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.
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