P.O.W #4: Triangle Theater
1. Problem statement:
Using the picture can you figure out what seat number would be in front of you if you sat in seat number 333?
2. Process:
The process I went through in order to solve this problem was pretty simple. The first way I did It was I drew the whole thing out. I started from 1 and just started making a pyramid until I got to 333. After I did this I tried to find the pattern so I found out that the numbers on the left get odd numbers added to them.
The second way I did it was the mathematical way which is pretty simple too. Since the pyramid goes by odd numbers you just have to keep adding by odd numbers. After you get to the row that has 333 in it you just write out the numbers and find out which is above seat #333.
3. Solution:
Whichever way I did it I still got the answer 297. I think there is only one solution because its not really estimation its more adding solid numbers. The reason I think my answer is the right one because ive tried multiple ways and ive gotten the same answer of 297.
4. Evaluation:
At first this problem was hard but once I figured out ways to solve it it became easy. I wouldn't change the problem if i had the chance because this problem was easy but hard at the same time.
Using the picture can you figure out what seat number would be in front of you if you sat in seat number 333?
2. Process:
The process I went through in order to solve this problem was pretty simple. The first way I did It was I drew the whole thing out. I started from 1 and just started making a pyramid until I got to 333. After I did this I tried to find the pattern so I found out that the numbers on the left get odd numbers added to them.
The second way I did it was the mathematical way which is pretty simple too. Since the pyramid goes by odd numbers you just have to keep adding by odd numbers. After you get to the row that has 333 in it you just write out the numbers and find out which is above seat #333.
3. Solution:
Whichever way I did it I still got the answer 297. I think there is only one solution because its not really estimation its more adding solid numbers. The reason I think my answer is the right one because ive tried multiple ways and ive gotten the same answer of 297.
4. Evaluation:
At first this problem was hard but once I figured out ways to solve it it became easy. I wouldn't change the problem if i had the chance because this problem was easy but hard at the same time.
P.O.W #12 Lattice Polygons
Problem statement:
For this problem you have to find the formula to solve any type of polygon. Figure out the formula using the number of boundary points, and the number of interior points.
Process:
Throughout this problem Jocelyn helped us and pretty much walked us through this problem. What we first did was Jocelyn made us draw 20 lattice polygons, we had to find the area, interior points, and boundary points. We did so we would have some data to collect. We each did this as separate individuals. Then then the next day we input all the data on a spreadsheet that the class made. We had over 100 pieces of information I mean some were repeats, but still. After that we started making a line of best fit using the area and boundary points. We did this all during class, but the formula we found only works if there are no interior points.
Now going more into depth about what we did. So at first we just drew three polygons, so we can get an idea of what we have to do. Then after for homework we had to draw 20 more polygons. The reason we made twenty polygons was for data. We made them so we could have data to collect. We also made them for background information. We had to find the area (area times width), interior points (any points that lie inside the figure), and the boundary points (any points that make up the figure or outline the figure). After that we made a graph, only using the area and boundary points as axis. After graphing them we drew the line of best fit and figured out the formula using the equation for slope-intercept form (y=mx+b). We compared all our answers and we all ended up getting y=1/2x-1. So we all agreed on this formula. But, this formula only works if your number of interior points is zero. We gathered all the data groups that had a number of zero and used the boundary points and area to get the numbers for the data points.
Then Jocelyn said the rest of this part of the problem we have to solve as a class without her help. So someone came around the class assigning each table pair a group, the ones with the number of interior points two, number of interior points being six, or Five, or twelve. Then as homework we had to figure out an equation for the number we go assigned. We did this the same way we did it the last time, using slope-intercept form, (y=mx+b). Then the next day as a class we all said the number of interior points we got assigned and the formula we found. Then Jocelyn went around the class assigning a formula, different from the one we found, and you would have draw a random figure with the same amount of interior points that the formula belonged to. If you found a figure in which the formula doesn't work, then the formula would be crossed out. The formulas that couldn't be proven wrong were kept up. We went through and re-listed the ones that were right. Then we got assigned new ones, just to double check, and same things apply. After that we re-listed them once again and noticed a pattern. The pattern was The formulas are all y=1/2x and then 1 number less than the number of interior points. So if the number of interior points is 10 then the formula would be y=1/2x+9. Then after that we all went around and tested it, it worked but we had to make it into an actual formula so we decided on y=1/2x=(i-1) or A=1/2B+(i-1), which is the same is just that y is replaced with a and x for b. Standing for Area and Boundary points.
Solution:
The solution we got was A=1/2B+(i-1)
ex:
For this problem you have to find the formula to solve any type of polygon. Figure out the formula using the number of boundary points, and the number of interior points.
Process:
Throughout this problem Jocelyn helped us and pretty much walked us through this problem. What we first did was Jocelyn made us draw 20 lattice polygons, we had to find the area, interior points, and boundary points. We did so we would have some data to collect. We each did this as separate individuals. Then then the next day we input all the data on a spreadsheet that the class made. We had over 100 pieces of information I mean some were repeats, but still. After that we started making a line of best fit using the area and boundary points. We did this all during class, but the formula we found only works if there are no interior points.
Now going more into depth about what we did. So at first we just drew three polygons, so we can get an idea of what we have to do. Then after for homework we had to draw 20 more polygons. The reason we made twenty polygons was for data. We made them so we could have data to collect. We also made them for background information. We had to find the area (area times width), interior points (any points that lie inside the figure), and the boundary points (any points that make up the figure or outline the figure). After that we made a graph, only using the area and boundary points as axis. After graphing them we drew the line of best fit and figured out the formula using the equation for slope-intercept form (y=mx+b). We compared all our answers and we all ended up getting y=1/2x-1. So we all agreed on this formula. But, this formula only works if your number of interior points is zero. We gathered all the data groups that had a number of zero and used the boundary points and area to get the numbers for the data points.
Then Jocelyn said the rest of this part of the problem we have to solve as a class without her help. So someone came around the class assigning each table pair a group, the ones with the number of interior points two, number of interior points being six, or Five, or twelve. Then as homework we had to figure out an equation for the number we go assigned. We did this the same way we did it the last time, using slope-intercept form, (y=mx+b). Then the next day as a class we all said the number of interior points we got assigned and the formula we found. Then Jocelyn went around the class assigning a formula, different from the one we found, and you would have draw a random figure with the same amount of interior points that the formula belonged to. If you found a figure in which the formula doesn't work, then the formula would be crossed out. The formulas that couldn't be proven wrong were kept up. We went through and re-listed the ones that were right. Then we got assigned new ones, just to double check, and same things apply. After that we re-listed them once again and noticed a pattern. The pattern was The formulas are all y=1/2x and then 1 number less than the number of interior points. So if the number of interior points is 10 then the formula would be y=1/2x+9. Then after that we all went around and tested it, it worked but we had to make it into an actual formula so we decided on y=1/2x=(i-1) or A=1/2B+(i-1), which is the same is just that y is replaced with a and x for b. Standing for Area and Boundary points.
Solution:
The solution we got was A=1/2B+(i-1)
ex: