Today, we continued looking at properties of polygons. I needed to cover regular versus irregular polygons, the exterior angle theorem for polygons, and the interior angle theorem for polygons.
To start things off, I gave the class a sheet of regular and irregular polygons. I asked them to name the polygons and to discuss whether they thought they were regular polygons or not. The sheet I used was an Identifying Regular and Irregular Polygons worksheet from Common Core Sheets - Shapes page. This was the first time I have used the Common Core Sheets site but I will probably use it again since there are a lot of options for sheets, reviews, and tests/quizzes.
I gave students time to work through naming the polygons, working with those students who needed help on an individual basis. Students discussed their ideas about which were regular polygons and which were irregular polygons at their tables and then we did a class share out.
On one board I wrote the title Regular Polygons and on another board I wrote the title Irregular Polygons. Students shared their ideas about what they though made a polygon regular or irregular. For regular polygons: common, well-known shape; all sides congruent. For irregular polygons: zi-zag shape, obtuse angles.
From here, I tackled the idea of regular polygons had congruent sides. I drew two hexagons, one regular and one irregular. The sides were congruent in each. Students could recognize the regular polygon but then wrestled with what else was needed to define regularity. Finally, a student said that all the interior angles were also congruent. Bingo, we had a definition for a regular polygon: A regular polygon is a polygon that has all sides and all interior angles congruent.
I next tackled the obtuse angle idea for irregular polygons. I asked students to look at the figures of regular octagons and decagons that they had on their worksheet. They could see that having an obtuse angle did not affect the condition of irregularity. We instead turned to the definition of regular polygons to guide the definition of irregular polygons: An irregular polygon is a polygon that has at least non-congruent side or angle.
Since we never formally defined what a polygon was, I took time to draw three closed figures on the board. The first had two straight sides and a curved side. The second had four sides, two of which crossed over each other. The third was a polygon. I asked students which of the figures were polygons. Students generally thought the first two figures were not polygons. They were correct for the first figure and they had noted it had a curved side.
The second figure was a bit trickier since it was a complex polygon. I pointed out how this was a closed plane figure formed by line segments (straight line segments) that intersected at single points. From this definition, the second figure did meet the criteria of being a polygon.
With that settled, I asked the class what an exterior angle was for a polygon. I drew a pentagon on the board. I asked students to think about how exterior angles were formed in triangles. A student said the exterior angle would be formed by extending a side. We discussed how many exterior angles could be formed and I drew side extension. I mentioned to the class that I liked to make the extensions look like a pin wheel.
We then moved into an investigation of exterior angles. Students drew different convex polygons and measured their exterior angles using a protractor. They then added their angle measurements together to find the exterior angle sum. I walked around helping students with using their protractor and with drawing exterior angles properly.
Students saw that the exterior angles were always summing to 360o. I then said we could use this information to answer questions such as, "What is the measure of exterior angles for a regular 18-gon?" Students came up with the answer and I checked by asking other students why they responded 20o. I also told students they should be able to answer, "How many sides does a regular polygon have if its exterior angles are all 18o?
From here I moved to interior angles. If the exterior angle is 18o, what must the interior angle measure? Students realized that the angles formed a linear pair and they just needed to subtract the angle measurement from 180o. I checked for understanding with other students and then moved to the next investigation.
I asked students to create a table that listed the interior angle measurement, the exterior angle measurement, and the sum of the interior angles. I asked the class to complete this table for polygons from triangles through decagons.
Some students were confused and asked for clarification. I used the triangle as an example. The exterior angles for a triangle summed to 360o so what does the size of an exterior angle have to be? Next, knowing that the exterior angle is 120o, what is the size of the interior angle? Knowing you have three interior angles, what is the sum of the interior angles. I then told students to do this for all the other polygons in the table.
Students were just finishing up this task when class ended. Their homework is to complete the table, if they hadn't finished, and then look for patterns in the sum of interior angles. Next class we'll look at what they come up with and then explore why the relationship is present.
This is a reverse from how I have taught the interior angle formula in the past. Most books break the inside of a polygon into triangles and then count how many triangles are present to determine how many times you multiply 180o.
My approach this year has a better feel. It ties together the exterior angle theorem to the size of interior angles which directly leads to the sum of the interior angles. We can then explore the result that the sum increases by 180o each time. The triangle breakdown helps to explain or justify the pattern that is found versus force fitting the pattern because of how we are breaking down the inside of a polygon.
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