parentheses and symbols for the power and the root—and therefore a preparation for the algebraic version. Students in the canton of Ticino begin this change into the algebraic form in the sixth grade and gradually develop it until the ninth grade.
It is useful to compare this lesson to C1, which examines the use of the Pythagorean theorem to solve problems with a plane figure (00:58–6:52) and a solid (7:17–12:59). See Pythagorean theorem for other lessons dealing with this topic. The lesson also serves as a good starting point for viewers to discuss the mathematical methods that would allow them to decide immediately which of the boxes given initially is the best choice—without having to calculate the length of the diagonal. (This could be done, for example, using the sum of the length of the sides. Viewers could organize computer-based research or make tables, etc.).
Balacheff, N., Cooper, M., Sutherland, R. & Warfield, V., Eds. & Trans. (1997). Theory of didactical situations in mathematics (Didactique des mathématiques, 1970–1990—Guy Brousseau. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Kieran, C. (Nov. 1998). Complexity and insight. Journal for Research in Mathematics Education, 29(5), 595-601 [this is a review of the previous reference].
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