It is the first in a sequence of four lessons working towards the more advanced concepts of polygons. The class has been together for two weeks. The class is in English—these students’ second language. There are 41 students in the class, and each student has been assigned a number. The teacher decides on whom to call by drawing from a stack of numbered cards. The lesson is 34 minutes long.
| 0:03–5:39 | (Whole class) The class reviews concave and convex polygons. The teacher writes “polygon” on the board. |
| 5:39–7:10 | Students come to the board individually. The teacher asks a student to draw a polygon on the board. The teacher labels it “convex hexagon” and asks for another student to draw different type of polygon. A second student comes to the board and draws a polygon. The teacher says it is a pentagon and convex. He asks another student to draw a different type of polygon. The teacher labels it “concave pentagon.” He asks students to discuss with their neighbor the difference between the concave and convex polygons. Students talk briefly. The teacher asks a student to describe the difference, and then asks the student’s neighbor to help. The neighbor mentions reflex angle. The teacher states that the rest of the lesson will focus only on convex polygons. |
| 7:10–13:14 | (Whole class and individual work) The teacher reviews the meaning of equilateral and equiangular. He asks students to draw 4- and 5-sided polygons that are (1) equilateral and not equiangular and (2) equiangular but not equilateral. |
| 13:14–21:35 | Four students come to the board to draw examples of above polygons, followed by a whole-class discussion. |
| 13:34–17:59 | Students draw 4- and 5-gons that are not both equilateral and equiangular. |
| 17:59–21:35 | Students draw 4- and 5-gons that are both equilateral and equiangular. The teacher says he is not satisfied with the examples of pentagons and asks two more students to draw examples. The teacher asks, “What do we call a polygon that is equilateral and equiangular?” |
| 21:35–23:42 | (Whole class) The teacher reviews the sum of the angles of a triangle. He draws a convex quadrilateral, draws a diagonal, and labels the interior angles. Students work individually to find the sum of the interior angles of the quadrilateral |
| 23:42–25:58 | (Whole class) The class discusses the sum of the interior angles of a (convex) quadrilateral. The teacher writes equations and asks students to come to the board to complete them. |
| 25:58–29:46 | Students work individually to find the sum of the interior angles of a (convex) pentagon. |
| 29:46–31:23 | Students work individually on finding the sum of interior angles of a pentagon. |
| 31:23–33:03 | (Whole class) The class discusses the sum of interior angles of a pentagon. |
| 33:03–34:00 | The teacher assigns homework. |
This 34-minute lesson covers a considerable amount of material. The teacher starts by having the students draw polygons rather than defining a polygon. For a different approach to the concepts of polygon, convex, and concave, see A1.
In his comments the teacher mentions one place where the lesson goes off track: He writes “a1+ a2 + b + c1+ c2+ d” rather than “a1+ d + c1 + a2 + b + c2,” and has to split the angles in a different way than into the triangles he wants.
(23:27) The national coordinator comments that when the teacher asks, “What is the sum?,” he has not yet discussed whether the sum is a constant or not.
Although all segments of the lesson touch on polygons, there is no flow between the different segments. For example, the purpose of talking about regular polygons before exploring the sum of the interior angles of a polygon is unclear. (This may have just been part of review.)
The students’ drawing capabilities are very impressive. Their notebooks look like printed books.
File name: NB11121456.
The lesson graph states that the teacher draws a hexagon on the board, when in fact a student draws the hexagon; the teacher labels it. Similarly, a student draws the concave pentagon and the teacher labels it.
The homework assignment on the lesson graph should be “Find out the relationship between the number of sides of a polygon. . . . ” rather than “Find out the relationship between the number of sides of a pentagon. . . . ”
Students need to know:
Students will use their knowledge of types of polygons and their properties throughout their study of geometry (e.g., when studying similar and congruent figures; solving problems that involve angle measure).
(21:35–29:46) This work is an example of solving mathematical problems by reducing them to simpler ones that have already been done (see Heuristics)—a very important idea.
Teachers should do the following problems before viewing the lesson:
If this is not possible, explain why not.
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