Lattice Polygons
This is a lattice polygon. Your job is to find a formula to find the area of this shape using the Boundary Points and Interior Points.
Process:
The way we started this problem really helped me in finding our answer. When we first received our POW I just started making random shapes on a lattice board and finding the number of interior points, boundary points, and Area by just counting them. I had only drawn shapes that did not have any interior points. Jocelyn told us that this would be a lot easier to start with this. So when we came back we had to have 20 shapes drawn and we had to have the boundary points, interior points, and area labeled for each one. After we added our information about the number of interior points, boundary points, and area and looked at their patterns. So we put them on a scatter plot and had to find the line of best fit for the graph. After I had found my line of best fit I had the equation of y=1/2x-1, which finds the area for any shapes with zero boundary points. The odd thing about my line of best fit was that there was a perfect line in between any of the random points. It might be that many of the shapes had a different shape or had slants. Or maybe people found the wrong boundary points or Areas.
After we worked as a class, we had 1 or 2 people work on certain sets of data with different numbers of interior points from different shapes, and to find their slopes, similar to what we did with the shapes with no interior points, and we had them work on their data sets over night, and to present them the next day. So we compiled all of our equations from the lines of best fit, together and we had to prove that the equations that each person had found, actually worked. After about 20 minutes of working with different equations to see if they worked we were left with very few working equations, but we had noticed that the remaining equations had some sort of pattern. The pattern was that the y intercept was one less than the number of interior points. After we saw this pattern we replaced many of our wrong equations, and we tested them and found that they worked. So we thought how we could find the main component to how each equation was able to find the area each time, so the equation we came up with was:
A=1/2B+(I-1)
We tried this equation with many shapes and found that this is the correct equation for finding the area of a lattice polygon. In class many people tried debunking the equation he had just found. But all of them worked with our equation. So we started to discuss how our equation actually worked. Our equation included adding the number of Interior Points and subtracting them by one so it would replicate the effect that the individual equations had.
This is all the work I had done in my journal. This shows me finding all of the BP’s, IP’s, and areas for different lattice polygons and proving equations right or wrong.
Solution: Using the equation and counting the IP and the BP of the shape we were given I came up with this… A=½(10)+(4-1) A=5+3 A=8 This equation proves useful because it is very hard to calculate the area of some shapes like the one we were given so this equation makes everything simple. Evaluation: In my opinion I really enjoyed working on this problem especially with my entire class. That made the problem fun and a bit less stressful and I enjoyed cooperating with my classmates. Also the problem wasn’t too difficult, but if I had to work alone with only a little help from my class I think that it would of taken a lot longer to find the final equation. Rate: 4.5/5 |