At this point, we are coming up on finals. I am not introducing any new material but trying to give students a chance to work through a variety of problems. Since the focus has been on building conceptual knowledge, students need some time to practice using the building blocks they have learned.
As for the quiz, about 1/3 of students did a fair to good job explaining the diagonal formula. One student did an exceptional job. The rest did not put thought into the situation and simply said that n represented the number of sides. Period. The biggest issue in the explanations was explaining why n multiples (n - 3).
I'll be back to go through progress second semester starting in January.
Monday, December 14, 2015
Thursday, December 10, 2015
Diagonals in Regular Polygons - Part 2
I started off the second day of looking into diagonals in polygons by writing two questions on the board:
- What is the number of diagonals in a regular 50-gon?
- What is a formula to calculate the number of diagonals in a regular n-gon?
With that, I told the class I was awaiting their answers.
Of course, this led to a lot of grumbling, staring, and not a lot of activity. I used this as an opportunity to tell the class that every one of them had the knowledge and ability to answer these questions. They needed to think about what was going on and use what they already knew to answer the questions. I also told them I wouldn't be asking them these questions if I didn't believe they could answer them.
The class settled down and started working. A few students still were sitting there looking distracted. I talked to each one about what they were doing. I told them that avoiding the problem was not going to help find a solution and re-directed them to start writing down what they knew and drawing figures to help them consider the dynamics that were occurring as we moved from a square to a pentagon to a hexagon.
After a while, students started observing and making notes of patterns. Most saw that the difference in the number of diagonals increased by one each time. A few students actually wrote out the sequence all the way to the 50-gon in order to determine the number of diagonals. One student noticed that the number of diagonals connected to a single vertex increased by one each time, i.e. in a square a single diagonal is connected to each vertex, in a pentagon two diagonals are connected to each vertex, etc.
It took a while for students to get to this point. I then tried to push them to answer the second problem. Here, students really struggled. A couple of groups were able to write out an informal recursive formula. No one was able to create the closed formula, although a couple of students did look up the formula online, saying D = n (n - 3) / 2.
I focused on the recursive formula first. Students don't get to see recursive formulas very often and they aren't presented in a more formal setting. I wanted to contrast the recursive formula with a closed formula in the hope of contrasting that the closed formula is a function solely based on the number of vertices versus a function that is based on the preceding value.
I took what students had and re-wrote it using subscripts: Dn = Dn-1 + n-2. We discussed what this showed, how to use it, and how you could work through each step to get to an answer, say for D50. I then said that, theoretically, every recursive formula could be re-written as a closed formula. Sometimes the conversion could be complex, but it could be done.
We turned our attention to the closed formula. I told everyone that if they couldn't explain the formula they couldn't use the formula. I then pushed them to consider what the n represented, why was three being subtracted from n, why were these two values being multiplied together and then divided by 2?
Students immediately said that the n in the formula represented the number of sides. I drew a pentagon and its diagonals on the board. I asked if n is the number of sides, that does n - 3 represent? I erased three of the sides. Students were baffled. I told the class the formula should connect back to reality.
After thinking about it some more, most students still believed that n still represented the number of sides. I asked, if this was the case, why are you multiplying the number of sides by three less than the number of sides? Again the class had no response.
I asked what else n could possibly represent. A few students said it could be the number of vertices rather than the number of sides. I told the class that when faced with evidence that contradicts or doesn't support their assumption of n representing the number of sides, that, perhaps, they should rethink their assumptions rather than ignore the evidence.
We focused on a single vertex. How many diagonals could not be drawn from this vertex? Well, you can't draw a diagonal to the vertex itself. You also cannot draw a vertex to the two adjacent vertices. This means that there are n - 3 vertices where diagonals can be drawn. For how many vertices total can this happen? Well, there are n vertices in total. So, how many total diagonals can be drawn? Well, you have to add up n - 3 each time for every vertex. What's a shorthand way to do repeated addition? Multiplication! So n (n - 3) is counting the number of diagonals that can be drawn.
But why do we divide this number by 2?
At this point time was running out on the class. I told students to focus on the physical process that was occurring here. Some started to see that by the time you went around all the vertices that you actually had drawn to diagonals between each pair of non-adjacent vertices.
I told the class we would have a quiz on Friday. The quiz is one question: explain how the formula n (n - 3) / 2 models the number of diagonals in a regular polygon.
I know that students can look up the result online and that is fine with me. I want them to see that formulas aren't just these arbitrary results. I want them to see and understand that mathematics is simply a way of describing reality. And that by describing reality, we can make use of the math to answer questions that would be difficult or impossible to answer otherwise, such as how many diagonals does a regular 500-gon contain?
Students immediately said that the n in the formula represented the number of sides. I drew a pentagon and its diagonals on the board. I asked if n is the number of sides, that does n - 3 represent? I erased three of the sides. Students were baffled. I told the class the formula should connect back to reality.
After thinking about it some more, most students still believed that n still represented the number of sides. I asked, if this was the case, why are you multiplying the number of sides by three less than the number of sides? Again the class had no response.
I asked what else n could possibly represent. A few students said it could be the number of vertices rather than the number of sides. I told the class that when faced with evidence that contradicts or doesn't support their assumption of n representing the number of sides, that, perhaps, they should rethink their assumptions rather than ignore the evidence.
We focused on a single vertex. How many diagonals could not be drawn from this vertex? Well, you can't draw a diagonal to the vertex itself. You also cannot draw a vertex to the two adjacent vertices. This means that there are n - 3 vertices where diagonals can be drawn. For how many vertices total can this happen? Well, there are n vertices in total. So, how many total diagonals can be drawn? Well, you have to add up n - 3 each time for every vertex. What's a shorthand way to do repeated addition? Multiplication! So n (n - 3) is counting the number of diagonals that can be drawn.
But why do we divide this number by 2?
At this point time was running out on the class. I told students to focus on the physical process that was occurring here. Some started to see that by the time you went around all the vertices that you actually had drawn to diagonals between each pair of non-adjacent vertices.
I told the class we would have a quiz on Friday. The quiz is one question: explain how the formula n (n - 3) / 2 models the number of diagonals in a regular polygon.
I know that students can look up the result online and that is fine with me. I want them to see that formulas aren't just these arbitrary results. I want them to see and understand that mathematics is simply a way of describing reality. And that by describing reality, we can make use of the math to answer questions that would be difficult or impossible to answer otherwise, such as how many diagonals does a regular 500-gon contain?
Tuesday, December 8, 2015
Diagonals in regular polygons - a first take
To preface this post, I didn't get as far as I hoped to as our school had a lock-down drill today that took up half the class period.
I asked students to consider how many diagonals were contained within different regular polygons. I drew a triangle, a square, and a pentagon on the board and drew the diagonals within each. As I drew, I defined a diagonal as a line segment that connects two non-adjacent vertices. I wrote the number of diagonals below each figure.
I told the class they were going to investigate the number of diagonals and look for patterns. Specifically,. they were to draw and determine the number of diagonals present in a hexagon, heptagon, and octagon. They were to use the patterns they were seeing to help determine the number of diagonals in a 50-gon. Finally, they were to use what they did to determine a formula for calculating the number of diagonals in any regular polygon.
I asked students to work on this task individually, what I call "engaging in individual think time." I re-visited some of the norms for working on an individual: explore, identify characteristics, connect to other work they may have done, and to check their work. I asked them to work individually for 5 minutes and then they could work with their table groups.
As I walked around the room, I could see that students were struggling with the basic idea of diagonals and that they wanted to jump straight to drawing the octagon and its diagonals.
I wrote the definition of a diagonal on the board and told students they needed to draw all the diagonals for all of the figures. I continued to walk around and help point out where diagonals were missing.
Once students started to discuss what they were seeing there was some progress. We confirmed that a hexagon has 9 diagonals and a heptagon has 14 diagonals. It took a while for students to confirm the heptagon value. Some students had counted 10 diagonals. I asked students to consider if this value made sense given all the information they had. The diagonal pattern developed so far was 0, 2, 5, 9. Does it look reasonable that the next diagonal count would be 10? Students admitted that 10 didn't look right.
Some students had counted 16 diagonals for the octagon. I repeated my question of does that look like a reasonable value? Students are supposed to try correctly counting diagonals in an octagon and to try to use this pattern to determine the number of diagonals in an enneagon and a decagon.
There continues to be a distinct lack of effort when it comes to pushing thinking and true problem solving. Next class may not progress very far as my intention is to push the class until they identify patterns that make sense to them and can then model their patterns using mathematics. If all goes well, students will push through and be able to calculate the number of diagonals in a 50-gon and have a formula they can use for any n-sided regular polygon.
I asked students to consider how many diagonals were contained within different regular polygons. I drew a triangle, a square, and a pentagon on the board and drew the diagonals within each. As I drew, I defined a diagonal as a line segment that connects two non-adjacent vertices. I wrote the number of diagonals below each figure.
I told the class they were going to investigate the number of diagonals and look for patterns. Specifically,. they were to draw and determine the number of diagonals present in a hexagon, heptagon, and octagon. They were to use the patterns they were seeing to help determine the number of diagonals in a 50-gon. Finally, they were to use what they did to determine a formula for calculating the number of diagonals in any regular polygon.
I asked students to work on this task individually, what I call "engaging in individual think time." I re-visited some of the norms for working on an individual: explore, identify characteristics, connect to other work they may have done, and to check their work. I asked them to work individually for 5 minutes and then they could work with their table groups.
As I walked around the room, I could see that students were struggling with the basic idea of diagonals and that they wanted to jump straight to drawing the octagon and its diagonals.
I wrote the definition of a diagonal on the board and told students they needed to draw all the diagonals for all of the figures. I continued to walk around and help point out where diagonals were missing.
Once students started to discuss what they were seeing there was some progress. We confirmed that a hexagon has 9 diagonals and a heptagon has 14 diagonals. It took a while for students to confirm the heptagon value. Some students had counted 10 diagonals. I asked students to consider if this value made sense given all the information they had. The diagonal pattern developed so far was 0, 2, 5, 9. Does it look reasonable that the next diagonal count would be 10? Students admitted that 10 didn't look right.
Some students had counted 16 diagonals for the octagon. I repeated my question of does that look like a reasonable value? Students are supposed to try correctly counting diagonals in an octagon and to try to use this pattern to determine the number of diagonals in an enneagon and a decagon.
There continues to be a distinct lack of effort when it comes to pushing thinking and true problem solving. Next class may not progress very far as my intention is to push the class until they identify patterns that make sense to them and can then model their patterns using mathematics. If all goes well, students will push through and be able to calculate the number of diagonals in a 50-gon and have a formula they can use for any n-sided regular polygon.
Monday, December 7, 2015
Meaningful formulas for interior and exterior angle sums
We had district testing last class, so it's been almost a week since we last looked at polygons, exterior angle values, and interior angle values. I figured I would need to spend some time easing into where we left off with the interior angle table for regular polygons and looking for a pattern for sums of interior angles in regular polygons.
I asked the class to pull out their tables and discuss what they had seen in their table groups. I projected a table on the board and asked for the class to share what they were seeing. Students mentioned that the size of the interior angle decreases as the number of sides increase and another stated that the interior angle size increases as the number of sides increase.
I wanted the class to be thinking about formulas and representations that model the work they are doing. I had thought about the process they were following to find angle measurements:
I asked the class to pull out their tables and discuss what they had seen in their table groups. I projected a table on the board and asked for the class to share what they were seeing. Students mentioned that the size of the interior angle decreases as the number of sides increase and another stated that the interior angle size increases as the number of sides increase.
I wanted the class to be thinking about formulas and representations that model the work they are doing. I had thought about the process they were following to find angle measurements:
- Exterior angle measurement = 360o / number of sides/angles
- Interior angle measurement = 180o - Exterior angle measurement
- Sum of interior angle measurements = number of sides/angles • Interior angle measurement
A colleague had shared some practice worksheets, some of which contained formulas. Specifically, the formula for the sum of interior angle measurements was given as S = 180(n - 2). It struck me that a formula like this would not make any sense to my class given the approach we had taken. And while the two formulas are equivalent, one might not be as meaningful as another.
This connected to a discussion we had in the math department about whether students should simplify. I noticed that we tend to set up arbitrary rules to always re-work expressions and equations to the "simplest" form. Yet, in simplifying the expression, we may have made the connection to what the math is modeling more complex.
I wrote the three statements on the board and asked the class how they might be able to shorten how to write out these equations. There is nothing wrong with writing equations like this except that it is a lot of writing and can be a bit cumbersome.
Students discussed this at their tables. Some students just wanted to use actually values, some wanted to just write the equations as they were. A few suggested using variables.
I re-wrote the equations as
- EA = 360o / NS
- IA = 180o - EA
- TIM = NS • IA
and then defined
- EA = exterior angle measurement
- IA = interior angle measurement
- TIM = total interior angle measurement
I explained that defining abbreviations enables me to use terms repeatedly without having to write out the whole definition each time. This is what happens when mathematicians write papers. Time is spent up-front defining terms and establishing notation that is then used throughout. I emphasized to the class that their formula should be meaningful to them. The mathematics is modeling reality and there should be a connection to the original problem situation.
I mentioned that sometimes they may see functional notation employed. Rather than writing TIM = NS • IA, they might see f(NS) = NS • IA. But that this may just use a generic variable label such as f(n) = n • IA.
At this point I could tell students were starting to become a bit confused or indifferent. I forged ahead believing I could bring everyone back to the same point.
I then pointed out that since IA = 180o - EA, we could substitute and get f(n) = n • (180o - EA). However, EA = 360o / NS, so we could substitute further to get f(n) = n • (180o - 360o / NS) or f(n) = n • (180o - 360o / n).
This last formula solely depends on how many sides the figure has. But we could now simplify the formula by using the distributive property to end up with f(n) = 180o (n - 2). This is the "traditional" formula that is given in geometry texts. However, how meaningful is this formula.
I asked the students to look at the table showing sums of interior angles. I asked students to focus on how the sum changes. They noticed it is going up by 180o each time. Why does this happen? Students were stumped. I asked them what connections could they make to 180o? Could they think of any figures that they associate with 180o? Students said that straight lines and triangles come to mind. Could they make connections with what they know to this situation?
After some thought, one student suggested that by adding two sides to the previous figure, another triangle was being added to the figure. I drew a triangle, a square, and a pentagon on the board. I placed an external vertex and drew the two new sides. It was clear to see that, in fact a new triangle was being added to the previous figure. This explained the increase of 180o each time.
But what about the formula. I asked students to pick a vertex and draw line segments to each of the non-adjacent vertices in the figure. For a triangle there were none, for a square there was one, and for a pentagon there were two. The results were one triangle, two triangles, and three triangles. Subtracting the number of triangles from the number of sides resulted in a constant value of two.
So, if I took the number of sides n and then subtracted 2 from it, I could determine how many triangles were contained in the figure. We had now circled back to the traditional formula. At this point, students could see why 180o was being multiplied by the number of sides minus two.
I wanted students to focus on creating meaningful formulas. The point being, that the math models the reality. When presented with a formula, they need to make connections back to the situation that the formula is modeling. There should be direct connections that they can make that will help make the formula more understandable. It isn't a matter of memorizing a formula, it is a matter of making sense of a formula that will help make the situations more understandable, not less.
With that, I asked students to capture their thinking about what they can do to make formulas more understandable.
Next class, we'll explore the number of diagonals and I will challenge the class to come up with a formula that describes the situation and makes sense to them. And then, we'll practice working through a series of problems where they can use different angle relationships to answer questions, in preparation for their final exam next wee.
At this point I could tell students were starting to become a bit confused or indifferent. I forged ahead believing I could bring everyone back to the same point.
I then pointed out that since IA = 180o - EA, we could substitute and get f(n) = n • (180o - EA). However, EA = 360o / NS, so we could substitute further to get f(n) = n • (180o - 360o / NS) or f(n) = n • (180o - 360o / n).
This last formula solely depends on how many sides the figure has. But we could now simplify the formula by using the distributive property to end up with f(n) = 180o (n - 2). This is the "traditional" formula that is given in geometry texts. However, how meaningful is this formula.
I asked the students to look at the table showing sums of interior angles. I asked students to focus on how the sum changes. They noticed it is going up by 180o each time. Why does this happen? Students were stumped. I asked them what connections could they make to 180o? Could they think of any figures that they associate with 180o? Students said that straight lines and triangles come to mind. Could they make connections with what they know to this situation?
After some thought, one student suggested that by adding two sides to the previous figure, another triangle was being added to the figure. I drew a triangle, a square, and a pentagon on the board. I placed an external vertex and drew the two new sides. It was clear to see that, in fact a new triangle was being added to the previous figure. This explained the increase of 180o each time.
But what about the formula. I asked students to pick a vertex and draw line segments to each of the non-adjacent vertices in the figure. For a triangle there were none, for a square there was one, and for a pentagon there were two. The results were one triangle, two triangles, and three triangles. Subtracting the number of triangles from the number of sides resulted in a constant value of two.
So, if I took the number of sides n and then subtracted 2 from it, I could determine how many triangles were contained in the figure. We had now circled back to the traditional formula. At this point, students could see why 180o was being multiplied by the number of sides minus two.
I wanted students to focus on creating meaningful formulas. The point being, that the math models the reality. When presented with a formula, they need to make connections back to the situation that the formula is modeling. There should be direct connections that they can make that will help make the formula more understandable. It isn't a matter of memorizing a formula, it is a matter of making sense of a formula that will help make the situations more understandable, not less.
With that, I asked students to capture their thinking about what they can do to make formulas more understandable.
Next class, we'll explore the number of diagonals and I will challenge the class to come up with a formula that describes the situation and makes sense to them. And then, we'll practice working through a series of problems where they can use different angle relationships to answer questions, in preparation for their final exam next wee.
Wednesday, December 2, 2015
Regular polygons and exterior/interior angle theorems
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.
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.
Tuesday, December 1, 2015
Introducing polygons and quadrilaterals
Our next unit focuses on quadrilaterals in the coordinate plane. There is some introductory work with polygons that also needs to be covered.
To start things off, I made use of Unexpected Riches from a Geoboard Quadrilateral Activity described in Rich and Engaging Mathematical Tasks Grades 5-9. Students were asked to construct as many quadrilaterals as possible on a geoboard and record the results. As students worked through their constructions, I encouraged them to be creative and think beyond just squares and rectangles.
Excluding rigid transformations, there are 16 shapes that can be created. As students worked, I called out how many different groups had found and told them to keep pushing. Students got to 16 shapes but some were either rotations or reflections of other shapes. Across the room, I did see all 16 shapes.
I displayed the shapes and gave students a chance to compare what they had completed. I numbered the 16 figures and wrote out the numbers for four of the figures on the board. I told the class these four figures all had something in common. I asked the class to describe what the common trait was.
The four figures were the only concave quadrilaterals. Students described how they looked like arrow heads. I pushed them for more clarification. They struggled some with how to describe what was going on. There were incorrect statements about all angles being acute, and the like, which I threw back at the class for verification. The discussion centered around the vertex which defined the concavity of the quadrilateral.
Finally, a student noted that the interior angle of the vertex was obtuse. I wrote down a definition:
A quadrilateral is concave if it has at least one interior angle that is obtuse.
I then noted that the other quadrilaterals were convex quadrilaterals. I wrote down a definition:
A quadrilateral is convex if all its interior angles are less than 180o.
I then asked the class if a concave triangle can exist. There was some good discussion around this idea. One student noted that he could draw a triangle with an obtuse angle and so it should be called a concave triangle. Other students objected since there wasn't any indentation in the figure and this would require a fourth side to accomplish that result.
I turned back to the definition of a concave quadrilateral and said that we might need a more precise definition. That having an obtuse angle was not enough to define concavity. I also note a couple of the convex quadrilaterals that had obtuse angles. At this, a student suggested that is should state the angle is greater than 180o and not just an obtuse angle. I re-wrote the definition:
A polygon is concave if it has at least one interior angle greater than 180o.
This provided a precise mathematical definition for concavity that works for more than just quadrilaterals. The discussion also built upon students' prior knowledge of interior angles of triangles and a natural transference of this idea to quadrilaterals.
With these ideas established, I moved on to polygon names. Most of the standard names through hexagons were known, as was octagon. Students did not know a 7-sided figure was called a heptagon, nor that a 9-sided figure was called an enneagon. With some connections to common words, such as 10 years is called a decade, students came up with a 10-sided figure is called a decagon.
I went through the naming of 11-sided through 19-sided figures as well, only to show how these names were built upon from earlier names, especially figures 13-19.
I then asked students to construct a convex and a concave pentagon on their geoboards. Next, I assigned table groups different figures, hexagon, heptagon, octagon, decagon and asked them to construct a concave figure on their geoboards. It was fun to see their creativity. For example, one group constructed what looked like a crown for the concave decagon.
I concluded with having students record their thoughts and capture definitions in their notebooks.
I still need to cover the ideas of exterior angles and regular polygons. Once these are done, I will move on to look at the sums of exterior angles, the sums of interior angles, and the number of diagonals in polygons.
To start things off, I made use of Unexpected Riches from a Geoboard Quadrilateral Activity described in Rich and Engaging Mathematical Tasks Grades 5-9. Students were asked to construct as many quadrilaterals as possible on a geoboard and record the results. As students worked through their constructions, I encouraged them to be creative and think beyond just squares and rectangles.
Excluding rigid transformations, there are 16 shapes that can be created. As students worked, I called out how many different groups had found and told them to keep pushing. Students got to 16 shapes but some were either rotations or reflections of other shapes. Across the room, I did see all 16 shapes.
I displayed the shapes and gave students a chance to compare what they had completed. I numbered the 16 figures and wrote out the numbers for four of the figures on the board. I told the class these four figures all had something in common. I asked the class to describe what the common trait was.
The four figures were the only concave quadrilaterals. Students described how they looked like arrow heads. I pushed them for more clarification. They struggled some with how to describe what was going on. There were incorrect statements about all angles being acute, and the like, which I threw back at the class for verification. The discussion centered around the vertex which defined the concavity of the quadrilateral.
Finally, a student noted that the interior angle of the vertex was obtuse. I wrote down a definition:
A quadrilateral is concave if it has at least one interior angle that is obtuse.
I then noted that the other quadrilaterals were convex quadrilaterals. I wrote down a definition:
A quadrilateral is convex if all its interior angles are less than 180o.
I then asked the class if a concave triangle can exist. There was some good discussion around this idea. One student noted that he could draw a triangle with an obtuse angle and so it should be called a concave triangle. Other students objected since there wasn't any indentation in the figure and this would require a fourth side to accomplish that result.
I turned back to the definition of a concave quadrilateral and said that we might need a more precise definition. That having an obtuse angle was not enough to define concavity. I also note a couple of the convex quadrilaterals that had obtuse angles. At this, a student suggested that is should state the angle is greater than 180o and not just an obtuse angle. I re-wrote the definition:
A polygon is concave if it has at least one interior angle greater than 180o.
This provided a precise mathematical definition for concavity that works for more than just quadrilaterals. The discussion also built upon students' prior knowledge of interior angles of triangles and a natural transference of this idea to quadrilaterals.
With these ideas established, I moved on to polygon names. Most of the standard names through hexagons were known, as was octagon. Students did not know a 7-sided figure was called a heptagon, nor that a 9-sided figure was called an enneagon. With some connections to common words, such as 10 years is called a decade, students came up with a 10-sided figure is called a decagon.
I went through the naming of 11-sided through 19-sided figures as well, only to show how these names were built upon from earlier names, especially figures 13-19.
I then asked students to construct a convex and a concave pentagon on their geoboards. Next, I assigned table groups different figures, hexagon, heptagon, octagon, decagon and asked them to construct a concave figure on their geoboards. It was fun to see their creativity. For example, one group constructed what looked like a crown for the concave decagon.
I concluded with having students record their thoughts and capture definitions in their notebooks.
I still need to cover the ideas of exterior angles and regular polygons. Once these are done, I will move on to look at the sums of exterior angles, the sums of interior angles, and the number of diagonals in polygons.
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