Jessica Waggener-
EMT715 Projects

Explorations on a Geoboard- aka Pick's Theorem

Definitions:

Investigate on the geoboard and GSP:
  1. How does the area change if one starts from a fundamental triangle (B=3, I=0), keeps B at constant, and subsequently increases I by one? Write down the results of your exploration (possibly in a table).
  2. How does the area change if one starts from a fundamental triangle (B=3, I=0), keeps I at constant, and subsequently increases B by one? Write down the results of your exploration.
  3. Discuss what you have found, that is, write a letter to a friend about your investigation and your results. Assume that your friend is not in EMT 515/715 and has no previous knowledge of a geoboard or the investigation. It might help you if you used the computational geoboard for visualization.
 
 

October 11, 1997

Dear Rebekah,

I'd like to share with you today something we've been learning in one of my mathematics education classes. The idea here is to stretch back to your early exposure to the concept of area, specifically, the area of a triangle. So, I want you to imagine that you don't know that the area of a triangle equals 1/2 base * height and think with me through this series of visual and numerical ideas until we can come up with several different "formulas" for finding the area of a triangle.

Let me begin by introducing you to the geoboard. Imagine a square piece of wood set with a square grid of nails (not hammered in completely).

The distance between any two adjacent nails in the horizontal or vertical direction is defined to be one. That definition implies that the distance between two adjacent diagonal nails is sqrt2. In the concrete version of the geoboard, rubber bands are used to create lines between these evenly spaced nails. Here, I will simply draw in lines between the evenly spaced points to represent the equivalent visual. You can begin to imagine how a student could easily compute the perimeter of some figures.

In our discussion of triangles, we need to make two definitions. Notice in the above picture that the lines touch 6 points on the board, but no points lie inside the triangle. Let the points which touch the lines be called border points. Let points inside the triangle be called interior points. So we would say the above triangle consists of 6 border points and zero interior points. Or, let (B,I) = (6,0).

Let's take a quick moment to list all the different ways we can measure or describe a triangle. We can: 1. measure the lengths of each side; 2. measure the length of the base and the height; 3. we can measure the length of two sides and one angle, and other variations of measuring sides and angles; or 4. we can count the number of border and interior points.

Now let's get to the real point of this discussion- how can we measure the area of a triangle? First, another definition: Let any triangle with zero interior points (I=0) and three border points (B=3) be known as a fundamental triangle. Any fundamental triangle always has an area of 1/2 units square. Consider the following examples.

Note: Measures of *.4 should be read as *.5. The error arises because of pixel variations in monitors.

From these examples we can observe some patterns, although it may not always be clear how some of the area measurements are derived. Following is a table of empirical measurements found using both the electronic geoboard above and a concrete geoboard we used in class:

Interior Points
Border pts. 0 1  2 3 4
3 0.5 1.5 2.5 3.5 4.5
4 1 2 3 4 5
5 1.5 2.5 3.5 4.5 5.5
6 2 3 4 5 6
We need to consider how some of these areas were found. Observe the triangle below.

This was relatively easy for a right triangle. Do these two methods work as well for more irregular shapes?
 

 

Thus far we have developed two intuitive ways of measuring the area of a triangle. The first, based on rectangles, is the foundation for our traditional formula of 1/2 (base *height). We saw with the right triangle how base and height there correlated to the sides of a rectangle. I am going to assume you are fairly familiar with this idea and let it rest at this point. The second method we've discussed is going to lead us to another formula, but we will develop it from a numerical pattern as opposed to trying to make a direct link between the figures and the formula. So now, let us consider a larger table of values of the areas of triangles measured by their border points and interior points.
 

B\I
0
1
2
3
4
5
6
7
8
9
10
11
12
3
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12.5
4
1
2
3
4
5
6
7
8
9
10
11
12
13
5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12.5
13.5
6
2
3
4
5
6
7
8
9
10
11
12
13
14
7
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12.5
13.5
14.5
8
3
4
5
6
7
8
9
10
11
12
13
14
15
9
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12.5
13.5
14.5
15.5
10
4
5
6
7
8
9
10
11
12
13
14
15
16
11
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12.5
13.5
14.5
15.5
16.5
12
5
6
7
8
9
10
11
12
13
14
15
16
17
During the following discussion it will be useful to keep the above table visible for easy reference. We will use the notation A(B,I) to imply the area of triangle with B border points and I interior points. For example A(6,3) = 5sq.un.. Note the darker shaded square in our table.

Recall that the basic unit for measuring area in this way is the area of the fundamental triangle, or A(3,0)=.5sq.un.. Let us see what we can discover by describing another cell in the table in terms of this particular cell. Consider A(7,10)=12.5sq.un., (the lightly shaded square). First, notice that A(7,10) is .5sq.un. greater than A(6,10). Extend that perception to see that A(7,10) is 2sq.un. greater than A(3,10). We can describe this 2.5sq.un. in terms of the distance between the two squares. There are four steps from A(3,10) to A(7,10), each measuring .5sq.un, for a total of 2sq.un.. Notice that 4 is the distance between 7 and 3. In general, we can describe the increase in area derived from the increase in border points as A(B,I)=A(3,I)+(B-3)*.5.

Similarly, notice that A(7,10) is 1sq.un. greater than A(7,9). And A(7,10) is 10sq.un. greater than A(7,0). This 10 sq.un. is equivalent to the distance 10-0 times the step-value of 1sq.un. Thus we can describe the increase in area derived from the increase in interior points as A(B,I)=A(B,0)+(I-0)*1. Take a moment now to verify these ideas for yourself. Pick a square on the table above and try to describe its value in relation to neighboring squares and eventually in relation to the square A(3,0). Do not continue reading until you are comfortable with this idea.

Now we can put all these ideas together to describe the area of any triangle in relation to the area of the fundamental triangle, given the number of points on the border and in the interior. Based on the previous two conclusions (in bold) we can make the following statement.

A(B,I) = A(3,0) + (B-3)*.5 + I*1

This is a somewhat cumbersome formula, so we will condense it some.

=.5 +(B-3)*.5 + I A(3,0)=.5 and I*1=I
= (B-2)*.5 + I .5*(1+B-3)=(B-2)*.5
= B*.5 -1 +I -2*.5=-1

To summarize,

A(B,I) = .5B + I -1.

This expression for finding the area of a triangle is known as Pick's formula.


MATHEMATICAL EXTENSION

Describe another alternative way in which to find the area of the triangle.

Credit goes to a class member for suggesting this method.

Suppose you only know the coordinate points of your vertices. Using only these, you can find the "lines" upon which the sides of your triangles lie. In order to do this, you must first find the slope of each side using the two endpoints and the relationship . Once you've found the slope, pick one endpoint and write the equation of the line using the point slope formula, y-y1=m(x-x1). This equation should then be simplified into slope-intercept form,

y = mx + b. You should do this for each side and label the functions by using the notation f(A) = mAx + bA, f(B) = mBx +bB, and f(C) = mCx + bC.

Now, pick one side, say A, and find the line which is the perpendicular to A, and which goes through the opposite vertex. See picture.

You can find the equation of this line by taking the negative inverse of the slope of A

(-1/mA) and using the point-slope equation again, this time through the point r. So, the equation of the altitude of the triangle becomes y-yr = -1/mA(x-xr). Simplify to f(H) in slope-intercept form.

The importance of this line for our purposes is that it marks the height of the triangle in our familiar 1/2 base * height formula. However, it is useless to use until we know the distance from p to r. In order to find this, we first need to know the coordinates of p (r is given). The point p marks the intersection of f(H) and f(A). And the way you find the intersection of two lines is to set their y-values equal (i.e. f(H)=f(A)) and solve for x in the corresponding equality (i.e. mHx +bH = mAx + bA).

Once the coordinates for p are known, the only steps left are in applying the distance formula. As a reminder, d(p1,p2) = sqrt[(x1 - x2)2 + (y1 - y2)2] Apply this formula to find the distance from p to r (height) and also the distance from q to s (base). Now you have all the vital information you need! Simplify calculate one-half the product of the base times the height!


PEDAGOGICAL EXTENSION

I'm not sure if I would push my students to learn Pick's formula, per se. What I might do, however, is to introduce them to the geoboard and make the assignment (within their normal working groups, of course) to develop TWO ways for finding the area of a triangle on the geoboard. It would be okay if their ideas weren't always good, because it would still force them to stretch their thinking and purposefully look for more than one solution path.

After exploring we would begin sharing by asking a group to only present its BEST method found. Once the first repetition arose, the group would then be asked to present the most UNIQUE solution. The most interesting part here would be discussions about two different solutions which may have appeared different, but represented the same basic idea. I would encourage students to identify similarities and differences in the approaches.

Afterwards, the lesson could be extended to GSP to see if they can come up with any additional ways to find area (not counting the "Area" macro on GSP!).

I'm not sure if this process of ideas would be best first with rectangles, triangles, regular polygons, or circles first. It might be interesting in different years, or with different classes to try this approach at these different levels and see which one yields the most variety and fruitfulness.

As a follow-up, the class could be asked to make a top 3 ranking as well as a listing of methods which were theoretically unsound (e.g. wrong). Of course, they would have to defend each selection and define criterion such as: correctness, efficiency, easiest to remember, makes the most sense, etc. It could become an excellent lesson in revealing the derivation of many of our common procedural rules. They would hopefully see how the mathematical community comes up with many possible procedures, but arrives at a consensus on the most elegant ones which become the most common due to their extensive usefulness.

Assessment for this exercise would actually appear very traditional. The main goal is to see if they can learn how to find the area of a figure, so assessment could be very straightforward and objective. You might ask for the students to use two different methods, or to explain why they used the method they did for a particular triangle.


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