This lesson focuses on investigating the sums of the exterior angles of convex polygons. It is the third in a six-lesson unit on “Working Mathematically with Space—Making and Testing Conjectures.” There are 32 students in the class. The lesson is 45 minutes long and is held in a computer lab with 15 computers.
Lesson outline
| 0:00–2:44 |
Students come into the classroom and sit down. |
| 2:44–6:15 |
(Whole class) The class reviews a student’s conclusion from the previous day about the sum of the angles of a five-pointed “star” inscribed in a circle. The teacher displays the student’s work on the overhead projector (see the Resources section of the CD for a copy of the student’s work). The previous lesson consisted of a Geometer’s Sketchpad exploration using a five-pointed star inscribed in a circle. The goal was to find the sum of the five angles of the star. The teacher passes out a worksheet and displays a copy on the overhead.
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| 6:15–8:59 |
Students read the worksheet “Exterior Angles in a Polygon” (shown in abbreviated form on the Lesson Graph), then work in pairs to identify words that are “important to understand.” The worksheet directs students to use Geometer’s Sketchpad to construct a convex pentagon, construct exterior angles, find the sum of their measures, and then to try similar constructions with other convex polygons. |
| 8:59–10:16 |
(Whole class) The teacher asks students for the words they have underlined on the worksheet (convex, quadrilateral, polygon, exterior angle, ray) and underlines or circles them on the overhead. |
| 10:13–16:19 |
(Whole class) The class discusses the meaning of the words (using a whiteboard), and the teacher gives directions for computer investigation:
| 10:16–11:20 |
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Meaning of polygon |
| 11:20–13:22 |
|
Meaning of exterior angle |
| 13:22–14:52 |
|
Constructing the diagram using the computer |
| 14:59–15:57 |
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Meaning of convex and concave |
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| 16:19–19:02 |
The teacher asks students to write their predictions about the sum of the exterior angles (the topic of the worksheet activity) and mentions the term “conjecture.” |
| 19:06–42:15 |
Students work in groups of two or three on the worksheet using Geometer’s Sketchpad. The teacher circulates and answers questions. |
| 42:15–43:00 |
(Whole class) The class checks its predictions against the results; most students have concluded that the sum of the exterior angles of the pentagon is 360 degrees. |
| 43:00–45:00 |
Students leave the classroom. |
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Mathematical and pedagogical comments
Mathematical reasoning
(2:44–5:35) The teacher gives no reasons why any of the claims are true.
Instructional decisions
(6:15–18:52) The class discusses the meaning of several geometry terms. HK3
(00:56–6:56) discusses some of these terms quite differently.
(16:19–19:02) The teacher departs from the curriculum materials without explanation.
The use of technology
(2:44–6:15) The teacher uses the overhead projector to display student work, but removes it and thus does not leave a record.
(10:13–16:19) The teacher uses the whiteboard to discuss meanings of terms underlined on the transparency displayed on the overhead projector. The teacher erases drawings made in the discussion, thus not leaving a record that students can refer to later in the lesson as in, for example, J3. (For a discussion of board work and possible reasons for different uses of the board and overhead projector, see Using technology).
(19:06–42:15) Students work in pairs or in groups of three at the computer using geometry software. In contrast, the J2 teacher uses geometry software only in whole-class discussion at the beginning (0:27–1:27) and at the end (46:36–49:47) of the class.
Missed opportunity
(36:13–37:58) A student measures the wrong angles of a pentagon—the interior rather than the exterior angles. The teacher asks him what to call these angles, and the student replies, “interior.” The teacher tells the student to use a computer, but before that to figure out what angles to measure. This error provided the teacher with an opportunity (which he does not take) to have the student use the relationship between interior and exterior angles to solve the problem. Since the student measured the interior angles correctly, he would have been able to solve the problem.
Since the connection is not related to the original problem, this is a good example of why a teacher needs a firm command of more mathematics than what is planned for the lesson.
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Technical details
File name: NB11121436. It is generally difficult to determine what is on the computer screens, except when the camera gets a close view.
The worksheets used in the previous lesson and the videotaped lesson come from materials provided with the Geometer’s Sketchpad software.
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Further Discussion
Relationship with previous learning
Students need to know:
- the concepts of an angle and angle measure
- the meanings of convex polygon, ray, interior angle, and exterior angle.
If the students are to prove the conjecture after the computer work, then they need to know that the sum of the angles in a triangle is 180 degrees and to have some experience with proof.
Students should have had experience using Geometer’s Sketchpad (e.g., the use of a text box, measuring angles, drawing rays, and placing points on a line).
Relationship with future learning
Experience with making and testing conjectures and drawing conclusions will be helpful in higher-level algebra and geometry classes, and increase general understanding of reasoning and proof. The generalization about the sum of the measures of the exterior angles of a polygon will be useful when studying other angle measure relationships in geometry and trigonometry.
Knowing how to use this technology might be helpful later. Knowing geometric definitions will be useful, and knowing that it is possible to define something is important.
Detailed mathematical and pedagogical comments
Mathematical reasoning
(2:44–5:35) The teacher gives no reasons why any of the claims are true. That is not part of the goal, but he should mention these since the students will need to engage in mathematical reasoning eventually.
Instructional decisions
(6:15–18:52) Students are asked to mention various words they do not know, and either students read descriptions from the book (e.g., for exterior angle) or the teacher draws pictures (e.g., convex and concave polygons). It would have been better to have students draw some polygons, as in HK3. This is especially true for convex and concave polygons but also true for figuring out a definition or description of a polygon. (For a different treatment of convex and concave, see HK3).
(10:13–10:51) The class discusses the question, “What is a polygon?” The teacher draws a round figure with squiggles. Another student says a polygon is any two-dimensional shape that is enclosed. The teacher remarks that the previous drawing is enclosed. The student says “a many-sided shape,” then adds “straight.” The teacher says that this is getting closer and says, “A shape that has a number of straight sides.” He never mentions that it needs to be closed. Pictures would have helped here: pictures of polygons and of figures that are not polygons, in addition to the round figure that was drawn.
(13:22–14:52) The teacher draws rays but does not discuss the meaning of “ray” separately.
Instructional decision: Departure from the curriculum materials
(16:19–19:02) The teacher asks students to write down their predictions about the sum of exterior angles before they move to the computers. In contrast, the worksheet asks students to construct the pentagon, investigate it, and then make a conjecture. Unlike the N2 teacher who also departs from the written materials, the A1 teacher does not give a reason for his action.
Asking students to make predictions before doing a simulated experiment on computers has been used in middle school science education to good effect (see Linn & Hsi, 2000, pp. 104–105) when teachers also request explanations for the predictions (see, e.g., p. 67).
Related to this is the meaning of “conjecture.” (16:19–18:24) The teacher says, ‘So we have a conjecture, but we haven’t really got any evidence yet.’ This a guess, not a conjecture, since you can make a conjecture only after you have evidence.
Uses of this lesson
Team members recommend using the following procedure with teachers:
- Read the assignment sheet and watch a portion of the video (15:01–18:50).
- Go through the investigation and make a conjecture about the sums of the measures of one set of exterior angles of convex polygons.
- Justify your conjecture.
- Write a definition of exterior angle that would allow your conjectures to apply to any polygon.
- Watch video segment (33:00–37:00). How can technology be used effectively with students?
- Watch lesson segment (15:45–18:52): setting up the problem.
- View video segment (10:13–10:51).
- Based on the discussion, which of the following figures might a student have difficulty deciding whether it is a polygon? Why?
- How would you introduce the concept of a polygon to prevent misconceptions about the meaning of “polygon”?
- Write a definition for polygon that would include figures A and E and exclude all the other figures.
For an alternative brief introduction to polygons see HK3.
- Before viewing this lesson, teachers should solve the five-pointed star problem (see above); ask if it is necessary to have the points of the star on a circle, and ask what this problem has to do with the main problem: finding the sum of the exterior angles of a convex pentagon, and, more generally, any convex polygon. Teachers should try to figure out if the theorem for convex polygons continues to hold for non-convex ones.
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Reference
Linn, M. C. & Hsi, S. (2000). Computers, teachers, peers: Science learning partners. Mahwah, NJ: Lawrence Erlbaum Associates.
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