How Many Diagonals Does a Polygon Have?

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Objective

SWBAT use an In/Out table in a geometric context to find a functional pattern. SWBAT prove why their pattern works.

Big Idea

Bringing some geometry into algebra! Students explore how many diagonals different sized polygons have and try to prove a relationship between the number of sides and the number of diagonals.

Opening

10 minutes

This lesson gives students the opportunity to apply their knowledge of algebra to explore the properties of geometric figures. I like making connections between algebra and geometry, so that my students understand that different math topics are not necessarily disconnected from each other.

The presentation of this class is outlined in the How Many Diagonals? PowerPoint. I begin class by asking students to remind me what a polygon is, at least to remind me in their own words. I write the word polygon on the board and ask if they know what the prefix "poly" means.  I usually tell my class that "gon" means angle, so the word polygon translates to "many angles."

Then, I ask a volunteer to come up and draw a polygon. I explain what a diagonal is, then I have my students figure out how many diagonals the students' particular polygon has. Finally, I introduce the "Diagonally Speaking" activity to my students. The version of this activity that I prefer to use is on page 42 of IMP's Year 1 textbook, but this activity, analyzing a table of data comparing the number of sides and the number of diagonals in a polygon is very common and easy to reproduce (see Slide 8 of How Many Diagonals?). 

 

Investigation

30 minutes

I like to let my students work in small groups on this investigation. As I circle the room, I make sure they are clear about organizing their results in an In/Out table with the In being the number of sides the polygon has and the Out as the number of diagonals. I also like to help them to use visual strategies to support their work on the investigation. 

Here are some things that I am watching for as my students work:

  • As the number of sides in the polygon increases, the number of diagonals can be hard to keep track of.  I like to encourage students to use colored pencils or markers to draw the lines from each vertex.  They can draw all the lines coming from one vertex in the same color and then use different colors for the other vertices.  This can also help students to see the underlying structure of the pattern.
  • Students may have trouble drawing the polygons themselves.  You can have some examples of hexagons, heptagons, and octagons printed if you don't mind if they don't draw them themselves.
  • If some students are working faster than others, you can push the students who have found a pattern to work on the closed formula.

Discussion + Closing

20 minutes

I make sure to leave plenty of time to discuss this activity.  One of the things I like about this problem is that there are a lot of different ways to see the pattern. I like to begin by having students who noticed the recursive pattern share out. I make sure to re-emphasize with students that in a recursive formula, each Out value is found by using the previous Out.  In this case, consecutive numbers are added to the Out values. I also make sure students understand that when they use a recursive pattern, they will have to fill in all of the In values to get the Outs.  For example, if they want to know how many sides a 20 sided polygon has, they will have to figure out how many sides all the polygons leading up to 20 sides has too.

Next, I will ask for other patterns or formulas that students found.  At this point I lead the discussion towards justifying why the pattern holds.  I might try to draw students attention to the number of diagonals that can be drawn from a given vertex.  For example, in a hexagon, three diagonals can be drawn from the first vertex.  For a heptagon, 4 diagonals can be drawn from the first vertex.  It is helpful to ask, "why this is so?" Questionning like this helps my students to see that no diagonal can be drawn to the vertex they are working from, or to the two vertices that are adjacent to that vertex. Therefore, the number of diagonals drawn from the first vertex will always be three less than the number of sides.

The closed formula for this problem Out = In (In -3) / 2 is sometimes hard for my students to find. When my students struggle, I give them a hint to move them in the right direction.  I might say something like, "If you can figure out the different numbers to multiply each In by, you might notice a pattern." Most of my students should be able to see that 7 * 2 = 14 pretty easily. I'll ask them what you have to multiply 4 by to get 2.  The idea of multiplying by 1/2 rather than dividing by 2 may be new to them, but it will come in handy often in different contexts.  From there, I might say something like, "Can you find a relationship between the In value and the value you multiply by?"

As usual, I make sure to allow time for other students to show different patterns if they have found them. I close class by asking students to think again about proving why their method for finding diagonals works.  For an Exit Ticket, I will ask students to write a paragraph responding to the following prompt:

Explain why your method for finding the number of diagonals must work. Whether you used a recursive formula or a closed formula, you will need to think about what a diagonal is and not just look at your In/Out table.  

Citations

This material is adapted from the IMP Teacher’s Guide, © 2010 Interactive Mathematics Program. Some rights reserved.