I chose Pick’s Theorem to do as a part of a Math Magic block of math circles. I have used this formula as
a party trick for several years. The next section introduces the properties of polygons and explains Pick’s
Theorem.
The Math
In the world of polygons, not all shapes are created equal. What could cause this inequality of forms
in two dimensional space? There are several properties, not all of which impact Pick’s Theorem. Understanding these properties gives a better appreciation for the theorem. Instructors may want to review some
of these properties with students.
One such property is convexity. A polygon may be concave or convex. Given any random polygon, one
can determine this property by trying to draw a line between any two points inside the polygon. If all these
lines are inside the polygon, it is convex. Otherwise it is concave. Some of the magic of Pick’s Theorem is
that convexity does not matter. Note that regular polygons such as squares and hexagons are convex. Often
students will think of only convex polygons. This is completely unnecessary. A good concave polygon to
consider may look like –
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However, a property that does matter is the placement of the vertices. A polygon is called a lattice polygon,
if on evenly spaced grid paper each vertex of the polygon is on a gird point. In other words, if a lattice
polygon was on a standard Cartesian plane, the vertices would occur at coordinates (a,b) where a and b are
integers. Thus all vertices of the polygons students draw should be on grid points.
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What is Pick’s Theorem? Pick’s Theorem establishes a relationship between the number of grid points
on the boundary of a polygon, the number of grid points inside the polygon, and area. Another part of the
magic of this theorem comes from the type of information it relates. Points are discrete values, yet using
them one can predict something which is not discrete – area.
Suppose B is the number of grid points on the boundary, I is the number of grid points inside, and A
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is the area of the polygon. Pick’s Theorem says that A=I+B/2-1. Here is an example –
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A = I + B/2 − 1 = 0 + 3.5 − 1 = 2.5
Sketch of Proof. To see why this may be true for all polygons, one can break any polygon up into
simpler polygons which can be considered the building blocks for larger polygons. Consider the example
for the polygon below and notice there are multiple way to disassemble the polygon. However, after all
decompositions are done, the result is a set of triangles.
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In fact any polygon can be broken into triangles. For this reason, any polygon can be represented as a sum
of triangles. Thus to show Pick’s Theorem works, it would suffice to show it for a single triangle and sums
of such triangles. This can be accomplished through induction on the number of triangles. For an excellent
and detailed proof of Pick’s Theorem by Tom Davis visit http : //www.geometer.org/mathcircles/pick.pdf .
Similarly there is a Pick’s Theorem for a polygon with n holes - A=I+B/2+n-1. Again Tom Davis covers the proof of the case with holes in his article. Here is an example of a polygon with holes. I have marked
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the boundary points in blue and the inside points in green.
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A = I + B/2 + n − 1 = 1 + 6 + 2 − 1 = 8
The Lesson
The lesson plan I developed was based on a Pick’s Theorem activity Dr. Matthias Beck had done with
the Oakland teachers circle. The original activity only included a table and some lattice polygons with
instructions to find the boundary points, internal points, and area of the polygons and then the relationship
between this data. It was clear to me the activity would need to be extended to include more exploration
of Pick’s Theorem and possibly related topics. I also knew my math circle usually worked as groups for the
entire hour and it could be extremely difficult to get the groups’ attention in the middle of the session. Thus
the lesson would need to be a series of questions they could do as a group, yet with some flexibility because
I wanted to try the activity on different math circles.
A challenge in developing a lesson plan is that one cannot be certain what students may or may not know.
For this reason, I developed the lesson plan first, and figured I would test it later. The first thing I did was
test the original problem to see how long it took me to complete. Needless to say I never finished it; the
tedium of finding areas and counting points nearly drove me crazy. It was then I decided I would also need
to find a way to split the activity up so there was not so much monotony of making the table. The way I
chose to do this was to have groups simply do 2 or 3 polygons each and then combine the results on the
board so that everyone could use them. This means students may be dependent on classmates’ accuracy
and work ethic, but it seemed better than them loosing interest. Another serious issue that occurred to
me was that finding the areas for each polygon could be challenging itself. It took me a little creativity to
find a method that would work effectively for almost any polygon. It involved considering a large rectangle
where farthest vertices in each direction were on the boundary; then I would find that area and subtract the
outside triangles. I realized that it may be nearly impossible for students to find that without guidance.
I immediately knew I wanted to have the last problem relate Picks Theorem with Farey fractions. A
classmate also working with the Mission math circle wanted to do Farey fractions the next session after my
lesson, and therefore, leading students in that direction. However, the problem would not detract from the
students experience in exploring Picks Theorem either. Fortunately for me Dr. Beck already had such a
problem written up, and all I had to do was copy it. However, I would realize later when testing on myself
and also on a classmate, this exercise needed a picture. For this reason, I added a picture for the case
when n=4. The drawing definitely made the problem more understandable, and illustrated the fractions and
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lattice points which were supposed to related.
The other major deficiency I noticed was that there would need to be a compelling first exercise. By
far finding a good beginning question was hardest. I decided to start with what I thought would give students a chance to see the magic of Pick’s Theorem in action. Unfortunately, choosing a number for boundary
points and interior points that was sufficiently challenging for students to compute was difficult. I settled on
6 and 4 which were moderately difficult for me at least. Some of the polygons I found which satisfied these
requirements are –
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In passing Dr. Beck had also mentioned that Pick’s Theorem could be extended to polygons with holes.
This makes perfect sense if you know the proof of Pick’s Theorem, but from a student perspective there
would be no reason to believe this. I consulted the Internet for both the formula for polygons with and
without holes. Just for reassurance I tested the formula for polygons with holes four times, and it seemed to
work. Feeling fairly confident Wikipedia – http : //en.wikipedia.org/wiki/P ick27st heorem – was correct
this time, I added an additional question for students to make their own polygons with holes. Then to
determine the boundary points, internal points, area, and number of holes to make a new table and find a
new formula. Before the lesson I confirmed the proof of the theorems from the aforementioned write up Tom
Davis did on Pick’s Theorem.
Different Students
I chose to test on at least two math circles – Mission High and Thurgood Marshall High. The two groups
would turn out to be very different, but with small modifications the lesson plan went well in both circles.
To my relief, I had anticipated most of the major sticking points, and was prepared to assist students with
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their problems.
Mission High has a fairly large circle consisting of about 25 to 30 students of which about 8 to 10 are
younger middle school or home schooled students. The remaining students attend Mission High. At the
beginning of class I briefly explained the activity, and made sure to assign each table two polygons to find
data on and then share with the class. Assigning polygons at the beginning would become problematic when
about five more people showed up 10 minutes late. However, even with some overlap the activity worked
out well. The groups with the same polygons simply had the first group put up the data.
As expected finding polygons with 6 interior points and 4 boundary points was difficult especially for students who did not like to experiment. For other groups, a parallelogram and an elongated triangle were
simple, but doing more was difficult. On average most groups seem to find at least two polygons which once
they found the area correctly, was enough for them to conjecture the area would always be 7.
Because of the way the activity had been photocopied, the page with the polygons was in an odd place.
This caused confusion for a lot of students about where the polygons I had assigned were. Once they found
them, they did fine until they had to find the area. Again area was challenging, and in fact, so challenging
that most would not finish the activity, and we would continue the following week.
I was excited to see students discussing and arguing different approaches to finding areas. These were
some of the most excited conversations math circles had generated so far that block. One of the most intense
discussions was between two girls discussing a parallelogram. One girl insisted that the area must be 5,
while the other patiently explained it was not. She offered geometric evidence by enclosing the parallelogram rectangle. After finding the area of this rectangle, she observed that two outer triangles could be put
together to made another rectangle. With this information the difference between the larger rectangle and
the smaller rectangle, they could find the area of the parallelogram. The other girl took some convincing,
but eventually agreed on the area.
We chose to continue the next week by giving students the data from the table, and having them work
on the formula. Because I knew the formula I had not put a lot of thought into how to deduce it from data.
That problem was quickly solved by an observant student who pointed out that only the polygons with an
odd number of boundary points had halves in their areas. Once students recognized they need to halve
the number of boundary points, the remainder of the formula followed quickly. For those unable to make
the connection between an odd number of boundary points and fractional area, the struggle of finding the
formula continued for a while. One student insisted I tell her the formula because she could not stand not
knowing, and another was frustrated enough to leave. However, most students did figure it out quickly and
moved on to drawing polygons with holes.
In the end, I would say that the activity was successful, and some valuable lessons were learned. The
students enjoyed the fact the activity was hands-on. The students also exchanged ideas. One suggestion
made after Mission math circle was that providing an information sheet for instructors with the formulas
and advise on how to do the activity may be helpful. I thought this would be worthwhile, and therefore,
included such a sheet on the activity before attempting it elsewhere.
The following week I attended the Thurgood Marshall circle to try the activity. I had emailed to the
graduate students and teachers a copy of the activity with the addition sheet for instructors. I feel fairly
confident in saying almost no one looked at it. However, I think this had no adverse effect on the math circle.
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Before the circle we decided that the full activity would be too much. Mission had not completed the
activity the first week, and the leaders of this math circle warned me the students may struggle a bit
more. Therefore, only the first three questions were included in the handout. Also we made the choice to
hand out the first page separately so students would not be distracted by the activities on the following pages.
The math circle that day was very small – about 8 students, 5 teachers, and 3 graduate students. Everyone got to work quickly, and despite some protests of students continued working. Because the teachers
had not looked at the advice sheet, their exploration added a sincerity to the activity. The students knew
that the teachers were thinking about it and working with them. Most people did seem to be working even
if slowly, with few complaints.
The first sheet was done about half way through the session. Most groups found at least two polygons
fitting the criteria. Students again struggled with thinking of such polygons, and with finding the area;
however, each group finished and concluded it was unusual they made two polygons with area 7.
I expected the second part to be more difficult, because attendance being so small meant that the polygons could not be easily broken up. Therefore, each group would have to find the boundary points, interior
points, and area for all of the pre-made polygons. This is when the difference between the Mission circle and
the Thurgood Marshall circle became quite clear here. Because the teachers were engaged and working on
the activity, the students were all taking part without feeling overwhelmed. Teachers helped lead students
through the more challenging polygons, and asked the students then to do the calculations. Occasionally
because the suggestion sheet had not been read, some one would have to offer advice on how to find an area
or consider the data. But this did not distract from the session, perhaps it even worked out better than if
the teachers had known.
At Thurgood Marshall there was one teacher who already knew the formula. We didn’t have extra copies of
remaining exercises so I simply asked him to prove it. He worked diligently for about half the class, and then
gave up. He was actually very close; he needed to apply some kind of induction argument, but could not
seem to figure out how. I need to remember to keep extra sheets even when we don’t expect to use them.
Future Directions
Even though the lesson went well there is definitely some room for improvement and adjustment. I see
several possible ways to improve and expand on the material presented in my original lesson. Certainly
there are more than the several directions I am proposing that could add value to this lesson, but these are
the ones I would pursue.
First it may be nice to take some more monotony out of the making of the table. One possible way
may be to develop the table on an interactive basis. For example, groups of students could make a polygon
on grid paper and calculate the area. They could then copy the polygon with the grid points onto a to
the board. This would allow the circle leader to demonstrate that there must be a quick way to calculate area on student polygons. Also the data from each student polygon could be recorded. In addition,
the instructor may want to keep a few example polygon in mind just in case students do not produce
diverse enough examples. For some classrooms this would be better suited than others surely; however, it
may allow students a little more creativity and illustrate a little better what is special about Pick’s Theorem.
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I would like to develop the final two questions further and in particular the final problem relating to Farey
fractions. I think relating Pick’s Theorem and Farey fractions could take an entire lesson plan themselves.
I have not thought much about what such a lesson may look like; however, the potential for interesting
activities relating these two topics and extending on the final question in this exercise set appears plausible.
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