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Polygon Angles
Intent
Building on previous work with patterns, In-Out tables, and functions, this activity asks students to generalize their observations about the relationship between the sum of interior angles and the number of sides of a polygon.
Mathematics
In this activity, students generalize the results from triangles and quadrilaterals to all polygons. This mathematical investigation is a valuable opportunity for them to learn about what doing mathematics is and to see themselves doing mathematics.
Progression
Students work in groups on a rather open task to explore and record what they notice about the sums of polygon angles. If they record their observations about different types of polygons in an In-Out table comparing number of angles to angle sum, they may observe another pattern.
Approximate Time
30 minutes
Classroom Organization
Groups
Materials
Protractors
Doing the Activity
This activity provides another opportunity for students to engage in a fairly open, unstructured exploration allowing them to approach the problem, structure their own time, and organize their data in their own way.
This activity immediately follows the observations made about angle sums for triangles and quadrilaterals. Tell students that now they will explore polygons with more than four sides.
As groups explore, you will have an opportunity to observe who may be having trouble with the protractor. Encourage group members to help each other use the tool properly.
While groups work, encourage them to gather data, make observations, and look for patterns. Ask, Think about different ways to organize your data to see if there might be patterns in your findings. By drawing polygons and measuring and adding their angles, students can build an In-Out table of conjectures, like the one below. What do you notice about your table?
| Number of Sides | Angle Sum |
| 3 | 180 |
| 4 | 360 |
| 5 | 540 |
| 6 | 720 |
| 7 | 900 |
Ask questions to help students progress from simple pattern identification to rule building.
Given these results, what might be a conjecture for the angle sum for a 10-sided polygon? A 12-sided polygon? A 100-sided polygon? An n-sided polygon?
Is there a general formula for connecting the In to the Out in this table?
Students may recognize that all angle sums are a multiple of 180 degrees, but what multiple? This generalization—if n is the number of sides, then (n – 180 is the angle sum—is the key underlying functional relationship. It is likely that several groups will notice the generalization, but may not have symbolic notation for the rule; it is not an easy step to recognize that (n – 2) can be written to represent “two less than the number of sides.” You might encourage students to write their rules as sentences.
The important challenge is the proof that this relationship must always hold, even beyond the data in the table. For groups that have achieved some confidence with this pattern, begin by reminding them it is only a pattern in the shapes they have seen—are they certain the pattern continues?—and then challenge them to prove the relationship they have conjectured.
Why must your rule be true for all polygons?
A slightly simpler question, to get a group started, is, Why should the triangle sum for quadrilaterals be exactly twice that for triangles?
Discussing and Debriefing the Activity
Bring the class together when you think that groups have made good progress exploring and observing patterns and a whole-class discussion can help them to move forward.
You may want to begin by asking students to review what they saw yesterday about angle sums for triangles and quadrilaterals. Then let volunteers state their conclusions about angle sums for polygons with more sides. Although their measurements will again be approximate, they will probably come up with conjectures that can be entered into an In-Out table. Take this table as far as students’ results lead, and then ask, Did anyone come up with a general formula expressing the angle sum as a function of the number of sides?
If you get a clear statement of the generalization, try to determine whether the class sees where the formula came from. If not, you can build up to the formula by asking students to guess what the angle sum would be for polygons with a specific number of sides not covered yet, based on information in the table.
For example, if the table goes up to a 7-sided polygon, ask students to use the data to formulate a conjecture for 10-sided polygons. What do you think is the sum of the angles in a 10-sided polygon?They should probably be able to extend the table by adding 180° three times to get additional rows. By now, they will probably have recognized that the Out values all seem to be multiples of 180°.
You can follow up with a large numerical case, such as a 100-sided polygon. What should you multiply 180° by to get the sum of the angles in a 100-sided polygon? At this point, the issue should be something like, What should you multiply 180° by to get the sum of the angles in a 100-sided polygon? Students should be able to confirm that the necessary factor seems to be found by subtracting 2 from the number of sides.
Add a row to the table to show this formula.
| Number ofsides | Angle sum |
| 3 | 180° |
| 4 | 360° |
| 5 | 540° |
| 6 | 720° |
| 7 | 900° |
| 8 | 1080° |
| 9 | 1260° |
| 10 | 1440° |
| 100 | 17640° |
| n | (n – 2)180° |
Ask if anyone can explain why the angle sum for quadrilaterals should be exactly twice that for triangles. They should be able to see that a diagonal can be constructed to split a quadrilateral into two triangles; this works even for concave quadrilaterals. Without getting into a lot of detail, use that fact to conclude that the angle sum for a quadrilateral is the sum of the angle sums for its two triangles. Emphasize that this argument does not prove that the angle sum for a triangle is 180° or even that every triangle has the same angle sum. It does prove that if every triangle has an angle sum of 180°, then every quadrilateral has an angle sum of 360°.
Finally, ask how the argument for quadrilaterals might be used to explain the formula for the general polygon. Here are two approaches students might use.
- They may see—using more examples, if needed—that a polygon with n sides can be divided, using diagonals, into n – 2 triangles. This is easy to see for convex polygons (all angles less than 180°), but can also be done for concave polygons, with slightly more effort.
- They may see that a single diagonal can be used to divide an n-sided polygon into a triangle and an (n – 1)-sided polygon. This explains why each side added to the polygon increases the angle sum by 180°. (This approach is another example of recursive reasoning; see the discussion of Diagonally Speaking.)
Key Questions
Think about different ways to organize your data to see whether there might be patterns in your findings.
What do you notice about your table?
What would be your conjecture for the angle sum for a 10-sided polygon? A 12-sided polygon? A 100-sided polygon?
Is there a general formula connecting the In to the Out in this table?
All angle sums are a multiple of 180 degrees, but what multiple?
What do you think is the sum of the angles in a 10-sided polygon?
What should you multiply 180° by to get the sum of the angles in a 100-sided polygon?
Why must your rule be true for all polygons?
Why should the triangle sum for quadrilaterals be exactly twice that for triangles?
Supplemental Activity
A Proof Gone Bad (reinforcement) asks students to explain the contradictions in another student’s proof.
Comments, questions, feedback, criticisms?
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- E-mail the author of the module, Polygon Angles
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This work is licensed by Interactive Mathematics Program under a Creative Commons Attribution License (CC-BY 2.0).
Last edited by Christine Osborne on Jun 3, 2008 8:39 pm GMT-5.